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p scales fromp' to the same cutoff contribute to the structure function D(p'). It is obvious, therefore, that the difference D(p') - D(p) contains the information about the turbulent scales from p to p'. 3. Velocity Fluctuations The inversion procedure above is rigorous only when velocity fluctuations can be disregarded. For HI studies in the galactic plane velocity fluctuations should be accounted t In Eq. (2.3) and further on it is assumed that the Aw and the frequency w are fixed.
98
Lazarian: Turbulence in HI
^ .' e
FIGURE 1. The schematic of lines of sight and the region under study. The slice of HI with the thickness L is observed from such a large distance R that lines of sight ei and e2 are nearly parallel within the slice. Various turbulence scales, e.g. l\, h, contribute to the correlation function of density for the fixed p = R9. h and I2 make angles ipi and rp2 with the lines of sight. Therefore the dependence of structure functions on tp is important for the inversion. For the isotropic densityfieldstructure functions do not depend on «/>.
for. Indeed, even in the absence of density inhomogeneities, intensity fluctuations in the velocity space can be produced by random velocity. Moreover, slicing of hydrogen may become ambiguous. We may recall, that the slicing assumes a monotonic dependence of velocity on distance. The random velocity uturb distorts HI motion, which otherwise would be determined by the Galactic rotation curve. The latter motion is characterized by the projection of the regular velocity to the line of sight (z-axis) V""69 and its spatial derivative / - 1 = (5Vre9/6z). Since the actual velocity along the line of sight is Vreg + uturb spatially distant regions may be mapped into the same slice, while adjacent regions with different velocities will be mapped into different slices. It is also obvious that the turbulence statistics in the velocity space is anisotropic even if the statistics is isotropic in galactic coordinates. Indeed, only the velocity component along the line of sight matters and this makes the direction towards the observer "the chosen direction". One may wander whether the statistical treatment described in section 2 is applicable to atomic hydrogen in the Galactic plane. Obviously, our analysis is not sensitive to velocity fluctuations when the integration over the whole 21 cm line is performed. It is also intuitively clear that when the thickness of the HI slice in the velocity space A y is much larger than the turbulent velocity dispersion 5uk at the scale under study, the fluctuations of velocity are marginally important. The quantitative treatment in Lazarian & Pogosyan (1998) (henceforth LP98) proves this. At the same time for AV < 6uk the velocity fluctuations may dominate the measurements. To distinguish the cases when velocity fluctuations are important and negligible it is convenient to talk about "thick" and "thin" slicing of data. Using this terminology we may say that L95, where velocity fluctuations were disregarded, dealt entirely with "thick" slicing. We may showf the difference between the "thin" and "thick" slicings assuming that t A rigorous treatment is given in LP98, while here we present simplified estimates.
Lazarian: Tiirbulence in HI
99
the Fourier modes of density are independent random numbers in the velocity space. The density at point (P,vz), where P is a two dimensional vector in xy plane, is 6n(P,vz)~
[dKdkzF1'2{K,kz)exp(iKP)exp(ikzvzf)
,
(3.7)
where F(K, kz) is the underlying 3D spectrum of HI random density. As 21 cm intensity is proportional to the integral of Sn over the thickness of the velocity slice, the correlation function of intensity is
f dKexp[iK{P
- Pi)]F 2 (K)
J
(3.8) where 51 and 8n are variations of intensity and density, respectively, while the two dimensional spectrum -^(K) is given by ,
F2{K) ~
,AV
dkz
dvz exp[ikzvzf]{l - vz/AV)F(K, kz)
,
(3.9)
where AV — \vz\ — vZ2\ is the thickness of HI slice in the velocity space. First consider "thick" slicing |K| >• 1 / / A V. The contribution to the integral (3.9) comes mostly from kz < 1/fAV, as for larger kz the exponent oscillates rapidly and the inner integral in Eq. (3.9) is small. Therefore F2(K) ~ F(K,kz)
,
(3.10)
where kz is a value in the interval [0,1//A V]. If the turbulence is isotropicf its spectrum in galactic coordinates, F{K,kz) = F{\/\K\2 + k2z) « F(|K|), and F 2 (|K|) ~ F(|K|) in agreement with L95. In the case of "thin" slicing |K| 0.5 are quite likely to be genuine superbubble detections (see MTB98 for details).
7. New results We have begun to apply the procedure described in Section 6 to the analysis of M33, NGC 300, and several nearby spirals from the sample of Braun (1995). Due to the significant computational burden of our method, we have only obtained a complete parameter space survey for NGC 2403, with more limited parameter sampling for the rest of the galaxies. Our results for all galaxies but NGC 2403 must therefore still be viewed as provisional. 7.1. Kinetic energy & dynamical age In contrast to earlier observational studies, we find that superbubbles having kinetic energy greater than 1053 erg are extremely rare. Most detected structures have E& ~
Thilker: Supershells in Spiral Galaxies
109
RIGHT ASCENSION <J3O00)
FIGURE
3. Continuum-subtracted Ha image of M101, including giant HII region NGC 5461. Each ellipse marks the observed in-plane size of a supershell cavity.
2-4 xlO51 erg, indicating a modest progenitor OB association. It is much too soon to guess whether this is a general result applicable to all spirals. Past studies have often estimated the dynamical age for expanding shells based on line-of-sight velocity splitting. Unfortunately this means that the derived dynamical age is coupled to the orientation of the galaxy in question (since radial expansion velocity varies within non-spherical bubbles). By fitting models to our data, we tightly constrain both the velocity and size, giving a highly accurate estimate of dynamical age. Typical ages range from 5-40 Myr. This suggests a substantial fraction of HI supershells will no longer contain an HII region, unless star formation is non-coeval. Incidently, we do not detect any very old shells (~75 Myr and up), although this could be a selection effect due to the fact our models don't account for shear. 7.2. Spatial distribution & SSF The spatial distribution of superbubbles in NGC 2403 is indicated by the images of Fig 2. We show the observed HI distribution in comparison to a velocity-integrated representation of our composite model cube. The similarity is remarkable on both large and small scales, as one might expect for a galaxy having a substantial fraction of its HI in the form of shells. Fig 2 shows all the structures included in our parameterized decomposition of NGC 2403. It is quite important to note that we exclude marginally resolved (and unresolved) bubbles. In order to more clearly illustrate the association of supershells with star forming regions we include Fig 3, showing an overlay of only the most significant cavity outlines with respect to a continuum-subtracted Ha image of M101. Note the two large supershell cavities apparently bordered by SSF. Three smaller bubbles within the giant HII region NGC 5461, still contain OB stars. 7.3. Velocity dispersion If there were no HI medium except for the gas contained in shells we detect, how would the average line profile be characterized? Given our composite model cube, it is simple
Thilker: Supershells in Spiral Galaxies
110
1
N5457 X
1000
-
X N059B x
•
N2403 x N3031
x N0300
100 • x N2366 i 1
.
i
1
0.1 Star Formation Rate (Mo / yr)
FIGURE
4. The total kinetic energy of supershells in a galaxy appears to scale roughly with the global star formation rate.
to compute an average profile and determine the second moment. For all the galaxies examined so far, we find that the velocity dispersion attributable to expanding shells is typically 3-4 km s" 1 , with localized regions approaching ~7 km s" 1 . This result is consistent with the analysis of Braun (1997), who found that the globally averaged lineprofile in many spiral galaxies can be described by a narrow core with broad Lorentzian wings. Supershells may be completely responsible for the presence of the low level wings. 7.4. Global quantities Assuming that HI shells are really formed as a consequence of energy deposition related to SNe and stellar winds from massive stars, one might conjecture that the total kinetic energy of superbubbles should be related to the global rate of star formation in a galaxy. We find that the total kinetic energy does appear to increase with SFR. Figure 4 illustrates this effect. It is premature to attempt fitting a power-law slope since only six galaxies have been considered. Our catalog for M33 (NGC 0598) is the most complete, while the M101 (NGC 5457) survey is certainly incomplete.
8. Summary New techniques for accurate, detailed comparison of 3D theoretical models and spectralline datacubes are beginning to emerge. This contribution described one such method. Our application of automated supershell recognition methods to a handful of spiral galaxies supports the following preliminary conclusions: (1) most superbubbles only require modest OB associations, (2) the spatial distribution of expanding structures often traces spiral arms, producing many characteristics of an integrated HI map, (3) the most significant detections are often clearly associated with progenitor HII regions or sites of triggered star formation, (4) supershells may be largely responsible for broad wings seen in globally averaged HI profiles, and (5) global SFR and the total kinetic energy associated with expanding bubbles seem to be correlated. The coming years hold much promise for favorable resolution of the issues raised in
Thilker: Supershells in Spiral Galaxies
111
Section 3. Indeed, a sample of 21 nearby galaxies is currently being analyzed with the procedure of MTB98 as part of the Las Cruces/Dwingeloo Supershell Survey (LCDSS). I am grateful for the essential advice and continuing support provided by Rene Walterbos and Robert Braun, my dissertation advisors. Thanks also to B. Benjamin, J. Bregman, B. Elmegreen, M. Mac Low, M.S. Oey, and S. Tufte for their insightful comments and stimulating conversation. This research has been supported by NASA in the form of a GSRP Fellowship (NGT-51640). REFERENCES BRAUN,
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D, BURTON, W 1997, Atlas of Galactic Neutral Hydrogen, Cambridge Univ. Press.
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The Size Distribution of Superbubbles in the Interstellar Medium By M. S. OEY AND C. J. CLARKE Institute of Astronomy, Madingley Road, Cambridge, CB3 OHA, UK We use the standard, adiabatic shell evolution to predict the size distribution N(R) for populations of SN-driven superbubbles in a uniform ISM. We derive N(R) for simple cases of superbubble creation rate and mechanical luminosity function. We then compare our predictions for N(R) with the largely complete HI hole catalogue for the SMC, with a view toward the global structure of the ISM in that galaxy. We also present a preliminary derivation for N(v), the distribution of shell expansion velocities.
1. Introduction Core-collapse supernovae (SNe) tend to be correlated in both space and time because of the clustering of the massive (> 8M G ) star progenitors. These clustered SNe, along with stellar winds of the most massive stars, produce superbubble structures in both the warm ionized (104 K) and atomic H I components of the interstellar medium (ISM) in star-forming galaxies. The hot, coronal component of the ISM is thought to originate largely from the shock heating of material interior to shells of superbubbles and supernova remnants (SNRs). Total kinetic energies deposited into the interstellar environment are in the range 1051 —1054 erg for OB associations, and > 1055 erg for starburst phenomena. Hence, the large-scale structure and kinematics of the multi-phase ISM could be largely determined by this superbubble activity. Likewise, this effect should influence turbulence on global, macroscopic scales, which then cascades to smaller scales. The standard model for understanding superbubble evolution is the adiabatic model for a continuous mechanical energy input (Pikel'ner 1968; Weaver et al. 1977; Dyson 1977), where the sequential SNe from the parent star cluster are treated as a continuous power source (McCray & Kafatos 1987). The evolution of the shell parameters can then be described by a set of simple, self-similar relations analogous to the Sedov (1959) model for a single point energy injection. How applicable is the standard, adiabatic model to the long-term evolution of superbubbles? It is possible to make some rudimentary assumptions about the global ambient ISM and shell creation history, and then use the analytic equations to derive a superbubble size distribution that is predicted by the model. Such a size distribution can then be compared to observed H I shell catalogues to gain insight into the global structure and kinematics of the ISM. Variations from the prediction can then point to important effects that have not been adequately treated. In this contribution, we summarize our derivation of the superbubble size distribution, which is described in greater detail by Oey & Clarke (1997). We compare our results to the Hi shell catalogue of the Small Magellanic Cloud (SMC). Finally, we present preliminary new results for an analogous derivation of the distribution of shell expansion velocities. 2. Assumptions Our purpose is to make the simplest feasible assumptions to see what the standard shell evolution predicts in the simplest conditions. Our assumptions are thus as follows. We assume coeval star formation in the parent clusters, which produce a constant 112
Oey & Clarke: The Superbubble Size Distribution
113
mechanical power L until the lowest-mass SN progenitors expire at an age te — 40 Myr. Studies of OB associations in the Magellanic Clouds and the Galaxy (Massey et al. 1995a, b) show stellar age spreads of < 3 Myr, motivating our adoption of coeval star formation. Likewise, stellar population synthesis modeling by Shull & Saken (1995) and Leitherer & Heckman (1995) suggests that the assumption of constant L appears to be reasonable. We also assume that the stellar initial mass function (IMF) remains fixed and universal, and we adopt a uniform and infinite ambient ISM. Based on the observed power-law form of the H II region luminosity function (H IILF; e.g. Kennicutt et al. 1989), we infer a similar power-law distribution of parent cluster masses. For a constant IMF, the nebular Ha luminosity scales directly with the number of stars, and likewise, L for the clusters is directly proportional to the number of corecollapse SN progenitors. We therefore take the mechanical luminosity function (MLF) for the cluster population to be a power-law as well:
4>(L)=d£; = AL-e ,
(2.1)
normalized such that f 4>{L) dL = 1. Ordinarily, the power-law index 0 of the MLF should be identical to that of the associated HII LF. We caution that evolutionary effects and small-number statistics in the stellar population can complicate this assumption (Oey & Clarke 1997, 1998), but essentially the power-laws are the same. We also consider one scenario with a single-valued MLF. In conjunction with these forms of the MLF, we consider a constant shell creation rate ip, and a single-burst creation model. The treatment of the endstage evolution for the superbubbles is crucial, but extremely uncertain. We assume that the shell growth stalls at an age U when the superbubble's internal pressure Pj < PQ, the ambient ISM pressure. Such a scenario is supported by numerical simulations (Garci'a-Segura k Franco 1996), in which radiative energy loss at this endstage suppresses further growth of the superbubble cavity. We then assume that the shell maintains this stall radius R{ until the input power stops at time te. However, objects that never achieve pressure equilibrium with the ambient ISM continue to grow until te. The subsequent destruction of the shells is even more uncertain. We simply assume that all objects survive for a nominal, universal period ts Le will never stall, and at some point before te will grow to radii R > Re. For an ambient number density n = 0.5 cm" 3 , mean particle weight fi = 1.25, and Po = 3 x 10~12 dyne cm" 2 , the adopted te = 40 Myr implies Re = 1300 pc and Le = 2.2 x 10 39 ergs~1 . These characteristic parameters are useful as scaling parameters, hence we have,
and
We now derive the differential superbubble size distribution Af(-R) for specific combinations of shell creation history and MLF. We define N(R) dR as the number of objects with radii in the range R to R + dR. 3.1. Continuous Creation, Single Luminosity For a continuous and constant superbubble creation rate ip and a single-valued MLF with <j)(L) = LQ, the size distribution for growing shells is given by, (3 7) ~m) • The size distribution for the stalled objects is clearly a (5-function at the stall radius associated with LQ, whose magnitude is determined by the length of the creation period:
,
(3.8)
where U{L0) and Rf(Lo) are the stall parameters for a shell powered by Lo. The distribution in R is therefore determined exclusively by the growing objects. Applying the relations for the standard evolution given above, equation 3.7 gives,
We define the power-law slope of the size distribution as a, analogously to that of the MLF (equation 2.1), such that N(R) a R~a. The case here is the only one for which we derive a positive power-law index in R, yielding —a = | . The positive index is induced by the single-valued MLF. 3.2. Single Burst Creation, Luminosity Spectrum We now consider the inverse case of instantaneous creation of all the objects, with a power-law MLF given by equation 2.1. Here, the size distributions for the growing and stalled objects are given by, (3.10) and
= Nh 105K), and found that the fractional amounts of these phases and their spatial distribution (topology) is coupled closely to the heating rate (the supernova rate). At low heating rates, the medium is largely neutral and this cold material is fairly continuous with a few low density regions with hot material, while at high heating rates, the hot gas occupies most of the volume, with the neutral gas being islands in a sea of hot gas. As the heating rate is increased, the scale height of all gas phases increases, so we adjust the heating rate until the observed scale heights are reproduced. For this
Bregman et al.: Large-Scale Motions in the ISM 0
in
125
400
100
. •e *
0.0
*" 0 for a < 1/3, the subset of space where a < 1/3 has to be smaller and smaller as a decreases. This led Mandelbrot (1974) to predict that the support of the dissipation is indeed a multifractal with a dimension D which depends on the singularity exponent a. Menevau & Sreenivasan (1991) deduced the dependence D(a) from flow experiments and showed that the D(a) function is universal. The main difficulty with such a description is that no physical process enters the description of the multiplicative cascade involved to build the self-similar subset of space on which dissipation is concentrated. The other description of intermittency is related to the observation that a large fraction of the dissipation (up to 20% in laboratory flows but possibly much larger at larger 5R) is found to be closely associated to coherent vortices. Numerical simulations of such flows (She, Jackson & Orszag 1990; Vincent & Meneguzzi 1991, 1994; Porter, Pouquet & Woodward 1994) and laboratory experiments (Douady, Couder & Brachet 1991; Cadot et al. 1995) now provide some firm elements to describe these vortices. They seem to be correctly described by a Gaussian vorticity distribution, the Burgers vortex, a solution of the Helmholtz equation for the evolution of the vorticity y
+ (i).V)w=(w.V)o + i'Aii;.
(2.1)
Ob
This solution corresponds to an equilibrium between the stretching of the vortex by advection in the ambient turbulent medium and the diffusion of vorticity. Experiments in gaseous helium at low temperature (Tabeling et al. 1997) and numerical simulations tend to show that the peak of azimuthal velocity in the vortex is equal to the rms velocity dispersion of the ambient turbulence. Vortices have a cross-sectional radius somewhere between the inner Kolmogorov scale Id (or the dissipation scale) and the Taylor microscale A = ij JL2//3 and lengths as large as the integral scale, L, the scale at which energy is injected. The lifetime of the largest vortices is observed to be of the order of the turnover timescale of the integral scale, i.e. up to about 100 times, or more, the vortex period, and that of the smallest vortices, of size close to the dissipation scale, may not exceed a few periods. Another important property of these vortices is their crowding. Experiments (Schwarz 1990) and numerical simulations (Porter et al. 1994; Vincent & Meneguzzi 1994) show that structures of intense vorticity in turbulence are not isolated and randomly distributed in space but bunched together. They are generated by the growth of instabilities developed in a larger scale shear layer. They later on attract each
134
Falgarone: Possible observational signatures of intermittency
n TK a I -pvi3/l A \v3/l
(i = V
Id
A=
1>J3L2'
(cm"3) (K) (kms- 1 ) (PC)
(erg cm 3 s 1) (erg cm"3 s"1) (L0/M0) (cm2 s-1) (AU) (PC)
CNM
molecular clouds
dense cores
30 100 «3.5 10 2xlO~25 5xlO" 24 1.5xlO~3 8.5 xlO17 6.7 0.45
200 40 1 3 1.7xlO-25 4xHT 24 l.lxlO- 4 5.3 xlO17 9 0.5
104 10 0.1 0.1 2.5 xlO- 25 3.5xHT24 3.2xlO-6 2.7 xlO17 13 0.55
TABLE 1. Characteristics of turbulence in the cold neutral medium (CNM), molecular clouds and dense cores. The velocity field is assumed Gaussian: a = 1.6w is the rms velocity dispersion, and v the mean velocity. A is the dominant radiative cooling term due to the C + fine structure line for CNM (with ionization fraction x — 10~3), and CO lines for clouds and cores. The mean mass per particle in the CNM is 1.45 /in and v is the kinematic viscosity.
other and merge into larger structures of braided small vortices. The viscous dissipation rate IQx • + dv • /dx)^
(2 2)
where r] is the dynamic viscosity, is therefore concentrated in the layers at the edge of the vortices where the shear of the velocity field reaches a maximum. Table 1 gathers a set of quantities relevant to turbulence and its dissipation in each of the three components which build up the cold interstellar medium. Although the numbers in this Table are crude estimates and the scales chosen just representative values of a full range of scales, the transfer rate of kinetic energy density ~pv3 /I (and therefore the dissipation rate at the smallest scales) seems to be the same in the three media, which supports the idea of a cascade pervading all the components of the cold medium. The comparison of the average value ^~pv~t3/l and A, the radiative cooling rate, shows readily that if the energy of the viscous dissipation (assumed to be equal to e) is deposited in a small fraction (< 10~2) of the whole volume, the corresponding heating rate locally exceeds the cooling rate and the gas heats up. In the next section, we show that significant changes may occur in the gas of a fluid particle of the CNM which experiences one of these dissipative bursts.
3. Chemical evolution of a fluid cell trapped in a Burgers vortex The following studies were motivated by the fact that in many respects, diffuse interstellar clouds appear more chemically active than anticipated, given their low gas density and temperature. In particular, the large observed abundances of CH + and OH imply formation routes for these molecules which require energy sources much in excess of the average energy density of diffuse clouds. Large fractional abundances of CH + and OH are commonly observed in clouds of temperature Tj< » 50 K while the formation of CH + proceeds through the endothermic reaction CH + + H2 —> CHj + H with AE/k = 4640 K, and that of OH via O + H2-> H + OH, which has an activation energy (AE/k = 2980 K).
Falgarone: Possible observational signatures of intermittency
300 200
time (yr) 100 50
135
0
c
5
c o 10 -12
o FIGURE 1. Fractional abundances of a set of species, as functions of the distance r of the fluid cell from the vortex axis, which decreases exponentially with time (the time is shown on the upper scale). The regions where the gas temperature exceeds 103 K and the ion-neutral drift 1 velocity exceeds 3 km s" are delineated by horizontal bars.
The idea which has been followed for a few years (Falgarone & Puget 1995; Falgarone, Pineau des Forets & Roueff 1995) is that the tiny regions heated by violent bursts of dissipation of the turbulent kinetic energy do become temporarily active chemically within cold diffuse clouds. In other words, the actual gas temperature which controls the thermal and chemical evolution of the gas is not close to the average temperature deduced from the observations, but has large excursions above this average. In our most recent study (Joulain et al. 1998), we chose to model the dissipative structures by a Burgers vortex. The vortex is entirely described by two independent parameters, ro = 1.2 x 1014 cm = 8 AU and u0 which determine the vorticity distribution w — w o exp(-r 2 /ro). These parameters uniquely determine the peak of the viscous dissipation rate and independently the peak of orthoradial velocity of the neutrals. The steady-state configuration includes a magnetic field mostly parallel to the vorticity with a small toroidal component which grows toward the ends of the vortex. We have studied the chemical evolution of a fluid particle of low density (nn ~ 30 cm"3) trapped in such a vortex, with a magnetic field intensity B — 10/zG. The ions, frozen to the field have very small velocities and large ion-neutral drift velocities are therefore generated in the layers where the neutrals have the largest tangential velocities. The chemical evolution is controlled by the sharp temperature rise following the passage through layers where viscous dissipation is intense, and by the ion-neutral drift velocities in the outer layers of the vortex. The values adopted in the standard model are T^max — 10~21 erg cm" 3 s" 1 and vo,max — 3.5 km s" 1 , a value dictated by the rms internal velocity dispersion of HI clouds (Crovisier 1981). Figure 1 displays the fractional abundances of a subset of molecules as a function of the distance of the fluid particle to the vortex axis. This distance decreases exponentially with time because the linear decrease with r of the radial velocity of the straining flow in the Burgers vortex. The corresponding time scale is shown in the upper scale of Fig. 1. The layers where the ion-
136
Falgarone: Possible observational signatures of intermittency
N(HCO+) (cm
2
10 )
2. The correlation between the column densities of OH and HC0 + . The observations of Lucas & Liszt (1996) are shown (solid squares). The error bars are smaller than the size of the symbols. The dotted line displays N(0U)/N(HC0+) = 25. Column densities predicted by the model are given by the two curves, for 10 and 3 x 10 standard vortices on the line of sight. Each point along these curves corresponds to a different value of Av from 0.1 mag to 1 mag. FIGURE
neutral drift velocity exceeds 3 km s" 1 and those where the kinetic temperature of the neutrals exceeds 103 K are delineated. The timescale for the vortex crossing is so short that thermal and chemical equilibria are never reached inside the vortex. For instance, the dissipative burst occurs in the layers where the velocity shear is maximum. Those layers are approximately the same as those where the drift velocity exceeds 3 km s" 1 . But the temperature maximum is reached much later, typically 200 yr later (see Fig. 1), because of the large thermal inertia of the gas. Similarly, almost everywhere the chemistry is out-of-equilibrium. The initial abundances are equilibrium abundances in the diffuse gas obtained for n\\ = 30cm~3 and TW = 50 K. Most of the species have their fractional abundance which increases sharply by several orders of magnitude as the fluid cell enters the vortex structure. As expected, the OH fractional abundance peaks in the region where the gas temperature attains a maximum. The abundance of H2O which forms mainly via OH +H2 —> H2O +H closely follows that of OH. At the opposite, the CH abundance (not shown) decreases as the temperature peaks because its main destruction paths have activation barriers AE/k « 2000 K. A direct consequence of the destruction of CH in the hot layers is a clear increase of the abundance of neutral carbon. In these layers, the fractional abundances of neutral and ionized carbon differ by no more than a factor « 2, leading to a very low ionization degree for carbon, while in the standard diffuse medium, the abundance of neutral carbon is expected to be negligible compared to that of C + . The abundances of most of the ions follow the time history of the ion-neutral drift velocity. CH3" is the most abundant ion after C + because its abundance follows that of CH + . A consequence of the large fractional abundance of CHjj" is that the formation of HCO+ is dominated by CH^ + O -»• HCO+ + H2 rather than by the reactions of C + with OH and H2O. The abundance of CO (not shown) is also enhanced by two orders of magnitude in the vortex, due mostly to the dissociative recombination of HCO + . The
Falgarone: Possible observational signatures of intermittency
137
fact that the two processes of viscous dissipation and large ion-neutral drift are closely associated in space and time, and that the amount of energy available in a small scale dissipative structure is large, enable us to reproduce, without fine-tuning the parameters of the model, the salient features of the observations of molecular species in diffuse gas: the large column densities of CH + , OH and HCO + , the remarkable proportionality of the OH and HCO+ column densities (see Figure 2), the similarity of the OH and HCO + (resp. CH and CH +) line centroids, and the fact that the OH-rich gas seen in absorption is not always detected in emission. A salient result is that only a few percents of the gas column density on any line of sight need to be in those chemically active regions to reproduce the observed column densities and that the turbulent energy dissipated in all these structures is, on average and at any time, significantly smaller than that available in the turbulent cascade of the diffuse medium. Last, the dependence of our results on the gas density confirms that this 'hot' chemistry has to develop in low density gas to meet the requirements provided by the observations. We have also investigated the chemical evolution of a fluid cell which would be trapped in a vortex for no more than one of its period, (P = 23 yr in the case of the model discussed here) and then, after some time spent in the inactive part of the cloud, would enter another nearby vortex, and so on. The thermal and chemical inertia of the gas are so large that the molecular enrichment of the fluid particle progressively builds up at each encounter. The same molecular enrichment as that obtained after one encounter with a long-lived vortex (Figure 1) is obtained after « 5 encounters with different vortices of the same neighborhood with several 100 yr of inactivity between each encounter.
4. The dissipation of supersonic turbulence in low mass dense cores Dense molecular cores with no embedded stars have very little suprathermal support. It is not stated yet whether dense cores are devoided of highly supersonic velocities because they are dense or whether it is the dissipation of turbulence which initiates the condensation, in the first place. This issue has been one of the motivations of the first IRAM key-project (Falgarone et al. 1998), a high angular resolution investigation of the velocity and density fields in the environment of nearby low mass dense cores. All selected fields contain a starless dense core of small internal velocity dispersion. Maps have been completed infivetransitions, 12CO(J=1-0) and (J=2-l), 13CO(J=1-0) and (J=2-l) and C 18 O(J=l-0), at high angular resolution (22" and 11") and velocity resolution of 0.05 km s" 1 . The spatial resolution of the high frequency maps is ~ 1700 AU. We have found that unresolved structure is still present in the velocity maps of all the fields and all the lines. The velocity gradients involved reach values as large as 10 km s" 1 pc" 1 , implying large accelerations never observed before at small scale in non star-forming clouds. But the most unexpected result is illustrated in Figure 3. In two of the fields, in addition to the dense core itself (not visible on the 12CO maps due to the large opacity of the line), there is an extended gas component bright in 12CO and barely detected in 13CO. Its texture exhibits filamentary structure with, in some cases, unresolved transverse dimensions, and aspect ratios larger than ~ 5. Its velocity dispersion is much larger than that of the dense cores. Unexpectedly, it is in the most opaque transitions and in the gas component of larger velocity dispersion that the smallest scale structure has been observed. Another result is the remarkable uniformity of the line temperature ratio of the two lowest CO rotational transitions: R(2-l/l-0)=0.65±0.15 for 80% of the data points in the three fields, across the whole profiles and for both 12CO and 13CO isotopes. From these
138
Falgarone: Possible observational signatures of intermittency POLARIS 12CO(2-1)
-5.50]
[ -5.50, -5.00]
[ -5.00, -4.50]
12 FIGURE 3. Velocity maps of the CO(J=2-1) line emission in the Polaris field, one of the three fields mapped. The velocity intervals are given at the top of each panel in km s~l. The linear size scale and the offsets in arcseconds appear only at the top left panel. The intensity scale, different for each map, is given in Kkms" 1 at the right of each panel.
well defined spectral properties, we infer that the lines have to form in very small cells, weakly coupled radiatively to one another, optically thick in the 12CO lines and that the line shapes are governed mostly by the spatial and velocity dilution of the emitting cells in the beam. Under the simple assumption that the cells are statistically independent, we estimate that they are smaller than ~ 200 AU with H2 densities riH2 ~ a few 103 cm" 3 in the gas component barely detected in 13CO, and are up to two orders of magnitude denser in the dense cores. We also notice an anticorrelation between the intensities of the 13CO lines and their linewidths which we interpret as a signature of a gradual loss of the non-thermal support which increases the phase-space radiative coupling of the cells. We speculate that the filaments and loops we have found in the 12CO lines are related to the dissipative process and are indeed collections of much smaller vortices, of diameter close to the dissipation scales, braided together into much larger filamentary structures. Observations planned in a near future with the Plateau de Bure interferometer may be able to detect substructure inside these filaments. Recent observations of one of the other fields at the Caltech Submillimeter Observatory telescope have revealed that the filaments extend straight over ~ 0.8 pc and keep a transverse size as small as ss 0.1 pc (Falgarone & Phillips in preparation). Last, the non-Gaussian wings of the PDFs of the 12CO and 13CO line centroid velocity increments are larger in the field which has the largest internal velocity dispersion, and therefore the largest 5R since the size scales are the same (Pety et al, in preparation). It has been shown (Lis et al. 1996) that these non-Gaussian distributions
Falgarone: Possible observational signatures of intermittency
139
trace that of one component of the vorticity. We therefore infer that the distribution of vorticity is more intermittent in the field of largest 5R than in the others. 5. Conclusion Although the smallest structures, and therefore those where the viscous dissipation is the most intense, cannot be detected individually with the present telescopes, it is possible that we are observing them indirectly for the following reasons: (i) a specific hot chemistry has been shown to develop in Burgers vortices chosen as templates of the regions of intense viscous dissipation. Magnetic field has been introduced so that the effects of viscous heating and ion-neutral drag add to each other. This chemistry reproduces remarkably well the large (and unexplained) C H + , OH and HCO + abundances observed in the diffuse interstellar medium, (ii) conspicuous filamentary structure has been observed in the vicinity of low mass dense cores which are almost thermally supported. These filaments are much thicker than the dissipation lengthscale in molecular clouds but they may correspond to a large number of much smaller filaments braided together, (Hi) constraints provided by spectral line observations of CO and isotopes in turbulent clouds tend to support that the velocity field in such clouds is correlated over small lengthscales « 200 AU, (iv) vorticity, if correctly traced by the line centroid velocity increments, is more intermittent in the most turbulent dense core environments.
CADOT
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Chemistry in turbulent flows By ROLAND GREDEL European Southern Observatory, Casilla 19001, Santiago 19, Chile The ubiquitous amount of interstellar CH + in translucent molecular clouds presents one of the outstanding problems of interstellar chemistry. The chemical pathways which lead to the formation and the destruction of the CH+ ion in the quiescent gas are well understood, yet the predicted abundances are orders of magnitudes below the observed values. This led to the suggestion that disturbances upon the quiescent material increase the CH+ formation rate via the reaction C + + H2 —> CH + + H, which is endothermic by 4650 K. Interstellar turbulence may very well provide the energy source to drive this reaction. The various formation scenarios of interstellar CH + are discussed, with an emphasis on processes which involve the dissipation of interstellar turbulence. The chemical properties of regions which are affected by the dissipation of turbulence are summarized.
1. Introduction Interstellar turbulence may affect the chemistry of translucent and dense molecular clouds in various ways. Turbulent mixing of material from dense cores to the surface of molecular clouds, and vice versa, may alter the abundances inferred from chemical networks. In particular, turbulent transport and diffusion was invoked to explain the large abundance of atomic carbon and that of complex organic molecules which is observed in dense molecular clouds (Boland & de Jong 1982; Chieze, Pineau des Forets & Herbst 1991; Xie, Allen & Langer 1995). The dissipation of turbulence in translucent molecular clouds is another physical process which has recently been considered to alter chemical abundances. It may create hot parcels of gas, where endothermic reactions which are inhibited in the cool gas proceed rapidly. Alternatively, the dissipation may create a fraction of ions, atoms and molecules which move at super-thermal velocities. In the latter case, the surrounding material remains cool, and the translational energy of the fast particles is used to overcome the reaction barriers of the endothermic reactions. The two scenarios may lead to differences in the chemistry, because in a hot material, all reactions proceed at their Maxwellian rates, whereas in a cool material, reactions which involve collision partners which move at super-thermal speeds are selectively enhanced. The present paper focuses on the problems related to the formation of interstellar CH + . The observational facts about CH + and the earlier attempts to explain its production in hot, post-shock regions are summarized in § 2. The effects of turbulent transport upon the chemistry of molecular clouds are briefly discussed in § 3.1. The main part of § 3 focuses on the dissipation of interstellar turbulence, and on the ways it affects the chemistry of molecular clouds. Formation scenarios of CH + which invoke hot turbulent boundary layers (Duley et al. 1992), low-density regions transiently heated by intermittent dissipation bursts of turbulence (Falgarone, Pineau des Forets & Roueff 1995), and non-Maxwellian velocity distributions in cool clouds generated Alfvenic turbulence (Gredel, van Dishoeck & Black 1993; Spaans 1995), are discussed in §3.2 to §3.4. 2. The C H + mystery The problem related to the formation of interstellar CH + was already realized by Bates & Spitzer (1951), who showed that large amounts of CH + can not be maintained 140
Gredel: Chemistry in turbulent
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141
by the chemical reactions which proceed in cold molecular clouds. The problem arises because the CH + radical is highly reactive and is efficiently removed by collisions with electrons, H and H2. The general remedy to this problem is to increase its formation rate via the endothermic reaction C+ + H2 -> CH+ + H
(2.1)
which has a reaction rate coefficient as listed in Millar, Farquhar & Willacy 1997 of k = 10"10 e~ 4640 / r cm3 s~1. Consequently, CH + can be efficiently produced in a hot gas of a few 1000 K temperature. Alternatively, CH+ may be produced in a cold gas, if a fraction of C + or H2 moves at super-thermal, non-Maxwellian speeds. 2.1. Translucent and dense molecular clouds CH is commonly observed in translucent molecular clouds. These clouds are characterized by visual extinctions of Ay = 1-5 mag, scaling factors Iyv — 0.5 — 2 of the interstellar ultraviolet radiation field, gas-kinetic temperatures of T = 10 — 50 K, gas densities of n = 102 - 103 cm""3, and electron abundances of xe = 10~4 - 10~6. Their chemistry is dominated by the ultraviolet photons of the ambient interstellar radiation field, thus the name translucent clouds. The UV photons cause a C+ —> C -> CO transition region, and van Dishoeck & Black (1988) demonstrated that its location in the cloud depends sensitively on various physical parameters such as IuvThe chemistry of dense molecular cloud cores differs significantly from that of translucent clouds or from outer layers of molecular clouds (see for example van Dishoeck 1998). This is because the ultraviolet photons from the interstellar radiation field cease to be important in the inner regions which are shielded by dust grains. Steady state models predict that all carbon is converted to CO at timescales of 105 years. The electron abundances are low, typically xe sa 10~8. +
2.2. CH+ in the interstellar medium The observational facts concerning interstellar CH + are readily summarized. The CH + column densities correlate well with the visual extinction Ay of the background stars. This is demonstrated in figure 1, which shows CH + measurements obtained toward lines of sight with visual extinctions ranging from Ay = 0.5-4.5 mag (from Gredel et al. 1993). The correlation coefficient for the N(CH+) - Ay relation for the full sample of data shown in figure 1 is 0.74, with a confidence level > 99.9%. Trends of increasing CH+ column density with Ay have been established by others such as Penprase (1993). Gredel (1997) showed that the correlation between N(CH+) and Ay also exists for lines of sight through single translucent clouds. The radial velocities of CH + agree well with those of other molecules such as CH. Observations obtained at very high spectral resolution such as those by Lambert, Sheffer & Crane (1990), Crawford (1995), and Crane, Lambert & Sheffer (1995), demonstrate that the CH+-CH velocity difference is generally less than 1 km s" 1 . The line profiles, on the other hand, show marked differences. The mean CH + column densities are of the order of « 4 x 10~8. The CH+ column density is correlated to that of CH, and the formation of CH + seems to be enhanced in low density regions (Gredel et al. 1993). 2.3. CH+ formation in shocks It was proposed by Elitzur & Watson (1978) and Elitzur & Watson (1980) that interstellar CH + is formed in the hot regions behind hydrodynamic shocks. In shocks, the kinetic energy of the bulk flow is thermalized in the shock front. The models of
142
Gredel: Chemistry in turbulent flows no saturation corrections
o
1. Inferred CH+ column densities plotted versus visual extinction Av • The left panel shows column densities inferred in the limit b —> oo, while the right panel shows column densities inferred for Doppler parameters of b = 1.5 — 2.5 km s" 1 which are typical for CH + . FIGURE
Elitzur & Watson (1980) show that low velocity, non-dissociating shocks with speeds up to 10 km s - 1 heat the material to temperatures of a few 1000 K and produce CH + column densities up to N(CH + ) « 1013 cm" 2 . Higher CH + column densities are formed in magneto-hydrodynamic (MHD) shocks, where the magnetic field introduces a differential streaming velocity u in between ions and neutrals. The differential streaming provides the additional energy which is required to drive reaction 2.1. Models of CH + production in MHD shocks were presented Pineau des Forets et al. (1986) and Draine (1986). Until recently, shocks were considered as the general and widespread formation scenario of interstellar CH + . However, the different streaming velocity of the ions and the neutrals in MHD shocks cause velocity differences of several km s" 1 between CH + and neutral species such as CH. This result is in conflict with the observations. In addition, the hot post-shock gas produces large amounts of OH as well. MHD shocks lessen the problem
Gredel: Chemistry in turbulent
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143
of overproduction of OH but they do not remove it. The large amounts of OH which are produced in a hot gas weaken the possibility that CH + is produced in hot material. 2.4. CH+ in circumstellar material CH+ has recently been identified in emission in the planetary nebula NGC 7027 by Cernicharo et al. (1997) and in the proto-planetary nebula surrounding HD 44179 by Balm & Jura (1992). CH + has been seen in absorption toward the post-AGB star HD 213985 by Bakker et al. (1997). In all three objects, CH+ column densities up to 1013 cm" 2 have been inferred, distributed over rotational partitions characterized by T rot = 150 - 250 K. This is in contrast to the interstellar case, where T rot = 0 K . The formation of CH+ close to central stars may very well occur in shocks as proposed by Bakker et al. (1997). Alternatively, Balm & Jura (1992) speculated that CH + forms in a photon-dominated region (PDR) which is produced by the ultraviolet radiation emission from the central object. The results of Sternberg & Dalgarno (1995) show that PDRs characterized by densities of n = 104 - 1 0 8 cm" 3 and radiation factors of Iyv = 103 - 1 0 6 produce CH + column densities up to 1014 cm" 2 . Because such high densities and radiation factors do not generally prevail in translucent clouds, a production of interstellar CH+ in PDRs is not favoured, although Snow (1993) suggest that exceptions may exist. 3. Chemistry in turbulent flows 3.1. Turbulent transport in molecular clouds The idea of turbulent mixing was introduced by Phillips & Huggins (1981) in an effort to explain the large abundance of atomic carbon which is observed in dense molecular clouds. The timescales at which carbon is converted to CO are comparable to the crossing time of a 10 pc cloud at velocities of a few km s" 1 . The replenished carbon atoms and ions may also maintain the large abundances of organic molecules in the dense clouds. Time-dependent chemical models generally predict a sharp decline of complex molecules after some 105 years, mainly because a very efficient adsorption of the gas onto grain surfaces. The mean lifetimes of molecular clouds are orders of magnitudes larger, thus some processes must be at work to prevent the cloud cores to reach chemical equilibrium. Boland & de Jong (1982) suggested that the evaporation and photo-desorption of the grain mantles transported to the outer parts of the clouds may release the heavy elements back to the gas phase. The chemistry of a molecular cloud with dynamical mixing was discussed in more detail by Chieze et al. (1991). The authors modelled the mixing by means of compression of parcels of diffuse gas at the surface into dense gas, which is then transported to the interior of the clouds where it mixes with the dense clumps. The diffuse gas attains chemical equilibrium before the compression and the mixing takes place. With this approach, Chieze et al. (1991) were able to maintain large abundances of complex molecules at steady state. Their approach was criticized by Williams & Hartquist (1991) who argued that turbulent mixing is a diffusive process, which occurs on timescales larger than those involved for the chemistry to reach equilibrium. In their models, mixing of ions from a fast stellar wind into the dense cores may explain the large abundances of molecules such as HC3N which are observed in dense cores such as TMC 1. The chemical abundances for dense clouds with Ay = 9 mag were calculated by Xie et al. (1995) who also treated mixing as a diffusive process. They obtained steady state abundances of molecules including C2, CH, C2H, OH, and H2CO which are in agreement with those inferred from observations. In their models, the chemistry is mainly affected by changes in the electron abundance. The electrons carried to the dense parts of the clouds by
144
Gredel: Chemistry in turbulent flows
turbulent diffusion drive electron recombination reactions, which are generally inhibited in the dense cores because of the low electron abundances which prevail at steady state. In a recent paper, Rawlings & Hartquist (1997) suggested that mixing may not be a global process at all, and that it may not operate over the entire clouds. They assert that turbulence driven transport is too slow to account for the high abundance of atomic carbon in dense molecular clouds. In their models, they restricted mixing to a thin (< 0.01 pc) turbulent boundary layer at the surface of the clouds. Turbulent boundary layers are discussed in more detail below. Turbulent mixing may have important consequences to the chemistry of molecular clouds but it is not a solution to the CH + problem. This is because mixing does not drive reaction (2.1). However, the required energy may very well be provided by the dissipation of interstellar turbulence. The CH + formation scenarios which involve the dissipation of turbulence are discussed in the following. 3.2. Turbulent boundary layers It was suggested by Duley et al. (1992) that interstellar CH + is formed in turbulent boundary layers at the molecular-cloud-intercloud surfaces which are produced by hot stellar winds. The authors presented models for gas densities of n = 60 - 600 cm" 3 and temperatures of T = 700 - 4000 K and obtained CH + column densities of a few 1013 cm" 2 for boundary layers with a thickness up to a few 0.1 pc. It was pointed out by Hartquist et al. (1998) that the general properties of such boundary layers are unknown as they are not subject to first principles. The boundary thickness Aj o y e r may nevertheless be approximated by &layer = Dobstacle /' ^rRe-
(3.2)
where Dobstacie is the size of the cloud and Re is the Reynolds number of the flow. Hartquist et al. (1998) suggest Re = 103 - 104, thus a thickness of the boundary layers of a few percent of the cloud diameter is to be expected. Because Aiayer is proportional to the size of a cloud, a correlation of N(CH + ) with N(CH) is expected, which is indeed observed. The turbulent heating rate per unit density is given by Hartquist et al. (1998) as T « 10- 23 er fl s- 1 (T c o r e /10/(')(^ n d /400 km s- 1 )(A, a2/er /0.003pc)- 1 .
(3.3)
The energy requirements to maintain boundary layers with a size of a few 0.1 pc at temperatures of several 1000 K which is needed to produce observed column densities as high as N(CH + ) « 1014 cm" 2 are very large. Contrary to the earlier work of Duley et al. (1992), Hartquist, Dyson & Williams (1992), and Nejad & Hartquist (1994), Rawlings & Hartquist (1997) considered only thin boundary layers (< 0.01 pc) at temperatures of 103 K. Such models result in very high abundances of molecules such as CH, OH, H 2 O, and HCO + . However, the new models produce too little CH + , compared to the observations. 3.3. Intermittent dissipation of interstellar turbulence The chemical properties of low-density regions transiently heated by the intermittent dissipation of interstellar turbulence was investigated by Falgarone et al. (1995). The authors incorporated thin layers of hot gas at low optical depths into their chemical models. For dissipation lengths of the order of ID « 20 AU, burst times of tburst — 3 x 103 - 3 x 104 yr, the authors obtained turbulent heating rates of T = 3 x 10~ 21 - 1 x 10~23 erg s~l cm'3.
(3.4)
Gredel: Chemistry in turbulent
flows
145
Resulting equilibrium temperatures range from 3500 K to 240 K. The material rapidly cools after timescales of 103 years. The models do reproduce the observed column densities of CH + very well, but fail to explain the observed correlations of N(CH + ) with Ay • Because of the stochastic nature of the bursts, and the location of the dissipation zones, correlations of the CH + column density with general parameters of the cloud, such as the visual extinction, are not expected. The models of Falgarone et al. (1995) suggest that large column densities of OH and HCO + are produced in low density regions of molecular clouds as well. The millimeter absorption line measurements of Liszt & Lucas (1994a) have shown that HCO + and HCN is generally very abundant in translucent clouds. The broad HCO + line profiles measured by Liszt & Lucas (1994b) toward £ Oph, at velocities where no CO is detected, were interpreted by Falgarone et al. (1995) in terms of a production of HCO + in hot and low-density material. An alternative explanation for the broad HCO + lines and the large abundance of HCN and HCO + in translucent clouds is given below. 3.4. Non-Maxwellian velocity distributions In a hot gas, large amounts of OH are produced by the endothermic reaction O + H2 -> OH + H
(3.5) 13 27
3160 r
which has a reaction rate coefficient of k = 3.4 x i o - r e / (Millar et al. 1997). Thus, if a hot gas is responsible for the production of interstellar CH + , the simultaneous production of large amounts of OH can not be avoided. Observed abundances of OH in translucent clouds are well explained by low-temperature chemical models. Because of the similar activation energies of reactions (2.1) and (3.5), a correlation between N(CH + ) and N(OH) is to be expected. This is in conflict with recent observations by Federman, Weber & Lambert (1996b) towards Per OB2. In addition, van Dishoeck (1998) point out that the sulfur chemistry which proceeds in a hot gas is characterized by ratios of SO/SO2 ~ 1500 which. This is in conflict with observed values of SO/SO2 = 1-15 in translucent clouds. It is thus questionable whether large amounts of hot gas prevail in translucent clouds. A non-thermal origin of CH + in a cool gas was suggested by Gredel et al. (1993). The authors speculated that if a fraction of C + attains a significant non-thermal motion as a result of Alfvenic turbulence, the CH + formation may proceed efficiently in regions where the surrounding material is cool. This scenario is not in conflict with the broad and Gaussian line profiles for CH + which are generally observed (e.g. Lambert et al. 1990). Because collisions of CH+ with H and H2 are reactive, CH + is not thermalized by collisions. Thus, the broad CH + line profiles may merely reflect the underlying velocity distribution of H2 or C + from which CH + forms. Because Alfvenic turbulence affects only the ions, endothermic reactions such as (3.5) are not affected. The problem with the overproduction of OH is thus avoided. In his PhD Thesis, Spaans (1995) has investigated in detail the production of CH + by a non-Maxwellian velocity distribution. He calculated the probability distribution P(vn) of the velocity increments in a turbulent cascade of kinetic energy from large to smaller scales. From P(vn), the CH + formation rate is readily obtained using R(CH+) = f P(v)va(v)d3v
(3.6)
where v is the velocity and o~(v) is the cross section of reaction (2.1). A CH + formation rate of 1 — 2 x 10~13 cm3 s" 1 which is required to maintain the mean observed CH + abundances is readily obtained for values of AD = 2-3 km s" 1 . At/2 is proportional to the total kinetic energy injected into the medium on the largest scales.
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Gredel: Chemistry in turbulent flows
The turbulent heating rates obtained by Spaans (1995) range from r = 1(T 25 - 1(T 23 erg s" 1 cm'3 -1
(3.7)
1
for values of Av — 2 km s and 5 km s" , respectively, and are typically an order of magnitude lower than the heating rates expected from photoelectric heating. Turbulent heating may thus be neglected. The non-thermal enhancements of ion-neutral reactions by Alfvenic turbulence in a cool gas were modelled by Federman et al. (1996a) by means of an effective temperature given by
-kTeff
= -kTkin + -nv1n
(3.8)
where Thin is the kinetic temperature of the gas, vin is the relative ion-neutral drift velocity, and fj, is the reduced mass of the system. For Alfven speeds of v& — 2 — 4 km s" 1 , resulting CH+ fractional abundances range from « 10~9 - 10" 7 , which is close the values inferred from observations. Hogerheijde et al. (1995) pointed out that large amounts of HCO + and HCN are produced via reactions which involve CH + . Consequently, broader line widths of HCO + and HCN are expected, as the CH + velocity distribution is inherited to HCO + and HCN. The authors showed that the line widths and the abundances of HCO + and HCN are consistent with formation in a cool medium and a turbulent energy input characterized a value of Av — 3.5 km s" 1 .
4. Summary Dissipation of interstellar turbulence in translucent molecular clouds may alter the chemistry by providing additional energy to drive endothermic reactions which are inhibited in cold molecular clouds. The dissipation of the non-thermal motions may result in thermalization, and create either hot boundary layers or temporarily heated parcels of gas in low-density regions and at the surface of molecular clouds. Alternatively, the dissipation may create a fraction of atoms and molecules with non-Maxwellian velocities distributions. Both scenarios provide new and interesting physical processes which may solve the long-standing problem related to the formation of interstellar CH + . REFERENCES BALM, S.P. & JURA, M. 1992, A&A, 261, L25
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Supersonic Turbulence in Giant Extragalactic HII Regions 23 By JORGE MELNICK1, GUILLERMO TENORIO-TAGLE 24 AND ROBERTO TERLEVICH '
European Southern Observatory, Casilla 19001, Santiago-19, Chile 2
INAOE, Apartado Postal 51, Puebla 72000, Mexico
institute of Astronomy, Madingley Road, Cambridge CB3 OHA, UK. 4
Royal Greenwich Observatory, Madingley Road, Cambridge CB3 OHA, UK.
The physical mechanism responsible for the supersonic broadening of the integrated emission lines of Giant HII Regions (GHR) to velocities well above the sound speed of the ionized gas is yet not clear. The observational evidence is reviewed and possible physical mechanisms discussed in this paper. It is shown that hydrodynamical turbulence and thermal motions dominate the kinematics of the gas at small scales while gravity and stellar winds are responsible for the width of the integrated line-profiles. The relative contribution of these two dominant mechanisms depends on age. Gravity dominates in young nebulae whereas expanding shells dominate when the most massive stars become supergiants.
1. Introduction More than their large sizes, the key defining property of Giant HII regions (GHIIRs), as a distinct class of objects, is the supersonic velocity widths of their integrated emissionline profiles (Smith & Weedman 1972; Melnick 1977; Melnick et al. 1987 and references therein). Since supersonic gas motions will rapidly decay due to the formation of strong radiative shocks, the detection of Mach numbers greater than 1 in the nebular gas poses an astrophysically challenging problem. Melnick (1977) suggested that the ionized gas is made of dense clumps moving in an empty or very tenuous medium, so that the integrated profiles reflect the velocity dispersion of discrete clouds rather than hydrodynamical turbulence. In this model, the relevant time scale for radiative decay of the kinetic energy is the crossing-time of the HII regions which turns out to be comparable to the ages of the ionizing clusters. This idea was refined by Terlevich & Melnick (1981) who showed that the sizes (R) and luminosities (L) of giant HII regions correlate with velocity dispersion (a) as R oc a2, and L ex a4, and that these two correlations extend to a similar but more luminous class of objects called HII galaxies. The fact that these correlations are similar to the relations exhibited by virialized self-gravitating stellar systems such as globular clusters, spiral bulges, and elliptical galaxies led Terlevich and Melnick to propose that GHIIRs and HII galaxies themselves are virialized systems and therefore that the velocity dispersion of the gas is a direct measure of their total mass. Although Gallagher and Hunter (1983) failed to confirm the relations in a particular sample of extragalactic nebulae,! the Terlevich and Melnick relations have been subsequently confirmed by Hippelein (1986) and Roy et al. (1986) and definitively established and calibrated by Melnick et al. (1987, 1988), albeit with^slopes slightly different to those found by earlier papers. It is interesting to note in this context that the analysis of more complete datasets for elliptical galaxies also yields f This result is now understood as due to the fact that their sample was dominated by HII regions with subsonic line widths. 148
Melnick et al.: Turbulence in Giant HII regions
149
slopes somewhat different from the scaling laws resulting from the simplest application of the Virial theorem (see Busarello et al. 1997 for a recent review). Tenorio-Tagle et al. (1993) proposed a model called the Cometary Stirring Model (CSM) to explain the origin and persistence of the supersonic gas motions. In the CSM kinetic energy is continuously injected to the interstellar medium by the bow shocks and wakes caused by low-mass stars undergoing winds while moving in the gravitational potential of the ionizing clusters. The gas thus stirred is subsequently ionized by massive stars which form at a later stage in the collapse. The CSM predicts that the nebular gas profiles in the densest regions of GHIIRs, should be smooth and well approximated by Gaussians of widths avir = y/GM/R. Thus, in both gravity driven models the global line profile is made of individual "clouds" (filaments) of gas moving at supersonic velocities. The main difference is that the CSM scenario is able to support and maintain the supersonic motions owing to the energetics of a large number of low-mass stars, while in the pure gravitational model collisions and dissipation in shocks will rapidly deplete the number of clouds. An entirely different broadening mechanism was investigated by Dyson (1979) who proposed that the observed line profiles reflect the combination of several unresolved stellar wind-driven expanding shells and filaments. In order to match the observed line profiles, Dyson's model required the expansion motions to be driven by the combined effects of groups of tens to hundreds of stars, all having essentially the same ages. In this model, the core of the global line-profiles is produced by an ionized but dynamically unperturbed medium with a thermal velocity. There is indeed ample observational evidence for the presence of expanding shells in nearby nebulae, fact that has led several authors to adhere to stellar wind driven shocks as the dominant line broadening mechanism in GHIIRs (Rosa and Solf 1984; Chu and Kennicutt 1994 and references therein). This scenario, however, has been recently analyzed in detail by Tenorio-Tagle et al. (1996), who modeled the integrated emission line profiles of collections of unresolved expanding shells for a wide range of input parameters. They concluded that, unless the age distribution of the ionizing stars is unusually strongly peaked, the resulting line profiles are always flat topped, contrary to what is observed. This conclusion was in fact anticipated by Dyson (1979), who concluded that star formation in GHIIRs had to be coeval and not sequential. The detailed numerical models, however, show that the age spread required to fit the observations is significantly shorter than the time scale for gravitational collapse of the proto-stellar cloud, result that rules out expanding shells as the sole line broadening mechanism. Thus, while shells can account for the wings of the integrated profiles, something else is needed to explain the supersonic line cores. In this contribution we review the status of the problem on the basis of a detailed study of the gas kinematics in the largest giant HII regions in the LMC (30 Doradus), and M33 (NGC 604).
2. Hydrodynamical Turbulence in Giant HII regions The first detailed investigation of the gas kinematics of a Giant HII region was presented by Smith and Weedman (1972). They used a single channel pressure scanned Fabry-Perot interferometer to map the nebula with a resolution of 13" (~ 3pc). These data were used by Melnick et al. (1987) to analyze the structure function of the gas turbulence in 30 Dor. They found that the structure function, < AV2 >, correlates with projected separation, A, as < AV2 >oc A 016 for A < 20 pc
150
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and, < AV2 > = constant for A > 20 pc From the fact that the structure function is much shallower than that predicted by turbulence theory, Melnick et al. concluded that not even at the smallest spatial scales is hydrodynamical turbulence present in the nebular gas. The number of pairs used in this analysis, however, was rather small (1265) and the error bars attached to each point were correspondingly large. Unfortunately, better sampled maps are not yet available for 30 Dor, but excellent 2D Fabry-Perot scans do exist for the next nearest first ranked giant HII region, NGC 604 in M33. Two groups have observed NGC 604 from La Palma using two different telescopes and two different interferometers (Yang et al. 1996 and Sabalisck et al. 1995). The latter data set was used by Medina-Tanco et al. (1997) to study the structure function. They find that the function resembles the Kolmogorov spectrum in 3 Dimensions but, because they observe a break in the function at a projected separation of about lOpc, they conclude that a double cascade of energy in two dimensions is a better interpretation of the data. This result, however, appears to be inconsistent with the observations of the line-profile widths. Yang et al. (1996) and Muiioz-Tunon et al. (1996) have analyzed the distribution of profile widths in NGC 604. The first group finds the profiles in the brightest regions of the nebula to be single Gaussians of widths similar to the global profile width of the nebula (a ~ 18km s" 1 ). This result has two immediate implications. The first is that the contribution of expanding shells to the overall profile is negligible, and the second that the radial velocity dispersion ( 10 km s" 1 (Reid et al. 1988, Gwinn et al. 1992). These motions are too great to result from measurement errors, and, at least in the case of Doppler velocity, must arise from the actual motions of gas molecules. As an example, figure 1 shows many H2O maser features in a small region in the W49N cluster of H2O masers. The random motions can justly be called turbulent. The random velocities vary over much smaller scales then the global velocity field of the maser cluster: features close together on the sky may have quite different velocities, as figure 1 shows (Gwinn 1994a). Moreover, Doppler velocities show power-law correlation functions of difference in velocity with position (Gwinn 1994a), as figure 2 shows. Such power-law correlation functions are characteristic of fluid turbulence ( Tatarski 1961, Goldreich & Sridhar 1997). The two-point correlation function of maser features also follows a power law (Walker 1984, Gwinn 1994a), similarly suggestive of maser emission from a turbulent fluid (Strelnitski 159
160
Gwinn: Hypersonic Turbulence of H2 0 Masers -40 -
10* 10 Flux Density (Jy)
-270
-275 Sky Position (mas)
-280
1. Left: Flux density plotted with Doppler velocity for maser features within a region 13 mas (140 AU) square, in W49N. Note that Doppler velocity runs vertically, increasing downward. Hatched areas indicate spectral regions not covered, because of instrumental limitations. Right: Positions of maser features within the 13 mas-square region. The connecting dotted lines relate maser features in the spatial and spectral plots. The maser features tend to cluster strongly: if features were distributed evenly over the ~1 arcsec2 cluster, the 15 features in this region would occupy more than 600 times this area. The spectrum shows flux densities of only compact emission; comparison with single-dish spectra indicates that a significant fraction of the flux density is resolved out at high angular resolution (Liljestrom et al. 1989). Reproduced from the Astrophysical Journal (Gwinn et al. 1994a). FIGURE
& Nedoluha 1997, Sobolev et al. 1998). Figure 2 shows this correlation function in right ascension. Masers with separation of a few AU must be nearly always physically associated, because the two-point correlation function is sharply peaked. For example, 3 features lie within 2 mas of the brightest feature shown in figure 1. Under the assumption that the correlation function in declination is similar to that in right ascension, the probability of finding 3 other features within this distance by chance is less than 10~ 9 : the neighboring features must be associated. Their Doppler velocities range over 22 km s" 1 . This is consistent with the velocity correlation function in figure 2, which shows a median velocity difference of 15 km s" 1 at separation 1.8 mas. The difference in Doppler velocity of masers separated by a few AU of ~ 20 km s" 1 is much less than the > 200 km s" 1 velocity of the wind required to drive the maser outflow, but it is much greater than the sound speed of 1 km s" 1 , or the typical Alfven speed of 0.8 km s" 1 (Liljestrom and Leppanen 1998). Indeed, it is greater than the maximum Alfven speed of 8.7 km s" 1 that Liljestrom and Leppanen derive. Thus, the inferred turbulence is hypersonic. The random motions probably take place in a relatively thin zone, ~ 300 AU thick, where wind meets ambient material. This 300-AU dimension is visible as the thickness of the "arcs" of maser features seen in images of major clusters. At these locations, the stellar wind transfers tremendous energy to ambient material, with specific momentum sufficient to drive masers, with expected densities of n ^ 109 cm" 3 , to velocities of ~ 200 km s" 1 . The region of maser activity is likely laced with many energy-dissipating shocks, only a small fraction of which produce observable maser activity. The 300-AU thickness of the regions of concentrated maser features appears as the "outer scale", or maximum spatial scale of the power law, for the correlation functions in figure 2. In
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FIGURE 2. Left: Velocity correlation functions with separation in right ascension, for spots of H2O maser emission in W49N. Points show median (Median: stars) and root-mean-square (RMS: circles) of the difference in Doppler velocity Vz, per logarithmic interval in right ascension Ax. The median and RMS were calculated from the observed distribution for each logarithmic interval. Both median and RMS decline smoothly with Ax to Ax « 300 ^as, where the median drops quickly to about 0.5 km s" 1 . The straight lines show best-fitting power laws for 300 /zas < Aar < 30 mas, with the forms med{Vz} oc |Ax| 0 3 2 5 and RMS{V2} oc |Ax| 0 2 3 4 . Right: Two-point correlation functions with separation in right ascension, for H2O maser features in W49N. Straight line shows the best-fitting power law, with the form n/ oc Ax~ 0 1 8 . Reproduced from the Astrophysical Journal (Gwinn 1994a).
this zone, fluid instabilities, perhaps including the instability of C-shocks to corrugation (Wardle 1990), could disrupt the interface between wind and ambient material to produce a highly corrugated surface, and rapidly-varying velocity field, that the observed powerlaw correlation functions characterize. Theoretical models for turbulence commonly predict power-law spectra. Such spectra commonly arise from turbulent cascades, where the fluid equations act to transfer energy from larger to smaller spatial scales (or from smaller to larger) in a self-similar fashion. The archetype of such models is the Kolmogorov spectrum, for which root-mean-square velocity difference AV varies with position difference Ax as AV oc Ax 1 / 3 . This theory assumes that the fluid is incompressible and isotropic. Highly-supersonic velocity differences over small distances, as seen in figures 1 and 2, should destroy turbulent motions in about a sound crossing time, "short-circuiting" any cascade. Supersonic turbulence thus usually denotes a random collection of shocks permeating a fluid volume. In the simplest case, the shocks are randomly positioned and have random velocity difference. In this case the velocity difference varies like a random walk, so that AV oc Ax 1 / 2 . Such a spectrum departs more from the correlation functions shown in figure 2 than does the Kolmogorov spectrum. A model for a random superposition of shocks does not describe the hypersonic turbulence of H 2 O masers. The distribution of velocity difference for H 2 O masers in W49N is highly non-Gaussian, with high wings extending to large velocity separation. Because of instrumental limitations of observing bandwidth, as shown in figure 1, the sampled distributions are not complete at large velocity difference. Because the rms velocity AV weights large velocity separations heavily, we adopt the median velocity difference, for each bin in Ax, as a better measure of the distribution. Interestingly, the median velocity difference follows the Kolmogorov index closely. A fully random collection of shocks would produce a steeper power-law inconsistent with observations. Other H 2 O maser clusters also show Kolmogorov-law correlation functions (Strelnitski &; Nedoluha 1997).
162
Gwinn: Hypersonic Tkirbulence of H2O Masers
A variety of factors might help to explain the observed nearly-Komogorov power-law spectrum, in the face of hypersonic differences in Doppler velocity. The masers very likely have densities greater than surrounding material, perhaps by 100 x, which would reduce dissipation. Maser features have lifetimes of only a few months, during which they show internal motions of order the sound speed, of about 1 km s" 1 . Thus, the turbulent motions are a short-lived phenomena for the participating gas, although a long-lived phenomenon for the source. The density of ambient material, as well as the momentum of the stellar wind, influence maser speed. If H2O masers form in a region of varying density, density variations (perhaps following a power law) will influence the velocity correlation function. High-dynamic-range mapping of maser features in regions like that shown in figure 1, and studies of extended structures surrounding masers (Gwinn 1994c), should help to distinguish among these possibilities.
3. Turbulence within Maser Features Individual clouds of masing gas are not quiescent, but have internal motions. These motions contribute a non-thermal component to maser linewidths (see Liljestrom & Leppanen 1998). Linewidths of low-and high-velocity features are different: for the sample of Gwinn (1994b), the median FWHM linewidth is 0.52 km s" 1 for features within the range of Doppler velocities of the molecular cloud (—2 to +18 km s" 1 ), 0.80 km s" 1 for features outside this range, and 1.20 km s" 1 for the highest-velocity features (Doppler velocity > 100 km s" 1 ). High-velocity features also show a variety of indications of greater internal motions, including shorter lifetime, lower flux density, and larger linear size. A feature at Doppler velocity 138 km s" 1 shows the broadest emission, over 6.2 km s" 1, for a flux-density weighted RMS linewidth of 4.7 km s" 1 . Internal motions of maser features are also responsible for small changes in the centroid of emission from a single features with Doppler velocity, barely visible in figure 1. They also result in differential motion within features, leading to distortion of features and their ultimate destruction (Liljestrom 1989). Velocities on the order of 0.8 km s" 1 characterize all of these internal motions. Liljestrom and Leppanen (1998) infer that this is the typical Alfven velocity in the cluster, suggesting that these effects arise from Alfven turbulence within the maser features. They suggest that the maximum Alfven velocity is about 8.7 km s" 1 , in the high-velocity features.
4. Conclusions H2O masers in star-forming regions present an excellent laboratory for studying supersonic turbulence. Correlation functions show the power-law signature of turbulence over 2i orders of magnitude in separation, from 300 AU to 1 AU. The index of the power law is approximately consistent with the Kolmogorov value; the index is not consistent with that expected for a random superposition of shocks. Random motion is also seen within maser features, indicating Alfvenic turbulence on these sub-AU scales.
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163
Water Masers Tracing Alfvenic Turbulence and Magnetic Fields in W51M and W49N By T. LILJESTROM AND K. LEPPANEN Metsahovi Radio Observatory Helsinki University of Technology, Otakaari 5. A, FIN-02150 Espoo, Finland We present sub-milliarcsecond linear polarization results of 22 GHz water masers in W51 M, and some statistically significant characteristics of water maser outbursts in W49 N. Two different methods are used to extract the fluctuating part of the preshock fluid velocities and magnetic fields in these dense high-mass star-forming regions.
1. Linear Polarization Observations of Water Masers in W51 M High-resolution polarization observations of water masers provide a powerful tool for studying Alfvenic turbulence and magnetic fields in dense circumstellar regions. Here we present some main results of the first 22 GHz linear polarization observations of water masers in the central low-velocity range of W51M, 54 < Visr < 68 km s" 1 , obtained with VLB A (Leppanen, Liljestrom, & Diamond 1998). The principal difference of polarimetric VLBI from total intensity VLBI is the need to calibrate the instrumental polarization parameters, which have been solved by Leppanen (1995) with a feed self-calibration algorithm (see also Leppanen, Zensus, & Diamond 1995). The uniformly weighted restoring (CLEAN) beam obtained was 0.71x0.26 mas; the velocity resolution was 0.2 km s" 1 . Figure la shows the spatial distribution of the maser spots. Superimposed on the spots are the linear polarization vectors with their lengths proportional to the degrees of polarization. The inset of Figure la is an enlargement of the compact maser concentration near the reference position (0,0) of W51M. The dotted line in the inset separates blueshifted (west of the dotted line) and redshifted (east of the dotted line) maser spots with respect to the velocity centroid, 61.5 km s" 1, of this maser concentration, hereafter called the protostellar cocoon. With a distance of 7.0 kpc to W51M, the inner and outer radii of this maser cocoon are approximately 5 AU and 66 AU, respectively. Figure la reveals also a 1200 AU long linear maser structure at a position angle of 200°, which is roughly aligned with the galactic magnetic field projection on the sky and the polarization position angle of these masers. Figure lb shows that these masers move longitudinally along this direction with a median velocity of 25 (±8.7) km s" 1 relative to the centroid of the cocoon. The proper motions exclude the interpretation of this streamer as a low-velocity bipolar outflow from W51 M. Most likely the stream is produced by shocks caused by, e.g., the nearby expanding HII region, W51IRS 1, which interacts with the dense molecular core of W51M on its western side. The fact that the proper motions are along the assumed shock front can be explained by the time evolution simulations of Boss (1995) for an "outside-in" collapse of a massive star. At later times (i.e., at 0.6tfree-fau) this model shows a large-scale linear stream, which moves along the initial planar shock front (the original triggering agent for the collapse). In contrast to the cocoon masers, which show a mean linear polarization of only 3% (maximum 13%), the masers in the streamer exhibit higher degrees of linear polarization (mean 12%; maximum 35%). The linear polarization results of our study are in good agreement with the classical maser polarization theory of Goldreich, Keeley, & Kwan (1973a, 1973b). 164
Liljestrom & Leppanen: Water Masers Tracing Alfvenic Turbulence
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FIGURE 1. Left (a): ThefirstVLBI linear polarization image of water masers (marked as circles) in W51 M. The lines show the direction of linear polarization; their lengths are proportional to the polarization degrees of the spots (1 mas — 1%). The inset is an enlargement of the protostellar cocoon near the reference position (0,0) of W51M (pol. lines: 1 mas = 2%). Right (b): Proper motion vectors of the observed water masers in W51 M (Leppanen, Liljestrom & Diamond 1998).
When the interstellar medium contains a well coupled magnetic field, disturbances propagate at the Alfven velocity, VA = (B^/Airp)1^2, where p is the mass density of the medium. Waves with super-Alfvenic fluid motions generate shocks and dissipate rapidly. Hence the fluctuating part of the fluid velocities will generally become sub-Alfvenic. Because the linear polarization vectors of the streamer in W51M have a well-defined mean direction, the turbulent motions in the medium are probably more wavelike than eddylike. Since the turbulent velocity fields produced by shocks induce turbulent magnetic fields, the level of magnetic fluctuations is related to the associated fluctuations in the kinetic energy by the principle of equipartition (McKee et al. 1993). A velocity perturbation, SV, perpendicular to the field distorts the field lines and induces a transverse magnetic field component, SB, which obeys the relation 5B/B = 5V/VA- The left-hand side of this relation determines the angular deviation, 6, of the linear polarization vectors from the magnetic field and can be replaced with it. If SV is random, as in turbulence, then the above relation can be averaged over all data points resulting to an Alfven velocity of VA =
5Vrms/drms.
The quantity Srms in the above relation is obtained from the angular deviations of the linear polarization vectors from the mean, and found to be 27.8° X7r/180° for the streamer masers. The flux density-weighted velocity dispersion of the streamer spots around the mean velocity of the spots is 0.54 km s" 1 . Thus an Alfven velocity of 1.1 (±0.23) km s" 1 results for a pure wavelike turbulence (see the arguments given in the previous chapter). In case the turbulent motions would be more eddylike, the thermal velocity dispersion should first be subtracted from the observed velocity dispersion in order to obtain the non-thermal velocity dispersion, which in this case would represent SVrms in the VA expression. In W51 M a temperature around 250 K is most likely (Jaffe et al. 1989). Thus, for an assumed eddylike turbulence an Alfven velocity of 0.86 (±0.23) km s" 1 would result at 250 K. These Alfven velocities yield a magnetic field parameter, b = V^/1.84 km s~1,
166
Liljestrom k Leppanen: Water Masers Tracing Alfvenic Turbulence
(Hollenbach k McKee 1989), of 0.47 - 0.60 (±0.13). The preshock magnetic field strength (perpendicular to the shock velocity) is obtained from Bo = b n ° 5 fiG (Hollenbach k McKee 1989), where the preshock hydrogen nuclei density of W51M, no = 3.8xlO6 cm" 3 , has been adopted from Plume et al. (1997). The observational parameters, VA and n o , yield thus a preshock field strength of 0.91 - 1.2 (±0.32) mG. Inside the masing regions the strength of the magnetic field is independent of the preshock field, since the magnetic pressure (which is determined by the ram pressure of the shock), dominates in the masing region (Hollenbach, Elitzur, k McKee 1993). Assuming that the median space velocity of the maser stream, 25 km s" 1 (with respect to the centroid of the cocoon), characterizes also the the shock velocity, we obtain (using eq. [4.6] of Elitzur, Hollenbach k McKee 1989) a typical total magnetic field strength around 38 mG inside the masing regions of the streamer.
2. Alfvenic Turbulence and Magnetic Fields in W49 N A long-term 22 GHz monitoring program of W49 N was carried out (Liljestrom et al. 1989) with the Metsahovi radio telescope during the same time and with the same velocity resolution as the 5-epoch VLBI observations of this source (Gwinn, Moran, & Reid 1992; Gwinn 1994a, 1994b, 1994c). The water maser outbursts were documented by fitting Gaussians to the individual line components of the H2O spectrum as function of time. Some 145 outbursts were covered reasonably well during the outbursts, and form the data base for statistical analyses (Liljestrom 1998). Here we report only results concerning the linewidth distribution of the outbursts and the flux excess associated with the outbursts. These, together with Gwinn's VLBI results, yield estimates for the characteristic preshock and postshock Alfven velocities as well as the magnetic fields in W49N. During the W49 N monitoring period, maser outbursts were found in the velocity interval of-240 to 180 km s" 1 . Inside the velocity range of the dense ambient medium (i.e., -2 to 18 km s"1) only 21 flares occured, whereas outside this velocity interval 124 outbursts were detected. The strongest outbursts occured within the velocity range of the ambient dense gas. For these the distribution of AF, i.e., the increase in flux density during an outburst, peaks at 104 Jy and extends over one order of magnitude. Outside the velocity range of the dense ambient gas, the A F distribution peaks at 103 Jy and extends over two orders of magnitude. Noteworthy, a flux excess, F - AF, is associated with all outbursts. This flux excess is clearly correlated with A F of the outbursts. The quite linear function between the flux excess and A F reveals that only some half of the measured flux density of an outburst feature corresponds to the actual outburst; the other half (i.e., the flux excess) is attributed to coherent scattering in ambient plasma, in accordance with Gwinn's detection (1994c) of maser halos in W49 N. He found that maser halos are associated with many compact maser features, and arise from coherent scattering within the maser cluster, most probably by ion sound waves. According to McKee k Hollenbach (1987) and Hollenbach k McKee (1989; see their Fig. 11) the ionization of the dissociated ambient H2 gas (i.e., the H gas) to H + gas increases rapidly when the shock velocities exceed 80 km s" 1 ; at shock velocities larger than 120 km s" 1 the upstream H2 gas is fully photoionized. The mean space velocity of the high-velocity masers in W49N is 100 km s" 1 (from Table 4 of Gwinn et al. 1992). Therefore, the presence of a dense ionized ambient medium is very likely in W49 N. The linewidth (FWHM) distribution of maser outbursts outside the velocity range of the dense ambient cloud is very symmetric and peaks at 1.10 km s" 1 . However, the VLBI data (in this same velocity range) yield a median FWHM linewidth of 0.80 km s" 1
Liljestrom & Leppanen: Water Masers Tracing Alfvenic Tkirbulence
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for the compact maser features (Gwinn 1998). Most probably the line broadening seen by the single dish is also caused by the coherent scattering of the maser radiation in the ambient dense plasma (this more extended emission may be partly resolved out with the VLBI). The difference in the single-dish and VLBI linewidths yields a linewidth component (FWHM) of 0.75 km s"1 for the scatterers, which is roughly the same as the linewidth of the compact maser emission. The linewidth of the scatterers implies a characteristic velocity dispersion of 0.32 km s" 1 . Thus an Alfven velocity, \/3crnon-th, of 0.55 km s"1 , and a preshock magnetic field parameter, b, of 0.30 results. This gives a preshock magnetic field (perpendicular to the shock velocity) of about 0.60 mG for a preshock hydrogen nuclei density of 4xlO 6 cm" 3 (Serabyn, Glisten, & Schulz 1993). We note that with the observational inputs of (1) b = 0.30 (this study), (2) the typical Vshock of 100 km s" 1 (Gwinn et al. 1992), and (3) the preshock hydrogen nuclei density of 4xlO 6 cm" 3 (Serabyn et al. 1993), the dissociative shock model of Hollenbach & McKee (1989, their eq. [3.4]) predicts a thickness of 1.1 AU for the postshock H2 formation plateau. The H2 formation heating is responsible for a temperature plateau of typically 350 K in W49 N providing a natural environment for the masing gas. The obtained thickness of the postshock H2 formation plateau is in excellent agreement with the typical diameter of a maser feature, 0.1 mas = 1.1 AU, found by Gwinn (1994a, see his Figs. 3 and 5 or Fig. 2 of Gwinn 1998) from two-point correlation functions of maser features. Since also the predicted (Hollenbach et al. 1993) typical total magnetic field strength inside the high-velocity masers, 126 mG, agrees with the observed one, 120 mG (Fiebig & Glisten 1989 [the F = 6 - 5 hyperfine transition]), it is evident that the dissociative shock model of Hollenbach & McKee (1989) and the maser model of Elitzur, Hollenbach, & McKee (1989) fit the conditions in W49N extremely well. Because the field value of Fiebig & Giisten (1989) yields only the component parallel to the line sight, we have estimated the total field strength by assuming that the inclination of the maser outflow axis (Gwinn et al. 1992) is roughly the same as the inclination of the field. Since the observational quantities obtained for W49N fix all other parameters in the above models, these observationally more hidden parameters of astrophysical interest can be straightforwardly determined (Liljestrom 1998). With the observational inputs of the typical preshock magnetic field parameter, b, the preshock density, and the shock velocity, the maximum magnetically supported postshock density of a typical water maser in W49N is 1.0 xlO9 cm" 3 (eq. [2.4] of Hollenbach & McKee 1989). The maximum value of the magnetic field parameter inside a high-velocity maser, bmaser, is thus 4.7 (eq. [4.9] of Elitzur et al. 1989), which corresponds to a maximum Alfven velocity, b m a s e r x 1-84 km s" 1 , of 8.7 km s" 1 in the masing region. In reality the postshock density may be lower than the theoretical maximum. Since the ratios bmaser/b, amaser/a, and V^(maser)/VOi all scale as (n m a s e r /n o ) 1 / 2 , the observed velocity dispersions of the masers and the preshock medium yield a characteristic VA (maser) of 0.8 km s" 1 and a maximum VA(maser) of 3.4 km s"1 , obtained from the maximum maser linewidth, 4.7 km s"1 , in the VLBI data (Gwinn 1998). This suggests, that the density enhancements in the high-velocity masers of W49 N are typically around one order of magnitude and extend at most to some 2 orders of magnitudes. On the other hand, the above VA (maser) estimates (based on amaser) may be lower limits, since infrared line radiation trapped between the molecular levels of H2O is capable to maintain narrow linewidths in saturated masers (Goldreich & Kwan 1974). REFERENCES Boss, A. P. 1995, ApJ, 439, 224
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Turbulence in the Ursa Major cirrus cloud By MARC-ANTOINE MIVILLE-DESCHENES1 2 , G. JONCAS 2 AND E. FALGARONE3 'institut d'Astrophysique Spatiale, Bat. 121, Universite Paris XI, F-91405 Orsay, Prance 2
Departement de Physique, Universite Laval and Observatoire du mont Me'gantic, Quebec, Quebec, Canada, G1K 7P4 3 Radioastronomie, Ecole Normale Superieure, 24 rue Lhomond, 75005, Paris, France
High resolution 21 cm observations of the Ursa Major cirrus revealed highly filamentary structures down to the 0.03 pc resolution. These filaments, still present in the line centroid map, show multi-Gaussian components and seem to be associated with high vorticity regions. Probability density functions of line centroid increments and structure functions were computed on the line centroid field, providing strong evidences for the presence of turbulence in the atomic gas.
1. Introduction Many statistical studies of the density and velocity structure of dense interstellar matter have been done on molecular clouds where turbulence is seen as a significant support against gravitational collapse that leads to star formation. Less attention has been devoted to turbulence in the Galactic atomic gas (HI). The cold atomic component (T ~ 100 K, n ~ 100 cm""3), alike molecular gas, is characterized by multiscale self-similar structures and non-thermal linewidths. A detailed and quantitative study of the turbulence and kinematics of HI clouds has never been done. Here we present a preliminary analysis of this kind based on high resolution 21 cm observations of an HI cloud located in the Ursa Major constellation. To characterize the turbulent state of the atomic gas, a statistical analysis of the line centroid field has been done. We have computed probability density functions of line centroid increments and structure functions. 2. HI Observations The Ursa Major cirrus (a(2000) = 9/i36m, Mj ~ 0.35 M©. Figure 3 shows local slopes for 400 different IMFs, one slope per IMF. Each box has 100 local slopes: the left and right boxes are for IMF models with 2500 and 104 stars, and the top boxes use H = 8 and N = 3, while the bottom boxes use H = 9 and N = 2. These local slopes were determined for one mass interval per IMF model, using a mass range that spans an order of magnitude and is randomly positioned within the total mass range of the calculation. They were derived from the cumulative mass functions to avoid the problem of empty bins in the histogram representation. In all cases, the slopes, x, are ~ 0 at low mass because of the flattening in the IMF, and they hover around x ~ 1.3 at intermediate mass; they are inaccurate at high mass because there are few stars in the high mass bins. Evidently, the fluctuations from case to case are large. The average
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Total Stellar Mass (Mo ) 0.1 0.3 1 3 10 30 100 300 Physical Mass (Mo) FIGURE 5. Model IMFs with different numbers of stars, equal to 2500, 5000, 10000, 25000, 50000, and 100000, showing an increase in the maximum stellar mass with increasing numbers of stars. Thefigureon the right shows the mass of the fifth largest star versus the total stellar mass, with an expected slope equal to 1/x = 1/1.3 and a measured slope equal to ~ 1/2.35. The difference between these two slopes is probably the result of a tendency for the mass function to begin to fall faster than x — 1.3 at high mass in the models as a result of the limited range in TV for average number of subclumps per clump TV and number of hierarchical levels H.
numbers of stars in the mass range 1 — 10 M© are 220 and 180 for the H = 8 and H — 9 2500-star models, respectively, and 930 and 500 in the H — 8 and H = 9 104-star models (most of the stars in the model IMFs are outside this mass range). The rms dispersion of the slopes is about 0.5 for the ~ 200 "observed" stars in the 2500-star models, and 0.27 for the 500-900 "observed" stars in the 104-star models. These results should be compared with the rms slopes and the actual numbers of observed stars in clusters whose IMFs are determined in this range. The model IMF fluctuations are consistent with the observed fluctuations, shown in Figure 4, from Scalo (1998), for about the same number of observed stars (Scalo, private communication). In this figure, the open symbols are for the Large Magellanic Clouds and the filled symbols are for the Galaxy. The abscissa is the average stellar mass from the observations, and the ordinate is the observed slope of the IMF. The level of fluctuations is large, around 0.5 rms in the slopes, prompting Scalo (1998) to question whether there is a "universal" IMF, as in the theoretical model discussed here, but in fact, the models in Figure 3 indicate that such large fluctuations are to be expected for this theory. Only the physical origin for the IMF may be universal, i.e., a universal fractal quality to the structure of interstellar clouds on pre-stellar scales. The actual IMF that results from random sampling in such a cloud can vary from region to region, with only the extremely populous clusters, or the averages over many clusters, revealing the underlying universality of the physics. The IMF fluctuations in our model result in part from random fluctuations in the structure of the hierarchical cloud, which, through the randomly varying number of subclumps per clump, has a lot of leverage to change the final IMF. Figure 5 shows the tendency for the largest stellar mass in a cloud to increase with cloud mass. This is an observed effect (Larson 1982), which is easily explained by random sampling from a distribution with a decreasing frequency of high mass stars (Elmegreen 1983, 1997).
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The effect of variations in Mj are shown in figure 6, which compares the IMF from figure 1 with an IMF generated in the same way, using different random numbers and a Jeans mass five times larger. The whole IMF shifts towards higher masses when Mj goes up, but the power law portion is the same. If there is an upper limit to the stellar mass in real clouds, then perhaps this would show up in cases where Mj is large. The model has no such upper limit at the present time, since observations do not demand it. The spike at a program mass of 1 is an artifact of the selection process; this is the lower limit to the mass of a clump that is taken to be a composite of smaller clumps. Note that when Mj is normally small, some brown dwarf stars are possible, but there will be a far smaller number than expected from simply extrapolating the Salpeter power law to the brown dwarf mass. 5. Conclusions The IMF could be universal and scale-free above the physical boundary determined by the thermal Jeans mass. It may have this property because turbulence generates cloud structure in a universal and scale-free way, and the IMF merely samples this structure as stars form. The model discussed here contains no physical gas properties aside from self-gravity in the relative selection rate and lower mass limit, and it contains no physics of star formation. Thus, it should not be considered a "theory" for the IMF, but only an illustration of a possible explanation. Nevertheless, it gives insight into what is likely to be an extremely important role for turbulence in the star formation process. It also illustrates how a universal process can have large fluctuations from region to region and within a single cloud without any significant physical cause. Helpful comments on the manuscript by John Scalo are appreciated.
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ELMEGREEN, B. G. 1983, MNRAS, 203, 1011 ELMEGREEN, B. G. 1997, ApJ, 486, 944 ELMEGREEN, B. G. k FALGARONE, E. 1996, ApJ, 471, 816 ELMEGREEN, B. G. k EFREMOV, Y. N. 1997, ApJ, 480, 235 GILL, A. G. k HENRIKSEN, R. N. 1990, ApJ, 365, L27 HEITHAUSEN, A., BENSCH, F., STUTZKI, J. ET AL. 1998, A&A, 331, 65 HENRIKSEN, R. N. 1986, ApJ, 310, 189 HENRIKSEN, R. N. 1991, ApJ, 377, 500
HILL; J. K., ISENSEE, J. E., CORNETT, R. H., ET AL. 1994. ApJ, 425, 122 HILL, R. S., CHENG, K.-P., BOHLIN, R. C , ET AL. 1995, ApJ, 446, 622 HOYLE, F. 1953, ApJ, 118, 513 HUNTER, J. H., FR. k FLECK, R. C , J R . 1982, ApJ, 256, 505 KENNICUTT, R. C., J R . , TAMBLYN, P. k CONGDON, C. W. 1994, ApJ, 435, 22 LARSON, LARSON, LARSON, LARSON,
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The structure of molecular clouds: are they fractal? By JONATHAN P. WILLIAMS Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA Molecular clouds are observed to be highly structured and fragmented but also follow simple power law relationships between, for example, their size and linewidth as first described by Larson. This self-similarity has led to a fractal description of cloud structure, but in recent years there have been a number of observations that indicate the existence of characteristic scales in molecular cloud cores and clusters of young stars. I present some observations of molecular clouds from large (1-10 pc) to small (0.1 pc) scales, and discuss whether a fractal description of cloud structure is universally appropriate.
1. Introduction The density and velocity structure within a molecular cloud is a remnant of its formation environment and the starting point for the creation of stars. It determines how deeply radiation can propagate through the cloud, and is a critical parameter for understanding the evolution of the ISM. How is it best described? Beginning with Larson (1981), correlations between cloud properties such as linewidth and size have been fit by power laws. Since a power law does not have a characteristic scale, the implication is that clouds are scale-free and self-similar. This has led to statements in the literature that clouds are best described as fractals (e.g. Falgarone, Phillips, & Walker 1991; Elmegreen 1997). On the other hand, other recent studies (Larson 1995; Simon 1997; Goodman et al. 1998; Blitz & Williams 1997) suggest that there are characteristic size and velocity scales in star-forming regions. Here, I show observations of molecular clouds over a variety of scales and discuss a simple technique to test for self-similarity and thereby attempt to answer the question that forms the title of this paper. I conclude that there is no single answer: a fractal description of cloud structure may be valid in certain applications, but in gravitationally bound, star-forming regions there are departures from self-similarity which indicate that an alternative description is more appropriate.
2. Self-similarity in molecular clouds Power law correlations between sizes, linewidths, masses, and numbers can be found from cloud to cloud in the Galaxy (Dame et al. 1986; Solomon et al. 1987) and from clump to clump within a single cloud (e.g. Stutzki & Giisten 1990). However, the range of parameters used for the fit is typically little more than an order of magnitude and to obtain a broader dynamic range it is necessary to combine observations of different objects and/or use observations at different resolutions. Using this approach, Kramer et al. (1998) find that a uniform power law clump mass spectrum, dN/dM oc M~169±0l°, applies to many different types of molecular cloud over almost 8 orders of magnitude in mass from 10~4 M© to 104 M©. Thus it appears that cloud structure really is self-similarf over a wide range of scales and environments. f In this paper, I assume that self-similarity is the same as being fractal. Note, however, that a multi-fractal is not self-similar. 190
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is independent of each other, but with the effect that the total number of pixels in the histogram decreases as the smoothing binsize increases (or as the effective resolution decreases). Therefore, the histograms for the more smoothed datasets have greater relative (Poisson) errors and their shape is less well determined. Another effect of smoothing is that, unless the map is resolved, the peak temperature decreases. The temperature histogram of a smoothed dataset, therefore, has a smaller number of pixels and, generally, a lower peak temperature than the unsmoothed histogram. That is, it shrinks towards the lower left hand corner in Fig. 3. To make a meaningful comparison between the shapes of the histograms, it is necessary to scale the axes. This is achieved by dividing the temperature axis by the peak temperature in the dataset so that it ends at 1 for all histograms. The number of pixels is mostly a function of the observational setup rather than being intrinsic to the cloud, and it is arbitrarily scaled so as to align the histograms as best as possible. Fig. 4 shows such "normalized" histograms for the Taurus dataset and for four other datasets from other clouds. The histograms have been stopped at T/Tpea.k — 0.1 — 0.3 (depending on the signal to noise in the dataset) to avoid the gaussian noise bump shown as the hashed area in Fig. 3. The datasets used in Fig. 3 are 13CO and CO maps of a variety of molecular clouds: the Rosette and Gem OBI are GMCs forming massive stars, NGC1499 and Taurus are smaller molecular clouds forming predominantly low mass stars, and G216-2.5 is a (mostly) starless GMC. There are two histograms for Taurus, one of the original Mizuno et al. map at a resolution of 0.12 pc, and another smoothed to a resolution of 0.8 pc, similar to the resolution of the other maps (0.5 — 0.8 pc). Apart from the high resolution Taurus histogram, the histogram shapes are very similar despite the quite different properties of the clouds (and the molecular tracer). This is another manifestation of the universally similar nature of (large scale) cloud structure. The common exponential shape should also be found in cloud simulations and may provide a simple test of how accurately they represent observations. However, the high resolution Taurus histogram is markedly different: there is a relative deficiency of pixels with T/Tpeay > 0.6. As the dataset is smoothed, the histogram asymptotically flattens to the shape seen in the other clouds (on the other hand, if the other cloud datasets are smoothed, the histograms do not significantly change).
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FIGURE 5. A fractal cloud and the effect of smoothing on its temperature histogram. The left panel shows a grayscale image of the model cloud from T = 0 to 1 K. The image size is 512 x 512 pixels. The right panel shows the temperature histogram for the original dataset (heavy solid line) and then for the same dataset after binning by 2, 4, 8, and 16 pixels. The histogram shape is independent of bin size (i.e. resolution), as expected for this self-similar structure.
So, does this mean that molecular clouds (or Taurus at least) are not fractal? Fig. 5 shows how the temperature histogram behaves for a model fractal cloud (provided by Chris Brunt of FCRAO). As expected for such a self-similar structure, the histogram shape is resolution invariant. Since the behavior of the Taurus dataset changes as it is smoothed, the structure is not self-similar.
4. Small scale structure in star-forming regions The Taurus histogram shape depends on the resolution, and the largest change occurs at a size scale of ~ 0.2 - 0.3 pc. This appears to be due to a rapid increase in column density within the central region of the dense cores within the cloud (Blitz & Williams 1997). There is other evidence for departures from self-similarity at similar size scales. Goodman et al. (1998) find that the size-linewidth relation in dense cores flattens in a central "coherent" region ~ 0.1 pc diameter as linewidths approach a constant, slightly greater than thermal, value. Also, Larson (1995) finds that the two-point angular cor-
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relation function of T Tauri stars in Taurus departs from a single power law at a size scale ~ 0.1 pc (see also Simon 1997). Larson points out that this size scale is close to the thermal Jeans length for gas of temperature, T = 10 K, and density n(H2) = 103 cm" 3 . There is considerable controversy over whether the thermal Jeans mass and length have any physical relevance for the formation of stars (consider also the break in the power law shape of the stellar IMF at M ~ 0.5 MQ). Whether this is true or not, the above studies suggest that cloud structure does change character at small scales and that this might be related to the properties of the stars that they form. To progress, it is necessary to obtain high resolution maps of molecular clouds and to characterize the structure within them in a similar manner as has been done with larger scale maps. Actually, such maps already exist (e.g. Falgarone, Puget, & Perault 1992) and do not show evidence for such a characteristic small scale. However, these studies were made in the parts of a cloud that are gravitationally unbound. The action of gravity, in competition with pressure, provides a natural scale, as in the case of the turbulent Jeans mass (and possibly the thermal Jeans mass) above. That gravity is the dominant force that leads to the creation of stars is understood: to find the scale at which gravity becomes important, and to see if there is a relation between cloud structure and star formation, it is necessary to observe the dense, gravitationally bound, star-forming regions within molecular clouds. Most star formation in the Galaxy occurs in large clusters within GMCs. With the notable exception of Orion, GMCs are several kilo-parsec away, too far for single-dish telescopes to resolve scales < 0.1 pc. It is therefore necessary to use interferometers and, so as to map a large area, to mosaic a number of fields together. Fig. 6 shows preliminary results from a study with the FCRAO and BIMA telescopes to map small scale structure in the Rosette molecular cloud. The observations and analysis are not yet complete, but the goal is to obtain a complete picture of cloud structure from large scales down to individual star formation scales and, eventually, to compare with cloud simulations.
5. Summary Power law fits to relationships between cloud parameters suggest that cloud structure is scale free. However, there may be limits over which such a self-similar description is valid. An upper limit is the turbulent Jeans scale, approximately equal to the largest, most massive, objects in the ensemble, for which their self-gravity balances their internal pressure. There may not be a lower limit in regions, such as cloud "edges", where gravity is not important and a fractal description may be appropriate in this case (e.g. Falgarone et al. 1991). In the case of the Taurus molecular cloud for which a large scale, high resolution, map exists, a simple test comparing structural properties at different resolutions shows that self-similarity breaks down at a size scale ~ 0.2—0.3 pc. Along with other results that show evidence for characteristic size scales in the properties of young stellar clusters and dense, star-forming cores, this demonstrates that self-gravitating systems are not self-similar. A revealing example is to compare the smallest clumps in the aforementioned Kramer et al. (1998) study to the low mass dense cores observed in a recent 1 mm continuum map of the p Ophiuchus cloud (Motte, Andre, & Neri 1998). The former, with masses, M ~ 103~4 MQ, cannot be star-forming and are too small to be gravitationally bound: their mass spectra follow the power law continuation, dN/dM oc M~17, as in more massive clouds, and can be described in the same self-similar manner. For the latter however, many are forming stars, and it is probable that those that are not currently forming stars will eventually do so: gravity is the principal force that guides their evolution and this is
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FIGURE 6. Small scale structure in the Rosette Molecular Cloud. Three images of this massive star-forming GMC are shown at sucessively higher resolution. The top left panel is a large scale map of the velocity integrated 13CO emission in the cloud made with the 7 m Bell Labs telescope. The grayscale ranges from 2.5 to 25 K km s" 1 and the angular resolution is 98", corresponding to a linear resolution of 0.7 pc. The solid box outlines a region of dense, gravitationally bound, but not yet star-forming gas that was observed at higher resolution. The lower left panel shows a map of velocity integrated C18O emission over this region made at a resolution of 49" (0.35 pc) with the FCRAO 14 m telescope. The grayscale ranges from 0.25 to 2.5 K km s - 1 . The circles represent the 113" diameter primary beam of the BIMA interferometer: four pointings are mosaiced together and a map of velocity integrated C18O emission is displayed in the right panel. The FCRAO single-dish data has been combined with the array data to fill in the missing low spatial frequencies. The map has been restored with a circular 10" (0.08 pc) beam in grayscale over the range 0.2 to 2 K km s" 1 . Note that the axes of this map are equatorial, and are rotated with respect to the axes in the left panels. The linear resolution of this map is about an order of magnitude higher than the Bell Labs 13CO map, and the spatial dynamic range is only slightly less. Future work will be to make a comparative study of the structure at the different scales.
reflected in their mass distribution. Motte et al. (1998) claim that the index of the mass spectrum of these cores is steeper than —1.7, and closer to the index, —2.3, of the MillerScalo stellar IMF. Clearly, more studies are needed: maps of the small scale structure in bound, star-forming regions will link the large scale properties of molecular clouds to the stars that they form and have the potential to provide a physical understanding of the stellar mass scale and even the origin of the IMF. Many thanks to Pepe Franco and Alberto Carraminana for inviting me to this most enjoyable meeting. Thanks also to Chris Brunt for educating me about various subtleties concerning fractals and for providing me with some datacubes to play with.
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Diagnosing Properties of Turbulent Flows from Spectral Line Observations of the Molecular Interstellar Medium By MARK H. HEYER Department of Physics and Astronomy and Five College Radio Astronomy Observatory, University of Massachusetts, Amherst, MA 01003 USA I describe the multivariate technique of Principal Component Analysis and its application to spectroscopic imaging data of the molecular interstellar medium. The technique identifies differences in line profiles with respect to the noise level at various scales. It is assumed that such differences arise from fluctuations within turbulent flows. From the resultant eigenvectors and eigenimages, a size line width relationship, (6v ~ r Q ), can be constructed which describes the relationship between the magnitude of velocity fluctuations and the angular scale over which these occur for a given region. From a sample of selected molecular regions in the outer Galaxy, I find the power law exponent varies from 0.4 to 0.7. Thus, the turbulent flows within molecular regions of the Galaxy do not follow the Kolmogorov-Obukhov relation for incompressible turbulence. Implications of these results are discussed with respect to the injection and dissipation of kinetic energy in molecular regions.
1. Introduction In the early, pioneering days of millimeter wave astronomy, the presence of turbulent flows within molecular regions of the Galaxy was inferred from the supersonic line widths of CO spectra. Since that time, telescope and detector technology has advanced such that one can now routinely construct detailed images of molecular emission from which the properties of interstellar turbulence can, in principle, be derived. In practice, statistical descriptions of the observations are required to fully exploit the available information. Examples of such descriptions are the structure and autocorrelation functions of the centroid velocity field (Scalo 1984; Kleiner k Dickman 1985; Kleiner & Dickman 1987; Miesch & Bally 1994) and the probability density function of the centroid velocity (Miesch & Scalo 1995). These investigations have searched for turbulent correlation lengths, the power spectrum, E(k), of kinetic energy fluctuations due to the turbulent velocity field, and signatures to intermittency. More recently, Heyer & Schloerb (1997) have applied the multivariate technique of Principal Component Analysis (hereafter, PCA) to spectroscopic data cubes to diagnose the relationship between velocity fluctuations and the scale over which these occur which can be related to E(k). In this contribution, I describe PCA, demonstrate its application upon 12CO observations of targeted molecular regions in the outer Galaxy, and discuss the results in context of hydrodynamic models of turbulent flows in the interstellar medium.
2. Description of Principal Component Analysis Spectroscopic imaging observations can be considered as a multivariate problem in which there are n samples each with p attributes of intensity at each velocity where n is the number of spectra and p is the number of spectroscopic channels. The data cube is 198
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199
formally represented as a nxp matrix X, where, /
f~iv \Jt
9 (i ^^z 1I , in IV
n in d LblbLb
' XI j JJ t\ Jo •^—
where r^ = (xi,yi) denotes the position on the sky. The goal of principal component analysis is to determine the set of orthogonal axes u, for which the data, X, when projected upon u, maximizes the variance. Operationally, this requires solving the eigenvalue equation, Su =
AM
where S is the covariance matrix of the data 1 v^ Sjk = — 2_.XijXik
where
j,k = l,p
and
E
if / = m
is the orthogonal condition. For a data cube with p velocity channels, there are p eigenvectors and eigenvalues. The eigenvalue for a given component is equal to the variance of projected values. In practice, one solves the eigenvalue equation for the covariance matrix and sorts the eigenvalues and associated eigenvectors from largest to smallest. The reordering ranks the axes according to the amount of variance contained within each component. To visualize the origin of variance for the fth principal component, an eigenimage is constructed by projecting each spectrum onto the /th eigenvector,
From the inspection of the eigenimages and eigenvectors, the sources of variance within the data cube can be localized in position and velocity. 2.1. Propagation of Errors Since PCA is a linear decomposition, the random noise of the input data can readily be propagated. The variance of projected values due to noise is
Assuming, that cr(Xij) is constant over the bandpass of the spectrum and recalling that u is orthonormal (53 U H = 1), the expression reduces to Thus, the variance of values projected onto the Ith component is equal to variance of the input data.
3. Application of PCA on 12CO Observations of Molecular Clouds The application of PCA on simplistic toy models of interstellar clouds shows that the technique identifies differences in line profiles over the image with respect to the noise level (Heyer &; Schloerb 1997). For observations of molecular clouds, such differences
Heyer: Diagnosing Turbulence in the Molecular ISM
200
-l
-l
109
108
109
I (deg.)
108
I (deg.)
1. Eigenimages for 1=1 to 6 from the decomposition of CO emission from the Sh 152 molecular region. For / > 2, the black and white halftones reflect positive and negative projected values respectively. All projected values with \PCi\ < lcr; have been set to zero and provide the uniform grey background. FIGURE
0.4 -
-0.4 -
-70
-60
FIGURE
-50
-40
-30
-70
-60
-50
-40
-30
2. Eigenvectors for / = ! to 6 for Sh 152.
presumably arise from dynamical processes in the interstellar medium such as rotation, outflow, and fluctuations of the density and velocity fields due to turbulent flows. To demonstrate the utility of PCA on real data, we have calculated the principal components of 12CO observations of the Sh 152 molecular cloud. The data are a subset of a much larger survey of the outer Galaxy (Heyer et al. 1998). In practice, the analysis of the velocity field is limited to regions of low opacity so one must consider that for certain cloud topologies, 12CO emission may probe only a limited volume due to the large optical depths. The first 6 eigenimages and eigenvectors from the decomposition are shown in Figure 1 and Figure 2 respectively. The first principal component identifies the variance generated between those channels with signal and those without any detected emission. Therefore, the resultant eigenim-
Heyer: Diagnosing Turbulence in the Molecular ISM
10
201
20
(PC)
FIGURE
3. The relationship between the magnitude of velocity differences and the size scale over which these occur for the Sh 152 cloud.
Cloud Sh 152 Cep OB3 W3
NGC 7538 G125+3.0
Distance (kpc)
a
5.0 0.7 2.4 3.5 0.5
0.53±0.01 0.55±0.02 0.69±0.08 0.70±0.01 0.43±0.04
TABLE 1. Size-Line Width Relationships
age, PCi, and eigenvector, u\, are similar to an image of integrated intensity and a summed spectrum respectively. The subsequent eigenimages and eigenvectors reveal a characteristic feature of the analysis on interstellar clouds. First, the eigenvectors identify smaller velocity differences with increasing I which is a direct consequence of using orthogonal functions to decompose the data. Similarly, the spatial granularity of the eigenimages decreases with increasing I. This latter feature is not inherent to the analysis but rather, reflects a basic property of interstellar gas observations. That is, there are no large differences of line profiles between nearby spectra. The differences which are identified and how these vary with spatial offset, density, and temperature, provide a powerful diagnostic to turbulent flows within the molecular interstellar medium. The mean velocity difference, Svi, and spatial granularity scale, TJ, are determined from each principal component with projected values above the noise. In Figure 3, the 6v, T points are plotted and define a power law relationship between the magnitude of velocity differences and the size scale over which these occur. Sv = cra
The analysis has been applied to other regions in the outer Galaxy, some of which are well known sites of massive star formation. The results are summarized in Table 1. On average, the derived power law indices are steeper than those measured by Miesch & Bally (1994) using the structure function of centroid velocities. For this limited sample, it appears that the giant molecular clouds, W3 and NGC 7538, follow a much steeper relationship than the diffuse molecular cloud G125+3.0. The Sv — T relationship identified by this analysis provides a quantitative, observational
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constraint to both phenomenological descriptions of turbulence and hydro-magnetic simulations. The derived power law indices shown in Table 1 for the sample clouds are much steeper than the 1/3 power law predicted for Kolmogorov-Obukhov incompressible turbulence. This result should come as no surprise given the dissipative nature of the molecular gas component of the interstellar medium. Simulations of supersonic and hydromagnetic turbulence also reveal power law indices greater than 1/3 and within the range of observed values shown in Table 1 (Porter, Pouquet, & Woodward 1992; Gammie & Ostriker 1996). While not strictly following Kolmogorov behavior, the simulations do exhibit a kinetic energy cascade from larger to smaller scales and in some cases, from smaller to larger scales. The absence of large incremental or decremental discontinuities within the observed Sv — r relationship suggests that there is no singular scale at which energy is injected into or removed from the gas. Of course, such processes may lurk at scales not resolved by these observations or are present within the cold, atomic gas component at equivalent or larger scales. Analyses of both higher resolution observations of molecular gas and HI 21cm line emission are required to complete this statistical description of turbulent flows within the dense interstellar medium. This work is supported by NSF grant AST 94-20159 to the Five College Radio Astronomy Observatory.
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Centroid velocity increments as a probe of the turbulent velocity field in interstellar molecular clouds By D. C. LIS1, T. G. PHILLIPS1 , M. GERIN 2 , J. KEENE1 , Y. LI1, J. PETY 2 AND E. FALGARONE2 'California Institute of Technology, MS 320-47, Pasadena, CA 91125, USA 2
CNRS URA 336, Ecole Normale Superieure, 24 rue Lhomond, 75005 Paris, Prance
We present a comparison of histograms (or PDFs) of CO (2-1) line centroid velocity increments in the p Ophiuchi and £ Ophiuchi molecular clouds with those computed for spectra synthesized from a three-dimensional, compressible, but non-star forming and non-gravitating hydrodynamic simulation. Histograms of centroid velocity increments in the two molecular clouds show non-Gaussian wings, similar to those found in histograms of velocity increments and derivatives in experimental studies of laboratory and atmospheric flows, as well as numerical simulations of turbulence. The magnitude of these wings increases monotonically with decreasing separation down to the angular resolution of the data. This behavior is consistent with that found in the phase of the simulation which has most of the properties of incompressible turbulence. This is consistent with the proposition that ISM velocity structure is vorticity dominated like that of the turbulent simulation.The p Ophiuchi molecular cloud contains some active star formation, as indicated by the presence of infrared sources and molecular outflows. As a result shocks may have important effects on the velocity field structure and molecular line shapes in this region. However, the £ Ophiuchi cloud represents a quiescent region without ongoing star formation and should be a good laboratory for studies of interstellar turbulence.
1. Introduction Early spectroscopic observations of interstellar lines of Hi, OH, and CO have revealed that observed line widths (or velocity dispersions) in interstellar clouds are larger than thermal line widths expected for these low-temperature regions (see e.g. Myers 1997 and references therein). These large line widths are indicative of supersonic motions of the gas, although the exact nature of these motions is still a subject of controversy. Proposed explanations generally involve turbulent motions of the gas, be it hydrodynamic or magneto-hydrodynamic (e.g. Scalo 1987; Falgarone 1997). One of the tools employed in studies of gas motions in the interstellar medium is the analysis of shapes of molecular line profiles. Falgarone & Phillips (1990) argued that the non-Gaussian CO line wings in inactive regions without associated star formation activity represent a direct observational signature of the turbulent nature of the gas flow within molecular clouds and of the existence of regions of intermittent turbulent activity. In a subsequent study Falgarone et al. (1994) showed that line profiles synthesized from a three-dimensional turbulent, compressible, but non-star forming and non-gravitating simulation (Porter, Pouquet & Woodward 1994) are in fact statistically similar to the CO line profiles observed in quiescent molecular clouds. However, Dubinsky, Narayan & Phillips (1995) showed that non-Gaussian line profiles can also be produced from a random velocity field with a Kolmogorov power spectrum. It is known from numerical simulations and atmospheric and laboratory measurements (e.g. Anselmet et al. 1984; Gagne 1987; Vincent & Meneguzzi 1991) that intermittency of turbulence manifests itself through non-Gaussian wings in probability density functions 203
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(PDFs or histograms) of velocity increments and derivatives. These kinds of measurements are not directly possible in the case of the interstellar medium where one has information integrated over a line-of-sight column defined by the passage of the telescope beam through the medium. However, in a recent study (Lis et al. 1996), we showed that histograms of the centroid velocity increments (or differences between line centroid velocities at positions separated by a given distance), for sections of the simulation data cube corresponding to the ISM columns, also show non-Gaussian wings, similar to those found in experiments and numerical simulations of incompressible turbulence. Due to line-of-sight averaging, the wings seen in PDFs of centroid velocity increments are not as pronounced as those in PDFs of velocity increments calculated over the whole data cube. Nevertheless, the effect is clearly present. We also demonstrated that the lines of sight contributing to the non-Gaussian wings of the PDFs of centroid velocity increments trace a filamentary structure, which follows the distribution of the two vorticity components involving cross-derivatives of the line-of-sight component of the velocity field. This suggests that the wings are a manifestation of the turbulent nature of the flow. In a subsequent study (Lis et al. 1998) we showed that the time evolution of the non-Gaussian wings in the histograms of centroid velocity increments in the simulation is consistent with the evolution of the vorticity in the flow, although we could not exclude the possibility that the shock interaction regions also contribute to the wings. 2. p Ophiuchi Results The increment method provides a useful new tool for studying velocity field in interstellar molecular clouds and may help ascertain the effects of intermittency of turbulence on physics and chemistry of the interstellar medium. In a recent paper (Lis et al. 1998) we applied this method to a large-scale map of the CO (2-1) emission from the p Ophiuchi molecular cloud. We found that histograms of centroid velocity increments in this region show clearly non-Gaussian wings, similar to those found in the simulation. The magnitude of these wings increases monotonically with decreasing separation, down to the angular resolution of the data (Fig. la). This behavior is similar to that found in the phase of the simulation which has most of the properties of incompressible turbulence and is consistent with the proposition that ISM velocity structure is vorticity dominated. However, the p Ophiuchi region contains active low-mass star formation as evidenced by the presence of infrared sources and molecular outflows. As a result, shocks associated with embedded young stellar objects can have their signatures in the velocity field. On the other hand, the energy injected into the cloud by the embedded sources may also play an important role in generating the turbulent cascade. A statistical comparison of the velocity field in both active and quiescent regions is thus required. To facilitate this comparison and to better separate the contribution of vorticity and shocks to the non-Gaussian wings in the histograms of centroid velocity increments, we started observations of quiescent regions without ongoing star formation. Preliminary results for the £ Ophiuchi molecular cloud are presented below.
3. C Ophiuchi Results The clouds on the line of sight toward the bright 0 star £ Ophiuchi have been widely discussed in terms of chemical models of diffuse interstellar clouds (e.g. van Dishoeck & Black 1986; Viala, Roueff k Abgrall 1992). Yet the detailed structure of the interstellar medium in the vicinity of the star is poorly known. The clouds extend over a large area on the sky, showing strong CO lines 30' N and S of the star (Liszt 1993; Kopp et
Lis et al.: Centroid velocity increments in molecular clouds
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al. 1996). The star itself is a runaway member of the Sco-OB2 association, located at a distance of 140 pc (Draine 1986). It is surrounded by an extended Hn region (Sivan 1974), approximately 20°, or 8 pc in diameter (Draine 1986). The clouds lie in front of the star, likely at the edge of the Hn region. The 12 CO/13 CO intensity ratio is ~30±5 toward the peak of the CO emission, indicating that 12CO lines have relatively low optical depths (r 2L/Lj — 2nj = 2ts/ts (i.e. ts > 2tf). The fact that observed clouds are known to have av/cs — 1.7nj (i.e. t% = 1.7£f), and may well have sufficient fluctuating field strengths to meet the criterion (2.10), has led to the hypothesis that GMCs are indeed supported (in all directions) over times £ i g by a combination of mean fields and fluctuating fields. Since nonlinear MHD waves are dissipative, any stabilizing effects from the time-dependent fields will decrease over time; thus, whether the quasilinear-theory expectation is realized or not must depend on the nonlinear dissipation rate of the turbulence. Over very long times, stabilization per force requires mechanical energy replenishment (which can be supplied in astronomical contexts by the star formation process). Direct numerical MHD simulations are required to evaluate these ideas quantitatively; some of our results are highlighted in the next section. 3. Numerical Simulations - Selected Results Our group has performed a variety of simulations to systematically explore compressible MHD turbulence in self-gravitating cloud models in 1 2/3, 2 1/2, and 3 dimensions.f X For self-gravitating magnetized cloud equilibria to exist, the ratio of the central column density to central field strength Y,/B must be smaller than the same value l/(27r\/G) (Mouschovias & Spitzer 1976, Tomisaka, Ikeuchi, & Nakamura 1988). f In "1 1/2 D" and "2 2/3 D" restricted geometry, there are respectively one or two independent spatial variables, but all components of v and B evolve in time (subject to V • B = 0).
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These models either start with a turbulent velocity field with a spectrum dE = (l/2)v2(k)dk oc k~2dk which freely decays, or else they include stochastic impulsive forcing on wavelengths small compared to the box, such that the kinetic energy input rate is constant. In the latter case, a quasisteady state is achieved. These numerical surveys have allowed us to address a number of questions about the fundamental nature of these turbulent flows, including: • How do the survival times for self-gravitating clouds depend on their initial kinetic energy, magnetic field strength, and size (described by the dimensionless parameters M = av/cs - ts/tf, j3 = (tA/ts)2, and nj = ts/te)? • Is it possible to support a cloud indefinitely, given sufficient mechanical energy inputs? • How do the temporal profiles for decaying turbulence, and the dissipation rates for quasisteady turbulence, depend on M and /?? • How do the density, velocity, and magnetic field distributions and power spectra depend on M, (3, and nj? Full presentations of our surveys and results to date appear in Gammie & Ostriker (1996), Ostriker (1997), and Ostriker, Gammie, & Stone (1998). Stone's contribution to this volume discusses recent results on comparative dissipation rates and power spectra with varying (3 in high-resolution 3D simulations. Other work in preparation analyzes the distributions of axis ratios and orientations (relative to the magnetic field) of 3D clumps and their 2D projections (Gammie, Stone, & Ostriker 1998). Here, we will highlight recent findings in two areas: comparisons of the survival times of cloud against gravitational collapse for varying A4, (3, and nj, and differences in the density structure that arise in clouds when varying the field strength. 3.1. Simulation results on cloud support Our first set of surveys, in 1 2/3D restricted geometry, confirmed the quasilinear-theory predictions for cloud support along the meanfieldby nonlinear analogues of Alfven waves. We found that low-turbulence model clouds (i.e. av/cs < 2nj) collapsed along the mean field in times t < £g, higher-turbulence models have delayed collapse, while in the highestturbulence cases, collapse was forced by the strong compression from nonlinear waves. Overall, survival times increased with the strength of the mean magnetic field, because the turbulent dissipation rate was lower. We also performed a survey of forced-turbulence self-gravitating experiments in 1 2/3 D, and found that quasi-steady, non-collapsed states can be attained as long as the input power (applied mainly at a scale 8 times smaller than the box) exceeded Ewave > 24/3°-24nj pc^. The corresponding energy replenishment timescale to maintain a steady state would be Ewave/E w 0.2/3~1//4ig ~ 0.3/3~1//4if. In 2 1/2D geometry, where the fluid is free to contract in the direction transverse to the mean field (this motion is restricted in 1 2/3D), we find that turbulent motions are unable to prevent cloud collapse unless the field is strong enough to make the clouds magnetically subcritical. Unmagnetized clouds collapse at « 0.5£g (corresponding to 5 Myr at typical conditions) regardless of the initial kinetic energy level; weakly-magnetized (supercritical) clouds generally last up to 0.5 - l£g before collapsing (some small, magneto-Jeans stable clouds survive up to 1.5tgbefore collapsing), while more-strongly magnetized (subcritical) clouds can last beyond 1.5tg with no signs of collapse. The subcritical models, which are most similar to the 1 2/3D models in that they are unable to collapse perpendicular to the mean field, confirm the conclusion that time-dependent fluctuating magnetic field can prevent gravitational collapse along the mean field for times 3> 0.3£g. However, since the majority of real molecular clouds are probably supercritical, our models imply that they are unlikely to last for times greater than 5 — 10 Myr before some parts of their
Ostriker: Numerical MET) Studies of Turbulent, Self-Gravitating Clouds
245
interior undergo collapse and initiate star formation. Further studies in 2 1/2D geometry are now in progress to asses the continuous energy input rate, as a function of /3, required to sustain clouds against collapse. We have also performed a number of self-gravitating simulations in 3D with varying (3 and nj, and found that, similar to the results in 2 1/2D, supercritical clouds collapse at times ~ 0.5ig even when their initial perturbation energies are large, whereas subcritical clouds can survive to later times. Further 3D simulations are planned to expand the parameter space, and further test these ideas. We can compare our results so far to the expectations of linear and quasilinear theory for cloud support. Our numerical experiments in 2 1/2D and 3D verify that magnetically supercritical (nj/?1 / 2 ^ > 1) clouds typically gravitationally collapse in times ,$, £g, whereas subcritical clouds can survive much longer without collapsing. In subcritical clouds, matter eventually slides along the field and forms thin sheets which do not themselves fragment (but may oscillate with respect to one another for some time). However, for lower field strengths the turbulent dissipation is rapid enough that long-term cloud support along the mean field is not possible. That is, even if clouds are magneto-Jeans stable and have initial turbulence levels exceeding crv/cs > 2nj, they have sufficient dissipation that they can collapse in all directions in times shorter than ts (contrary to what quasilinear theory would predict if dissipation is ignored). 3.2. Simulation results on density distributions An important way to make comparisons between simulated clouds and real clouds, so as to discriminate among the possible values of the input model parameters, is to analyze density distributions. In all of our models, we find that the cloud density structure becomes very clumpy and filamentary, due to the combination of turbulent Reynolds (i.e. ram pressure) and magnetic stresses. Figure 1 shows examples of snapshots of cloud structure in two 2 1/2 turbulent-decay models. Both models have initial turbulent energies with M2 = (av/cs)2 = 200. One has a stronger (3 — 0.01 mean field; the other has a weaker /? — 0.1 mean field (corresponding to BQ = 14/^G and B$ — 4.4/iG for fiducial GMC parameters, cf. eq. 2.7). The two models' snapshots are taken when the
FIGURE 1. Snapshots of density (contours, starting at log(p/p) = 0 with logarithmic increments 0.1), velocity field (vectors), and magneticfieldlines in ft = 0.01 (left) and P = 0.1 (right) 2.5D decay simulations when crv/cs ss 10. nj = 3 for both models.
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Ostriker: Numerical MHD Studies of Turbulent, Self-Gravitating Clouds
kinetic energy has declined to half of the initial value, i.e. to M = 10 (corresponding to 3D velocity dispersion 2-3km sounder GMC conditions); this is before self-gravity has become very important (the specific gravitational binding energy is only -l.lCg, -1.2Cg for the (3 — 0.01, 0.1 models, respectively).
FIGURE 2. Density distributions for snapshots shown in Figure 1. Solid-line histograms show fractional volume as a function of log(p/p); dotted-line histograms show fractional mass as a function of log(p/p);
For each of these snapshots, we have tabulated the fraction of the total volume, and the fraction of the total mass, as a function of log(p/p), the logarithmic density contrast relative to the mean (p = Mtot/L3). These volume and mass distributions are shown in Figure 2. In both models, the mass distribution is centered at log(p/p) > 0, while the volume distribution is centered at log(p/p) < 0, as a consequence of matter clumping: most of the mass is in clumps which have densities higher than the mean for the whole cloud, whereas most of the volume is filled with matter at less-than-average densities. The (3 = 0.01 model has mass-averaged mean contrast (A-)M = 0.49 (with median contrast 0.52), and volume-averaged mean contrast (A-)y = ~0-50 (with median contrast -0.60). The (3 = 0.1 model has mass-averaged mean contrast (A-)M = 0.33 (with median contrast 0.31), and volume-averaged mean contrast (ir)v = -0.28 (with median contrast -0.34). The distributions are very roughly log-normal, which accounts for the values of (ir)v and (ZT)M being almost equal and opposite. We note that the density contrast in the (3 = 0.01 models is closer to the values ~ ±0.4 - 0.6 inferred for real clouds based on studies of clumping in 13CO (e.g. Bally et al 1987, Williams, Blitz, & Stark 1995). The difference shown in these examples in density contrast between models with different (3 at the same Mach number is characteristic of all of the models examined so far: we generally find greater contrast in models with /3 = 0.01 than in the corresponding models with /3 = 0.1. We have also found that the density contrast in (3 = 1 models is intermediate between the /? = 0.1 and /? = 0.01 cases. Physically, we believe that the increase in contrast toward high (3 can be attributed to stronger compressions arising directly from the compressive part of the velocity field (V • vne0) when the magnetic pressure B2/8n is smaller, whereas the increase in contrast toward low /? can be attributed to stronger compressions arising nonlinearly from the shear part of the velocity field - at kinks in
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2
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Super-Alfvenic Turbulent Fragmentation in Molecular Clouds By PAOLO PADOAN1 AND AKE NORDLUND2 x
Instituto Nacional de Astroffsica, Optica y Electronica, Apartado Postal 216, Puebla 72000, Mexico 2 Astronomical Observatory and Theoretical Astrophysics Center, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
The dynamics of molecular clouds are often described in terms of magneto-hydro-dynamic (MHD) waves, in order to explain the super-sonic line widths and the fact that molecular clouds do not seem to be efficiently fragmenting into stars on a free-fall time-scale. In this work we discuss an alternative scenario, where the dynamics of molecular clouds are super-Alfvenic, due to a lower magnetic field strength than usually assumed (or inferred from observations). Molecular clouds are modeled here as random MHD super-sonic flows, using numerical solutions of the three-dimensional MHD equations. A Monte Carlo non-LTE radiative transfer code is used to calculate synthetic spectra from the molecular cloud models. The comparison with observational data shows that the super-Alfvenic model we discuss provides a natural description of the dynamics of molecular clouds, while the traditional equipartition model encounters several difficulties.
1. Introduction Molecular clouds (MCs) are recognized to be the sites of present day star formation in our galaxy. The description of their dynamics is an essential ingredient for the theory of star formation. A lot of work has been devoted to understand i) how super-sonic random motions in MCs can persist for at least a few dynamical times and ii) why MCs do not collapse, or fragment gravitationally into stars, on a free-fall time-scale. The magnetic field has been advocated as the solution for both problems. Magneto-hydrodynamic (MHD) waves were believed to dissipate at a significantly lower rate then super-Alfvenic and super-sonic random motions. Moreover, the magnetic pressure could at the same time support a cloud against its gravitational collapse. Therefore, models of magnetized clouds have been proposed, where the magnetic energy is of the order of the internal kinetic or gravitational energy of the cloud. If the motions observed in MCs have velocities of the order of the Alfven velocity, the magnetic field strength should be about 25 fiG ubiquitously. However, many upper limits on the field strength in MCs are available from OH Zeeman splitting, which seem to indicate in most cases a value significantly lower than 25 fxG (eg Crutcher et al. 1993), although high density regions, favorable to the field detection, are usually selected by OH observations. The Zeeman splitting can only detect the field component in the direction along the line of sight, but the field orientation cannot be claimed to explain the majority of the low upper-limits, and statistically can account only for a factor 2 in the field strength. Field tangling is often used as a possible explanation for the lack of detection through OH Zeeman splitting, but field tangling is not significant in cloud models where the magnetic energy is in approximate equipartition with the kinetic energy of the random motions. In such models, the magnetic field is too strong to be significantly tangled by the flow. It has also been recently shown that MHD waves dissipate at almost the same rate as 248
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super-Alfvenic and super-sonic random motions (Mac Low et al. 1988a, 1998b, 1998c, Padoan & Nordlund 1998), and so the magnetic field is not anymore a particularly good candidate for explaining the molecular spectral line-width. In recent years, compelling evidence for fragmentation of MCs have accumulated. Structures are found in MCs down to extremely small scales (Falgarone et al. 1992, Langer et al. 1995). Such a fragmented structure is apparently not due to the gravitational fragmentation (see the next section), and should be explained by a model for the dynamics of MCs, together with the nature of the super-sonic random motions. While the old question was: "Why do not MCs collapse or fragment on a free-fall time-scale?", the more modern question should rather be: "Why are MCs so strongly fragmented?". In the context of a traditional equipartition model, where the magnetic field is dynamically strong, it is rather hard to understand the origin of the fragmentation, and the explanation must rely on the combined effect of gravity and ambipolar diffusion. Clearly, the existence of a density related time-scale and length-scale for the process of ambipolar diffusion, generates a great difficulty. On the other hand, in a model where the random motions are super-Alfvenic the answer to our question ("Why are MCs so strongly fragmented?") is trivial: the random super-sonic and super-Alfvenic motions generate a complex system of criss-crossing shocks, and therefore a network of strong density enhancements, due to the highly radiative nature of the gas. Since gravity enters this mechanism of fragmentation only in a second stage (collapse of dense unstable fragments) we refer to this process as turbulent fragmentation. In a series of papers we have discussed the turbulent fragmentation from different points of view, using numerical simulations of super-sonic MHD turbulence (Padoan et al. 1997a, 1998b, Padoan & Nordlund 1998). The description of the method and the setup of different numerical experiments can be found in those works, while here, for reason of space, we focus on the latest results. The idea that the observed random super-sonic motions might be at the origin of the complex structure of MCs and might play a role in the star formation process, is found in previous papers (Larson 1981, Hunter 1979, Hunter & Fleck 1982, Leorat et al. 1990, Falgarone et al. 1992, Elmegreen 1993, Vazquez-Semadeni 1994, Vazquez-Semadeni et al. 1995, 1997, Scalo et al. 1997). The present work concentrates on the discussion of the basic assumption of the turbulent fragmentation mechanism, that is the super-Alfvenic nature of the random motions.
2. Turbulent Fragmentation Once the hypothesis of a strong magnetic field is abandoned, super-sonic random motions can generate large density contrasts, due to the highly radiative nature of the gas. The super-Alfvenic model thus offers a simple solution to the problem of the observed fragmentation of MCs. We have run many numerical simulations of super-Alfvenic and super-sonic MHD turbulent flows, and studied their statistical properties. The probability density function (pdf) of the gas density is well approximated by a Log-Normal distribution (Padoan et al. 1997b, Scalo et al. 1997, Nordlund & Padoan 1998). A Log-Normal distribution implies that most of the mass concentrates in a small fraction of the volume, reminiscent of the small volume rilling fraction of the dense gas in MCs. A significant fraction of the mass of a typical MC model is in cores and filaments with densities as high as 105-106 cm" 3 , as found in real clouds. The turbulent fragmentation also offer an explanation for the filamentary and cobwebby morphology of the gas in MCs, since the natural topology of random compressible flows is made of filaments and cores distributed around voids.
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Turbulence and gravity can act together, in the sense that the turbulent fragmentation can produce dense regions that are large enough to be gravitationally unstable and collapse into protostars. On the other hand, turbulence can also slow down the process of star formation, by producing dense fragments that are too small to be gravitationally bound and collapse (Padoan 1995). Observations of MCs on very small scale seem to confirm the presence of a fragmentation mechanism alternative to the gravitational instability. Falgarone et al. (1992) found gravitationally unbound and probably transient structures on very small scale in different MCs, that could not be the consequence of gravitational instability. Langer et al. (1995) found that the Taurus MC is fragmented into clumps with mass of 0.01 MQ and density of about 105 cm" 3 , also too small to be gravitationally bound. Turbulent fragmentation is certainly a candidate to explain the presence of such small unbound clumps in MCs. Is the turbulent fragmentation efficient enough to be the main fragmentation mechanism also inside dense star forming cores? If the answer is affirmative, then gravitational instability would be only the ultimate cause of the collapse/accretion of single protostars, and the study of the statistical properties of turbulent flows could be a viable way to formulate a theory for the origin of the stellar initial mass function (IMF) (Padoan et al. 1997b). This is the physical motivation behind mathematical models of the stellar IMF based on the assumption of the existence of scaling properties in the mass distribution inside MCs (Elmegreen 1997, 1998).
3. Stellar Extinction Stellar extinction measurements can be used to map the column density distribution of dark clouds. It can also be used to infer some statistical properties of the 3-D structure
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of dark clouds. Lada et al. (1994) proposed to use the plot of dispersion of extinction versus mean extinction to discriminate between different models of the structure of MCs. Padoan et al. (1997a) used the same plot to show that the gas density distribution in MCs is consistent with a Log-Normal distribution, with the same properties as the density distribution in super-sonic turbulent flows of an isothermal gas. Here we reproduce the plot of dispersion of extinction versus mean extinction using our numerical simulations as models of the mass distribution inside MCs. We simply project onto a 2-D plane the 3-D density field of a snapshot of one of our runs, select randomly a number of points on the 2-D plane, as to simulate the random positions of stars, and superpose a 2-D regular grid. On each cell of the grid a few "stars" are found, and the mean surface density and the dispersion around the mean are measured, using only the value of surface density where the "stars" are found. The results are plotted in Fig. 1, where the observational plot by Lada et al. (1994) is compared with the plots for a super-Alfvenic model and for an equipartition model. While the super-Alfvenic model compares rather well with the observations, the equipartition model does not. The reason is that the super-Alfvenic model is able to develop a very large density contrast with very low density regions (voids). The equipartition model instead, behaves much more like an elastic medium, due to the large magnetic pressure, and therefore is not able to produce a sufficient density contrast, with randomly distributed deep voids. Moreover, in the equipartition model, the high density regions tend to accumulate over 2-D structures perpendicular to the magnetic field direction, since significant gas compressions are possible only along magnetic field lines. Such more regular and sheet-like structure found in the equipartition model also contributes to the low contrast in the density field projected along a random direction. Stellar extinction measurements show therefore that the turbulent fragmentation mechanism is a good candidate to explain the observed fragmentation of MCs, and that the observed random motions are more likely to be significantly super-Alfvenic, rather than MHD waves.
4. Distribution of Magnetic Field Strength and B — n relation For conditions typical of MCs, the magnetic field is well coupled to the neutral gas through ion-neutral collisions. Only in the densest regions, on small scales, and assuming a low fractional ionization, can the ambipolar diffusion time-scale be comparable with the dynamical time-scale. The hypothesis of flux-freezing might therefore be thought to allow an approximate description of the evolution of the magnetic field topology and statistical distribution. Under this hypothesis, and assuming isotropic compressions, the magnetic field strength should depend on the local density a s B a n 2 / 3 . In our numerical simulations of the super-Alfvenic model, we indeed find a correlation between B and n, but the B - n relation has a very large dispersion. In 1283 runs, we find a range of values of B covering 2 or 3 orders of magnitude, at any given value of n. On the other hand, the B — n scatter plot has a well defined power law upper envelope. The slope of the upper envelope is initially close to unity, B oc n, and later decreases until B oc n 0 3 " 0 - 4 . The B — n relation is initially close to linear because 1-D compressions perpendicular to the direction of the magnetic field are initially dominant. Later on the magnetic field tends to align with the velocity field, and therefore the gas density tends to grow due to motions parallel to the field lines, that do not affect B. This causes the flattening of the B — n relation. The magnetic field and velocity vectors align because of the stretching of field lines by the flow. This partial alignment of the field lines with the flow, expressed by the flattening of the B — n envelope, is an important effect, because it allows even
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random isotropic kinematics to concentrate gravitational energy faster than magnetic energy, in regions of high density. In other words, the turbulent fragmentation in the super-Alfvenic model proceeds in such a way that dense cores tend to accrete mass along magnetic field lines and reduce their magnetic flux to mass ratio efficiently, even in the absence of ambipolar diffusion. The distribution of magnetic field strength in the super-Alfvenic model is characterized by a long exponential tail, which means that regions with field strength much larger than the mean exist with a finite probability. Because of the intermittent distribution of B, Zeeman splitting measurements may detect a field strength much larger than the mean field strength. This is the reason why we suggest that the detection of a 10-50 \xG field strength in some dense cores does not mean that the mean field in MCs is close to the equipartition value (about 25 fiG). In fact many low upper limits on the field strength have been obtained (eg Crutcher et al. 1993) in favor of a lower mean field strength. In Fig. 2 we present a compilation of observational results that seems to confirm the existence of a B - n relation, but also the presence of a scatter of about 2 orders of magnitude in B, for density values between 10 and 104 cm" 3 . Both the slope and the scatter are consistent with the prediction of the super-Alfvenic model (thick contour lines). Instead, the equipartition model (thin contour lines in Fig. 2) has an almost flat B - n envelope, and covers a too small range both in density and in magnetic field strength. Note that in 1283 simulations the largest density that can be reached in a MC model is about 105 cm" 3 , due to the limited resolution. With a larger resolution and the same rms Mach number of the model flow, the largest density could reach 106 cm" 3 , even without the assistance of gravity. The discrepancy between the equipartition model
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4
FIGURE 3. Left panel: line width versus integrated antenna temperature in the equipartition model, for a line of sight parallel to the magnetic field direction. Right panel: 12CO mean spectra; the J=2—)•! spectra are divided by 0.62.
and the observations would then be more obvious, while the super-Alfvenic model would account for the latest CN Zeeman splitting measurements (Crutcher 1998). The super-Alfvenic model and the turbulent fragmentation mechanism are therefore good candidates to interpret the Zeeman splitting observations of dark clouds as well, while the equipartition model would predict a too small range of values of B and n, and almost no correlation between the two.
5. Synthetic Molecular Spectra We have used our 1283 MHD runs to compute grids of 90 x 90 spectra of different molecular transitions. The radiative transfer calculations are performed with a non-LTE Monte Carlo code, described in Juvela (1997). Several statistical properties of the synthetic spectra are discussed in Padoan et al. (1998b) and compared with observational data in Padoan et al. (1998a). Here we present some recent results concerning the relation between the line width and the integrated antenna temperature, and the line width and line intensity ratios. Heyer et al. (1996) proposed to use the observed growth of line width with integrated temperature as a test for the magnetic field strength in MCs. Using synthetic spectra from the super-Alfvenic model we find that the line width grows with integrated temperature, and also its dispersion around the mean, as in the observations (Heyer et al. 1996, Padoan et al. 1998a). In the equipartition model, instead, for lines of sight perpendicular to the magnetic field, the growth of the line width is very limited, and its dispersion does not grow with integrated temperature. For a line of sight parallel to the magnetic field, the line width in the equipartition model does not grow at all with integrated temperature, as shown in Fig. 3, left panel. Falgarone k Phillips (1996) find the line intensity ratio RCo (2-1/1 - 0 ) = 0.62 ±0.08, constant in space and also across line profiles, in a region situated at the edge of the Perseus-Auriga complex. In Fig. 3 (right panel) the average 12CO J=l->0 and J=2-»1 spectra are plotted for two super-Alfvenic models representative of MCs on the scales of 5 and 20 pc. The J=2-»l spectra, divided by 0.62, are almost perfectly coincident with the J=l—>0 spectra, in agreement with the observations. The left panel of Fig. 4 shows the i ? c o ( 2 - l / l - 0 ) ratio, but for velocity integrated temperature of single lines of sight. The plot is again consistent with the observations. A good agreement with the results of
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Falgarone & Phillips (1996) is also found for the ratio of 12CO J=2->1 to 13CO J=2->1 line widths, plotted in the right panel of Fig.4 The comparison of synthetic spectra with the observations confirms again the validity of the super-Alfvenic model, and therefore supports the scenario of the turbulent fragmentation of MCs.
6. Conclusions In this work we have argued that the internal dynamics of MCs are super-Alfvenic and that MCs are primarily fragmented by the observed super-sonic random motions, rather than by the gravitational instability. We have referred to this process as turbulent fragmentation. Using numerical simulations of highly super-sonic MHD turbulent flows, and solving the radiation transfer problem through the numerical datacubes with a non-LTE Monte Carlo radiative transfer code, we have shown that the turbulent fragmentation process provides a natural explanation for i) the highly fragmented and filamentary structure of MCs; ii) the statistical properties of the mass distribution in MCs, as probed by stellar extinction measurements; iii) the Zeeman splitting measurements of the magnetic field strength; iv) the slope and the dispersion of the B — n relation; v) the molecular line intensity ratios; vi) the line width ratios; vii) the relation between the line width and the integrated antenna temperature. On the other hand, the equipartition model cannot easily account for i) the stellar extinction results; ii) the many low upper limits on B from Zeeman splitting measurements; iii) the slope and the scatter of the B — n relation; iv) the growth of the line width with integrated temperature. Moreover, it has recently been confirmed that equipartition MHD turbulence is approximately as dissipative as super-Alfvenic motions (Mac Low et al. 1988a, 1998b, 1998c, Padoan & Nordlund 1998), even if the highly dissipative ion-neutral friction is not considered. We conclude that the super-Alfvenic model we have proposed offers a reasonable description of the dynamics of MCs, since it provides natural explanations for all the observed properties of MCs that we have analyzed so far. The same is not true for the equipartition model. We thank all participants to the conference for useful discussions, and Jan Johannes Blom for reading carefully the manuscript and suggesting corrections.
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Decay Timescales of MHD Turbulence in Molecular Clouds By MORDECAI-MARK MAC LOW1, RALF S. KLESSEN1, ANDREAS BURKERT 1, AND MICHAEL D. SMITH2 1
Max-Planck-Institut fur Astronomie, Konigstuhl 17, Heidelberg, Germany
2
Astronomisches Institut der Universitat Wiirzburg, Am Hubland, Wiirzburg, Germany
We compute 3D models of supersonic, sub-Alfvenic, and super-Alfvenic decaying turbulence, with initial rms Alfv^n and Mach numbers ranging up to five, and an isothermal equation of state appropriate for star-forming interstellar clouds of molecular gas. We find that in 3D the kinetic energy decays as t~v, with 0.85 < 77 < 1.2. In ID magnetized turbulence actually decays faster than unmagnetized turbulence. We compared different algorithms, and performed resolution studies reaching 2563 zones or 703 particles. External driving must produce the observed long lifetimes and supersonic motions in molecular clouds, as undriven turbulence decays too fast.
1. Introduction Molecular cloud lifetimes are of order 3 x 107 yr (Blitz & Shu 1980), while free-fall gravitational collapse times are only iff = (1.4 x 106 yr)(n/10 3 cm" 3 )" 1 / 2 . In the absence of non-thermal support, these clouds should collapse and form stars in a small fraction of their observed lifetime. Supersonic hydrodynamical (HD) turbulence is suggested as a support mechanism by the observed broad lines, but was dismissed because it would decay in times of order t^. A popular alternative has been sub- or trans-Alfvenic magnetohydrodynamical (MHD) turbulence, which was first suggested by Arons & Max (1975) to decay an order of magnitude more slowly. (Also see Gammie & Ostriker 1996). However, analytic estimates and computational models suggest that incompressible MHD turbulence decays as t~v, with a decay rate 2/3 < r) < 1.0 (Biskamp 1994; Hossain et al. 1995; Politano, Pouquet, & Sulem 1995; Galtier, Politano, & Pouquet 1997), while incompressible HD turbulence has been experimentally measured to decay with 1.2 < T) < 2 (Comte-Bellot & Corrsin 1966; Smith et al. 1993; Warhaft & Lumley 1978). The difference in decay rates between incompressible HD and MHD turbulence is clearly not as large as had been suggested for compressible astrophysical turbulence. In this work we compute the decay rates of compressible, homogeneous, isothermal, decaying turbulence with supersonic, sub-Alfenic, and super-Alfvenic root-mean-square (rms) initial velocities vrms, and show that the decay rates in these physical regimes, 0.85 < 77 < 1.2, strongly resemble the incompressible results. These results are also presented in Mac Low et al. (1998).
2. Numerical Techniques We use both a finite difference code and an SPH code for our HD models, while for our MHD models we use only the finite difference code. Thisfinite-differencecode is the well-tested MHD code ZEUS (Stone & Norman 1992a, 1992b), which uses second-order Van Leer (1977) advection, and a consistent transport algorithm for the magnetic fields (Evans & Hawley 1988). It resolves shocks using a standard von Neumann artificial viscosity, but otherwise includes no explicit viscosity, relying on numerical viscosity to provide dissipation at small scales. This should certainly be a reasonable approximation 256
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for shock-dominated flows, as most dissipation occurs in the shock fronts, where the artificial viscosity dominates in any case. Our resolution studies show that our major results are, in fact, independent of the resolution, and thus of the strength of numerical viscosity. SPH is a particle based approach to solving the HD equations described, for example, by Benz (1990) and Monaghan (1992), in which the system is represented by an ensemble of particles, each carrying mass, momentum, and fluid properties. We used the specialpurpose processor GRAPE (Ebisuzaki et al. 1993), to accelerate computation of nearestneighbor lists for the SPH algorithm (Steinmetz 1996). We chose initial conditions for our models inspired by the popular idea that setting up velocity perturbations with an initial power spectrum P(k) oc ka in Fourier space similar to that of developed turbulence would be in some way equivalent to starting with developed turbulence, as adopted by, among others, Padoan & Nordlund (1997), and Porter, Pouquet, & Woodward (1992, 1994). Observing the development of our models, it became clear to us that, especially in the supersonic regime, the loss of phase information in the power spectrum allows extremely different gas distributions to have the same power spectrum. For example, supersonic, HD turbulence has been found in simulations by Porter, Pouquet, & Woodward (1994) to have a power spectrum a = - 2 . However, any single, discontinuous shock wave will also have such a power spectrum, as that is simply the Fourier transform of a step function, and taking the Fourier transform of many shocks will not change this power law. Nevertheless, most distributions with a — — 2 do not contain shocks. After experimentation, we decided that the quickest way to generate fully developed
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FIGURE 2. Comparison of density and velocity profiles of ID hydro and MHD models demonstrates why ID MHD models dissipate faster: they have more dissipation regions, due to their more complex physics.
turbulence was with Gaussian perturbations having a flat power spectrum a = 0 with a cutoff at kmax — 8. In all of our models we take c s = 0.1, initial density p0 = 1, and we use a periodic grid with sides L = 2 centered on the origin.
3. One-Dimensional Results To verify our numerical methods, we reproduced the ID, MHD results of Gammie & Ostriker (1996). The first panel of Fig. 1 shows the results of a resolution study comparable to their Figure 1, with initial rms Mach number M = 5, initial uniform field parallel to the z-axis, and initial rms Alfven number A = urms/i>A = 1, where v\ = B2/47T/90- Note that t — 20 in our units corresponds to t = 1 in theirs. Aside from a rather faster convergence rate in our study, attributable to the details of our choice of initial conditions, we reproduce excellently their result: a decrease in wave energy -Ewave = EK + (By + B%)/8n by a factor of five in one sound-crossing time L/cs. We then extended our study by examining the equivalent HD problem, as shown in the third panel of Fig. 1, only to find that the decay rate of HD turbulence in ID is significantly slower than that of MHD turbulence. This appears to be due to the sweeping up of slower shocks by faster ones in the HD case, resulting in the pathological case of pure Bergers turbulence, as shown on the left in Fig. 2, as predicted by, e.g. Lesieur (1997). As a result there are very few dissipative regions, and energy is only lost very slowly. In contrast, multiple wave interactions occur in the MHD case shown on the right in Fig. 2, producing many dissipative regions and so faster dissipation.
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0.01 10.0 FIGURE 3. 3D resolution studies for M—5, isothermal models. Linear resolutions vary by a factor of two between lines for the ZEUS runs, while particle number varies by seven. "Weak field" corresponds to A = 5, while "strong field" corresponds to A — 1.
Finally we compared ID models with 256 zone resolution to equivalent 3D models with 2563 zones. The 3D model loses energy far faster than the ID model in both the the MHD and HD cases, as shown on the right in Fig. 1. The increased number of degrees of freedom available in 3D presumably produces more shocks and interaction regions, again resulting in increased energy dissipation.
4. Three-Dimensional Results We next performed resolution studies using ZEUS for three different cases with M — 5 and no field, weak field and strong field as shown in Fig. 3. The initial ratio of thermal to magnetic pressure is /3 = 2 for the weak field and /? — 0.08 for the strong field. We ran the same HD model with the SPH code to demonstrate that our results are truly independent of the details of the viscous dissipation. We also ran HD and MHD models with adiabatic index 7 = 1.4, as well as an isothermal model with initial M = 0.1. For each of our runs we performed a least-squares fit to the power-law portion of the kinetic energy decay curves shown in Fig. 3. A full description of the results is given by Mac Low et al. (1998). We find the results for the power laws appear converged at the 5-10% level, and, reassuringly, that the different numerical methods converge to the same result for the HD case, r] ~ 1. We find that highly compressible, isothermal turbulence decays somewhat more slowly, with rj = 0.98, than less compressible, adiabatic turbulence, with r/ = 1.2, or than incompressible turbulence, with 77 = 1.1 (also see Smith et al 1993; and Lohse 1994).
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Adding magnetic fields decreases the decay rate somewhat further in the isothermal case to 1) ~ 0.9, with very slight dependence on the field strength or adiabatic index. Even strong magnetic fields, with the field in equipartition with the kinetic energy, cannot prevent the decay of turbulent motions on dynamical timescales far shorter than the observed lifetimes of molecular clouds. The significant kinetic energy observed in molecular cloud gas must be supplied more or less continuously. If turbulence supports molecular clouds against star formation, it must be constantly driven. Some computations presented here were performed at the Rechenzentrum Garching of the MPG. ZEUS was used courtesy of the Laboratory for Computational Astrophysics at the NCSA. MDS thanks the DFG for financial support. REFERENCES ARONS, M., & MAX, C. 1975, ApJ,
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BENZ, W. 1990, in The Numerical Modelling of Nonlinear Stellar Pulsations, edited by J. R. Buchler (Dordrecht: Kluwer), 269 BISKAMP, D. 1994, Nonlinear Magnetohydrodynamics (Cambridge University Press) BLITZ, L., & SHU, F. H. 1980, ApJ, 238, COMTE-BELLOT,
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S. 1993, Publ. Astron. Soc. Japan, 45, 269 GALTIER, S., POLITANO, H., & POUQUET, A. 1997, Phys. Rev. Lett., 79, 2807 GAMMIE, C. F. & OSTRIKER, E. C. 1996, ApJ, 466,
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M., GRAY, P., PONTIUS, D., MATTHAEUS, W. & OUGHTON, S. 1995, Phys. Fluids, 7, 2886 LESIEUR, M. 1997, Turbulence in Fluids, 3rd Edition, (Dordrecht: Kluwer), 239 LOHSE, D. 1994, Phys. Rev. Lett., 73, 3223 MAC LOW, M.-M., KLESSEN, R. S., BURKERT, A., & SMITH, M. D. 1998, Phys. Rev. Lett., in press MONAGHAN, J. J. 1992, ARA&A, 30, 543 PADOAN, P. & NORDLUND, A. 1998, ApJ, in press, (astro-ph/9703110) POLITANO, H., POUQUET, A. & SULEM, P. L. 1995, In Small-Scale Structures in Fluids and MHD, Lecture Notes in Physics, eds. M. Meneguzzi, A. Pouquet, & P. L. Sulem, (Berlin: Springer-Verlag), 462, 281 PORTER, D. H., POUQUET, D. & WOODWARD, P. 1992, Phys. Rev. Lett., 68, 3156 PORTER, D. H., POUQUET, D. & WOODWARD, P. 1994, Phys. Fluids, 6, 2133 SMITH, M. R., DONNELLY, R. J., GOLDENFELD, N., & VINEN, W. F. 1993, Phys. Rev. Lett., 71, 2583
HOSSAIN,
STEINMETZ, M. 1996, MNRAS, 278, 1005
J. M. & NORMAN, M. L. 1992a, ApJ, 80, 753 J. M. & NORMAN, M. L. 1992b, ApJ, 80, 791 VAN LEER, B. 1977, J. Comput. Phys., 23, 276 WARHAFT, Z. & LUMLEY, J. 1978, J. Fluid Mech., 88, 659 STONE, STONE,
Numerical Magnetohydrodynamic Studies of Turbulence and Star Formation By D. S. BALSARA1, A. POUQUET 2 , D. WARD-THOMPSON3 AND R. M. CRUTCHER 1 'N.C.S.A., University of Illinois at Urbana-Champaign, Illinois, U.S.A. 2
Observatoire de la Cote d'Azur, France
3
Royal Observatory, Blackford Hill, Edinburgh, U.K.
In this paper we examine two problems numerically. The first problem concerns the structure and evolution of MHD turbulence. Simulations are presented which show evidence of forming a turbulent cascade leading to a self-similar phase and eventually a decay phase. Several dynamical diagnostics of interest are tracked. Spectra for the kinetic and magnetic energies are presented. The second problem consists of the formation of pre-protostellar cores in a turbulent, magnetized molecular clouds. It is shown that the magnetic field strength correlates positively with the density in keeping with observations. It is also shown that the density and magnetic fields organize themselves into filamentary structures. Through the construction of simulated channel maps it is shown that accretion onto the cores takes place along the filaments. Thus a new dynamical process is reported for accretion onto cores. We have used the first author's RIEMANN code for astrophysical fluid dynamics for all these calculations.
1. Introduction The conference for which this paper is being written has been instrumental in opening the eyes of astronomers to the need for understanding turbulent processes in astrophysics. While several astrophysical environments where turbulent processes could be important were identified by numerous contributors in this conference, the pulsar scintillation measurements and the study of lines in molecular clouds provide two environments where the need for magnetohydrodynamic (MHD) turbulence is observationally well-founded. Since the MHD equations are highly non-linear analytical approaches sometimes prove to be of limited utility. As a result, the numerical study of MHD turbulence in non-selfgravitating and self-gravitating environments becomes a very useful tool. This has been aided by the availability of very accurate and reliable numerical methods that use higher order Godunov methodology for numerical MHD, see Roe and Balsara (1996) and Balsara (1998a,b) . Such methods have been implemented in the first author's RIEMANN code for astrophysical fluid dynamics. Several non-self-gravitating and self-gravitating simulations have been carried out by these authors. In Section II we discuss MHD turbulence decay. In Section III we discuss pre-protostellar core formation. In Section IV we give some conclusions.
2. Decay of MHD Turbulence The computation is done on a uniform grid of 2563 points, with periodic boundary conditions, adequate for homogeneous flows. Initial conditions are centered in the large scale, with a random distribution of Fourier modes with an exponential fall-off in the smallest scales. The initial ratio of longitudinal to transverse velocity fluctuations is ~ 0.08%. In the computation that is reported on here, there is no uniform magnetic field, and the turbulent magnetic energy is initially in statistical equipartition with the 261
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FIGURE 1. (a) shows the evolution of the kinetic , "V" , and magnetic energies, "M" , as a function of time that is normalized by the turn-over time, (b) does the same for the corresponding enstrophies.
kinetic energy. Density is normalized to unity, the initial sound speed is 0.24 and the initial r.m.s. Mach number is equal to unity. The computation is done for roughly one and a half acoustic (and nonlinear eddy turn-over)times r ac (r oc = 4.0 in code units). In this study we : (i) examine the overall temporal dependence of relevant variables and (ii) examine the statistical properties of the fluid at the level of Fourier mode spectra. The Mach number, initially unity, decreases to a final value of 0.36 based on the r.m.s. velocity. The speed of sound has increased by 9% because of heating due to shocks. Fig la shows the evolution of the kinetic and magnetic energies as a function of time that is normalized by the turn-over time. Fig lb does the same for the corresponding enstrophies. Thus the line denoted by "M" in Figlb indicates the magnetic enstrophy which is just the current and is, therefore, a measure of the dissipation in the field. Similarly, the line denoted with "V" is the vorticity. In this evolution, a plateau can be observed which is due to acoustic exchanges between kinetic and internal energy. We observe that shocks develop rapidly as exemplified by the growth of the second moment of the compressible part of the velocity field, with a maximum at t = 1 . ( Times are either given in code units of roughly 0.25rac or as times that are normalized by rac . ) The existence of magnetic shocks imply the early development of small-scale currents as well, as observed. The development , through mode-coupling and cascading to small scales, of the vorticity follows but is slower, with a maximum of the enstrophy reached around t = 3 which corresponds to 0.75rac in Fig lb . During that second phase (1. < t < 3.), the compressible excitation at small scale decreases, whereas the electric current, subject to mode coupling through the induction equation, continues to grow and reaches its maximum at t = 3 , with a total increase by a factor of 16 (and of 10 for the vorticity). In the last phase of development (3. < t < 6.) the self-similar decay of energy begins, with the spectra preserving their power-law shape. The flow initially dominated by vortices remain in that regime, with a peak in the ratio of compressible to solenoidal
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FIGURE 2. (a) and (b) show stacked sequences of the kinetic energy and magnetic energy spectra. A unit offset in the vertical direction is put between successive spectra which are shown at equal intervals of time.
energy of 0.19 at t = 1. This peak is stronger when stressing small scale properties, measuring the ratio of r.m.s. divergence to vorticity, which equals 0.7 at t=l. In apparent contradiction with these observations - since the current is the dissipation of the magnetic energy - the magnetic energy grows during that phase. This can be seen from Fig la. This is due to the nonlinear coupling between velocity and magnetic field, with a net transfer to the magnetic mode. Indeed, at the final time of the computation, the kinetic energy Ev has decreased by 80%, whereas the magnetic energyEM has decreased by only 52%. The end result is an excess of magnetic energy with EM/Ev = 1.76 at t ~ Tac, a feature commonly observed in computations of incompressible MHD also. The spectra, as expected, develop in time with the formation of small scales. Density fluctuations develop as well at all scales. The density contrast is still sizeable at the final time. Figs 2a and 2b show stacked sequences of the kinetic energy and magnetic energy spectra. A unit offset in the vertical direction is put between successive spectra which are shown at equal intervals of time. The energy spectra appear to follow a —2 law for the compressible part of the velocity and for the magnetic field as well. As pointed out by Balsara, Crutcher and Pouquet (1997) this spectral index is an implicit validation of Larson's laws.
3. Formation of Pre-Protostellar Cores We have carried out several simulations of pre-protostellar core formation in a magnetized, self-gravitating patch of molecular cloud. The size of the simulation is arranged so that it has several Jeans lengths in it. As demonstrated in Crutcher et al (1993) there is a positive correlation between the density and the magnetic field. In Fig 3 we show the mean magnetic pressure as a function of density, denoted by " 0", from one of our simulations. When possible we also show the one sigma fluctuation bounds on the mean, denoted by "1" , as obtained from our simulations. The mean density is normalized to unity. It becomes apparent from Fig 1 that at lower densities the magnetic
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3. shows the mean magnetic pressure as a function of density, denoted by "0". When possible we also show the one sigmafluctuationbounds on the mean denoted by "1" .
FIGURE
pressure shows a very tight positive correlation with the density. The higher densities probe regions where cores form. They too show a positive correlation though they show a considerable amount of scatter. The very highest densities could be poorly sampled. However, the correlation of magnetic field with density seems to be inverted. On examining the data we found that several of the densest cores form in regions with magnetic null points. Those are regions with the least magnetic pressure support and would, therefore, be most prone to gravitational collapse. Thus a dynamically consistent reason is found that would explain this inverted correlation at the highest densities. Examination of the data has shown that the cores are often interconnected by filamentary structures in both the density and the magnetic field. Cuts through the data, presented in Balsara, Crutcher and Pouquet (1996) , also support this view. Extensive volumetric rendering has shown that that cores are often not threaded by a magnetic field with a single polarity. This would explain why the cores often have multiple magnetic and density filaments emerging from them. A single polarity of magnetic field threading a core would imply that each core has just two filaments emanating from it. Furthermore, it would imply that the core assumes an hourglass morphology. The cores that form in our simulations, by and large, do not display such an idealized morphology. The complexity of the morphologies of the cores and the filaments that connect to them is consistent with a more complicated magnetic field topology. We have, in fact, carried out a JCMT study of the morphologies of fifty cores and found that only one has the idealized hourglass morphology! In Figs 4a - 4f we show simulated channel maps from a quadrant of one of our simulations. These have been done assuming optically thin radiative transfer and should correlate with isotopic lines. Fig 4a corresponds to line center. We invite the reader to focus on the most prominent core in the simulation which is in the center and towards the top in Fig 4a. Fig 4b shows that on shifting away from line center the maximal intensity comes from a location around the core but not on the core. Fig 4c shows a filament going off to the right as also an intensity peak going off to the north-west of
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4. shows a mosaic of simulated channel maps with velocity shifts from the mean shown on the top of each plot.
that core. Fig 4d shows that that trend continues. Fig 4e shows the original right-going filament bending and forming an arc. The fact that there is a variation in velocity along filaments is very interesting because it provides evidence for accretion onto the cores. Remember that these filaments are also regions of strong magnetic field. The magnetic field constrains the gas to flow along the filament and onto the core that it joins. Thus the simulations have provided a new insight into the nature of accretion onto cores and show that filaments of strong magnetic field play an important role in modulating the accretion. In Balsara et al (1988) we have made intercomparisons of this process with actual channel maps for S106 and found that the observational data shows the same trend as the simulated data thus lending support to our claim that magnetized filaments modulate accretion onto the cores.
4. Conclusions We have analyzed the spectra for MHD turbulence and found that the kinetic and magnetic energies follow a power law with a spectral index of — 2 . Diagnostics have
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been developed to show that a true inertial range has been achieved in our simulations. We also show that we obtain a self-similar decay. We have also shown the positive correlation between density and magnetic pressure in self-gravitating turbulence simulations. We show that the morphology consists of density and magnetic field filaments interconnecting the cores. Using simulated channel maps we show that the strongly magnetized filaments serve to channel the accretion onto the cores. Thus a new mechanism is found for accretion onto cores. It has been compared with actual observations for SI06.
REFERENCES D. S. 1998a, ApJS, 116, in press D. S. 1998b, ApJS, 116, in press D. S., CRUTCHER, R.M. & POUQUET, A. 1996, in Star Formation Near and Far, ed. S. Holt & L. G. Mundy, 89
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BALSARA, D. S., CRUTCHER, R.M.
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Direct Numerical Simulations of Compressible Magnetohydrodynamical Turbulence By JAMES M. STONE Department of Astronomy, The University of Maryland, College Park, MD 20742-2421 [email protected] We report the results of three-dimensional, direct numerical simulations of compressible MHD turbulence relevant to the internal dynamics of molecular clouds. Models of both driven and decaying turbulence are considered. The decay rate of driven supersonic MHD turbulence is found to be large, of order of one eddy turnover time at the driving scale. Non-ideal MHD effects can increase this decay rate by a factor of about two. In models where the magnetic field is strong (strong enough that the velocity dispersion in the saturated state is less than the Alfven speed), the power spectrum of the turbulence is remarkably similar to the expectations of the theory of incompressible MHD turbulence.
1. Introduction Numerical tools are likely to play an important role in the investigation of MHD turbulence in cold molecular clouds if for no other reason than because the observed linewidths are highly supersonic, and as of yet there does not exist a comprehensive analytic theory of compressible MHD turbulence. Our group (C. Gammie, E. Ostriker, and myself) has begun a project to study systematically the internal dynamics of magnetized, self-gravitating molecular clouds in two- and three-dimensions. Our motivations are two-fold: not only do we wish to understand the dynamics of compressible MHD turbulence as a well-defined physics problem, but also we would like to use the dynamical models as a basis with which to interpret the enormous collection of astronomical observations of molecular clouds that have been collected over the past several decades. The issues being addressed by this project, along with some results from a campaign of two-dimensional simulations, are summarized by Ostriker (this volume) and (Ostriker, Gammie, & Stone 1998). In this paper, I briefly summarize some of the results of three-dimensional calculations. The simulations all use a version of the ZEUS compressible MHD code (Stone &; Norman 1992). The initial conditions consist of a cubic box of size L which contains a plasma with uniform density p0 threaded by a uniform magnetic field in the a;-direction, B = (Bx, 0,0). The sound speed in the fluid Cs is constant, so that the relevant timescales for the dynamics are the sound crossing time ts = L/Cs, and the Alfven crossing time tA = L\Z4irpo/Bx. The dynamics is computed using an isothermal equation of state. We use periodic boundary conditions in each direction, and grid resolutions which vary between 323 and 5123. Most of the simulations assume the magnetic field is perfectly coupled to the fluid (the ideal MHD approximation), but in some models we include ambipolar diffusion in the strong coupling limit; this is described more fully below. In some simulations we include self-gravity computed using Fourier transform techniques. Additionally, in all models we evolve a passive contaminant which initially fills a cylindrical volume in the center of the grid orientated with the symmetry axis parallel to B and with diameter and axial length equal to L/2. Studying the distribution of this contaminant at later times not only allows us to follow field line tangling (at least for 267
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the ideal MHD simulations), but also allows study of the mixing rate of passive chemical species in compressible MHD turbulence. Following (Gammie & Ostriker 1996), we study three kinds of turbulence models: (1) randomly driven turbulence, (2) decaying turbulence from saturated initial conditions, and (3) decaying turbulence with self-gravity. We present results from the first kind of models in the next section, and from the second kind in section 3. Results from the third kind of models are given by Ostriker (this volume) and Ostriker, Gammie & Stone (1998) for 2-D simulations, and by Gammie, Stone, & Ostriker (1998) for 3-D simulations.
2. Results from Driven Turbulence Models To drive turbulence in our simulations, we add random velocity perturbations 5v at discrete time intervals At with At/ts = 0.001. The velocity perturbations are chosen to satisfy several constraints. First, we set the power spectrum of the fluctuations so that \6vl\ oc k6 exp(-8/c/fcpfc). This form gives a steeply rising spectrum at small wavenumbers with an exponential cut-off near k — kpk- We set kpk = 8. Second, the perturbations are drawn from a Gaussian random field and normalized so that the kinetic energy input rate is a constant value E. Third, we constrain the fluctuations so that no net momentum is added to the box. Finally, the fluctuations are chosen to be incompressible, i.e. k-<W = 0. There are three free parameters which describe our driven turbulence models. First we have the energy injection rate E. We study models with E/poL3Cg = 1000, which corresponds to about 10L© for typical cloud parameters (Gammie & Ostriker 1996). We find this value gives a saturated energy comparable to observed values. Second, we specify a magnetic field strength by choosing the ratio of the square of the sound to Alfven speeds /? = C^/V^. We study values of @ — 0.01,1.0 and oo (corresponding to pure hydrodynamics). Finally, the driving spectrum chosen above must be considered as a free parameter; we study how variation of both the form of the spectrum, and the wavenumber at which the spectrum is peaked, affect the saturated turbulence in our simulations. One motivation for this study is to measure the decay rate of saturated MHD turbulence. We define the wave energy Ew as the sum of the kinetic and magnetic energy associated with velocity and magnetic field fluctuations respectively. Studying how the wave energy varies with time reveals when and at what amplitude saturation occurs: at saturation the decay rate is then equal to the driving rate E. Figure 1 (left panel) plots Ew{t) for three simulations using different magnetic field strengths. Each model is computed at a resolution of 2563. The plot reveals that saturation occurs quickly, within only a few hundredths of a sound crossing time. Saturated amplitudes for Ew are around 15 - 20 in units of p(,L3Cg. The RMS sonic Mach number is about 5 in all models, whereas the RMS Alfvenic Mach number is roughly 0.5 in the /3 = 0.01 model, but is about 3 in the /3 = 1.0 model. Note this implies the time for the turbulence to saturate is roughly one eddy turnover time at the scale at which the turbulence is being driven. Through a large number of simulations which surveyed parameter space, (Gammie & Ostriker 1996) found an empirical saturation predictor for driven MHD turbulence in 1-2/3 D. Using this result (see eq. 19 in Gammie & Ostriker 1996), we would expect the amplitude of Ew at saturation to be 50 - 100 in our models, which is 3 - 5 times larger than actually observed in Figure 1 (left panel). Equivalently, this indicates the decay rate of compressible MHD turbulence is 5 - 10 times larger in three-dimensions than in one-dimension. We find the dissipation time in units of the sound crossing time
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FIGURE 1. left Wave energy versus time for the strong field 0 = 0.01 model (solid line), the weak field 0 = 1.0 model (dotted line), and the pure hydrodynamic 0 = oo model (dashed line). right Power spectrum for the 0 — 0.01 model (solid line), and the 0 = 1.0 model (dashed line). Also shown is the spectrum of velocity perturbations with which the turbulence is driven.
is tdiss/ts ~ Ew/E = 0.02. However, in units of the eddy turnover time at the driving wavenumber kpk, the decay time is of order unity. Several interesting results are revealed by studying the power spectrum of the velocity and magnetic field fluctuations in the saturated state at different magnetic field strengths. Figure 1 (right panel) shows the three-dimensional power spectrum (averaged over spherical shells) for f3 = 0.01 and 1.0 at a numerical resolution of 2563, as well as the spectrum with which the turbulence is driven. At high k, the spectra are clearly affected by numerical dissipation, evident as an increase in slope above k — 50 - 60 (corresponding to length scales of about 4 A i ) . At low k the spectra are flat, which is remarkable given that virtually no energy is fed into these wavenumbers by the forcing. Instead, the energy at large scales (small k) must come through an inverse cascade from smaller scales. At intermediate scales, we identify an inertial range. Although this inertial range does not span a large domain of wavenumbers, it is large enough to measure a slope of the power spectrum in each model. Interestingly, the best-fit slope measured from the weakly magnetized turbulence model is steeper compared to that measured from the strongly magnetized case. In fact, we find a slope for the /? = 1.0 model which is roughly —4, consistent with the expectation that strongly supersonic turbulence is dominated by shocks. On the other hand, the slope of the /? = 0.01 model is roughly -11/3, consistent with a Kolmogorov spectrum. In fact, theories of strong incompressible MHD turbulence predict the slope of the power spectrum should be nearly Kolmogorov (Goldreich & Sridhar 1995). It would therefore seem that compressibility effects are reduced if the velocity dispersion is comparable to or less than the Alfven speed, even though it may greatly exceed the sound speed. Figure 2 shows two-dimensional power spectra for the strongly and weakly magnetized turbulence models plotted as functions of wavenumber perpendicular and parallel to the mean field (k± and fc|| respectively). The contours are clearly circular in the weak field case, indicating the turbulence is isotropic. However, in the strong field case, the contours are elliptical; elongated perpendicular to the mean field. In fact, anisotropic turbulence with a cascade perpendicular to the field has emerged as one of the most
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characteristic properties of incompressible MHD turbulence. Thus, we find important similarities between our compressible MHD simulations and the theory of incompressible MHD turbulence (e.g., Goldreich k Sridhar 1995).
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FIGURE 2. Two-dimensional power spectra of driven models plotted along wavenumbers parallel fc|| and perpendicular k± to the mean magnetic field, left Result for weakly magnetized turbulence, right Result for strongly magnetized turbulence. Note the spectrum is anisotropic in this case.
By plotting the distribution of the passive contaminant along directions parallel and perpendicular to the field, it is also possible to determine mixing rates. Our preliminary analysis indicates that the mixing rate is isotropic in the weak field case, but 10 times larger along field lines than across them in the strong field case. Finally, we have studied the effect of ambipolar diffusion in the strong coupling limit on the decay rate of driven turbulence. In this limit, the inertia of the ions is neglected, allowing one to write the induction equation as
^
+ V x (B x un) = V x ( — ? — x [B x (V x B)]l,
(2.1)
where un is the velocity of the neutrals, pi and pn are the density of the ions and neutrals respectively, and 7 is the ion-neutral collision coupling constant. The ion density can be eliminated from the expression if ionization equilibrium is assumed, so that p, = Cpn where C is a constant. The importance of ambipolar diffusion to the dynamics can be measured in terms of an effective "Reynolds" number expressed as the ratio of the Alfven crossing time to the ion-neutral collision time. In real clouds, we expect this ratio to be (Re) A D =
tA
45
B
-1
(2.2)
We have computed driven MHD turbulence models including ambipolar diffusion in the strong coupling limit (i.e. we solve equation 2.1) using effective Reynolds numbers of 25 and 100. We find that the decay rate of driven turbulence in these models can be increased by a factor of about two above the ideal MHD case. Preliminary investigation of the power spectra in the models including ambipolar diffusion indicate no additional power at large k in the magnetic field, which might be expected if sharp current sheets were being enhanced by the ambipolar diffusion (Brandenburg & Zweibel 1994).
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3. Results from Other Models We have studied the properties of decaying supersonic MHD turbulence using the saturated state computed in the driven models as an initial condition. At some late time t0 we simply turn off the driving mechanism, and let the turbulence decay. Because these decay models begin from a self-consistent saturated initial condition (at least for the forcing scheme used here); this technique avoids any questions about what initial power spectrum to choose to describe the initial state. We find the wave energy in decaying turbulence models asymptotically falls as a power law in time with a slope near -0.85, in agreement with previous results (e.g. see MacLow et al, this volume). The slope shows only small variation with the strength of the mean magnetic field in the model. Because the wave energy decreases roughly as t~l, this means the dissipation rate, which is proportional to Ew/E\v, also is proportional to t~l. Therefore, the decay rate in these models is not well defined as it depends on the initial amplitude of the turbulence, and the time in the problem. 4. Summary We have used direct numerical simulations of driven compressible MHD turbulence to study questions of direct relevance to the dynamics of the cold ISM. We find the decay rate of supersonic turbulence is large, roughly one turnover time at the scale at which the turbulence is driven. The power spectrum of velocity and magnetic field fluctuations at saturation shows several interesting features, including evidence for an inverse cascade at large scales. Strongly magnetized turbulence (defined as a velocity dispersion in the saturated state less than the Alfven speed) shows an anisotropic power spectrum with more power perpendicular to the field than parallel to it. Moreover, the 3-D spectrum appears to have a slope close to Kolmogorov, whereas weakly magnetized and hydrodynamic compressible turbulence have a slightly steeper spectrum consistent with a flow dominated by shocks. These latter results are in accordance with the expectations of studies of strong incompressible MHD turbulence (see Sridhar, this volume, and Goldreich & Sridhar 1995). Much work has yet to be done in further studying the gas dynamics revealed by these simulations, and in directly comparing the properties of the turbulence to observations of molecular clouds. Support for this work is provided by NASA under contract number NAG53840. Computations were performed on the Cray/SGI Origin 2000 system at the National Center for Supercomputing Applications. I thank my colleagues Charles Gammie and Eve Ostriker who contributed significantly to the work presented here.
REFERENCES BRANDENBURG, A., & ZWEIBEL, E.G. GAMMIE, C.F., GAMMIE,
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M.L., 1992, ApJS, 80, 791
ApJ
Fragmentation in Molecular Clouds: The Formation of a Stellar Cluster By RALF KLESSEN AND ANDREAS BURKERT Max-Planck-Institut fur Astronomie, Konigstuhl 17, 69117 Heidelberg, Germany The isothermal gravitational collapse and fragmentation of a molecular cloud region and the subsequent formation of a protostellar cluster is investigated numerically. The clump mass spectrum which forms during the fragmentation phase can be well approximated by a power law distribution dN/dM <x M~15. In contrast, the mass spectrum of protostellar cores that form in the centers of Jeans unstable clumps and evolve through accretion and iV-body interaction is best described by a log-normal distribution. Assuming a star formation efficiency of ~10%, it is in excellent agreement with the IMF of multiple stellar systems.
1. Introduction Understanding the processes leading to the formation of stars is one of the fundamental challenges in astronomy and astrophysics. However, theoretical models considerably lag behind the recent observational progress. The analytical description of the star formation process is restricted to the collapse of isolated, idealized objects (Whitworth & Summers 1985). Much the same applies to numerical studies (e.g. Boss 1997; Burkert et al. 1997 and reference therein). Previous numerical models that treated cloud fragmentation on scales larger than single, isolated clumps were strongly constrained by numerical resolution. Larson (1978), for example, used just 150 particles in an SPH-like simulation. Whitworth et al. (1995) were the first who addressed star formation in an entire cloud region using high-resolution numerical models. However, they studied a different problem: fragmentation and star formation in the shocked interface of colliding molecular clumps. While clump-clump interactions are expected to be abundant in molecular clouds, the rapid formation of a whole star cluster requires gravitational collapse on a size scale which contains many clumps and dense filaments. Here, we present a high-resolution numerical model describing the dynamical evolution of an entire region embedded in the interior of a molecular cloud. We follow the fragmentation into dense protostellar cores which form a hierarchically structured cluster. 2. Numerical Technique and Initial Condition To follow the time evolution of the system, we use smoothed particle hydrodynamics (SPH: for a review see Monaghan 1992) which is intrinsically Lagrangian and can resolve very high density contrasts. We adopt a standard description of artificial viscosity (Monaghan &; Gingold 1983) with the parameters av = 1 and j3v = 2. The smoothing lengths are variable in space and time such that the number of neighbors for each particle remains at approximately fifty. The system is integrated in time using a second order Runge-Kutta-Fehlberg scheme, allowing individual timesteps for each particle. Once a highly condensed object has formed in the center of a collapsing cloud fragment and has passed beyond a certain density, we substitute it by a 'sink' particle which then continues to accrete material from its infalling gaseous envelope (Bate et al. 1995). By doing so we prevent the code time stepping from becoming prohibitively small. This procedure implies that we cannot describe the evolution of gas inside such a sink particle. For a 272
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detailed description of the physical processes inside a protostellar core, i.e. its further collapse and fragmentation, a new simulation just concentrating on this single object with the appropriate initial conditions taken from the larger scale simulation would be necessary (Burkert et al. 1998). To achieve high computational speed, we have combined SPH with the special purpose hardware device GRAPE (Ebisuzaki et al. 1993), following the implementation described in detail by Steinmetz (1996). Since we wish to describe a region in the interior of a molecular cloud, we have to prevent global collapse. Therefore, we use periodic boundaries, applying the Ewald method in an PM-like scheme (Klessen 1997). The structure of molecular clouds is very complex, consisting of a hierarchy of clumps and filaments on all scales (e.g. Blitz 1993). Many attempts have been made to identify the clump structure and derive its properties (Stutzki & Giisten 1990, Williams et al. 1994). We choose as starting conditions Gaussian random density fluctuations with a power spectrum P(k) oc l/kN and 0 < N < 3. Thefieldsare generated by applying the Zel'dovich (1970) approximation to an originally homogeneous gas distribution: we compute a hypothetical field of density fluctuations in Fourier space and solve Poisson's equation to obtain the corresponding self-consistent velocity field. These velocities are then used to advance the particles in one single big timestep St. We present simulations with 50 000 and 500 000 SPH particles, respectively. 3. A Case Study As a case study, we present the time evolution of a region in the interior of a molecular cloud with P(k) oc \/k2 and containing a total mass of 222 Jeans masses determined from the temperature and mean density of the gas. Figure 1 depicts snapshots of the system initially, and when 10, 30 and 60 per cent of the gas mass has been accreted onto the protostellar cores. Note that the cube has to be seen periodically replicated in all directions. At the beginning, pressure smears out small scale features, whereas large scale fluctuations start to collapse into filaments and knots. After t^O.3, the first highlycondensed cores form in the centers of the most massive and densest Jeans unstable gas clumps and are replaced by sink particles. Soon clumps of lower mass and density follow, altogether creating a hierarchically-structured cluster of accreting protostellar cores. For a realistic timing estimate, the Zel'dovich shift interval St = 2.0 has to be taken into account. In dimension-less time units, the free-fall time of the isolated cube is r^ = 1.4. 3.1. Scaling Properties The gas is isothermal. Hence, the calculations are scale free, depending only on one parameter: the dimensionless temperature T = E-mt/\Epot\, which is defined as the ratio between the internal and gravitational energy of the gas. The model can thus be applied to star-forming regions with different physical properties. In the case of a dark cloud with mean density n(H2) ^ 100 cm" 3 and a temperature T ~ 10 K like Taurus-Auriga, the computation corresponds to a cube of length 10 pc and a total mass of 6 3 0 0 M Q . The dimensionless time unit corresponds to 2.2 x 106yrs. For a high-mass star-forming region like Orion with n{H2) ^ 105cm~3 and T ~ 30 K these values scale to 0.5 pc and IOOOM0, respectively. The time scale is 6.9 x 104yrs. 3.2. The Importance of Dynamical Interaction and Competitive Accretion The location and the time at which protostellar cores form, is determined by the dynamical evolution of their parental gas clouds. Besides collapsing individually, clumps stream towards a common center of attraction where they merge with each other or undergo
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t = 1.3 FIGURE 1. Time evolution and fragmentation of a region of 222 Jeans masses with initial Gaussian densityfluctuationswith power law P(k) oc 1/fc2. Collapse sets in and soon forms a cluster of highly-condensed cores, which continue to accrete from the surrounding gas reservoir. At t = 0.7 about 10% of all the gas mass is converted into "protostellar" cores (denoted by black dots). At t = 1.3 and t — 2.0 these values are 30% and 60%, respectively. Initially the cube contains 50000 SPH particles.
further fragmentation. The formation of dense cores in the centers of clumps depends strongly on the relation between the timescales for individual collapse, merging and subfragmentation. Individual clumps may become Jeans unstable and start to collapse to form a condensed core in their centers. When clumps merge, the larger new clump continues to collapse, but contains now a multiple system of cores in its center. Now sharing a common environment, these cores compete for the limited reservoir of gas in their surrounding (see e.g. Price & Podsiadlowski 1995, Bonnell et al. 1997). Furthermore, the protostellar cores interact gravitationally with each other. As in dense stellar clusters, close encounters lead to the formation of unstable triple or higher order systems and alter the orbital parameters of the cluster members. As a result, a considerable fraction of "protostellar" cores get expelled from their parental clump. Suddenly bereft of the massive gas inflow from their collapsing surrounding, they effectively stop accreting and their final mass is determined. Ejected objects can travel quite far and resemble the weak
Klessen & Burkert: Fragmentation in Molecular Clouds 2.0 1.5
r
0%
b)
10%
275
1.5
1.0 ]
0.5 0.0 -0.5
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30%
60%
'iri/1.
-0.5 -2.0-1.5-1.0-0.5 0.0 0.5 1.0-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 log M/M, log M/M,
-1.5
-1.0
-0.5
0.0 0.5 log M/M,
FIGURE 2. a) - d) Mass distribution of gas clumps (thin lines) and of protostellar cores (thick lines) at times t = 0.0, 0.7, 1.3 and 2.0 when 0%, 10%, 30% and 60% of the total gas mass is condensed in cores, respectively. The vertical lines indicate the resolution limit of the simulation with 500000 particles (Bate & Burkert 1997), and the dashed lines illustrate the observed clump mass spectrum with dN/dM oc M~ (Blitz 1993). e) Comparison of the final core mass spectrum (thick line) with different observationally based models for the IMF. The thick dashed line denotes the log-normal form for the IMF, uncorrected for binary stars as proposed by Kroupa et al. (1990). In order for the peaks of both distributions to overlap, a core star formation efficiency of 10% has to be assumed. The agreement in width is remarkable. The multiple power-law IMF, corrected for binary stars (Kroupa et al. 1993) is shown by the thin solid line. As reference, the thin dashed line denotes the Salpeter (1955) IMF. Both are scaled to fit at the high-mass end of the spectrum. All masses are normalized to the overall Jeans mass in the system. line T Tauri stars found via X-ray observation in the vicinities of star-forming molecular clouds (e.g. Neuhauser et al. 1995).
3.3. Mass Spectrum - Implications for the IMF Figures 2a - d describe the mass distribution of identified gas clumps (thin lines) and of protostellar cores (thick lines) that formed within unstable clumps in a simulation analogous to Fig. 1, but with 10 times higher resolution. To identify individual clumps we have developed an algorithm similar to the method described by Williams et al. (1994), but based on the framework of SPH. As reference, we also plot the observed canonical form for the clump mass spectrum, dN/dM oc M~1-5 (Blitz 1993), which has a slope of —0.5 when plotting N versus M. Note that our initial condition does not exhibit a clear power law clump spectrum. The Zel'dovich approximation generates an overabundance of small scale fluctuations. However, in the subsequent evolution, these small clumps are immediately damped by pressure forces and non-linear gravitational collapse begins to create a power-law like mass spectrum. A common feature in all our simulations is the broad mass spectrum of protostellar cores which peaks slightly above the overall Jeans mass of the system. This is somewhat surprising, given the fact that the evolution of individual cores is highly influenced by complex dynamical processes. In a statistical sense, the system retains 'knowledge' of its (initial) average properties. The present simulations cannot resolve sub-fragmentation in condensed cores. Since detailed simulations show that perturbed cores tend to break up into multiple systems (e.g. Burkert et al. 1997), we can only determine the mass function of multiple systems. Our simulations predict an initial mass function with a log-normal functional form. Figure 2e compares the results of our calculations with the observed IMF for mul-
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tiple systems from Kroupa et al. (1990). Assuming a typical Jeans mass Mj ss 1 M 0 and a star formation efficiency of individual cores of 10%, the agreement between the numerically-calculated mass function and the observed IMF for multiple systems (thick dashed line; from Kroupa et al. 1990) is excellent. For comparison, also the IMF corrected for binary stars (Kroupa et a. 1993) is indicated as thin solid line, together with the mass function from Salpeter (1955) as thin dashed line.
4. Discussion Large-scale collapse and fragmentation in molecular clouds leads to a hierarchical cluster of condensed objects whose further dynamical evolution is extremely complex. The agreement between the numerically-calculated mass function and the observations strongly suggests that gravitational fragmentation and accretion processes dominate the origin of stellar masses. The final mass distribution of protostellar cores in isothermal models is a consequence of the chaotic kinematical evolution during the accretion phase. Our simulations give evidence, that the star formation process can best be understood in the frame work of a probabilistic theory. A sequence of statistical events may naturally lead to a log-normal IMF (see e.g. Zinnecker 1984; Adams & Fatuzzo 1996; also Price & Podsiadlowski 1995; Murray & Lin 1996; Elmegreen 1997).
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MNRAS, 277, 727
Accretion Disk Turbulence By CHARLES F. GAMMIE1-2 1 2
Isaac Newton Institute, 20 Clarkson Rd., Cambridge, CB3 OEH, UK
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA, 02138, USA
I review recent developments in the theory of turbulence in centrifugally supported astrophysical disks. Turbulence in disks is astrophysically important because it can transport angular momentum through shear stresses and thus allow disks to evolve and accrete. Turbulence can be initiated by magnetic, gravitational, or purely hydrodynamic instabilities; I give an abbreviated review of the linear and nonlinear theory of each of these possibilities, and conclude with a list of problems.
1. Introduction Spiral galaxies, quasars, active galactic nuclei, X-ray binaries, cataclysmic variables, and young stars: these are a few of the astronomical objects that contain disks. Disks are common in astrophysics because it is usually difficult to change the specific angular momentum of gas, but easy to radiate away its thermal energy. Gas injected into in a spherically symmetric potential thus naturally shocks, radiates, and settles down into a plane normal to its mean angular momentum. Because they are so common, disks occupy a lot of the astronomical community's time and energy (that would otherwise be entirely dissipated in attempting to measure f2o). Although there are enormous differences between individual disk systems in global structure and observational appearance, there are a number of fluid dynamical processes common to all disks. These processes are worth understanding in detail. The most fundamental process in disks, analogous to nuclear reactions in stars, is angular momentum transport. The disk cannot evolve unless gas in the disk can be persuaded to give up some of its angular momentum and spiral down the gravitational potential. Accretion disks (e.g. CV disks, but not spiral galaxies) are mainly heated by frictional processes associated with this gradual inflow; we would not see them at all absent some process for redistributing angular momentum. The central role of angular momentum transport is evident if we write down an equation for the evolution of the disk surface density E, obtained directly from the angular momentum and continuity equation in the limit that the disk is thin: r, t)
1
0 / 1 0 ,
2__,
,
here Cl = orbital frequency ~ r~q and t,w is the mass lost in a wind. Angular momentum is either redistributed (diffused) through the disk by the height-integrated and azimuthally averaged shear stress Wrip = J dzd<j>wr(f,/(2TV) (using cylindrical coordinates centered on the disk) or else removed directly from the disk by an external torque r per unit area, provided perhaps by a magnetohydrodynamic (MHD) wind. Without angular momentum transport (or a wind) the disk does not evolve. The shear stress wr = pVrdvt - ^BrB,/, 277
+ ^Q^9r9-
(1-2)
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Here 5 implies departures from the mean value and g is the gravitational field. One of the main goals of accretion disk theory is to calculate wr^,. Disk turbulence is interesting because of its importance to accretion disk evolution, but it is also interesting from a fluid-dynamical viewpoint as a model, self-sustained turbulent system. In this review I will discuss turbulence initiated by magnetohydrodynamical instabilities (Section 2), gravitational instabilities (Section 3), and purely hydrodynamical instabilities (Section 4). In conclusion (Section 5) I discuss directions for future research. Finally, there are a number of other recent reviews that focus on similar issues: Balbus & Hawley (1998) gives a complete discussion of the literature; Gammie (1998a) focuses on recent numerical experiments; Stone et al. (1998) discuss transport processes in protostellar disks; Brandenburg (1998) gives a different perspective on the numerical experiments with an emphasis on the connection to dynamo theory.
2. Magnetohydrodynamic turbulence 2.1. Balbus-Hawley instability: Linear Theory Of all the unstable modes discovered to date one stands out as having the largest growth rates in the largest part of disk "phase space." It is a local, linear, MED instability (here local means that it does not depend on the global radial or vertical structure of the disk) first understood in the context of accretion disks by Balbus & Hawley in 1991 (although it was discovered earlier in the context of magnetized Couette flow by Velikhov 1959). The BH instability is most easily explained using a mechanical analogy first developed by Balbus & Hawley (1992). Consider two equal masses in coplanar orbit in a Keplerian potential. The masses are close together in the sense that | 5_£. (STABLE) (2.6) dz IV This is equivalent to the Schwarzschild criterion written as if the magnetic field were absent (it is, however, implicitly present in the equilibrium). I will only quote the stability criterion for the Parker mode, since it always goes unstable before the interchange mode. Notice that a large radiative diffusivity, as in accretion disks, can erase the stabilizing effects of stable stratification (Acheson 1978, Hughes 1985), driving the Parker mode stability criterion back to that for an adiabatic atmosphere: dlnB/dz > 0. Brandenburg et al. (1995) and Stone et al. (1996) have studied the nonlinear outcome of the Balbus-Hawley instability in a stratified disk, which is then potentially subject to magnetic Rayleigh-Taylor instabilities. In the experiments of Stone et al. (1996) there was no significant vertical Poynting flux, suggesting that magnetic buoyancy was not dynamically important. The vertical run of magnetic pressure was consistent with marginal stability to the Parker mode. Radiation-dominated disks, such as might be found in disks around neutron stars and black holes accreting near the Eddington limit, are subject to yet another type of quasi-global instability called "photon bubbles" (Arons 1992, Gammie 1998b). The instability occurs in radiation dominated regions where the magnetic pressure exceeds the gas pressure. The nonlinear outcome of the instability is not yet known, although it has been investigated in the context of neutron star polar cap accretion by Hsu et al. (1997), where it greatly enhances the vertical transport of energy. 2.4. Global MHD Instabilities Disks are potentially subject to an enormous variety of global instabilities. Global or quasi-global linear analyses of model astrophysical disks may be found in Papaloizou & Szuszkiewicz (1992), Gammie & Balbus (1994), Ogilvie & Pringle (1996), Terquem & Papaloizou (1996), Curry & Pudritz (1996), and Ogilvie (1998), to name a few. There is also an extensive literature on global instabilities in contexts other than disks (e.g. Couette flows); see Balbus & Hawley (1998) and references therein. Global analyses can exhibit the effects of boundaries and background gradients in the flow, but they lack the generality of local analyses since they depend on particular choices of the equilibrium. The three-dimensional nonlinear outcome of Balbus-Hawley and other global MHD instabilities in disks has not yet been studied, except in an idealized Couette flow model (Armitage 1998). 3. Gravitational Instability In the outer parts of AGN disks and protostellar disks gravitational instability may compete with or completely dominate the BH instability. Local stability of the disk is determined by Toomre's Q = cs«;/(7rGE), where K = epicyclic frequency. For Q < 1 the
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disk is axisymmetrically unstable.(the precise value depends on the vertical structure of the disk; 0.676 is the critical value for an isothermal disk finite thickness disk, 1 for a zero thickness disk). In an a disk, the instability criterion can be rewritten in the form r3
/
r3
\3
s (3.7) M > 3a-?- ~ 7 x 10~3a —r M 0 yr~\ (UNSTABLE) G \ lkms / a condition easily satisfied in the outer parts of AGN disks for a ~ 1, and in YSO disks for a « l . Disks that are grossly unstable do not exist in nature, so the nonlinear theory of such systems is a mathematical exercise. Instead it is likely that disks are driven unstable, either by cooling (lowering cs) or by mass-loading (raising S, possibly via infall), and that stability is partially recovered in the nonlinear outcome either by dissipation (raising cs, possibly by shock heating) or by mass-shedding (lowering E, in AGNs possibly by star formation). For an a disk, cooling gives d\nQ/dt ~ ad. If infall is to dominate this cooling, then it is easy to show that the infall accretion rate per logarithmic interval in radius must exceed the accretion rate within the disk by a factor of order (R/H)2. It thus seems likely that cooling is the main driver of gravitational instability in most circumstances. The nonlinear outcome of gravitational instability with cooling has been studied in the context of a thin, local model of a gaseous disk by Gammie (1998c). I find that the disk goes unstable due to cooling and that, if certain conditions are satisfied, it then shock heats and returns to marginal stability. In the outcome the disk contains fluctuating surface density variations of order unity, and the density correlation length is of order 2nQH. The density structure transport angular momentum through both Reynolds stress and through gravitational stresses. Finally, a note on linear theory: it is somewhat under appreciated that self-gravitating disks with constant dynamic viscosity are secularly unstable, a point first noticed by Lynden-Bell and Pringle (1974) and later discussed in the context of differentially rotating disks by Safronov (1991), Willerding (1992), and Gammie (1996). The instability grows on the viscous timescale ("viscosity" is here a proxy for smaller-scale turbulence; molecular viscosity is negligible). In the limit of weak viscosity, the growth rate s of an axisymmetric mode in a zero thickness disk is
For an a disk (uturb ~ acsH), in the limit that Q > 1 and a 0, i.e. specific angular momentum should
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increase outwards. Most disks, and in particular Keplerian disks, satisfy the Rayleigh criterion. It has been suggested that Keplerian disks are locally nonlinearly unstable because of their high Reynolds number (e.g. Shakura & Sunyaev 1973, Lynden-Bell & Pringle 1974). This idea has been developed in some detail by Dubrulle & Zahn (1991), Dubrulle (1993), and Kato & Yoshizawa (1997). Numerical experiments in the local model (Balbus et al. 1996), however, fail to find any evidence of nonlinear instability in Keplerian shear flows. Nonlinear instability is found in a narrow band near dlnil/dlnr — —2, i.e. in disks that are marginally stable by the Rayleigh criterion. While one can always ask whether the numerical experiments achieve sufficiently high Reynolds number, Balbus et al. (1996) present an argument based on moments of the momentum equations that suggests, but does not prove, that Keplerian disks are nonlinearly stable. Disks can also suffer quasi-global instabilities such as convection (see Ruden et al. 1988 for the axisymmetric linear theory). One point that is not generally appreciated is the degree to which ordinary convective instabilities are damped by radiative diffusion in disks (although there are other, inertial, oscillations that become overstable in the presence of radiative diffusion). Workers had long thought that convection might lead to enhanced turbulent transport of angular momentum in disks, the idea being that turbulence always implies transport. An early sign that this expectation might be incorrect was a quasi-linear calculation (Ryu & Goodman 1992) of the angular momentum flux associated with linear, nonaxisymmetric convective motions; the direction of the flux was found to be inwards rather than outwards. Subsequent numerical experiments (Stone & Balbus 1996; Cabot 1996) showed that in the nonlinear regime the angular momentum flux was small and inwards. This nonintuitive result is a nice illustration of the value of numerical experiments. Finally, disks are susceptible to a wide variety of global hydrodynamic instabilities. One example is the Papaloizou-Pringle (1984, 1985) instability, subsequently elucidated by Narayan et al. (1986); see Savonije & Heemskerk (1990) for a readable physical account of this and allied global instabilities. A different type of global instability has been discovered by Goodman (1993). It requires a tidal field capable of distorting the disk streamlines into an oval shape. The instability grows from the free energy available in this oval distortion, causing it to decay by parametric instability into small scale inertial oscillations.
5. Conclusions Great progress has been made in the last few years in understanding the origins and development of turbulence in accretion disks. We know that under a broad range of conditions the BH instability can initiate turbulence that transports angular momentum outwards. We also have strong numerical evidence that other types of turbulence in disks, such as convective turbulence, do not necessarily provide the angular momentum transport required for disk evolution. But there are still many interesting open questions about turbulence in disks; I will conclude with three of particular current interest. 1. Is angular momentum transport local? Numerical studies of the three dimensional nonlinear outcome of the BH instability have so far been restricted to regions of the disk of order H in size (but see Armitage 1998). It is always found that most of the energy, and angular momentum flux, is contained in structures that are as large as allowed in the experiments. Thus the outcome is limited by the experiment size. What will happen in more realistic, larger-scale experiments? One possibility is that largest scale structures will have a small fraction of the turbulent
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energy, with most the turbulent energy being concentrated at scales of order H. In this case angular momentum transport would be truly local. But another possibility is that most of the energy is always contained in the largest scale structures allowed. Then angular momentum transport would be mainly due to structure much larger than the disk scale height: it is nonlocal. This would be inconsistent with current approaches to modeling disk evolution embodied in the a model. Numerical experiments may be able to decide between these, and intermediate, alternatives in the near future. 2. Are unmagnetized Keplerian disks nonlinearly stable ? Numerical experiments have diligently sought nonlinear instability in Keplerian disks and not found it (Balbus et al. 1996). But there remains a pool of skeptics who point out that the numerical experiments do not reach astrophysical Reynolds numbers, and so there is still the possibility of nonlinear instability. Since astrophysical Reynolds numbers will never be computationally accessible, what is needed is either a proof of nonlinear stability- a mathematically challenging problem- or an explicit demonstration of nonlinear instability. But for now the bulk of the evidence seems to favor the nonlinear stability of Keplerian shear flows. 3. How do waves and turbulence interact in disks? It is common to model the effect of turbulence on waves as a viscosity. This is done in studies of the tidal interaction between planets and protostellar disks (e.g. Lin & Papaloizou 1993), and in studies of warped disks (e.g. Pringle 1996); in these examples the turbulent viscosity completely governs the evolution of the disk. But the viscous model is completely untested. It could be quite misleading if, for example, it amplifies certain modes, or couples together linear modes of the laminar disk, or even gives the disk gas elastic properties. Numerical experiments that are immediately practical could measure the effects of turbulence on large-scale waves and settle this issue. I am grateful to Jim Stone, Eve Ostriker, and Gordon Ogilvie for their comments and suggestions. This work was supported in part by NASA grant NAG 52837.
ACHESON,
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LIST OF PARTICIPANTS
1. Alvarez, Cesar 2. Antnez Garcia, Joel 3. Arias, Lorena 4. Arthur, Jane 5. Avila, Remy 6. Balsara, Dinshaw 7. Ballesteros-Paredes, Javier 8. Barrera, Pablo 9. Benjamin, Robert 10. Bensch, Prank 11. Bhattacharjee, Amitava 12. Blackman, Eric 13. Blom, Jon J. 14. Braun, Robert 15. Bregman, Joel 16. Brunt, Christopher 17. Cardona, Octavio 18. Carraminana, Alberto 19. Carrasco, Luis 20. Carrasco, Esperanza 21. Castafieda, Lizbeth A. 22. Colombon, Laura 23. Cordes, James 24. Crutcher, Richard 25. Cuevas, Salvador 26. Chatterjee, Tapan 27. Chavez, Miguel 28. Chavira, Enrique 29. Dalessio, Paola 30. Desai, Ketan 31. Dottori, Horacio 32. Duschl, Wolfgang J. 33. Elmegreen, Bruce 34. Falgarone, Edith 35. Flores, Aaron 36. Franco, Jose 37. Gammie, Charles 38. Garcia, Nieves 39. Garcia, Jose 40. Garcia-Segura, Guillermo 41. Gazol, Adriana 42. Gehman, Curtis 43. Gibson, Carl H. 44. Goldreich, Peter 45. Gomez-Reyes, Gilberto 46. Gonzalez, Alejandro
285
[email protected] j ag@bufadora. astrosen. unam. mx [email protected] j ane@astroscu. unam. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] blackman@ast .cam .ac.uk [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] cordes@spacenet. tn. Cornell, edu [email protected] chavoc@astroscu. unam. mx mchavez@inaoep. mx [email protected] dalessio@astroscu. unam. mx [email protected] [email protected] [email protected] [email protected] edith. falgarone@ensapb. ens. fr [email protected] [email protected] cgammie@cfa. harvard. edu nieves@astroscu. unam. mx [email protected] ggs@bufadora. astrosen. unam. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
286 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96.
List of Participants Gredel, Roland Guichard, Jose Gulati, Ravi Kumar Gwinn, Carl Ross Heithausen, Andreas Heyer, Mark Jokipii, Randy Joncas, Gilles Kessel, Olaf Klessen, Ralf Stephan Korpi, Maarit Lacey, Christina LaRosa, Ted Lazarian, Alex Lazio, Joseph Liljestrom, Tarja Lis, Darek Liszt, Harvey Lucas, Robert Luna, Abraham Mac Low, Mordecai-Mark MacLeod, Gordon Magnani, Loris Maron, Jason Martinez-Bravo, Oscar Mario Mayya, Divakara McKee, Christopher Melnick, Jorge Mendoza-Torres, Jose-Eduardo Minter, Anthony Miville-Deschenes, Marc-Antoine Muders, Dirk Munch, Guido Myers, Phil NG, Chung-Sang Nordlund, Ake Oey, Sally Orlov, Valeri Ortega, Maria Luisa Ossenkopf, Volker Ostriker, Eve Owocki, Stan Padoan, Paolo Palacios, Maria Norma Pereyra, Antonio Perez, Enrique Piccineli, Gabriela Pichardo, Barbara Pineda, Leopoldo Pouquet, Annick
[email protected] [email protected] [email protected] [email protected] [email protected] heyer@fermat .phast. umass.edu [email protected] [email protected] [email protected]. de [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] omartin@inaoep. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] dmuders@as. arizona. edu [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] .uni-koeln.de [email protected] [email protected] [email protected] palacios@inaoep. mx [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]
List of Participants 97. Puerari, Ivanio 98. Rajagopalan, Ramachandran 99. Recillas, Elsa 100. Reyes-Ruiz, Mauricio 101. Romano, Emilio 102. Romano, Roberto 103. Rosa, Daniel 104. Roth, Miguel 105. Santillan, Alfredo 106. Segura, Juan 107. Serrano, Alfonso 108. Seshadri, Sridhar 109. Silantev, Nikolai 110. Spangler, Steven 111. Stone, James 112. Sunada, Kazuyoshi 113. Tedds, Jonathan A. 114. Terlevich, Elena 115. Terlevich, Roberto 116. Thilker, David 117. Tovmassian, Hrant 118. Trinidad, Miguel Angel 119. Tufte, Stephen L. 120. Valdes-Parra, Jose Ramon 121. Valdez, Margarita 122. Valenzuela, Octavio 123. Van Atta, Charles 124. Vazquez, Gerardo 125. Vazquez-Semadeni, Enrique 126. Walterbos, Rene 127. Wall, William 128. Williams, Jonathan 129. Yam, Joel Omar 130. Zweibel, Ellen
287
[email protected] [email protected] [email protected] maurey @bufadora. astrosen. unam .mx eromano@inaoep. mx rromano@inaoep. mx danrosa@inaoep. mx [email protected] [email protected] j segura@inaoep. mx [email protected] [email protected] [email protected] srs@ vest a. physics. uiowa. edu [email protected] sunada@nro. nao. ac .j p [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] mago@inaoep. mx [email protected] c vanat ta@ames. ucsd. edu gerar@astroscu. unam. mx [email protected] rwalterb@nmsu .edu [email protected] [email protected] [email protected] [email protected]