Advances in Space Research Journal ~ Volume 43, Issue 10, Pages 1471-1594, 15 May 2009

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Advances in Space Research 43 (2009) 1471–1478 www.elsevier.com/locate/asr

The acceleration of Anomalous Cosmic Rays by stochastic acceleration in the heliosheath L.A. Fisk *, G. Gloeckler Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109-2143, USA Received 5 November 2008; received in revised form 9 February 2009; accepted 19 February 2009

Abstract Stochastic acceleration in the heliosheath appears to be a likely mechanism by which Anomalous Cosmic Rays (ACRs) are accelerated. However, most stochastic acceleration mechanisms are not appropriate. The energy density in the ACRs and in the interstellar pickup ions out of which the ACRs are accelerated greatly exceeds the energy density in the turbulence in the heliosheath. Thus, a traditional stochastic acceleration mechanism in which particles are accelerated by damping the turbulence will not work. A stochastic acceleration mechanism has been developed in which the total energy of the pickup ions and the ACRs is conserved. Energy is redistributed from the core pickup ions into a suprathermal tail to create the ACRs. A model for the acceleration of the ACRs in the heliosheath, based on this stochastic acceleration mechanism, is presented. The model provides reasonable fits to the spectra of suprathermal particles and ACRs observed by Voyager. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Anomalous Cosmic Rays; Acceleration; Heliosheath

1. Introduction Both Voyager spacecraft have now crossed the termination shock of the solar wind, where the supersonic solar wind flow goes subsonic to begin the process of merging with the local interstellar medium (Stone et al., 2005, 2008; Decker et al., 2005, 2008; Burlaga et al., 2005, 2008; Gurnett and Kurth, 2005, 2008). The Voyagers are now penetrating into the heliosheath. Voyager 1 crossed in December 2004 at 94 AU from the Sun. Voyager 2 crossed in August 2007 at 83.7 AU. The two spacecraft were at substantially different heliographic latitudes; Voyager 1 crossed at +34.3° and Voyager 2 at 27.5°. The shock crossings produced a major surprise. There was no evidence at either crossing of the acceleration of Anomalous Cosmic Rays (ACRs). Interstellar neutral gas is swept into the solar system by the motion of the Sun relative to the local interstellar medium. Once near the Sun *

Corresponding author. E-mail address: lafi[email protected] (L.A. Fisk).

the neutrals are ionized and then picked up by the outward flowing solar wind, acquiring energies 1 keV nucleon1. The pickup ions are then convected into the outer solar system, where they are accelerated to energies of tens of MeV nucleon1, and form the ACRs (Fisk et al., 1974). It was expected that the ACRs would be accelerated at the termination shock for the simple reason that shocks are known to accelerate particles, and the termination shock is expected to be a relatively strong shock that surrounds the solar system (Pesses et al. (1981), Jokipii, 1990 and reference therein; Zank, 1999 and references therein). It is possible of course that the ACRs are accelerated at other locations on the termination shock, besides where the two Voyagers crossed. McComas and Schwadron (2006) have proposed a model in which the ACRs are accelerated along the flanks of the termination shock, remote from the Voyagers. The Voyagers crossed closer to the nose of the heliosphere, in the direction the solar system is moving relative to the local interstellar medium. Given the turbulent nature of the heliospheric magnetic field, and the random directions that result, it is not obvious why the termination

0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.02.010

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L.A. Fisk, G. Gloeckler / Advances in Space Research 43 (2009) 1471–1478

shock on the flanks should be a preferred acceleration site compared to the nose. It is also necessary in this model that the accelerated ACRs move freely along the magnetic field in the heliosheath, with limited cross-field diffusion, to account for the Voyager observations (Kota and Jokipii, 2008). It will be important to build a model that is consistent with the propagation properties both of ACRs and of comparable-rigidity galactic cosmic rays, which are believed to propagate in the heliosheath primarily by cross-field diffusion. The alternative method for accelerating the ACRs is stochastic acceleration in the heliosheath. However, most stochastic acceleration mechanisms are not appropriate. In traditional stochastic acceleration mechanisms, and there are many in the literature, particles are accelerated by diffusing in velocity space (diffusion in velocity space is described in many basic plasma physics books, e.g., Bellan, 2006). The source of energy is the energy in the turbulence with which the particles are interacting; traditional stochastic acceleration mechanisms are a damping mechanism. In the heliosheath, however, the energy density in the turbulence is very much less than the energy density in the pickup ions and in the ACRs that are created from the pickup ions. The energy density or pressure in the pickup ions is dominant in the supersonic solar wind upstream from the termination shock. Voyager 2, which has a working solar wind plasma detector, observed that the flow energy of the solar wind is converted into energy in the pickup ions at the termination shock, not to solar wind thermal energy (Richardson et al., 2008). Thus, the pickup ions are by far the dominant internal energy in the heliosheath, and, as we shall discuss, the energy density in the ACRs must be comparable to that of the pickup ions to explain the observed ACRs. There is simply not enough energy in the turbulence in the heliosheath to account for the energy in the ACRs, and a traditional stochastic acceleration mechanism in which the ACRs are accelerated by damping the turbulence will not work. Fisk and Gloeckler (2006, 2007, 2008) have developed a stochastic acceleration theory that seems ideally suited to accelerate the ACRs in the heliosheath. The theory is based on the premise that there are circumstances, spatial homogeneity, where it is reasonable to assume that the total energy in the core pickup ions and in the suprathermal tails, which are stochastically accelerated from the core, is a constant. This stochastic acceleration mechanism is thus a redistribution mechanism; energy is redistributed from the core into the tail particles. Energy is not extracted from the turbulence. In the heliosheath, where the dominant energy is contained in the pickup ions, not in the turbulence, only a redistribution mechanism will work. In addition, the stochastic acceleration mechanism of Fisk and Gloeckler (2006, 2007, 2008) applies in compressional turbulence, where there are random compressions and expansions of the thermal plasma, which contains the mass. The subsonic heliosheath contains ample compressional turbulence (Burlaga et al., 2005, 2008).

In Fisk and Gloeckler (2008) the equation describing the time evolution of their stochastic acceleration mechanism is derived. The solutions to the equation yield an equilibrium spectrum for the suprathermal tail on the distribution function that is a power law in particle speed with spectral index of 5. This is a robust result, independent of the governing parameters such as the particle spatial diffusion coefficient. This result accounts for the fact that this spectral shape is observed to occur in many different circumstances in the solar wind, in the quiet solar wind and in disturbed conditions downstream from shocks (e.g., Gloeckler et al., 2008). Of particular relevance to the discussion here, this is the spectral shape of the suprathermal tails on the pickup ion distributions observed by Voyager 1, now deep in the heliosheath (Decker et al., 2006; Hill et al., 2006). Voyager 2, at the time of the writing of this paper, still appears to be relatively close to the termination shock, where the conditions and the suprathermal particle spectra remain time varying (Decker et al., 2008). However, in all conditions in the supersonic solar wind and in the heliosheath where it is reasonable to consider that spatial homogeneity prevails, conditions that should particularly prevail in the deep heliosheath, the spectral shape of the suprathermal tails are observed to be power laws with spectral index of 5, consistent with the results of Fisk and Gloeckler (2006, 2007, 2008). Fisk and Gloeckler (2008) applied their stochastic acceleration theory to the suprathermal tails observed in the supersonic solar wind, where adiabatic deceleration due to the expansion of the mean solar wind flow is important. They found energy and mass-to-charge dependencies of the rollovers in the spectra that occur at higher energies that agree well with observations from the Advanced Composition Explorer. They also found that the competition between the stochastic acceleration and the adiabatic deceleration limits the energy that suprathermal tail particles can achieve in the supersonic solar wind to about a few MeV nucleon1, consistent with the observations of Voyager in the outer heliosphere, upstream from the termination shock. In the subsonic heliosheath, there should be no or limited adiabatic deceleration. There is thus no competition to the stochastic acceleration mechanism of Fisk and Gloeckler (2008). The suprathermal tails on the pickup ions observed by Voyager should attain ever higher energies as the particles are convected into the heliosheath. The accelerated particles will then diffuse back in to be seen as the ACRs by Voyager, and eventually by other spacecraft in the inner heliosphere. In this paper, we apply the stochastic acceleration mechanisms developed in Fisk and Gloeckler (2008) to the acceleration of the ACRs in the heliosheath. We consider a simple model for acceleration in the heliosheath and show that the resulting spectra of the ACRs are in good agreement with the spectra observed by Voyager 1. We use the observations of multiple charge states of the higher energy ACRs to constrain the acceleration time, and thus our results are consistent with the multiple charge-state obser-


vations (Mewaldt et al., 1996). Finally, we comment on whether this theory can account for the observed behavior of the ACRs during the solar cycle. 2. The model Suprathermal tails on the pickup ion distributions, extending up to energies of about a few MeV nucleon1, are present in the supersonic solar wind in the outer heliosphere (Decker et al., 2005; Gloeckler et al., 2008). The spectra, expressed as distribution functions, are consistent with power laws in particle speed with spectral index of 5. Equivalently, the spectral index is 1.5 when the spectra are expressed as differential intensity, which is in more common use for higher energy particles. These particles are accelerated in crossing the termination shock, but the maximum energy of the suprathermal particles does not increase much beyond its upstream value. In the model of Fisk et al. (2006) for the termination shock, the pressure of the suprathermal tails behaves according to the Rankine–Hugoniot relationships, as do the core pickup ions. Initially behind the termination shock, there are time variations in the intensity and the spectral index of the suprathermal tails. However, as Voyager 1 has penetrated further into the heliosheath, in part because the termination shock at Voyager 1 appears to be moving inward, the spectra of the suprathermal tails settles into a remarkably constant power law spectrum with spectral index of 1.5 when expressed as differential intensity (Decker et al., 2006; Hill et al., 2006). Our model is then that the suprathermal tails present in the heliosheath are accelerated to higher energies to form the ACRs. We propose below that there is a prime acceleration region near the heliopause, where the suprathermal tails attain the highest energies, and that this is the source region of the ACRs. The particles accelerated in the prime acceleration region diffuse back into the heliosheath and ultimately into the inner heliosphere to be seen as ACRs. We use a simple spherically symmetric model for the heliosheath. The solar wind flow is subsonic, and there is no adiabatic deceleration competing with our stochastic acceleration. The pressure in the interstellar pickup ions is dominant, and in this subsonic medium this pressure must be constant. The suprathermal tails and ultimately the ACRs are accelerated out of the pickup ions, and so the total pressure or energy density in the pickup ions, the suprathermal tails and the ACRs must be constant. We take the spatial diffusion coefficient for all energetic particles, both the suprathermal tails and the ACRs, to be of the standard form, particle speed times a power law in particle rigidity, which for singly-charged pickup ions and in terms of particle kinetic energy, becomes j ¼ jo Aa Eðaþ1Þ=2 :

ð1Þ

Here, jo is a constant, i.e. we neglect here for simplicity any spatial dependence in the diffusion coefficient; A is the mass number, and E is particle kinetic energy, measured in units

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of MeV nucleon1. The power law index of the rigidity dependence of j is a, which is an adjustable parameter that will be chosen to fit the observations. We also take the mean square speed of the compressional turbulence in the heliosheath, du2, to be constant. 2.1. The basic equation and the resulting spectra We use the equation for stochastic acceleration derived in Fisk and Gloeckler (2008). The steady state equation for the distribution function, f, for the stochastic acceleration in a spherically symmetric heliosheath, is then @ 5 @ du2 @ 5 v ðv f Þ : ð2Þ u ðv f Þ ¼ v @r @v 9j @v Here, u is the radial component of the solar wind speed; v is particle speed; r is heliocentric radial distance. Eq. (2) has a straightforward solution " # 9j 1 5 : ð3Þ f ¼ fo v exp 2 ð1 þ aÞ du2 strans Here, the quantity strans satisfies Z r dstrans 1 1 ¼ or strans ¼ dr; dr u ro u

ð4Þ

and represents the transit time at the solar wind speed from where the particles are first able to be accelerated to where they obtain their highest energies. We can also express our resulting spectrum in terms of the differential intensity, j, which is more common in dealing with energetic particle spectra, or " # 9j 1 : ð5Þ j / E1:5 exp ð1 þ aÞ2 du2 strans We should note that there are various means by which strans could be limited. For example, particles could leak from the acceleration region by diffusion, in which case we can readily show that there is a maximum value for strans equal to strans;max ¼

3g ; ð1 þ aÞdu

ð6Þ

where g is the characteristic scale length for the diffusive escape. 2.2. The solar wind speed in the heliosheath Consider the solar wind speed in the heliosheath. In a simple spherically symmetric heliosheath, the solar wind velocity flows radially and falls off as 1/r2, yielding a constant solar wind density and no adiabatic deceleration. However, if the heliopause is not infinitely distant, then the solar wind flow needs to turn and flow parallel to the heliopause. The flow can still remain relatively incompressible ($ u = 0), with no adiabatic deceleration. However, the radial component of the flow goes to zero at the

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heliopause. If the accelerated particles flow freely along the primarily azimuthal magnetic field, or are capable of rapid spatial diffusion, the gradients of the accelerated particles should be small in the azimuthal direction, and Eq. (2), which is for spherically symmetric conditions, should still apply. The solar wind speed in Eq. (2) has then to be interpreted only as the radial component of the solar wind flow, and to go to zero at the heliopause. For simplicity we take the radial solar wind speed in the heliosheath to be 2 rts k : ð7Þ ur ¼ uts 2 exp r ðrhp rÞ Here, ts denotes termination shock and hp, the heliopause; k is the characteristic distance at which the radial solar wind speed begins to decline to zero at the heliopause. Consider, for example, that rts = 90 AU (the termination shock seen by Voyager 1 has moved inward); uts = 135 km s1, as seen on Voyager 2 (Richardson et al. 2008); rhp = 140 AU, and k = 8 AU. Then the transit time from the termination shock in years, given in Eq. (4), plotted versus the radial distance in AU, is shown in Fig. 1. Clearly, the particles spend most of their time near the heliopause. This is where the prime acceleration occurs, and where the highest energy particles are created; where the ACRs are created.

mum energy spectrum near the heliopause can be relative to the local suprathermal tail. Fisk and Gloeckler (2007) showed that the pressure in the tail relative to the pressure in the core particles should be given by P tail 2 b ¼ ; P core 5 ð1 þ bÞ

where b is the maximum relative spatial variation in the core or in the tail pressure. They found observational support for this expression; in the slow solar wind in the inner heliosphere, b 0.66, and Ptail/Pcore 0.16. Fisk et al. (2006) found that Ptail/Pcore 0.32 immediately upstream of the termination shock, as observed by Voyager 1. However, Gloeckler and Fisk (2006) then showed that the suprathermal particles upstream of the termination shock are due to beams of downstream particles, and the background spectrum which is actually accelerated at the shock is a factor of 3 lower than the average of the beam spectra. That being the case, a value of Ptail/Pcore 0.16 is probably a reasonable result upstream of the termination shock, and since the core and the tail pressures are each increased the same, according to the Rankine–Hugoniot relationship, this ratio should hold downstream as well. We take then the total pressure in the pickup ions immediately downstream from the termination shock to be P tot;ts ¼ 7:3P tail;ts ;

2.3. The normalization of the maximum energy spectrum

ð8Þ

ð9Þ

Consider the normalization factor for the spectrum in Eq. (5), evaluated at the maximum value of strans, i.e. the spectrum that attains the highest energy particles, the ACRs. This spectrum should occur near the heliopause. The basic requirement in the heliosheath is that the pressure in the pickup ions remains constant. The pickup ions are the dominant pressure and in the subsonic heliosheath the pressure must be constant. This requirement places a constraint on how much larger the intensity of the maxi-

where the subscript ts refers to near the termination shock. The criterion in Eq. (8) was derived by assuming that there was a fixed threshold between the core and the tail, and that only those core particles that were adiabatically compressed and obtain energies above the threshold could flow into the tail (Fisk and Gloeckler, 2007). The location of this threshold was established in Fisk and Gloeckler (2008) (see Eq. (19)) to be j r du ; ð10Þ l2 3

Fig. 1. Transit time in years versus the distance from the termination shock.

where l is the characteristic scale length of the compressional turbulence; e.g., the correlation length of the compressional turbulence. Note that |$ du| du/l. In principle, then, the threshold energy can decrease with decreasing l, provided that j does not scale as l. In our model for the heliosheath, the radial solar wind speed decreases as the flow approaches the heliopause. The scale length l is in the direction of du, which compresses the magnetic field and thus is in a direction normal to the magnetic field and in the radial direction. The scale length l should thus decrease as the radial solar wind speed decreases near the heliopause, and the threshold in the source region of the ACRs, given by Eq. (10), will be lower. That is, a larger fraction of the pressure in the core can be placed in the tail than is predicted by Eq. (8). In Eq. (8), compressions raise the energy of the particles to exceed the threshold, and only a certain fraction of the energy can be raised above the threshold. However, if the thresh-


old is lowered, there is no limit to how much core pressure can be deposited in the tail. In principle the core could vanish. In the source region of the ACRs there is only the core and the ACRs; the suprathermal tail is the ACRs. We take n to be the ratio of the tail to the core pressure, and thus the total pressure in the pickup ions in the source region s is P total;s ¼ ð1 þ nÞP ACR;s :

ð11Þ

The pressure of the ACRs near the termination shock also enters into the pressure balance required in the heliosheath. However, the pressure in the ACRs near the termination shock can be neglected since these particles are modulated, which also reduces the roll-over energy and the intensity. Our requirement of constant pressure in the pickup ions throughout the heliosheath thus becomes 7:3P tail;ts ¼ ð1 þ nÞP ACR;s :

ð12Þ

Some of the pressure increase in PACR in the source region, compared to Ptail,ts, results because the ACRs in the source region obtain higher energies than does the tail nearer to the termination shock. With a spectral index of 1.5 (or 5 when expressed as a distribution function) the pressure increases as the log of the energy. We also concluded that the low-energy threshold in the source region is lower than near the termination shock; e.g., let us take it to be a factor of 2 lower. Typical values that we will use below to fit the spectra observed by Voyager 1 are Eo = 160; Eo,tail,ts = 17; Eth = 0.008 (recall, E is in units of MeV nucleon1). Thus, the pressure increase in the ACRs in the source region due solely to the increase in energy is ln½2Eo =Eth ¼ 1:4: ln½Eo;tail;ts =Eth

ð13Þ

The remaining increase in pressure in the ACRs in the source region, required to satisfy Eq. (12), is due to an increase in the normalization factor on Eq. (5). There is also a possibility that the normalization factor is different for different species. Since the diffusion coefficient in Eq. (10) is a function of rigidity, the threshold energy can be lower for higher mass-to-charge particles, suggesting that the normalization factor will be larger for these particles as well. The above argument is best applied in reverse: we use our model to fit the observations of Voyager 1 and from these fits we infer the normalization factors in the source region for the different species of ACRs. Gloeckler et al. (2009) performed this exercise using a model similar to the one developed here and found that the increase in the normalization factor for ACRs in the source region, as compared to the normalization factor on the suprathermal tails seen at the location of Voyager 1, is a factor 2.5, and not strongly dependent on ACR species. This result is similar to the fits to the ACR spectra presented below. Thus, from Eqs. (12) and (13), we infer that the pressure in the ACRs in the source region is comparable to the pressure in the core pickup ions. This result

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strongly supports our conclusion that there is insufficient energy in the turbulence in the heliosheath to accelerate the ACRs, and thus a stochastic acceleration mechanism in which particles are accelerated by damping turbulence is inappropriate. 2.4. Conditions in the acceleration region near the heliopause and constraints from observations of multiple charge states We expect that the normalization factor on the ACR spectrum in the source region near the heliopause is a factor of 2.5 larger than the normalization factor on the spectrum of the local suprathermal tail. We also expect that the spectrum near the heliopause obtains the highest possible energies since this region, with its slow radial solar wind speeds, can contain the longest transit times. We need, however, to consider only the transit time near the heliopause since this is the region where the threshold is lowered and where there is the extra injection of particles by a factor 2, above the lower energy threshold, i.e., most of the particles accelerated near the heliopause were injected in this region.There is a constraint on the maximum allowable acceleration time, or in our case the maximum allowable transit time for particles injected near the heliopause. The presence of multiple charge states of O at energies above 20 MeV nucleon1 as seen by SAMPEX, and their absence below these energies, allowed Mewaldt et al. (1996) to conclude that the acceleration time of 10 MeV nucleon1 O had to be 1 year. All particles being accelerated by our mechanism experience the same transit time. We thus have a constraint that the transit time in the prime acceleration region, from where the threshold on the core is lowered and a factor of 2 times more particles are injected, to where the maximum energies are achieved, is 1 year. From Fig. 1, the prime acceleration region could be within a few AU of the heliopause. In either case the limit on the transit time could be determined by Eq. (6). With this constraint on the transit time we also have a constraint on the diffusion coefficient and the turbulent speeds in the prime acceleration region near the heliopause. We assume that this region near the heliopause may be different than the rest of the heliosheath, e.g., since there are strong shear flows at the heliopause, the region could experience Kelvin-Helmholtz instabilities and be more turbulent. We are free then to choose the diffusion coefficient and the turbulent speeds in the prime acceleration region, separate from the diffusion coefficient and turbulent speeds that prevail elsewhere in the heliosheath. We could also adjust the rigidity dependence of the diffusion coefficient, i.e., the value of a in Eq. (1), to be different near the heliopause, but we will not introduce this additional complication. We have then a single source region for the ACRs. It is near the heliopause where the radial solar wind speed is very low, the threshold between the core and the tail has been lowered, and diffusion coefficient and turbulent speeds are conducive to acceleration.


j ¼ jo E1:5

uj ¼ j

dj : dr

ð14Þ

Or with j given in Eq. (1), u given in Eq. (7), and the source spectrum given in Eq. (5), we find that the intensity spectrum of the ACRs at r is given by j ¼ jo E1:5

" ðaþ1Þ=2 # uts rv a E exp A exp : Eo jo Aa Eð1þaÞ=2

ð15Þ

Here, v¼

1 r

Z

rhp r

2 rts k dr0 ; exp ðrhp r0 Þ r 02

ð16Þ

and 2

Eoðaþ1Þ=2 ¼

ð1 þ aÞ du2hp strans;acc : 9jo;hp

ð17Þ

We have allowed for the fact that the conditions near the heliopause may be different from those elsewhere in the heliosheath, and labeled the parameters describing conditions near the heliopause by the subscript hp. We note also that the different choice for the diffusion coefficient near the heliopause does not affect the modulation since the radial solar wind speed is effectively zero there. The value of jo can be determined by fitting the observations and in turn will set the threshold between the core and the tail in the source region of the ACRs. Note that in determining the radial dependence of j in Eq. (16), the radial dependence of v needs to be taken into account. 2.6. Fitting the observations We first use the gradient of He observed by Voyager in the heliosheath to specify the spatial diffusion coefficient. At 16 MeV nucleon1, Stone et al. (2008) finds the gradient of He to be 5% per AU. We assume that this gradient exists at a reference location, rref, equal to 100 AU; we take the solar wind speed at the reference location to be uref. We can then use Eq. (14) to specify jo as jo ¼

uref rref 54

ð2aþ1Þ

:

ð18Þ

Eq. (18) can then be substituted into Eq. (15) to yield the final result for j for ACRs as

# " ð1þaÞ=2 # 5 4ð2aþ1Þ rv uts a E exp a ð1þaÞ=2 exp A : rref uref Eo AE ð19Þ

We will consider the Voyager 1 observations at r = 100 AU, or equivalently at rref. The solar wind speed, by Eq. (7), should decrease between rts and rref, or if rts = 90 AU for Voyager 1, and r = 100 AU, then, uts/uref = 1.23. If we then use the same parameters as in Fig. 2, we find that v = 0.14. Finally, Eq. (19) becomes at 100 AU, j ¼ jo E1:5

"

# " ð1þaÞ=2 # 0:85 4ð2aþ1Þ a E exp a ð1þaÞ=2 exp A : Eo AE

ð20Þ

In Fig. 2, we compare the solutions predicted by Eq. (20) with the ACR spectra observed by the LECP instrument on Voyager 1 (Decker et al., 2005, 2008; Gloeckler et al., 2008, 2009). The parameters are Eo = 160 and a = 0.87; the normalization constants are chosen to fit the observations, and all elements are a factor of 2 above the local suprathermal tails at 100 AU. Also shown in Fig. 2 are the local suprathermal tail spectra predicted by Eq. (3). Here the parameters are Eo = 17 and the same a = 0.87. The combined spectra yield a reasonable fit to the Voyager observations. Note that the spectra of 6 different elements are well fit with, in the case of the ACRs, only two parameters, Eo and a. If we use uts = 135 km s1, the value observed at the Voyager 2 crossing (Richardson et al., 2008), and the radial dependence of the radial solar wind speed in Eq. (7), we find from Eq. (18) that

Heliosheath 100

H

Voyager 1 LECP 2005.0 - 2008.0

He 10-2

10-4

10-6

N

O

Ne

Fig 2 dj/dE Sum 2005.0-2007.200 ACR paper.qpc

Since the ACRs come from a single source region, the modulation problem becomes very simple. The modulation in the heliosheath is simple convection-diffusion modulation, without adiabatic deceleration. The governing equation is

"

-1

2.5. Modulation of the ACRs

Differential Intensity (s cm 2 sr MeV/nuc)

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Ar

10-8

10-10 10-1

100

101

102

Energy/nucleon (MeV/nuc) Fig. 2. Model fits to the spectra of H, He, O, N, Ne, and Ar observed by the LECP instrument on Voyager 1 at 100 AU in the heliosheath. Data are from Decker et al. (2005, 2008) and Gloeckler et al. (2009).


jo ¼ 5:3 1019 cm2 s1 :

ð21Þ

Also, we assume that the prime acceleration region near the heliopause is more turbulent than elsewhere in the heliosheath and take du 100 km s1, as opposed to 35 km s1 observed by Voyager 2 (Richardson et al., 2008). Then, with the constraint that the transit time in the prime acceleration region should be 1 year, and with Eo = 160 (in MeV nucleon1) and a = 0.87, we find from Eq. (17) that jo;hp ¼ 1019 cm2 s1 :

ð22Þ

Thus, the diffusion coefficient in the prime acceleration region near the heliopause is a factor of 5 times smaller than elsewhere in the heliosheath. Again, we are free to choose a smaller value for the diffusion coefficient in the acceleration region, without regard to the impact on the modulation, since the radial solar wind speed is effectively zero there. Finally, from Eq. (6), with du 100 km s1 and the transit time of 1 year, we find that g = 12 AU, i.e., the principal escape by diffusion is across the heliopause. It should be emphasized that all of these values for the parameters describing the conditions near the heliopause depend on the constraint that the acceleration time of the ACRs is limited to 1 year. The general concepts of the theory for the acceleration of the ACRs would not change if there is a different limitation of the acceleration time; however, the final values for the governing parameters would be different. 2.7. Dependence on the solar magnetic cycle There is another constraint on acceleration of ACRs that is worth commenting upon: the dependence of the ACRs on the solar magnetic cycle. Full consideration of this constraint will require more sophisticated models than the simple analytic formulae that we have developed. We will leave that to later work. ACRs are known to vary with the solar magnetic cycle, with the largest intensities occurring in the so-called A < 0 portion of the solar magnetic cycle, when the heliospheric magnetic field is inward toward the Sun in the northern hemisphere. Jokipii and co-workers (Jokipii, 1982, 1986; Jokipii and Giacalone, 1998) have argued that this dependence is the result of gradient drifts along the termination shock. During the A > 0 portion of the solar magnetic cycle, the ACRs drift toward the pole along the termination shock, then inward into the heliosphere, and downward in latitudes, and then outward again along the current sheet. The maximum of the intensity of the ACRs is then near the poles. For the A < 0 cycle, when the heliospheric field is inward in the northern hemisphere, the directions are opposite, and the peak in the ACR intensity should be at low latitudes, which is seen as higher ACR intensities by Voyager and at Earth. Particles drift along the termination shock because of the sharp gradient in the magnetic field. The heliospheric

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magnetic field declines in the supersonic solar wind, as 1/r. The field strength always increases when the shock is crossed. Hence the gradients are in opposite direction, and the resulting drifts in the supersonic solar wind and along the termination shock are always opposite to each other. In our model for the heliosheath, the radial velocity of the solar wind slows down as the flow approaches the heliopause, and the flow turns to flow parallel to the heliopause. The flow could remain incompressible, and this is the normal assumption since the gas is subsonic. However, some compression is allowed. The actual requirement is that the pressure is constant, but the pressure is all in the pickup ions. Thus, it is possible that the density of the solar wind, and thus the frozen-in magnetic field increases near the heliopause. We have then exactly the same gradient in the field strength as occurs near the termination shock. The drifts are slower since the field increase occurs over a longer distance than the abrupt increase that occurs at the termination shock. Nonetheless, the drifts will carry the ACRs towards the pole in the A > 0 cycle and visa versa in the A < 0 cycle. 3. Concluding remarks We have developed a theory for the stochastic acceleration of ACRs in the heliosheath that is consistent with the required source of energy, viz. the pickup ions themselves, and that provides a reasonable fit to the spectra of ACRs observed by Voyager 1. We have limited our fits to Voyager 1 data, since Voyager 1, unlike Voyager 2, is now deep into the heliosheath, and a steady-state model, such as the one we have constructed, should be valid. Clearly, as Voyager 2 penetrates further into the heliosheath, it will provide further tests of our model. There are of course limitations to our analytic model, which can only be addressed with a full numerical simulation. One of these limitations is the assumption of spherical symmetry. We assumed that the accelerated pickup ions have only small azimuthal gradients in the heliosheath, and thus only the radial solar wind flow is important in determining the transit time and thus the acceleration time of the ACRs. This assumption, as well as our assumption that the acceleration of the ACRs occurs primarily near the heliopause, should be valid in the direction of motion of the solar system relative to the local interstellar medium, the so-called nose region, which is generally in the region where the Voyagers are located. Here, the heliosheath should be its narrowest, and thus radial gradients are more important than azimuthal gradients. Further, there is more likely to be a preferred acceleration region near the heliopause, where the drop in the solar wind speed occurs in a relatively narrow band, creating a region of preferred injection and acceleration. On the flanks of the heliosheath, however, conditions are likely to be different; in particular, the heliosheath will be wider, resulting in a more gradual decline in the radial solar wind flow, and perhaps different

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conditions near the heliopause. In many ways, we might expect that the heliosheath in the nose region is a better location for accelerating ACRs to high energies than on the flanks. This will of course introduce azimuthal variations in the production of ACRs. Such variations are unlikely to be observable in the inner heliosphere, since the Parker spiral of the heliospheric magnetic field in the supersonic solar wind circles the Sun many times between the inner heliosphere and the termination shock, and should smooth out any variations in the production of ACRs in the heliosheath. Acknowledgement This work was supported in part by the Voyager Guest Investigation grant NNX07AH, and by ACE data analysis contract, 44A-1080828. References Bellan, P.M. Fundamentals of Plasma Physics. Cambridge University Press, Cambridge, 2006. Burlaga, L.F., Ness, N.F., Acunã, M.H., Lepping, R.P., Connerney, J.E.P., Richardson, J.D. Magnetic fields at the solar wind termination shock. Nature 454, 75–77, 2008. Burlaga, L.F., Ness, N.F., Acunã, M.H., Lepping, R.P., Connerney, J.E.P., Stone, E.C., McDonald, F.B. Crossing the termination shock in the heliosheath: magnetic fields. Science 309, 2027–2029, 2005. Decker, R.B., Krimigis, S.M., Roelof, E.C., Hill, M.E., Armstrong, T.P., Gloeckler, G., Hamilton, D.E., Lanzerotti, L.J. Voyager 1 in the foreshock, termination shock and heliosheath. Science 309, 2020–2024, 2005. Decker, R.B., Krimigis, S.M., Roelof, E.C., Hill, M.E., Armstrong, T.P., Gloeckler, G., Hamilton, D.E., Lanzerotti, L.J. Mediation of the solar wind termination shock by non-thermal ions. Nature 454, 67–70, 2008. Decker, R.B., Roelof, E.C., Krimigis, S.M., Hill, M.E. Low-energy ions near the termination shock, in: J. Heerikhuisen et al. (Eds.), AIP Conf. Proc. 858, Physics of the Inner Heliosheath, Danvers, MA, AIPC, p. 73–78, 2006. Fisk, L.A., Gloeckler, G. The common spectrum for accelerated ions in the quiet-time solar wind. Astrophys. J. 640, L79–L82, 2006. Fisk, L.A., Gloeckler, G. Thermodynamic constraints on stochastic acceleration in compressional turbulence. Proc. Natl. Acad. Sci. 104, 5749–5754, 2007. Fisk, L.A., Gloeckler, G. Acceleration of suprathermal tails in the solar wind. Astrophys. J. 686, 1466–1473, 2008. Fisk, L.A., Gloeckler, G., Zurbuchen, T.H. Acceleration of low-energy ions at the termination shock of the solar wind. Astrophys. J. 644, 631– 637, 2006. Fisk, L.A., Kozlovsky, B., Ramaty, R. An interpretation of the observed oxygen and nitrogen enhancements in low energy cosmic rays. Astrophys. J. Lett. 190, L35–L37, 1974.

Gloeckler, G., Fisk, L.A. Anisotropic beams upstream of the termination shock of the solar wind. Astrophys. J. 648, L63–L66, 2006. Gloeckler, G., Fisk, L.A., Mason, G.M., Hill, M.E. Formation of power law tail with spectral index 5 inside and beyond the heliosphere, in: Li, G., Hu, Q., Verhoglyadora, O., Zank, G.P., Lin, R.P., Luhmann, J. (Eds.), AIP Conf. Proc. 1039, Particle Acceleration and Transport in the Heliosheath and Beyond, 367–374, 2008. Gloeckler, G., Fisk, L.A., Geiss, J., Hill, M.E., Hamilton, D.C., Decker, R.B., Krimigis, S.M. Composition of interstellar neutrals and the origin of anomalous cosmic rays. Space Sci. Rev. 143, 163– 175, 2009. Gurnett, D.A., Kurth, W.S. Electron plasma oscillations upstream of the solar wind termination shock. Science 309, 2025–2027, 2005. Gurnett, D.A., Kurth, W.S. Intense plasma waves at and near the solar wind termination shock. Nature 454, 78–80, 2008. Hill, M.E., Decker, R.B., Roelof, E.C., Krimigis, S.M., Gloeckler, G. Heliosheath particles anomalous cosmic rays and a possible ‘‘third source” of energetic ions, in: Heerikhuisen, J. et al. (Eds.), AIP Conf. Proc. 858 Physics of the Inner Heliosheath. AIPC, Danvers, MA, pp. 98–103, 2006. Jokipii, J.R. Particle drift, diffusion and acceleration at shocks. Astrophys. J. 255, 716–720, 1982. Jokipii, J.R. Particle acceleration at a termination shock. I – application to the solar wind and the anomalous component. J. Geophys. Res. 91, 2929–2932, 1986. Jokipii, J.R. The anomalous component of cosmic rays, in: Grzedielski, S., Page, D.E. (Eds.), COSPAR Colloq., Physics of the Outer Heliosphere. Pergamon, Elmsford, pp. 169–178, 1990. Jokipii, J.R., Giacalone, J. The theory of anomalous cosmic rays. Space Sci. Rev. 83, 123–136, 1998. Kota, J., Jokipii, J.R. 2008. Anomalous cosmic rays in the heliosheath: simulation with a blunt termination shock, in: Li, G., Hu, Q., Verhoglyadora, O., Zank, G.P., Lin, R.P. Luhmann, J. (Eds.), AIP Conf. Proc. 1039, Particle Acceleration and Transport in the Heliosheath and Beyond, 397–403, 2008. McComas, D.J., Schwadron, N.A. An explanation for the Voyager paradox: particle acceleration at a blunt termination shock. Geophys. Res. Lett. 33, L04102, doi:10.1029/2005GL025437, 2006. Mewaldt, R.A., Selesnick, R.S., Cummings, A.C., Stone, E.C. Evidence for multiply charged anomalous cosmic rays. Astrophys. J. Lett., L43– L46, 1996. Pesses, M.E., Jokipii, J.R., Eicher, D. Cosmic ray drift, shock wave acceleration, and the anomalous component of cosmic rays. Astrophys. J. Lett. 246, L85–L88, 1981. Richardson, J.D., Kasper, J.C., Wang, C., Belcher, W., Lazarsu, A.J. Cool heliosheath plasma and deceleration of the upstream solar wind at the termination shock. Nature 454, 63–66, 2008. Stone, E.C., Cummings, A.C., McDonald, F.B., Heikkila, B.C., Lal, N., Webber, W.R. Voyager 1 explores the termination shock region and heliosheath beyond. Science 309, 2017–2020, 2005. Stone, E.C., Cummings, A.C., McDonald, F.B., Heikkila, B.C., Lal, N., Webber, W.R. An asymmetric solar wind termination shock. Nature 454, 71–74, 2008. Zank, G. Interaction of the solar wind with the local interstellar medium: a theoretical perspective. Space Sci. Rev. 89, 413–688, 1999.



Coronal fast wave trains of the decimetric type IV radio event observed during the decay phase of the June 6, 2000 flare H. Me´sza´rosova´ a,*, H.S. Sawant b, J.R. Cecatto b, J. Ryba´k c, M. Karlicky´ a, F.C.R. Fernandes d, M.C. de Andrade b, K. Jirˇicˇka a a Astronomical Institute, Czech Academy of Sciences, CZ-25165 Ondrˇejov, Czech Republic National Space Research Institute (INPE), Ave. dos Astronautas 1758, 1221-0000 Saõ Jose´ dos Campos, SP, Brazil c Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranska´ Lomnica, Slovak Republic d Institute of Research and Development (IP&D – UNIVAP), Ave. Shishima Hifum 2911, Urbanova, 12244-000 Saõ Jose´ dos Campos, SP, Brazil b

Received 27 October 2008; received in revised form 23 January 2009; accepted 27 January 2009

Abstract The 22 min long decimetric type IV radio event observed during the decay phase of the June 6, 2000 flare simultaneously by the Brazilian Solar Spectroscope (BSS) and the Ondrˇejov radiospectrograph in frequency range 1200–4500 MHz has been analyzed. We have found that the characteristic periods of about 60 s belong to the long-period spectral component of the fast wave trains with a tadpole pattern in their wavelet power spectra. We have detected these trains in the whole frequency range 1200–4500 MHz. The behavior of individual wave trains at lower frequencies is different from that at higher frequencies. These individual wave trains have some common as well as different properties. In this paper, we focus on two examples of wave trains in a loop segment and the main statistical parameters in their wavelet power and global spectra are studied and discussed. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Sun; Corona; Flares; Radio radiation; MHD waves

1. Introduction It has been theoretically predicted (Roberts et al., 1983, 1984) that periodicity of fast magnetoacoustic modes can be modified by the time evolution of an impulsively generated signal. An obvious source of such an impulsive disturbance is a flare (providing either a single or multiple source of disturbances). These fast magnetoacoustic waves are trapped in regions (e.g. coronal loop) with a high density (i.e. with a low Alfveń speed). These regions are acting as waveguides. The impulsively generated (propagating) wave

*

Corresponding author. Tel.: +420 323 620155; fax: +420 323 620210. E-mail addresses: [email protected] (H. Me´sza´rosova´), sawant@das. inpe.br (H.S. Sawant), [email protected] (J.R. Cecatto), [email protected] (J. Ryba´k), [email protected] (M. Karlicky´), [email protected] (F.C.R. Fernandes), [email protected] (M.C. de Andrade), [email protected] (K. Jirˇicˇka).

in a density coronal waveguide exhibits several phases: (1) periodic phase (long-period spectral components arrive as the first at the observation point), (2) quasi-periodic phase (as both long and short-period spectral components arrive and interact), and finally (3) decay phase (as the signal passes). The quasi-periodic phase is generally much stronger in amplitude and shorter in ‘periodicity’ than the earlier periodic phase. The numerical simulation of characteristic time (wavelet) signatures of impulsively generated fast magnetoacoustic wave trains propagating along a coronal loop with different ratios of the density contrast has been studied by Nakariakov et al. (2004). It was found that the dispersive evolution of fast wave trains leads to the appearance of characteristic tadpole wavelet signature where a narrow-spectrum tail precedes a broadband head. Such tadpole signatures were observed in solar eclipse data (Katsiyannis et al., 2003; Williams et al., 2001). Now, for


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the first time, these tadpole wavelet signatures of impulsively generated fast magnetoacoustic wave trains observed in decimetric type IV radio event are presented. 2. Observation and results The June 6, 2000 flare (classified as X2.3, GOES X-ray maximum at 15:25 UT) was observed during 14:58– 17:00 UT in the active region NOAA AR 9026. In Ha the flare has the importance 2B. We have observed a 22 min long (15:40:05–16:02:00 UT) decimetric type IV radio event during the decay phase of this flare recorded simultaneously by the Brazilian Solar Spectroscope (BSS, frequency range 1200–1700 MHz, time and frequency resolution is 50 ms and 5 MHz, respectively) and by the Ondrˇejov radiospectrograph (frequency range 2000– 4500 MHz, time and frequency resolution is 100 ms and 10 MHz, respectively). The radio spectra for the whole time interval are shown in Fig. 1. The high time and frequency resolutions of both instruments enable us to investigate the spectra in detail. Time series of these spectra have been analyzed in their power and global wavelet spectra at individual frequencies in the range 1200–4500 MHz.

For the analysis, we have divided the interval under study (15:40:05–16:02:00 UT) into four subintervals A–D (Table 1) and we have recognized tadpole wavelet signatures in all these subintervals. An example of the characteristic tadpole pattern (time subinterval C) is shown in Fig. 2. The top panel shows the time series at the frequency 1395 MHz and the middle panel exhibits the corresponding wavelet power spectrum with tadpole pattern as the signature of a coronal fast wave train. The lighter area indicates greater power in the wavelet power spectrum and the hatched region belongs to the cone of influence (COI) where edge effects become important due to dealing with finite-length time series. The solid contour shows the 95% confidence level. In the analysis of each of time series only these contoured regions which exist outside the COI have been considered as valid. The long-period spectral component (tadpole tail) has characteristic period P = 63.8 s. The range of periods at the point of the maximal extension of the tadpole head is 39.5–70.1 s. The bottom panel shows the global wavelet spectrum with the characteristic period P = 63.8 s above the 95% global significance level (horizontal line). Some examples of tadpoles at different frequencies are presented in Figs. 3 and 4.

Fig. 1. Left panels: 22 min long decimetric type IV radio spectrum (the highest intensity in black) observed during the decay phase of the June 6, 2000 flare recorded simultaneously by the BSS (1200–1700 MHz) and by the Ondrˇejov radiospectrograph (2000–4500 MHz). Right panels: characteristic flux time series at six selected frequencies.


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Table 1 Time subintervals of broadband pulsations observed during the decay phase of the June 6, 2000 flare in the frequency range 1200–4500 MHz. Time subinterval

Start (UT)

End (UT)

A B C D

15:40:05 15:45:34 15:51:03 15:56:32

15:45:33 15:51:02 15:56:31 16:02:00

Fig. 2. Example of a tadpole wavelet pattern of a coronal fast wave train. Top panel: time series at the frequency 1395 MHz. Middle panel: corresponding wavelet power spectrum with a tadpole signature. The lighter area indicates a greater power in the wavelet power spectrum and the hatched region belongs to the cone of influence (COI). The solid contour shows the 95% confidence level. The range of powers (grey scale) is presented on right side of the spectrum. The long-period spectral component (tadpole tail) has characteristic period P = 63.8 s. Bottom panel: global wavelet spectrum with the characteristic period P = 63.8 s above the 95% global significance level (horizontal line).

Series of two tadpoles (time subintervals C and D) between frequencies 1300 and 1400 MHz can be seen in Fig. 3. The long-period spectral components (tadpole tails) of these wave trains with the characteristic period P = 63.8 s propagate faster than the medium and shortperiod ones. This characteristic period P has on average 2.3 and 1.2 wave oscillations for the tadpoles in left and right column in Fig. 3, respectively. When the duration of long-period spectral component was calculated, any portion within the COI was discarded. The average range of periods in the place of the maximal extension of the tadpole head is 36.9–70.9 and 26.0–65.5 s for tadpoles in the left and right column in Fig. 3, respectively. Some basic tadpole parameters (time subinterval C, Fig. 3) are shown in Table 2 and they are very similar at different frequencies. Tadpoles occurring between frequencies 3980 and 4070 MHz during time subinterval A are present in Fig. 4. The long-period spectral components of these wave trains have the characteristic period P = 54.5 s with 4.5

Fig. 3. Tadpole wavelet signatures of coronal fast wave trains and their changes at individual frequencies in the range 1300–1400 MHz (time subintervals C and D). The lighter area indicates a greater power in the wavelet power spectrum and the hatched region belongs to the cone of influence (COI). The solid contour shows the 95% confidence level. The ranges of powers (grey scales) are presented on right side of the spectra (common for spectra in both columns). The long-period spectral components (tadpole tails) of all wave trains have characteristic period P = 63.8 s.

wave oscillations on average. The average range of periods in the place of the maximal extension of the tadpole head is 26.7–63.1 s. These basic tadpole parameters are shown in Table 3 and they differ at individual frequencies more than in the previous case (Table 2). There are some differences between individual wave trains (tadpoles) and their behavior at higher frequencies and at lower frequencies. Furthermore, the tadpoles at the same frequency but in different time subinterval (Table 1) have some common as well as different properties. The two tadpole series in Fig. 3 (time subintervals C and D) occur in the same frequency range 1300–1400 MHz with the same characteristic period P = 63.8 s (long-period spectral component) but with the different wave train duration: average 2.3 wave oscillations for the earlier wave train


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Fig. 4. Tadpole wavelet signatures of coronal fast wave trains and their changes at individual frequencies in the range 3980–4070 MHz (time subinterval A). The lighter area indicates a greater power in the wavelet power spectrum and the hatched region belongs to the cone of influence (COI). The solid contour shows the 95% confidence level. The ranges of powers (grey scales) are presented on right side of the spectra (common for spectra in both columns). The long-period spectral components (tadpole tails) of all wave trains have characteristic period P = 54.5 s.

Table 2 Basic tadpole parameters for the time subinterval C (left panels in Fig. 3). All wave trains have characteristic period P = 63.8 s (Duration = duration of the tadpole, Oscillations = number of wave oscillations of the period P, Range of periods = range of periods at the point of the maximal extension of the tadpole head).

Table 3 Basic tadpole parameters for the time subinterval A (see Fig. 4). All wave trains have characteristic period P = 54.5 s (Duration = duration of the tadpole, Oscillations = number of wave oscillations of the period P, Range of periods = range of periods at the point of the maximal extension of the tadpole head).

Frequency (MHz)

Duration (s)

Oscillations

Range of periods (s)

Frequency (MHz)

Duration (s)

Oscillations

Range of periods (s)

1350 1355 1360 1365 1370 1375 1380 1385 1390 1395 1400

150.6 150.0 150.0 150.0 149.4 151.3 148.7 150.0 151.4 149.4 151.8

2.4 2.3 2.3 2.3 2.3 2.4 2.3 2.3 2.4 2.3 2.4

48.9–69.3 52.9–68.5 50.0–70.1 49.5–68.5 44.7–69.3 37.7–69.3 37.7–70.9 40.8–70.9 43.2–70.1 39.5–70.1 36.9–70.9

3980 3990 4000 4010 4020 4030 4040 4050 4060 4070

205.0 187.5 281.3 217.6 250.5 206.7 188.2 235.8

3.8 3.4 5.2 3.4 4.6 3.8 3.4 4.3

26.7–59.1

234.4

4.3

32.2–58.4

(time subinterval C) and 1.2 wave oscillations for the later one (time subinterval D). Moreover, the appearance of both wave trains is different. In the case of the earlier wave train (left column of panels in Fig. 3) we can see a full tadpole structure (tail + head) to the frequency 1370 MHz. Then, in direction to the lower frequencies, the head is fading and from the frequency 1345 MHz the whole tadpole is

38.8–63.1 27.9–61.8 32.9–62.4 47.9–54.4 31.2–59.1

decayed. In the case of the later wave trains (right column of panels in Fig. 3) we can see a full tadpole structure to the frequency 1340 MHz where the tadpole is decayed and at lower frequencies we can see only a rest of the tadpole head. The tadpole patterns of the individual wave trains (in the same time interval) at lower frequencies (Fig. 3) are very similar to each other i.e. their changes from frequency to frequency are slow.


Different tadpole behavior can be seen at higher frequencies (3980–4070 MHz) in Fig. 4 (time subinterval A). There is a set of different tadpoles but with the same characteristic period P = 54.5 s (long-period spectral components) and with a similar duration (on average 4.5 wave oscillations). The process of tadpole appearance and fading (see tadpoles at frequencies 4030–4070) is significantly more rapid and repeating. Individual tadpoles have different heads (set of middle and short-period spectral components) and in some cases the head is absent (see tadpoles at frequencies 3990 and 4020 MHz). Sometimes, the whole tadpole is fully absent (for example at 4061 MHz). 3. Conclusions We have investigated the 22 min long decimetric type IV of radio event during the decay phase of the June 6, 2000 flare observed simultaneously by the BSS and the Ondrˇejov radiospectrograph (1200–4500 MHz). For the first time, the tadpole structures of dm-radio event are evident in their wavelet power spectra in the whole time interval and at all frequencies. We have distinguished the tadpole wavelet signatures in our observational data and it allows us to identify the corresponding waves as fast magnetoacoustic wave trains. These waves are probably trapped in a waveguide (e.g. loop) and formed by an impulsive source (e.g. flare or reconnection process). We present two examples of tadpoles (Figs. 3 and 4) with different behavior but similar characteristic period P (long-period spectral component of all wave trains) of about 60 s. The tadpoles in Fig. 3 show relatively slow changes of the same tadpole at different frequencies. It may reflect that at the lower frequencies the plasma density is more smooth (no abrupt changes in density). These tadpoles have short duration of long-period spectral component (low number of wave oscillations of the characteristic period) i.e. wave damping is strong here. Furthermore, these tadpoles can decay at the lowest frequencies under study (Fig. 3). This may represent less density contrast (density ratio between external and internal plasma density of a loop) which causes that such a segment of a loop is a worse waveguide. Tadpoles at higher frequencies (Fig. 4) have longer duration of long-period spectral component (higher number of wave oscillations of their characteristic period P). On the other hand, the individual tadpoles show very rapid changes of the same tadpole at different frequencies. Thus, individual tadpoles are different (mainly with respect to their heads, i.e. their middle and short-period spectral components). Sometimes, we can observe only the long-period

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spectral component (tail) of the wave train that arrives as the first at the observable point. This can represent a higher diversity in plasma density of a loop (more rapid changes of plasma density inside such a loop segment). The wave damping is strong also in this case. We have distinguished two groups of individual tadpoles with different properties: the first tadpole group was detected in the frequency range 1200–1600 MHz (time subintervals C and D, Fig. 3) and the second group in the frequency range 3800–4500 MHz (time subinterval A, Fig. 4). Thus, each group can belong to a different radio emission source. If one emission source is dominant we can detect individual tadpoles with similar properties. On the other hand, the wavelet power spectra are more complex (chaotic) in the frequency range 1600–3800 MHz (i.e. we cannot see individual tadpoles). It can happen when the time series in this frequency range are influenced for example, by more than one emission source. The tadpoles can provide an evidence for MHD waves in the corona. They may provide the basics for determination of the transverse structure, density distribution and other properties of the waveguide (coronal loop). Acknowledgements H.M. acknowledges the FAPESP support for the project 2006/50039-7. H.M. and M.K. acknowledge the support from the Grant IAA300030701 of the Academy of Sciences of the Czech Republic. F.C.R.F. thanks CNPq for scholarship (proc. 310005/2005-1). J.R.C. acknowledges the CNPq support for project 475723/2004-0. J.R. acknowledges the support of the grant agency VEGA 02/6195/26. The wavelet analysis was performed using the software based on tools provided by C. Torrence and G. Compo at http:// paos.colorado.edu/research/wavelets/. References Katsiyannis, A.C., Williams, D.R., McAteer, R.T.J., et al. Eclipse observations of high-frequency oscillations in active region coronal loops. A&A 406, 709–714, 2003. Nakariakov, V.M., Arber, T.D., Ault, C.E., et al. Time signatures of impulsively generated coronal fast wave trains. Mon. Not. Roy. Astron. Soc. 349, 705–709, 2004. Roberts, B., Edwin, P.M., Benz, A.O. Fast pulsations in the solar corona. Nature 305, 688–690, 1983. Roberts, B., Edwin, P.M., Benz, A.O. On coronal oscillations. Atrophys. J. 279, 857–865, 1984. Williams, D.R., Phillips, K.J.H., Rudawy, P., et al. High-frequency oscillations in a solar active region coronal loop. Mon. Not. Roy. Astron. Soc. 326, 428–436, 2001.



Viewing radiation signatures of solar energetic particles in interplanetary space S.W. Kahler *, B.R. Ragot 1 Air Force Research Laboratory, RVBXS, 29 Randolph Road, Hanscom AFB, MA 01731, USA Received 18 August 2008; received in revised form 14 November 2008; accepted 6 January 2009

Abstract A current serious limitation on the studies of solar energetic particle (SEP) events is that their properties in the inner heliosphere are studied only through in situ spacecraft observations. Our understanding of spatial distributions and temporal variations of SEP events has come through statistical studies of many such events over several solar cycles. In contrast, flare SEPs in the solar corona can be imaged through their radiative and collisional interactions with solar fields and particles. We suggest that the heliospheric SEPs may also interact with heliospheric particles and fields to produce signatures which can be remotely observed and imaged. A challenge with any such candidate signature is to separate it from that of flare SEPs. The optimum case for imaging high-energy (E > 100 MeV) heliospheric protons may be the emission of p0-decay c-rays following proton collisions with solar wind (SW) ions. In the case of E > 1 MeV electrons, gyrosynchrotron radio emission may be the most readily detectible remote signal. In both cases we may already have observed one or two such events. Another radiative signature from nonthermal particles may be resonant transition radiation, which has likely already been observed from solar flare electrons. We discuss energetic neutrons as another possible remote signature, but we rule out c-ray line and 0.511 MeV positron annihilation emission as observable signatures of heliospheric energetic ions. We are already acquiring global signatures of large inner-heliospheric SW density features and of heliosheath interactions between the SW and interstellar neutral ions. By finding an appropriate observable signature of remote heliospheric SEPs, we could supplement the in situ observations with global maps of energetic SEP events to provide a comprehensive view of SEP events. Published by Elsevier Ltd. on behalf of COSPAR. Keywords: Solar energetic particles; Interplanetary magnetic fields; Coronal mass ejections

1. Introduction Forecasting the occurrence of SEP events has become increasingly important as we consider their impact on the human exploration of space (Turner, 2006). At the current time we must rely on solar flare and coronal mass ejection (CME) signatures to predict the temporal, spatial, and energetic variations of heliospheric SEP events, but the presumed SEP production in CME-driven shocks can be only loosely connected to those solar signatures (Kahler, 2001). Furthermore, we rely on statistical studies of in situ observations to *

1

Corresponding author. E-mail address: [email protected] (S.W. Kahler). NRC Senior Research Associate.

0273-1177/$36.00 Published by Elsevier Ltd. on behalf of COSPAR. doi:10.1016/j.asr.2009.01.013

determine the characteristics of the heliospheric SEP events. Although the 1 AU in situ observations give us detailed information on heliospheric SEP spectra and composition, the lack of a complementary global context for SEP production and the loss of SEP source information imposed by particle scattering on magnetic fluctuations during SEP transport in the inner heliosphere are clearly severe impediments for our characterization and understanding of those SEP events. The low ambient densities and weak magnetic fields of the heliosphere restrict any radiative SEP signatures to low levels not yet observed. In contrast to these limitations, the SEPs accelerated in solar coronal flare structures are remotely observed and diagnosed with microwave, optical, X-ray, and c-ray emission and neutron detections. The RHESSI spacecraft is a solar observatory that provides

S.W. Kahler, B.R. Ragot / Advances in Space Research 43 (2009) 1484–1490

dedicated solar flare observations in the X-ray and c-ray range up to 20 MeV with good spatial, temporal and spectral resolution (Lin et al., 2002). Here we consider some possible remote signatures of heliospheric SEPs that might serve as future observable diagnostics. Our recent calculations of c-ray line and continuum emissions from SEP interactions with solar wind (SW) ions (Kahler and Ragot, 2008) are briefly reviewed. We also examine several other SEP interactions that might serve to produce observable heliospheric SEP signatures. It is important that we consider all possible forms of remote SEP information – radiative, magnetic, electric, particle, or other. We point out that at the relatively low energies of SW particles, remote observations, perhaps not thought possible several decades ago, are now becoming a reality and will complement various in situ observations in the heliosphere and at the termination shock and heliosheath.

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We have recently considered (Kahler and Ragot, 2008) the possibility of remote detection of c-rays produced in the near-Sun heliosphere by the interaction of very intense SEPs with SW particles, as shown in Fig. 1. This work was motivated by three factors. First, in the galaxy c-ray line (Tatischeff and Kiener, 2004) and high-energy (E > 70 MeV) continuum emission (Strong and Mattox, 1996) resulting from cosmic ray collisions with interstellar gas and dust has been observed. Second, SEP spectra and temporal variations for relatively large gradual events have been well characterized during the past solar cycle (Mewaldt et al., 2005), providing a SEP data base for selecting events to model in the near-Sun heliosphere. Finally, the GLAST

spacecraft (Bhattacharjee, 2008), launched on 2008 June 11, will provide a new capability for sensitive measurements of c-rays from heliospheric SEPs. Two c-ray regimes were explored (Kahler and Ragot, 2008). First, we considered c-ray line emission resulting from excitation by 3–30 MeV nuc1 ions on SW gas and dust in the 5–15 R region using the estimated peak 3–30 MeV proton spectrum of the large gradual SEP event of 2003 October 28 and the excitation cross sections of Kozlovsky et al. (2002). The calculated intensities for the strongest lines (1.37, 4.44, and 6.13 MeV of 24Mg, 12C, and 16O, respectively) were slightly lower than the combined observed diffuse extragalactic component background (Strong et al., 1996) and the weaker calculated background from the inverse Compton scattering of cosmic ray electrons on the solar photon halo (Moskalenko et al., 2006). Intensities consistent with the inverse Compton scattered component have recently been detected in the two higher energy ranges of 100–300 MeV and >300 MeV by Orlando and Strong (2008). However, over the short (1–10 h) timescale of the peak of a gradual SEP event less than a single count would be recorded in the GLAST Burst Monitor, rendering this approach hopeless. The prospects for an event detection were more favorable, however, for p0-decay c-rays resulting from collisions of high energy (E J 300 MeV) SEP protons on SW ions. We used an E2 differential energy spectrum to represent the 2005 January 20 SEP event peak (Fig. 2) and calculated not only that the p0-decay c-ray intensity was three orders of magnitude above background but that over the event peak hour the Large Area Telescope (LAT) on GLAST would have detected J 104 counts (using the correct 0.6 cm2 s1 sr1 value for the c-ray intensity of I p0 , which we erroneously took as 0.3 cm2 s1 sr1). The SEP differential fluence spectrum for the January 20 event has also

Fig. 1. c-Ray imaging of both flare (dashed-dot line and wiggly arrow) and interplanetary (thin dashed lines and wiggly arrows) SEP populations. The CME drives a shock (thick dashed line) from which SEPs propagate outward (straight arrows) to interact with solar wind and dust ions (gray area) and produce c-rays. From Kahler and Ragot (2008).

Fig. 2. Time profiles of the E > 100 MeV intensities of the largest SEP events of the last 30 years. The 2005 January 20 event used in our c-ray calculation had the fastest rise of any of the events. From Mewaldt et al. (2005).

2. Candidate SEP signatures 2.1. c-Ray continuum emission

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been calculated from neutron monitor data and is steeper and less intense than the intensity spectrum we used. When the calculated neutron-monitor integral fluence above 300 MeV matches that of our normalized intensity spectrum, the c-ray flux and total LAT counts are reduced by a factor of 3 (R. Murphy and A. Tylka, private communication), but still produce a very detectible LAT signal. Details of the line and p0-decay emission calculations are given in Kahler and Ragot (2008). A major problem in detecting any near-Sun heliospheric SEP event signature is to distinguish that signal from a similar signature of solar flare SEPs. The low signal intensities and the very high (610 4 R) detector angular resolution required to make that distinction combine to make a formidable observing challenge. We (Kahler and Ragot, 2008) suggested that the flare and heliospheric SEP signals could be distinguished from each other either temporally, when a later heliospheric signal follows or dominates an earlier flare signal, or spatially, when the flare region lies over the solar limb and only the heliospheric component is observed. Ryan (2000) considered in detail how protons accelerated in antisunward propagating shocks might precipitate back to the lower corona or chromosphere as an explanation for observed long-duration solar c-ray flares. His concept is similar to our suggestion of a temporal separation between flare and heliospheric SEP sources, except that he would have the later collisions and c-ray production from the heliospheric SEPs occurring in much lower and denser coronal regions rather than in the SW. 2.2. Positron-decay 0.511 MeV line emission We expect that a corresponding heliospheric SEP signature resulting from collisions of SEP ions with SW ions should match each observed signature of flare SEP ions with solar ambient atmospheric ions. Collisions of high energy (E J 300 MeV) SEP protons on SW ions will produce not only the p0-decay c-rays considered in our earlier work (Kahler and Ragot, 2008) but also p+ and p mesons, which decay into l+ and l mesons and then into positrons and electrons. Positrons are also produced as products of collisions of lower energy (E J 10 MeV) SEPs with ambient ions to produce radioactive b+-emitting nuclei (Ramaty et al., 1975). Production of those nuclei can be greatly enhanced if the SEP composition has a high 3He abundance in the 1–10 MeV nuc1 energy range (Kozlovsky et al., 2004). The positrons decay into a pair of 0.511 MeV c-rays when they annihilate directly with ambient electrons or after forming positronium. From positronium only 25% decay into the 0.511 MeV c-rays, while the remaining 75% undergo a three-c-ray decay. Could the 0.511 MeV line also be observed from the heliospheric SEP-generated positrons as it is in some solar flares? The p+-decay positrons are formed with energies of 30 MeV and for annihilation they require thermalization times of 1013/n1 s, where n1 is the ambient density in cm3 (Ramaty and Murphy, 1987). In heliospheric regions of

density n < 108 cm3 those relativistic positrons will leave the inner heliosphere with essentially no 0.511 MeV line emission. The b+-emission positrons have lower characteristic energies of 1 MeV and shorter thermalization times of 4 1012/n1 s, but they too should rapidly propagate away from the inner heliosphere before decay into the 0.511 MeV line is possible. Thus, the long positron thermalization times preclude the possibility of observing heliospheric 0.511 MeV line emission. Direct detection of the positrons themselves at 1 AU is not precluded here, but as charged particles scattered by magnetic field fluctations, they lose the spatial information of their sources that we are seeking. 2.3. Neutrons and neutron-capture 2.23 MeV c-ray emission Since energetic neutrons travel directly from their source regions to an observer, directional information on neutrons produced by heliospheric SEP collisions with SW ions could produce information on the SEP temporal–spatial distribution. Neutrons are produced primarily by p–a interactions above 30 MeV nuc1, where the production cross sections are only slightly energy dependent (Ramaty et al., 1975). In the case of heliospheric SEPs the thin-target model of interactions is appropriate for neutron production calculations, rather than the more efficient thick-target model usually assumed (i.e., Murphy et al., 1987) for flare SEPs, which are stopped in the dense flare target region. The b decay of the neutrons en route to the observer will modify the observed neutron energy spectrum, but this could provide model-dependent information on the SEPsource distance from the observer that would not be available from the heliospheric p0-decay c-rays. Neutron emission in the 36–100 MeV range associated with a solar flare located 60–90 behind the solar east limb was observed on 1991 June 1 (Murphy et al., 1999). Comparison of that event with a second neutron and c-ray flare event on 1991 June 4 indicated that a thin-target neutron source was possible only if the SEP spectrum was extremely hard. Although the June 1 flare was an extremely energetic event (Kane et al., 1995), it suggests that energetic neutron emission could, with a sufficiently sensitive detector, serve as a remote signature of heliospheric SEP events. The energetic neutrons produced by p–a collisions between energetic heliospheric SEPs and the SW ions will produce 2.23 MeV c-ray line emission from capture by SW protons. However, two neutron timescales limit this capture process: the 15-min lifetime against b decay and the required thermalization by elastic scattering before SW proton capture. In solar flares the 2.23 MeV emission is generated in the n > 1016 cm3 density region of the photosphere after a thermalization time of 100 s for 1–100 MeV neutrons (Wang and Ramaty, 1974; Ramaty et al., 1975). However, the SW densities, lower by orders of magnitude, preclude any neutron thermalization, and detectible 2.23 MeV line emission from heliospheric SEPs is therefore not expected. We note, however, that


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2.23 MeV emission was observed on 1989 September 29 in association with a GLE, a fast CME, and a solar flare located 50–150 behind the solar west limb. Vestrand and Forrest (1993) and Cliver et al. (1993) suggested that a fraction of the E > 30 MeV protons at the coronal shock may have precipitated back to the solar atmosphere and then produced the observed emission via neutron generation. Our consideration here of neutron production by heliospheric SEPs suggests the alternative possibility of direct atmospheric precipitation of the energetic neutrons from the SEP–SW interactions, which, although unlikely, avoids the problem of achieving charged particle precipitation through converging coronal magnetic fields. 2.4. Electron synchrotron emission Bursts of gyrosynchrotron radiation from E J 100 keV electrons in solar flare loops are commonly observed in the microwave range (Bastian et al., 1998). Can synchrotron emission also be observed from transient energetic electron populations that escape the Sun? Bastian et al. (2001) described a fast CME on 1998 April 20 in which the expanding CME loops were imaged directly in radio wavelengths with the Nancay radioheliograph out to 3.5 R. Their interpretation was that the emission was synchrotron emission from nonthermal electrons with energies from 0.5 to 5 MeV in fields of 0.1 to several G. The longduration phase of an X-class flare event on 2003 November 3 was also interpreted in terms of electron gyrosynchrotron emission from a large coronal structure with B 2 G (Dauphin et al., 2005). Matching modulations were observed in hard X-rays by RHESSI and in the decimetric/metric continuum at the Nancay Radioheliograph during a fast CME. Another event, in which radio loops were observed in a CME out to >2.1 R on 2001 April 15 (Maia et al., 2007), provides a third example. The inferred electron high-energy cutoff ranged from 1 to 10 MeV and the estimated field B was 1 G. These events were imaged at the four Nancay observing frequencies from 432 to 164 MHz, and their cospatial emission at different frequencies indicated synchrotron, not plasma emission, as the source. The 1998 and 2001 events originated from the southwest solar limb and were accompanied at 1 AU by large E J 100 keV electron and E > 10 MeV proton events, suggesting that the escaping nonthermal electrons may have been imaged near the Sun. Can we see synchrotron emission from electrons propagating even farther from the Sun? An interplanetary type-II-like burst was observed with the Wind/WAVES instrument on 2003 June 17–18, which Bastian (2007) interpreted as synchrotron emission. The burst followed a strong type III burst and preceded a type II burst and was observed when the CME height was J 10 R (Fig. 3). He argued that the frequency band width, drift rates and the single smoothly varying emission lane of the burst were inconsistent with plasma emission from a propagating shock. Acceptance of a new class of low-frequency ( 0.75 occurred with CR A/M 30 m2/kg and 40 m2/kg, respectively (Fig. 13). Therefore, depending on the initial conditions, a maximum ‘‘relatively stable” eccentricity of 0.7 would be obtained with 25 m2/ kg 6 CR A/M 6 90 m2/kg.

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Concerning the evolution of the orbit plane, as mentioned in Section 4, for A/M 6 1 m2/kg it would be still dominated by the interplay between the Earth’s oblateness and luni-solar perturbations, with the typical nodal regression period of approximately 26 years and an inclination oscillation amplitude of about 1.5°, superimposed on a longer term trend driven by third body attraction. An increase in A/M would result in a faster nodal regression and wider amplitude of the inclination oscillation, even though, for any given value of A/M, the nodal rate and the inclination excursion would depend on the initial conditions. In the E2 case, for example, CR A/M 12 m2/ kg would induce a Di ±3.5° and a nodal precession period of 20.6 years, CR A/M 30 m2/kg would induce a Di ±10° and a nodal precession period of 11.7 years, and CR A/M 48 m2/kg would induce a Di ±15° and a nodal precession period of 6.6 years. Note that, for sufficiently high A/M values and specific initial conditions, the orbit would become periodically retrograde (i.e. with i > 90°) for some time. For instance, this situation would occur with CR A/M P 48 m2/kg in the B1 case (Fig. 22) and with CR A/M P 54 m2/kg in the C4 case.

6. Conclusions This study has revealed and clarified several qualitative and quantitative aspects concerning the long-term evolution of a new, and still hypothetical, class of orbital debris, characterized by very high area-to-mass ratios and released with negligible DV in the orbits used by GPS Block II satellites. Among many other results, mainly summarized in Section 5, it was found that also in this case, as with similar objects released in geostationary orbit, very long orbital lifetimes are possible, even with extremely high A/M values. Depending on the initial conditions, eccentricities as high as 0.7 could be attained with 25 m2/kg 6 CR A/M 6 90 m2/kg, maintaining, however, a semi-major axis and mean motion close to the semi-synchronous values. The inclination would also be subject to a wider excursion, linked to a faster regression of the ascending node, with increasing values of the area-to-mass ratio. In conclusion, if high A/M objects, such as those discovered in near synchronous orbits, have been released in GPS trajectories, they would have long enough lifetimes, in high eccentricity semi-synchronous orbits, to be detected by optical observers using appropriate search strategies.

Acknowledgments The results described in this paper were presented at the 37th COSPAR Scientific Assembly, held in Montreál, Canada, on 13–20 July 2008. They were obtained within the framework of the ASI/CISAS Contract No. I/046/07/0.

References Agapov, V., Biryukov, V., Kiladze, R., Molotov, I., Rumyantsev, V., Sochilina, A., Titenko, V. Faint GEO objects search and orbital analysis, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, ESA Publications Division, Noordwijk, The Netherlands, pp. 119–124, 2005. Anselmo, L., Pardini, C. Orbital evolution of geosynchronous objects with high area-to-mass ratios, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, ESA Publications Division, Noordwijk, The Netherlands, pp. 279–284, 2005. Anselmo, L., Pardini, C. Long-Term Evolution of High Earth Orbits: Effects of Direct Solar Radiation Pressure and Comparison of Trajectory Propagators, Technical Report 2007-TR-008, ISTI/CNR, Pisa, Italy, 29 March 2007. Beutler, G. Methods of Celestial Mechanics, Volume II: Application to Planetary System, Geodynamics and Satellite Geodesy. Springer, Berlin, Germany, 2005. Chao, C.C. Analytical investigation of GEO debris with high area-to-mass ratio, Paper No. AIAA-2006-6514, Presented at the 2006 AIAA/AAS Astrodynamics Specialist Conference, Keystone, Colorado, USA, 2006. Ely, T.A. Impact of eccentricity on East–West stationkeeping for the GPS class of orbits, Paper No. AAS-99-389, Presented at the 1999 AAS/ AIAA Astrodynamics Specialist Conference, Girdwood, Alaska, USA, 1999. Hugentobler, U.. Astrometry and Satellite Orbits: Theoretical Considerations and Typical Applications Geoda¨tisch-geophysikalische Arbeiten in der Schweiz, vol. 57. Schweizerische Geoda¨tische Kommission, Zu¨rich, Switzerland, 1998. Hughes, S. Earth satellite orbits with resonant lunisolar perturbations. I. Resonances dependent only on inclination. Proc. Royal Soc. Lond. A 372, 243–264, 1980. Ineichen, D., Beutler, G., Hugentobler, U. Sensitivity of GPS and GLONASS orbits with respect to resonant geopotential parameters. J. Geodesy 77, 478–486, 2003. Kwok, J.H. The Artificial Satellite Analysis Program (ASAP), Version 2.0, JPL NPO-17522, Jet Propulsion Laboratory (JPL), Pasadena, CA, USA, 20 April 1987. Liou, J.-C., Weaver, J.K. Orbital evolution of GEO debris with very high area-to-mass ratios. The Orbital Debris Quarterly News 8 (3), 6–7, 2004. Liou, J.-C., Weaver, J.K., Orbital dynamics of high area-to-mass ratio debris and their distribution in the geosynchronous region, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, ESA Publications Division, Noordwijk, The Netherlands, pp. 285–290, 2005. Pardini, C., Anselmo, L. SATRAP: Satellite Reentry Analysis Program, Internal Report C94-17, CNUCE/CNR, Pisa, Italy, 30 August 1994. Pardini, C., Anselmo, L. Long-term evolution of geosynchronous orbital debris with high area-to-mass ratios. Trans. Jpn. Soc. Aero. Space Sci. 51, 22–27, 2008. Rossi, A. Resonant dynamics of medium Earth orbits: space debris issues. Celest. Mech. Dyn. Astr. 100, 267–286, 2008. Schildknecht, T., Musci, R., Ploner, M., Beutler, G., Flury, W., Kuusela, J., de Leon Cruz, J., de Fatima Dominguez Palmero, L. Optical observations of space debris in GEO and in highly-eccentric orbits. Adv. Space Res. 34, 901–911, 2004. Schildknecht, T., Musci, R., Flohrer, T. Properties of the high area-tomass ratio space debris population at high altitudes. Adv. Space Res. 41, 1039–1045, 2008. Valk, S. Global dynamics of geosynchronous space debris with high areato-mass ratios, Ph.D. Thesis in Mathematics, University of Namur, Belgium, 2008. Valk, S., Lemaıˆtre, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios. Adv. Space Res. 41, 1077–1090, 2008.



Global dynamics of high area-to-mass ratios GEO space debris by means of the MEGNO indicator S. Valk, N. Delsate *, A. Lemaıˆtre, T. Carletti University of Namur (FUNDP), De´partement de Mathe´matique, Unite´ de Syste´mes Dynamiques, 8, Rempart de la Vierge, B-5000 Namur, Belgium Received 21 April 2008; received in revised form 6 February 2009; accepted 23 February 2009

Abstract In this paper we provide an extensive analysis of the global dynamics of high-area-to-mass ratios geosynchronous (GEO) space debris, applying a recent technique developed by Cincotta and Simo´ [Cincotta, P.M., Simo´, C.Simple tools to study global dynamics in nonaxisymmetric galactic potentials–I. Astron. Astrophys. (147), 205–228, 2000.], Mean Exponential Growth factor of Nearby Orbits (MEGNO), which provides an efficient tool to investigate both regular and chaotic components of the phase space. We compute a stability atlas, for a large set of near-geosynchronous space debris, by numerically computing the MEGNO indicator, to provide an accurate understanding of the location of stable and unstable orbits as well as the timescale of their exponential divergence in case of chaotic motion. The results improve the analysis presented in Breiter et al. [Breiter, S., Wytrzyszczak, I., Melendo, B. Long-term predictability of orbits around the geosynchronous altitude. Advances in Space Research 35, 1313–1317, 2005] notably by considering the particular case of high-area-to-mass ratios space debris. The results indicate that chaotic orbits regions can be highly relevant, especially for very high area-to-mass ratios. We then provide some numerical investigations and an analytical theory that lead to a detailed understanding of the resonance structures appearing in the phase space. These analyses bring to the fore a relevant class of secondary resonances on both sides of the wellknown pendulum-like pattern of geostationary objects, leading to a complex dynamics. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Solar radiation pressure; Space debris; MEGNO; Detection of chaos; Long-term evolution; Geosynchronous orbit; High area-to-mass ratios; Secondary resonances

1. Introduction Recent optical surveys in high-altitude orbits, performed by the European Space Agency 1 m telescope on Tenerife (Canary islands), have discovered a new unexpected population of 10 cm sized space debris in near geosynchronous orbits (GEO). These objects sometimes present highly eccentric orbits with eccentricities as high as 0.55 (Schildknecht et al., 2004; Schildknecht et al., 2005). Following the initial guess of Liou and Weaver (2004) who suggested that this new population may be constituted by GEO objects with high area-to*

Corresponding author. E-mail address: nicolas.delsate@ fundp.ac.be (N. Delsate).

mass ratios, recent numerical and analytical investigations were performed to support this assumption (Anselmo and Pardini, 2005; Liou and Weaver, 2005). In addition, these authors and others, such as Chao (2006) and later Valk et al. (2008), presented some detailed results concerning the short- and long-term evolution of high area-to-mass ratios geosynchronous space debris subjected to direct solar radiation pressure. More specifically, these latter authors mainly focused their attention on the long-term variation of both the eccentricity and the inclination vector. Moreover, some studies concerning the effects of the Earth’s shadowing effects on the motion of such space debris were given in Valk and Lemaıˆtre (2008). However, nobody ever dealt with the question to know whether these orbits


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are really predictable or not on the time scales of their investigations. The objective of this paper is twofold. The first goal is the investigation of the long-term stability of high areato-mass ratio space debris subjected to direct solar radiation pressure, by means of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) criterion. Second, still considering high area-to-mass ratios, we bring to the fore a relevant class of additional secondary structures appearing in the phase space. The paper is organized as follows. In Section 2, we focus our attention to the specification of the underlying model and we give some details about the numerical aspects of the method. In Section 3, for the sake of completeness, we dwell upon the detailed definition of the MEGNO indicator, also providing a review of its main properties, in order to understand the behavior of the chaos indicator. Then in Section 4, in the framework of the validation of our implementation, we retrieve the results obtained by Breiter et al. (2005). We also discuss the significance of the time of integration, recently reported by Barrio et al. (in press). In Section 5, we apply the MEGNO technique in order to give a insightful understanding of the stability of high area-tomass ratio space debris. More specifically, we show that the orbits of such peculiar space debris are extremely sensitive to initial conditions, especially with respect to the mean longitude and the semi-major axis. Second, we perform extended numerical analyses, showing that the related 2dimensional phase space is dominated by chaotic regions, in particular when the area-to-mass ratio is large. In addition, we also provide some results presenting the importance of the initial eccentricity value in the appearance of chaotic regions. Finally, in Section 6, we present extensive numerical and analytical investigations of the additional patterns which will be identified as secondary resonances. 2. The model For the purpose of our study, we consider the modeling of a space debris subjected to the influence of the Earth’s gravity field, to both the gravitational perturbations of the Sun and the Moon as well as to the direct solar radiation pressure. As a consequence the differential system of equations governing the dynamics is given by €r ¼ apot þ

þ a þ arp ;

where apot is the acceleration induced by the Earth’s gravity field, which can be expressed as the gradient of the following potential U ðr; k; /Þ ¼

1 X n n lX Re Pmn ðsin /ÞðC nm cos mk þ S nm sin mkÞ; r n¼0 m¼0 r

ð1Þ

where the quantities C nm and S nm are the spherical harmonics coefficients of the geopotential. The Earth’s gravity field adopted is the EGM96 model (Lemoine et al., 1987). In Eq.

(1), l is the gravitational constant of the Earth, Re is the Earth’s equatorial radius and the quantities ðr; k; /Þ are the geocentric spherical coordinates of the space debris. Pmn are the well-known Legendre functions. It is worth noting that the potential of Eq. (1) is subsequently expressed in Cartesian coordinates by means of the Cunningham algorithm (Cunningham, 1970). and a result from the gravity Both the accelerations interaction with a third body of mass m , where ¼ and ¼ , and can be expressed with respect to the Earth’s center of mass as a ¼ l

r r kr r k3

þ

r kr k3

! ;

where r and r are the geocentric coordinates of the space debris and of the mass m , respectively. The quantity l is the gravitational constant of the third-body. In our implementation, we chose the high accurate solar system ephemeris given by the Jet Propulsion Laboratory (JPL) to provide the positions of both the Sun and the Moon (Standish, 1998). Regarding direct solar radiation pressure, we assume an hypothetically spherical space debris. The albedo of the Earth is ignored and the Earth’s shadowing effects are not taken into account either. The acceleration induced by direct solar radiation pressure is given by 2 a A r r ; arp ¼ C r P r kr r k m kr r k where C r is the adimensional reflectivity coefficient (fixed to 1 further on in this paper) which depends on the optical properties of the space debris surface; P r ¼ 4:56 106 N=m2 is the radiation pressure for an object located at the distance of 1 AU; a ¼ 1 AU is a constant parameter equal to the mean distance between the Sun and the Earth and r is the geocentric position of the Sun. Finally, the coefficient A=m is the so-called area-to-mass ratio where A and m are the effective cross-section and mass of the space debris, respectively. 3. The mean exponential growth factor of nearby orbits For the sake of clarity we present in this section the definition and some properties of the MEGNO criterion. Let Hðp; qÞ, with p 2 Rn ; q 2 Tn , be a n-degree of freedom Hamiltonian system and let us introduce the compact notation x ¼ ðp; qÞ 2 R2n as well as f ¼ ð@H=@q; @ H=@pÞ 2 R2n , then the dynamical system is described by the following set of ordinary differential equations d xðtÞ ¼ f ðxðtÞ; aÞ; dt

x 2 R2n ;

ð2Þ

where a is a vector of parameters entirely defined by the model. Let /ðtÞ ¼ /ðt; x0 ; t0 Þ be a solution of the flow defined in Eq. (2) with initial conditions ðt0 ; x0 Þ, then it has


associated the Lyapunov Characteristic Number (hereafter LCN), defined by (Benettin et al., 1980) k ¼ lim

t!1

1 kd/ ðtÞk ; ln t kd/ ðt0 Þk

ð3Þ

where d/ ðtÞ, the so-called tangent vector, measures the evolution of an initial infinitesimal deviation d/ ðt0 Þ d0 between /ðtÞ and a nearby orbit, and whose evolution is given by the variational equations (terms of order Oðd2 Þ are omitted) d d_ / ¼ d/ ðtÞ ¼ Jð/ðtÞÞ d/ ðtÞ; dt

with

Jð/ðtÞÞ ¼

@f ð/ðtÞÞ; @x ð4Þ

where Jð/ðtÞÞ is the Jacobian matrix of the differential system of equations, evaluated on the solution /ðtÞ. Let us note that the definition of LCN, given by Eq. (3), can also be written in an integral form Z 1 t d_ / ðsÞ ds; k ¼ lim t!1 t 0 d/ ðsÞ where d/ ¼ kd/ k; d_ / ¼ d_ / d/ =d/ . The Mean Exponential Growth factor of Nearby Orbits Y / ðtÞ is based on a modified time-weighted version of the integral form of LCN (Cincotta and Simo´, 2000). More precisely 2 Y / ðtÞ ¼ t

Z

t 0

d_ / ðsÞ s ds; d/ ðsÞ

as well as its corresponding mean value, to get rid of the quasi-periodic oscillation possibly existing in Y / ðtÞ Z 1 t Y / ðtÞ ¼ Y / ðsÞ ds: t 0 In the following we will omit the explicit dependence of Y and Y on the specific orbit /, when this will be clear from the context. Actually, Y ðtÞ allows to study the dynamics for long time scales, where generically Y ðtÞ does not converge, while limt!1 Y ðtÞ is well defined (Cincotta et al., 2003). Consequently, the time evolution of Y ðtÞ allows to derive the possible divergence of the norm of the tangent vector dðtÞ, giving a clear indication of the character of the different orbits. Indeed, for quasi-periodic (regular) orbits, Y ðtÞ oscillates around the value 2 with a linear growth of the separation between nearby orbits. On the other hand, for chaotic (irregular) motion, the norm of d grows exponentially with time, and Y ðtÞ oscillates around a linear divergence line. Cincotta et al. (2003) showed that, for the quasi-periodic orbits, Y ðtÞ always converges to 2, that is a fixed constant. Moreover, it has been shown that ordered motions with harmonic oscillations, i.e. orbits very close to a stable periodic orbit, tend asymptotically to Y ðtÞ ¼ 0. These latter properties can also be used to compute efficiently a good estimation of LCN, or similarly the Lyapu-

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nov time T k ¼ 1=k, by means of a linear least square fit of Y ðtÞ. Indeed, in the case of an irregular orbit, the time evolution of Y ðtÞ may be easily written as Y ðtÞ ’ aH t þ d;

t ! 1;

where aH is simply related to LCN by the relation aH ¼ k=2 and d is small for chaotic motion. But for regular orbits, the best fitted linear regression, after a transient time, can provide a d not necessarily close to zero. Thus, the value of d may be considered as the measure of the time during which the orbits stick to a regular torus before getting chaotic (Cincotta and Simo´, 2000). Regarding the numerical computation of the MEGNO indicator, we adopt the same strategy as in Goz´dziewski et al. (2001). To be specific, in addition to the numerical integrations of both the equations of motion and the first order variation equations, we consider the two additional differential equations d d_ d y¼ ; dt dd

d y w¼2 ; dt t

ð5Þ

which allow to derive the MEGNO indicators as Y ðtÞ ¼ 2 yðtÞ=t;

Y ðtÞ ¼ wðtÞ=t:

The MEGNO criterion, unlike the common Lyapunov variational methods, takes advantage of the whole dynamical information for the orbits and the evolution of its tangent vector, which results in shorter times of integration to achieve comparable results. Moreover, a couple of applications found in the literature (e.g. Goz´dziewski et al., 2001; Goz´dziewski et al., 2008; Cincotta and Simo´, 2000; Breiter et al., 2005) justify and confirm that MEGNO is relevant, reliable and provides an efficient way for the investigation of the dynamics by detecting regular as well as stochastic regimes. 3.1. MEGNO and numerical integrations As previously mentioned, in order to evaluate the MEGNO indicator, we have to integrate the differential system of Eq. (2), the linear first order variational system of Eq. (4), as well as the two additional differential Eq. (5). We choose to write both the expressions of the perturbing forces and the variational system, i.e. the Jacobian matrix, in rectangular coordinates positions-velocities. In such a way we can overcome both the null eccentricity and the null inclination singularity present in the dynamics of space debris (Valk et al., 2009). Moreover, the explicit analytical expressions of the vector fields allow us to avoid the difficulties inherent in the classical method of neighboring trajectories (two particles method). In order to numerically integrate the two differential systems of equations, we adopt the variable step size Bulirsh–Stoer algorithm (see e.g. Bulirsh and Stoer, 1966; Stoer and Bulirsch, 1980). Let us note that, for the purpose

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of validation, the numerical integrations were also made with a couple of other numerical integrators. However, the Bulirsh–Stoer algorithm seems to be the best compromise between accuracy and efficiency. Moreover, as quoted by Wisdom (1983): What is more important for this study, Benettin et al. (1980) found that the maximum LCE1 did not depend on the precision of their calculation. It appears likely that as long as a certain minimum precision is kept, maximum LCE’s may be accurately computed, even though it is not possible to precisely follow a specified trajectory for the required length of time. Although this latter observation was formulated in the framework of both Lyapunov variational method and Hamiltonian systems, it seems that it remains relevant in the computation of the MEGNO criterion, at least in the particular case of our analysis.

effects induced by the 1:1 resonance, the MEGNO evolution no longer depends on the random choice of the initial tangent vector. In this latter case, the intrinsic stability of the chosen orbits seems also to dictate the evolution of MEGNO as reported in Cincotta et al. (2003). More specifically, the stability of the orbit seems to influence the time evolution of MEGNO the more the orbit is closer to a stable or unstable equilibrium point. For instance, regarding the orbits extremely close to a stable equilibrium point, MEGNO generally approaches slowly the limit value 2 from below, even though some infrequent orbits present a MEGNO convergence from above. Conversely, the orbits initially close to the separatrices generally present a MEGNO approaching the value 2 from above.

3.2. Influence of the initial tangent vector d0

In this section we will study the MEGNO indicator for integrable Hamiltonian systems and we will show that generically (if the system is not isochronous) it always converges to 2, moreover the way Y ðtÞ reaches this limit value, say from higher or lower values, depends only on the choice of the initial tangent vector and not on the orbit itself. So let us consider an integrable Hamiltonian system write in action-angle variables, H ¼ HðpÞ, where p 2 B Rn denotes the action variables and q 2 Tn denotes the angle variables. Then the Hamiltonian equations are

By construction MEGNO depends on the initial value of the tangent vector d0 as the LCE (Benettin et al., 1980). This is why we preferred to adopt the strategy of randomly initialize the initial tangent vectors in order to avoid some parts of the artificially created zones of low MEGNO due to the proximity of d0 to the minimum Lyapunov exponent direction (Breiter et al., 2005). Moreover, as pointed out by Goz´dziewski et al. (2001), the random sampling of d0 is relevant in the sense that different initial tangent vectors can lead to different behaviors of the MEGNO time evolution while considering the same orbit. This observation has been reported in the framework of extra-solar planetary systems and seems to remain similar in the case of Earth orbiting objects and more generally for high-dimensional dynamical systems (having more than 3 degrees of freedom). Regarding the impact of the choice of the initial tangent vector d0 , we performed a set of exhaustive numerical investigations of regular orbits. More specifically, we compared the time-evolution of MEGNO using different initial tangent vectors and identical generic initial conditions. The results confirm that the random choice of the initial tangent vector induces a significant random behavior in the way MEGNO approaches the limit value 2, hence preventing this information from being useful to check the stability/instability character of regular orbits. Actually, when considering a slightly perturbed two-body problem (such as the central attraction disturbed by the oblateness of the Earth), the way MEGNO converges to 2 is completely unpredictable, leading to more or less 50% of convergence of Y ðtÞ to 2 from above and the other remaining 50% from below. This result is formally discussed in the following subsections. However, when the order of magnitude of the perturbation is larger, the result does not completely hold anymore. In particular, when considering the perturbing

3.3. MEGNO for integrable systems

p_ ¼ 0; @H ¼ xðpÞ: q_ ¼ @p The tangent space (to a given orbit) can be split into the action and angle direction, namely d ¼ ðdp ; dq Þ, thus the variational system can be written as d_ p ¼ 0; 2

@ H dp ¼ MðpÞ dp : d_ q ¼ @p2 If the system is isochronous then M 0, thus dp and dq are constant and Y ðtÞ ¼ 0 for all t. On the other hand, if the system is non-isochronous we get dp ðtÞ ¼ dp ð0Þ and dq ðtÞ ¼ dq ð0Þ þ Mðpð0ÞÞ dp ð0Þt. To simplify the notations, let us introduce Mðpð0ÞÞ ¼ M 0 ;

Using the definition of MEGNO, we get

Y ðtÞ ¼ 1

Lyapunov Characteristic Exponent.

dp ð0Þ ¼ n0 and dq ð0Þ ¼ g0 :

1 t

Z

t 0

2

ðM 0 n0 Þ s þ M 0 n0 g0 ðn0 Þ2 þ ðg0 Þ2 þ 2M 0 n0 g0 s þ ðM 0 n0 Þ2 s2

s ds;


@H @V ¼ @q @q @H @V ¼ xðpÞ þ ; q_ ¼ @p @p

and this integral can be explicitly computed, obtaining h i M 0 n0 g0 2 2 Y ðtÞ ¼ 2 log 1 þ 2M n g t þ ðM n Þ t 0 0 0 0 0 2 tðM 0 n0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðM 0 n0 Þ ðM 0 n0 g0 Þ 2 t ðM 0 n0 Þ 2

p_ ¼

and a similar decomposition can be done for the variational system @2V @2V d p 2 dq d_ p ¼ @p@q @q 2 2 @ H @ V @2V d þ þ dq : d_ q ¼ p @p2 @p2 @p@q

2

M 0 n0 g0 þ ðM 0 n0 Þ t2 6 ffi 4arctan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðM 0 n0 Þ ðM 0 n0 g0 Þ 3 M 0 n0 g0 7 ffi5: arctan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðM 0 n0 Þ ðM 0 n0 g0 Þ

ð6Þ

One can check that the square root is well defined, i.e. positive, and thus one can cast Eq. (6) into Y ðtÞ ¼ 2

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M 0 n0 g0 1 F 1 ðtÞ F 2 ðtÞ; t t

where F 1 and F 2 are positive functions and F 2 is bounded. We can then conclude that (see Fig. 1) (1) if M 0 n0 g0 > 0 then Y ðtÞ approaches 2 from below; (2) if M 0 n0 g0 < 0 then Y ðtÞ approaches 2 from above, in fact for large t the first contribution dominates the bounded term F 2 . In this last part we will consider if and under which assumptions the previous results concerning the convergence Y ! 2 are still valid, for a quasi-integrable Hamiltonian system of the form H ðp; q; Þ ¼ H 0 ðpÞ þ V ðp; qÞ. The main idea is the following: fix > 0, but small, and consider a ‘‘non-chaotic orbit” / , namely an orbit without a positive Lyapunov exponent (or with a bounded MEGNO), then if is sufficiently small this orbit is a perturbation of an orbit existing also for ¼ 0; /0 , and we can check that Y / ¼ Y /0 þ OðÞ, hence the smallness of such -correction cannot change ‘‘the way Y goes to 2”. More precisely, the Hamilton equations are now

Looking for dp and dq as -power series, i.e. dp ¼ dp;0 þ dp;1 þ . . . and dq ¼ dq;0 þ dq;1 þ . . ., and collecting together, in the definition of MEGNO, terms contributing to the same power of , we can thus get 1 Y / ðtÞ ¼ t

Z

ðM 0 dp;0 Þ2 s þ M 0 dp;0 dq;0

t 0

2

ðdp;0 Þ þ ðdp;0 Þ2 þ 2M 0 dp;0 dq;0 s þ ðM 0 dq;0 Þ2 s2

s ds

þ OðÞ ¼ Y /0 ðtÞ þ OðÞ:

4. Validation of the method To validate our method we first apply the technique on a simplified model, considering only the Earth’s gravity field expanded up to the second degree and order harmonics, namely, J 2 ¼ C 20 ; C 22 and S 22 . For the purpose of the analysis, we followed a set of 12 600 orbits, propagated over a 30-year time span, that is the order of 104 fundamental periods (1 day) empirically required by the method (Goz´dziewski et al., 2001). As reported in Breiter et al. (2005), a 30-year time span seems to be relatively small for long-term investigations of geosynchronous space debris. However, the numerical integration of variational equations in addition to the extrapolation of the orbit is quite time consuming. Indeed, the simulation with an entry level step size of 400 s takes approximately 20 s per orbit when including only the Earth’s gravity field, whereas it takes 42 s with a complete model. Thus, the examination of large sets of initial conditions can take a lot of time

Fig. 1. MEGNO for quasi-integrable adimensional Hamiltonian system. We consider the evolution of Y / for the system H ¼ p21 =2 þ p2 þ cos q1 þ cosðq1 q2 Þ. On the left panel ¼ 104 , while on the right panel ¼ 103 . In both cases is small enough to confirm the theoretical predictions; let 1 0 observe that in this case the matrix M is given by and thus the sign condition reads Mdp;0 dq;0 ¼ d1p;0 d1q;0 . The unit of time corresponding to 1/10 0 0 of period of the orbit.

(typically 5 days for 104 orbits). On the other hand, the analysis of the following section will bring to the fore some indications about the Lyapunov times (smaller than 30 years). As a consequence, the integration time can be considered as sufficiently large in the particular case of our study. For the purpose of this validation study, we consider a set of initial conditions defined by a mean longitude k grid of 1°, spanning 90° on both sides of the first stable equilibrium point and a semi-major axis a grid of 1 km, spanning the 42164 ± 35 km range. The other fixed initial conditions are e0 ¼ 0:002 for the eccentricity, i0 ¼ 0:004 rad for the inclination, X0 ¼ x0 ¼ 0 rad for the longitude of the ascending node and the argument of perigee, respectively. These values have been fixed to compare our results for the nearly-geosynchronous orbits with the ones of Breiter et al. (2005). As pointed out by Breiter et al. (2005), due to the 1:1 resonance, good variables to present our results will be ða0 ; r0 Þ, where a0 is the osculating initial semi-major axis and r is the so-called resonant angle, i.e. r ¼ k h, where h is the sidereal time. Fig. 2 (left panel) shows the MEGNO values computed using 30 years of integration time. We identify clearly a blow-up of the typical double pendulum-like pattern related to the 1:1 resonance. Here, we plot only over a horizontal range of 180°, i.e. only one eye. The existence of both the stable and the two unstable equilibrium points can be easily inferred. We observe that the phase space seems to be essentially filled in with MEGNO values Y ðtÞ ’ 2, that is plenty of regular orbits. Moreover, the two separatrices are also identifiable and are associated with neighboring MEGNO values 2 < Y ðtÞ 6 4. Therefore, following the properties defined in Section 3, one could consider that these orbits are chaotic. However, we will show that this conclusion is false. Indeed, a careful identification of the MEGNO time evolution shows that the latter always approach slowly the limit 2 from above. The closer to the separatrices, the slower the convergence. More precisely, orbits close to the separatrix integrated over long

30 20 10 0 −10 −20 −30 0

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time span present a bounded MEGNO evolution. Hence they should be considered as non-chaotic. To clarify this point, we performed a similar study, but using a significantly longer time-span, namely 300 years. The results are shown in Fig. 2 (right panel). For the sake of comparison, the color bars have been taken identical on both plots. Let us observe that the maximum value reached by the MEGNO is 4 in the left panel and 2.5 in the right one. In the 300 years simulation (Fig. 2, right), the MEGNO values, associated with orbits close to the separatrices, turn out to be, on average, smaller than in Fig. 2 (left panel), reaching almost the limit Y ðtÞ ! 2, due to the longer time of integration. Similarly, the dark zone in the neighborhood of the stable equilibrium point, corresponding to MEGNO values close to zero, is strongly shrunk, supporting the result that, in the limit of infinitely large t, only the orbit originating from the exact stable equilibrium point leads to Y ¼ 0, whereas the neighboring trajectories converge slowly to Y ðtÞ ¼ 2. Let us note that the importance of the integration time has been recently reported by Barrio et al. (in press) in the framework of applications of the MEGNO method. We confirm that a too short time of integration can give wrong conclusions about the dynamical behavior. Moreover, the latter paper also underlines some spurious structures appearing in the maps of the variational chaos indicators, explaining the presence of the sine wave of lower MEGNO with a bulge at the center of Fig. 2, ‘‘suggesting that the same periodic orbit is more or less regular depending on the initial conditions choice”. Actually, according to the latter authors and to our analysis, this conclusion is wrong because this spurious structure is related to numerical artifacts. 5. High area-to-mass ratios analysis The study of the long-term stability of near-geosynchronous objects has recently prompted an increasing interest


of the scientific community. In the particular case of classical near-geosynchronous objects, the problem has been solved by computing the MEGNO indicator for a family of simulated geostationary, geosynchronous and super-geosynchronous orbits. A classical near-geosynchronous object has a period close to one sidereal day and is subjected to the main gravitational effects of the Earth, including the 1:1 resonance, luni-solar perturbing effects, as well as solar radiation pressure associated to a small area-tomass ratio ðA=m 1m2 =kgÞ. According to Breiter et al. (2005) and Wytrzyszczak et al. (2007), the near-geostationary region presents chaotic orbits only very close to the separatrices, due to the irregular transits between the libration and the circulation regimes. Regarding the super-geostationary orbits, all of them seem to be entirely regular on the time scale of the investigations, that is a few decades. The aim of this section is to provide a more extensive analysis of the dynamics of near-geosynchronous space debris with high area-to-mass ratios ðA=m 1m2 =kgÞ, subjected to direct solar radiation pressure. Our main objective is to study the effects of high area-to-mass ratios on the stability of the principal periodic orbits and on the chaotic components. This analysis is divided into three parts. First, in Section 5.1, we focus our attention on the sensitivity to initial conditions; then, in Section 5.2, we report the results of dedicated numerical analyses which emphasize the importance of the area-to-mass ratio value. Finally, in Section 5.3, we study the influence of both the initial eccentricity and time at epoch. Let us recall that for large area-to-mass ratios ðA=m P 10m2 =kgÞ, the solar radiation pressure may become the major perturbation, by far larger than the dominant zonal gravity term J 2 (Valk et al., 2008). In this particular case, the larger the area-to-mass ratio, the more affected the dynamics of the near-geosynchronous space debris, leading to daily high-amplitude oscillations of the semi-major axis, yearly oscillations of the eccentricity as well as long-term variations of the inclination. As an illustration, Fig. 3 shows the orbital elements histories of the first 210 years of a geosynchronous high area-to-mass ratio space debris ðA=m ¼ 10 m2 =kgÞ. The yearly variation of the eccentricity reaches 0.2, which confirms the expected values predicted (e.g. Anselmo and Pardini, 2005; Liou and Weaver, 2005). The inclination evolution presents a well known long-term variation whose period is directly related to the area-to-mass ratio value. Regarding the longitude of ascending node as well as the argument of perigee, they both present a libration due to the chosen set of initial conditions. For further details, we refer to Valk et al. (2008) as well as Chao (2006), where a full description of the longterm motion of high area-to-mass ratios space debris is given. 5.1. Sensitivity to initial conditions To start with, we follow the evolution of two high areato-mass ratio space debris ðA=m ¼ 10 m2 =kgÞ defined by

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two sets of very close initial conditions, differing only in the 10th digits in mean longitude. Fig. 3 shows the first one and Fig. 4 shows the second nearby orbit. We observe that the most difference (in the behavior) take place in the semi-major axis and resonant angle panels. We notice that there are some differences in the dynamics of the semimajor axis already after 20 years. This is the same for resonant angle, confirming the hypothesis that the sensitivity to initial conditions is especially relevant for the semimajor axis and resonant angle whereas the difference, in the behavior, between the other orbital elements remains small. We first focus our attention on the time evolution of the semi-major axis and resonant angle. As a complement to Fig. 3, we numerically computed two orbits for two space debris with different area-to-mass ratios, A=m ¼ 1 m2 =kg and A=m ¼ 10 m2 =kg, whose initial conditions have been chosen near the separatrices, to emphasize their chaotic behaviors. Fig. 5 shows a blow-up of the evolution of the semi-major axis (top panels) and resonant angle (middle panels) over the time span of 250 years. It is clear that the semi-major axis presents some irregular components over its evolution, related to some transitions between different regimes of motion, clearly identifiable in the resonant angle plots. In addition, we also computed the corresponding MEGNO time evolution. The bottom panel in each graph shows the time evolution of the MEGNO indicator as well as its corresponding mean value. First, we see that the time evolution of Y ðtÞ presents a quasilinear growth almost since the beginning of the integration process, leading to the conclusion that these orbits are clearly chaotic over that time scale. Therefore, we also computed the linear fit Y ðtÞ ’ aH t þ d in order to evaluate the Lyapunov time T k :T k is the inverse of the LCN ðkÞ calculated by the linear regression coefficients aH ¼ k=2. Let us remark that to avoid the initial transient state, the least square fits were performed on the last 85% of the time interval. This latter analysis brings to the fore the fact that larger area-to-mass ratios lead to smaller Lyapunov times, i.e. larger Lyapunov Characteristic Numbers. Indeed, for A=m ¼ 1m2 =kg, the Lyapunov time turns out to be on the order of 11 years, whereas it reaches the value T k ’ 3:7 years for A=m ¼ 10 m2 =kg. Second, let us also remark that the behavior of the MEGNO indicator is of particular interest in these cases. A careful analysis of Y ðtÞ underlines some irregular patterns directly related to the evolution of r, in particular when the orbits seem to transit across the separatrices. Finally, we can also highlight the fact that the sudden changes between libration and circulation regimes occur mainly when the inclination changes its sign of variation, especially at the maximum value for A=m 1 m2 =kg and at the minimum for A=m 6 1 m2 =kg (Fig. 5, top panels, dashed line), with an empirical long-term periodicity of T X , that is the long-term periodicity of the longitude of the ascending node, which is all the more smaller when A=m is large (Valk et al., 2008).

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Fig. 3. Time-evolution of high area-to-mass ratio space debris. Orbital elements over 210 years for a A=m ¼ 10 m2 =kg; initial conditions are: a0 ¼ 42166:473 km; e0 ¼ 0:002; i0 ¼ 0:004 rad, X0 ¼ x0 ¼ 0 rad and M 0 ¼ 4:928 rad. Time at epoch is 25 January 1991.

5.2. Extended numerical analyses We considered a set of 12,600 simulated orbits with various initial semi-major axes and mean longitudes. We took into account the following perturbing effects: second degree and order harmonics (J 2 ; C 22 and S 22 ), the luni-solar interaction as well as the perturbing effects of the solar radiation pressure with four values of the area-to-mass ratio ðA=m ¼ 1; 5; 10; 20 m2 =kgÞ. The results are reported in Fig. 6.

In the case with A=m ¼ 1 m2 =kg (top left panel) we recognize the same pendulum-like pattern as in Fig. 2. Considering the same integration time (30 years), we notice that the MEGNO values tend to be slightly larger than in Fig. 2 (left). Moreover, some irregularly distributed MEGNO values are clearly visible close to the two saddle unstable stationary points. These results completely agree with those presented by Breiter et al. (2005), where the solar radiation pressure was taken into account, but only for very small area-to-mass ratios (typically 0:005m2 =kg). Indeed, our lat-

Semi−Major Axis 4.222e+07

m

4.219e+07 4.216e+07 4.213e+07 4.210e+07 0

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Time [years] Eccentricity 0.255 0.185 0.115 0.045 −0.025 0

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ter analysis shows that in addition to the luni-solar perturbations, solar radiation pressure with small to moderate area-to-mass ratios, that is 0 6 A=m 6 1 m2 =kg, do not change considerably the phase space pattern. On the other hand, the remaining panels of Fig. 6 show that the phase portrait becomes significantly more intricate with increasing area-to-mass ratios. Indeed, the width of the stochastic zone in the neighborhood of the separatrices becomes relevant, with a large displacement of the separatrices on the phase plane. The larger chaotic region can

readily be explained by the osculating motion of the separatrices due to the before-mentioned daily variations of the semi-major axis with respect to some mean value as well as by the increasing amplitudes of the eccentricities. These variations lead inevitably to transits through both the regions separating libration and circulation motion for orbits initially close to the separatrices. Moreover, it is also clear that the usual double pendulum-like phase space shows a tendency to be distorted with an apparent displacement of the unstable equilibrium

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points, whereas the stable equilibrium points remain almost fixed. This last result is however quite awkward insofar as there is no physical interpretation to this phenomenon. Indeed, direct solar radiation pressure does not depend explicitly on the (mean) resonant angle with respect to the long-term investigations after averaging over short periodic terms. Therefore, it cannot induce a displacement of the equilibrium points in the phase space. Actually, a clever explanation can be found regarding the way the sampling is considered in the elaboration of the graphics. More specifically, it is worth noting that, at first, the sampling is carried out with respect to osculating initial conditions.

Second, within the framework of mean motion theory, it is well-known that, due to the short-period oscillations, the mean and the osculating initial conditions cannot be considered to be equal. In other words, for the same fixed value of the initial osculating semi-major axis and for various initial mean longitude, we obtain different values for the mean semi-major axis; as explained with Fig. 7. Actually, the different initial mean longitudes induce a phase difference in the corresponding evolution of the semi-major axis, leading to different mean initial semi-major axes. Let us remark that the maximum difference between both the mean semi-major axes is directly related to the order of magni-

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tude of the short-period variations, and, as a consequence, is also directly related to the area-to-mass ratio. More rigorously, the difference between osculating and mean initial conditions is a well-defined transformation, depending on the generating function used within the averaging process allowing to change from mean to osculating dynamics. For further details concerning this explicit trans-

formation, we refer to the Lie algorithm discussed in Deprit (1969) and Henrard (1970). However, because we bound our analysis mainly to numerical simulations, we cannot access such generating function; we can nevertheless overcome this problem by numerically computing, for each semi-major axis osculating initial condition, the related mean initial semi-major axis, by considering the

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average over a short time span of 10 days. As an illustration, in Fig. 8, we give the relation between the mean semi-major axis and the resonant angle for various values of the osculating semi-major axis ðA=m ¼ 10 m2 =kgÞ. The first difference is related to a semi-major axis sampling taken above the libration region, the second is related to a semi-major axis sampling which crosses the libration region and finally, the third sampling is taken below this region. In conclusion, we clearly see that the order of magnitude of the differences is, as previously mentioned, the order of the amplitudes of the daily variations observed in the semi-major axis dynamics. Let us note that in the latter case, i.e. A=m ¼ 10 m2 =kg, the differences reach at most 27 km, which correspond exactly to the difference between the stable and unstable equilibrium points, as shown in Fig. 6(bottom, left). We can thus apply numerically the transformation as a post-treatment process, that is considering the MEGNO val-

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amean − 42164 [km]

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5.3. Initial time at epoch and importance of the mean eccentricity One should also recall that solar radiation pressure leads to a theoretical equilibrium defined both in eccentricity e0 and longitude of perigee -0 . The conditions leading to such an equilibrium can be written as ( e0 ¼ 32 C r P r mA n a1n cos2 2 ’ 0:01 C r mA ; -0 ¼ k ð0Þ:

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ues not in the osculating initial conditions phase space, but in the mean initial conditions phase space. For the sake of comparison with Fig. 6, we show the results once such a transformation has been applied (Fig. 9): it is clear that now the vertical gaps between both the stable and unstable equilibrium points are almost completely eliminated, hence these points have almost the same mean semi-major axis, getting rid of the what we called the ‘‘short-period artefact”. The thin light waves crossing the Fig. 9 are due to gaps in the set of initial conditions and have no dynamical significance (also valid for the only one light wave crossing Fig. 13 in Section 6). Let us also remark that, from now on, all the results will be shown in the mean initial conditions phase space.

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Fig. 8. Relation between the mean semi-major axis and the resonant angle for various values of the osculating semi-major axis. The first osculating semi-major axis is taken above the libration region, the second is related to an osculating semi-major axis sampling which crosses the libration region and, finally, the third sampling is taken below this region.

where n and n are the angular motions of both the space debris and the Sun, respectively, is the obliquity of the Earth with respect to the ecliptic and k ð0Þ the initial ecliptic longitude of the Sun. If these conditions are fulfilled, it has been shown (Chao, 2006; and later Valk et al., 2008, 2009), that the eccentricity vector ðe cos -; e sin -Þ remains constant, leading to a fixed value of both the eccentricity and longitude of perigee. As an illustration, Fig. 10 shows the mid-term variations of the eccentricity for a fixed value of the area-to-mass ratio ðA=m ¼ 10 m2 =kgÞ and fixed initial conditions, namely, a0 ¼ 42; 164 km;e0 ¼ 0:1; i0 ¼ 0 rad, X0 ¼ x0 ¼ k0 ¼ 0 rad. It is clear that, apart from a phase difference, the amplitudes of the variations of the eccentricities are

Fig. 9. The MEGNO computed as a function of initial mean longitudes k0 and initial mean semi-major axis a0 . The model is the same as in Fig. 6. The areato-mass ratio is A=m ¼ 5 and 10 m2 =kg for the left and for the right graph, respectively.

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qualitatively the same, except when adopting an initial time at epoch equal to 21 March. In this latter case, the eccentricity remains almost constant, as expected by the theory. Fig. 11 shows the phase space in mean semi-major axis and longitude for A=m ¼ 10 m2 =kg and fixed values of the initial conditions, namely e0 ¼ 0:1; i0 ¼ 0:004 rad, X0 ¼ x0 ¼ 0 rad. The differences between the two graphs only depends on the initial time at epoch parameter t0 . We could actually expect that different initial times at epoch, namely, different initial ecliptic longitudes of the Sun k ð0Þ, will reveal a quite rich collection of behaviors, depending on the different states with respect to the before-mentioned eccentricity equilibrium. Actually, assuming an initial time at epoch of 21 December 2001, we see clearly that the phase space is filled by a large number of

chaotic orbits (Fig. 11, left). On the contrary, starting with an initial time at epoch of 21 March 2000, that is adopting a Sun pointing longitude of perigee (k ð0Þ ¼ 0 rad), the MEGNO values tend to be smaller and associated with significantly narrower chaotic regions, always located close to the separatrices (Fig. 11, right). In the latter case, the eccentricity presents only small yearly variations due to the proximity of the theoretical equilibrium. Therefore, these results seem to suggest that high amplitude variations of the eccentricity increase considerably the extension of chaotic regions close to the separatrices and, conversely, small eccentricity variations seem to minimize considerably the extent of chaotic regions. To justify this assumption, we performed a dedicated numerical simulation with the same set of parameters used in the one reported in Fig. 11, but considering higher values of the initial eccentricity. The results are reported in Fig. 12, the chosen time at epoch is 21 December 2000 and the initial eccentricities are, e0 ¼ 0:2 (left panel) and e0 ¼ 0:4 (right panel). In the latter case, the huge variations of the perigee altitude, induced by the large variations of the eccentricity as well as by the variations of the semi-major axis, leads to even more complicated dynamics. These results thus confirm the importance of the initial eccentricity in the appearance of chaos. 6. Secondary resonances It is worth noting that inspecting Figs. 9, 11 and 12 we clearly note the presence of some additional patterns located on both sides of the separatrices in the phase space. These never seen before regions, hence unexplained so far, are actually characterized by very low MEGNO values. Indeed, this observation underlines the fact that the dynamics of high area-to-mass ratio space debris is even more intricate than expected. In the following two sections we will provide some numerical results and an analytical

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Fig. 12. MEGNO computed as a function of initial mean longitude k0 and semi-major axis a0 . The equations of motion include the central body attraction, the second degree and order harmonics J 2 ; C 22 and S 22 , the luni-solar interaction as well as the perturbing effects of solar radiation pressure. The mean longitude grid is 1° and the semi-major axis grid is 500 m spanning the 42,164 ± 35 km range. The initial conditions are i0 ¼ 0:004 rad, X0 ¼ x0 ¼ 0 rad with A=m ¼ 10 m2 =kg. Time at epoch is 21 December 2000. The patterns have been obtained using two initial eccentricities, e0 ¼ 0:2 (left) and e0 ¼ 0:4 (right).

theory, based on a simplified model, to better understand such zones. 6.1. Numerical investigations We followed a large set of near-geosynchronous space debris, related to an extremely large set of initial conditions taken on both sides of the pendulum-like pattern, and for each one of the 72,000 orbits we computed the related MEGNO indicator. The initial conditions have been fixed by a mean longitude grid of 1°, spanning 360°, and a semi-major axis grid of 1 km, spanning the 42,164 ± 100 km range, while the remaining orbit parameters and time at epoch are the same as in Fig. 6. Moreover, as in the previous extended analyses, the model of forces also includes the central body attraction, the second degree and order harmonics J 2 ; C 22 and S 22 as well as the combined attractions of the Sun and the Moon. The perturbing effects of direct solar radiation pressure are also taken into account for a high area-to-mass ratio fixed at 10 m2 =kg. The results are reported in Fig. 13, which is nothing but an extensive enlargement of the phase space presented in Fig. 6(bottom, left). This phase space widening clearly underlines the before-mentioned additional structures located at 40 km on each side of the resonant area. Furthermore, besides these patterns, what is of special interest is that this Figure also brings to the light supplementary structures located at approximately 80 km on both sides of the main resonance, suggesting that the phase space is actually foliated by a larger set of secondary structures. Moreover, the width of these additional patterns and the numerical values of the MEGNO both seem to be directly related to the inverse of the distance with respect to the resonant area. In addition, we also performed a set of similar numerical investigations, in order to distinguish qualitatively the relative relevance of some parameters such as the initial mean

Fig. 13. MEGNO computed as a function of initial mean longitude k0 and semi-major axis a0 . The equations of motion include the central body attraction, the second degree and order harmonics J 2 ; C 22 and S 22 as well as the luni-solar perturbations. The mean longitude grid is 1° and the semimajor axis grid is 1 km, spanning the 42,164 ± 100 km range. The initial conditions are e0 ¼ 0:002; i0 ¼ 0:004 rad and X0 ¼ x0 ¼ 0 rad. The areato-mass ratio is 10 m2 =kg. Time at epoch is 25 January 1991.

eccentricity, the value of the area-to-mass ratio, as well as the importance of the 1:1 resonance and of the third-body perturbations in the occurrence of such secondary structures. Even though these results are not presented here in detail, we can draw the following preliminary conclusions: the second order harmonic J 2 , as well as the third-body perturbations, do not seem to be really relevant and crucial in the appearance of these additional patterns. In other words, the unexpected patterns occur only when taking into account the combined effects of both the second order and degree harmonic and direct solar radiation pressure. As a matter of fact, the extended numerical investigations performed in Fig. 6(top, left), or similarly those shown in


Breiter et al. (2005), also present these structures, even though they are difficult to perceive. Actually, the extension and chaoticity indicator of the secondary patterns seem to be directly proportional to the area-to-mass ratio value or, equivalently directly proportional to the mean value of the eccentricity. To get even more concluding results, we considered a blow-up of the phase space (dashed line rectangle in Fig. 13) with really high resolution sampling (150 m in the semi-major axis a and 0.3° in the resonant angle r). Fig. 14(top) shows this phase space widening wherein we defined a so-called resonant angle section (horizontal black solid line), that is the subset of orbits having the same initial resonant angle value. This resonant angle section spans the complete range in semi-major axis and passes close to the stable equilibrium point. For each orbit defined on this section, we computed the MEGNO indicator and in Fig. 14(middle) we report this value at the end of the simulation as a function of the semi-major axis. To double check our results, we performed a frequency analysis investigation (see Laskar, 1990; Laskar, 1995; Noyelles et al., 2008) aimed to study the behavior of the proper frequency of the resonant angle r, whose results are reported in Fig. 14(bottom). Here one can clearly notice the distinctive characteristics regarding the wellknown 1:1 resonance between the mean longitude and the

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sidereal time. Indeed, both MEGNO and the fundamental period show distinctively a minimum close to the stable equilibrium point. In this case, as previously mentioned in Section 4, MEGNO should slowly converge to Y ðtÞ ¼ 2 everywhere, except at the equilibrium point where the limit value is Y ðtÞ ¼ 0, that is why, using a finite integration time, we obtain such V-shaped curve, close to 0 in the center of the resonance and to 2 on the borders. It is also worth noting that the fundamental period of r is reported to be close to 2.25 years, which is in good agreement with the well-known 818 days libration period of a typical uncontrolled near-geosynchronous object. Near the separatrices, MEGNO clearly presents some obvious high values which confirms the presence of chaotic orbits. Here, the fundamental period reaches significant values and, as a matter of fact, is not well determined, once again supporting the result of the existence of a chaotic zone. Moreover, the use of frequency analysis allows us to support strongly the hypothesis that the additional patterns are actually related to secondary resonances. Indeed, if we look at the evolution of the fundamental period with respect to the semi-major axis, it is clear that the so-called secondary resonances are associated, regarding the angle r, with periods which are commensurate with 1 year. More precisely, the major secondary resonances, located at approximately 40 km on both sides of the pendulum-like

Fig. 14. Blow-up of the phase space with the specification of a resonant angle section (horizontal black solid line), that is the set of orbits having the same ¼ 81:67 (top panel). Evolution of MEGNO with respect to the (osculating) initial resonant angle value, near the first stable equilibrium point, namely rsection 0 initial semi-major axis a0 for the specified section (middle panel). The fundamental period of r with respect to the initial semi-major axis a0 , computed by means of frequency analysis for the specified section (bottom panel). The estimation of the periods are made over a 20 years period of time.

2σ + λ

sun

Resonant angle σ

pattern, are related to a 2-year fundamental period of r. Concerning the farther patterns located at 80 km, the fundamental period of r turns out to be very close to 1 year. As a consequence, we can presumably assume that these secondary resonances are actually related to a commensurability between r and the 1 year period angle k , that is the ecliptic longitude of the Sun. To justify this assumption, we focused our attention to the major secondary resonances located at 40 km on both sides of the pendulum-like pattern, considering the time evolution of various linear combinations of r and k . For this purpose, we considered various initial semi-major axes in the phase space. The results are shown in Fig. 15. At first glance, it is apparent that three propagations stand apart from the others. In the first row of Fig. 15, that is regarding the evolution of the resonant angle r, we clearly identify the well-known characteristics related to the primary resonance. In particular, in Fig. 15a, that is when considering an initial semi-major axis inside the primary resonant ða0 ¼ 42; 188 kmÞ; r shows the well-known longperiod libration (2.25 years), whereas r circulates outside this region. Furthermore, what is of special interest is the time evolution of both 2r þ k and 2r k , shown in the second and third row, respectively. It is clear that most of the time these angles show a circulation regime. However, when considering an initial semi-major axis inside the major lower secondary resonance for 2r k or, similarly inside the major upper secondary resonance for 2r þ k , both these angles show a significant long-term evolution (Fig. 15b and c).

The presence and the location of these secondary resonances can be studied using an appropriate simplified model. Hence we model the averaged geostationary motion by a pendulum-like system, given by its Hamiltonian formulation (Valk et al., 2009) up to order e2 in the series expansion l2 l4 2 5 2 _ H ¼ 2 hL þ 3 6 Re 1 e S 2200 ðX; x; M; hÞ; 2 2L L where pffiffiffiffiffiffi L ¼ la

and S 2200 ðX; x; M; hÞ ¼ C 22 cos 2r þ S 22 sin 2r:

In the context of direct solar radiation pressure, we can introduce the factor Z proportional to A=m through the eccentricity e (for further details, we refer to the averaged simplified analytical model developed in Valk et al., 2008, 2009). As a first approximation, the time evolution of both the eccentricity e and the longitude of perigee - were found to be (neglecting the obliquity of the Earth with respect to the ecliptic) Z cos k þ a0 ; L n Z e sin - ¼ sin k b0 ; L n e cos - ¼

which introduces k in the Hamiltonian. The quantity n is the mean motion of the Sun and both a0 and b0 are related

6

6

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2σ − λsun

6.2. Analytical investigation – simplified model

10

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to the initial conditions with respect to the eccentricity and the longitude of perigee). The resulting Hamiltonian takes the generic form l2 _ þ F cos ð2r 2r0 Þ G 2 cos ð2r hL 2 L6 L6 2L 2r0 Þ cos ðk þ dÞ;

H¼

where d; F ; G; r0 are constants. A suitable transformation is then necessary to introduce action-angle variables (w; J Þ in the libration and in the circulation region of the double pendulum, in such a way any trajectory of the double pendulum is characterized by a constant action J and a corresponding _ Rewriting the perturbed system (beconstant frequency w. cause of k terms) by means of these new variables and then using the expansions in Bessel functions, we could isolate any resonance of the type kw k in the circulation region, for any jkj, and in the libration region, for jkj P 3, which corresponds to our frequency analysis. This analysis is surely promising, but it is outside the goals of this paper. Further investigations will be detailed in a forthcoming publication (Lemaıˆtre et al., accepted for publication). 7. Conclusions The predictability of the trajectory high area-to-mass ratio space debris located near the geosynchronous region was investigated by means of a recent variational chaos indicator called MEGNO. Thanks to this technique, we clearly identified the regular (stable) and irregular (chaotic) orbits. This efficient method allowed us to obtain a clear picture of the phase space, hence showing that chaotic regions can be particularly relevant, especially for very high area-to-mass ratio objects. Moreover, we discussed the importance of both the initial eccentricity and time at epoch in the appearance of chaos. Finally, we brought to the fore a relevant class of additional unexpected patterns which were identified as secondary resonances, that were numerically studied by means of both the MEGNO criterion and frequency map analysis, to eventually conclude that they involve commensurabilities between the primary resonant angle and the ecliptic longitude of the Sun. We also presented an analytical scheme that could explain their existence. It will be the subject of further work. Acknowledgements The authors thank S. Breiter for helpful discussions about both the MEGNO criterion as well as numerical issues which led to substantial improvement of the present paper, also providing some useful references. We are also grateful for the opportunity to use the frequency analysis tools developed by B. Noyelles and A. Vienne. Finally, the authors warmly thank the two referees for their suggestions that allowed us to strongly improve the paper.

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References Anselmo, L., Pardini, C. Orbital evolution of geosynchronous objects with high area-to-mass ratios, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris, (ESA SP-587). ESA Publications Division, Noordwijk, The Netherlands, pp. 279–284, 2005. Barrio, R., Borczyk, W., Breiter, S. Spurious structures in chaos indicators maps. Chaos Soliton. Fract., doi:10.1016/j.chaos.2007.09.084, in press. Benettin, G., Galgani, L., Giorgilli, A., et al. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Part 1: Theory. Meccanica 15, 9–20, 1980. Breiter, S., Wytrzyszczak, I., Melendo, B. Long-term predictability of orbits around the geosynchronous altitude. Adv. Space Res. 35, 1313– 1317, 2005. Bulirsh, R., Stoer, J. Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8, 1–13, March, 1966. Chao, C.C. Analytical investigation of GEO debris with high area-to-mass ratio. AIAA Paper No. AIAA-2006-6514, Presented at the 2006 AIAA/AAS Astrodynamics Specialist Conference, Keystone, Colorado, August 2006. Cincotta, P.M., Giordano, C.M., Simo´, C. Phase space structure of multidimensional systems by means of the mean exponential growth factor of nearby orbits. Physica D 182, 151–178, 2003. Cincotta, P.M., Simo´, C. Simple tools to study global dynamics in nonaxisymmetric galactic potentials – I. Astron. Astrophys. (147), 205– 228, 2000. Cunningham, L.E. On the computation of the spherical harmonics terms needed during the numerical integration of the orbital motion of an artificial satellite. Celest. Mech. 2, 207–216, 1970. Deprit, A. Canonical transformations depending on a small parameter. Celest. Mech. 1, 12–30, 1969. Goz´dziewski, K., Bois, E., Maciejewski, A.J., et al. Global dynamics of planetary systems with the MEGNO criterion. Astron. Astrophys. 378, 569–586, 2001. Goz´dziewski, K., Breiter, S., Borczyk, W. The long-term stability of extrasolar system HD37154, numerical study of resonance effects. Mon. Notices RAS 383, 989–999, 2008. Henrard, J. On a perturbation theory using lie transforms. Celest. Mech. 3, 107–120, 1970. Laskar, J. The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones. Icarus 88, 266–291, 1990. Laskar, J. Introduction to frequency map analysis, in: Proceedings of 3DHAM95 NATO Advanced Institute, vol. 533, S’Agaro, 134–150, June 1995. Lemaıˆtre, A., Delsate, N., Valk, S. A web of secondary resonances for large A=m geostationary debris. Celestial Mechanics and Dynamical Astronomy, accepted for publication. Lemoine, F.G., Kenyon, S.C., Factor, J.K., et al. The development of the joint nasa gsfc and nima geopotential model EGM96. Tech. Rep., NASA, TP-1998-206861, 1987. Liou, J.-C., Weaver, J.K. Orbital evolution of GEO debris with very high area-to-mass ratios. The Orbital Quarterly News, vol. 8(3), The NASA Orbital Debris Program Office, 2004. Liou, J.-C., Weaver, J.K. Orbital dynamics of high area-to-mass ratio debris and their distribution in the geosynchronous region, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris (ESA SP-589). ESA Publications Division, Noordwijk, The Netherlands, pp. 285–290, 2005. Noyelles, B., Lemaıˆtre, A., Vienne, A. Titan’s rotation. A 3-dimensional theory. Astron. Astrophys. 475, 959–970, 2008. Schildknecht, T., Musci, R., Flury, W., et al. Optical observations of space debris in high-altitude orbits, in: Danesy, D. (Ed.), Proceedings of the Fourth European Conference on Space Debris. ESA SP-587. ESA Publications Division, Noordwijk, The Netherlands, pp. 113–118, 2005.

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Schildknecht, T., Musci, R., Ploner, M., et al. Optical observations of space debris in GEO and in highly-eccentric orbits. Adv. Space Res. 34, 901–911, 2004. Standish, E.M. JPL planetary and lunar ephemeris, DE405/LE405. JPL Interoffice Memorandum IOM 312.D-98-048, August 1998. Stoer, J., Bulirsch, R. Introduction to Numerical Analysis. SpringerVerlag, New York, 1980. Valk, S., Lemaıˆtre, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios influenced by solar radiation pressure. Adv. Space Res. 41, 1077–1090, 2008.

Valk, S., Lemaıˆtre, A. Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including earth’s shadowing effects. Adv. Space Res. 42 (8), 1429–1443, 2008. Valk, S., Lemaıˆtre, A., Deleflie, F. Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence. Adv. Space Res. 43, 1070–1082, 2009. Wisdom, J. Chaotic behavior and the origin of the 3/1 Kirkwood gap. Icarus 56, 51–74, 1983. Wytrzyszczak, I., Breiter, S., Borczyk, W. Regular and chaotic motion of high altitude satellites. Adv. Space Res. 40, 134–142, 2007.



Faster algorithm of debris cloud orbital character from spacecraft collision breakup Li Yi-yong *, Shen Huai-rong, Li Zhi Academy of Equipment Command & Technology, Huairou City, Beijing, China Received 18 June 2008; received in revised form 3 March 2009; accepted 6 March 2009

Abstract Space debris is polluting the space environment. Collision fragment is its important source. NASA standard breakup model, including size distributions, area-to-mass distributions, and delta velocity distributions, is a statistic experimental model used widely. The general algorithm based on the model is introduced. But this algorithm is difficult when debris quantity is more than hundreds or thousands. So a new faster algorithm for calculating debris cloud orbital lifetime and character from spacecraft collision breakup is presented first. For validating the faster algorithm, USA 193 satellite breakup event is simulated and compared with general algorithm. Contrast result indicates that calculation speed and efficiency of faster algorithm is very good. When debris size is in 0.01–0.05 m, the faster algorithm is almost a hundred times faster than general algorithm. And at the same time, its calculation precision is held well. The difference between corresponding orbital debris ratios from two algorithms is less than 1% generally. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Orbital character; Collision breakup; USA 193; EVOLVE software

1. Introduction Research on space debris has been intensified in recent years as the number of space debris objects grows. The number of all objects in Earth orbit officially cataloged by the U.S. Space Surveillance Network (SSN) is more than 12,000 at the end of 2007 (The NASA Orbital Debris Program Office, 2008). The population of debris has now reached the level that orbital debris has become an important design factor for spacecraft. They induce serious space environment question and make potential threat to orbital spacecraft by the large impact energy. Although the current near-Earth space debris environment is dominated by explosion fragments, it is predicted that more debris will be generated by collisions rather than explosions in the future (Liou, 2006). Therefore, a fundamental issue in trying to limit the growth of future debris populations is to understand the nature of the predicted *

Corresponding author. Tel.: +86 010 66364384. E-mail address: [email protected] (L. Yi-yong).

orbital collisions. A key question is to have a high fidelity breakup model and a faster and effective algorithm to simulate collision activities. Since the 1970s, the NASA Orbital Debris Program Office has modeled the debris clouds generated by on-orbit explosions and collisions in terms of fragment size and velocity distributions. Up to now, the last major changes to these breakup models occurred in the 1990s. In the past ten years, NASA standard breakup model has been used to describe the outcome of spacecraft fragments widely (Michael et al., 2006). This paper presents a new algorithm of debris cloud orbital character from spacecraft collision breakup based on NASA standard breakup model (Reynolds et al., 1998). The USA 193 test is calculated based on the new algorithm and general algorithm separately, and the result validates that the new algorithm is faster and effective. 2. Collision breakup model and algorithm A collision breakup model, at a minimum, should define the size, area-to-mass ratio, and ejection velocity of each


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generated fragment. Since these parameters are not constant for all debris, distributions as a function of a given parameter, e.g., mass or characteristic length, are necessary. In addition, the initial conditions of the breakup, e.g., the total mass of the parent objects or the collision velocity, can be highly influential. Since multiple breakup events of the same type of object, for instance a Delta second stage, will not produce exactly the same debris cloud each time, the breakup model should also address variances about the derived distributions. NASA standard breakup model was adopted in recent versions of EVOLVE (Johnson et al., 2001) and LEGEND (Liou et al., 2004). The new model was founded on broader experimental databases. The single-valued functions for number, area-to-mass ratio, and velocity were to be replaced by more representative distributions. And the new NASA approach employed ‘characteristic length’, Lc, as independent variable replacing previous mass. This leads to a fundamentally different data flow (Bendisch et al., 2004). But the new model has also drawbacks, for example, the masses assigned to objects below 1 mm diameter are too large, leading to very dense particles, and the additional velocity distribution for small fragments which does not match measurement data. Michael et al. (2004) found these shortcomings and presented the approach for the improvement of the breakup model fortunately, then applied it for MASTER 2005. Based upon NASA standard breakup model, the spacecraft collision breakup algorithm can be given (see Fig. 1) (Li et al., 2008). For convenient contrast, we can name this algorithm for general algorithm. The steps are as follows:

(1) To calculate size of every debris. Firstly, the type of collision (catastrophic or non-catastrophic) is decided by the target mass mt, projectile mass mp, and collision velocity v. Size distribution is calculated based on size distribution function. (2) To calculate A/M of every debris. Area-to-mass distribution is made certain for a given characteristic length based on area-to-mass ratio distribution function. (3) To calculate mass of every debris. Mass of a given size and A/M debris is calculated. A/M of every debris is independent and submits to certain distribution, so random calculated mass of debris is often uncertain and mass sum of all created debris likely does not meet mass of parent objects. Thus, recalculation to return (2) is required. (4) To calculate DV of every debris. Delta velocity distribution is got for a given A/M based on delta velocity distribution function. The validation of conservation of momentum and kinetic energy is needed. (5) To calculate orbit of every debris. Position, velocity and orbit parameters of every debris are obtained based upon DV of debris, breakup position and velocity. These parameters can be applied for calculating debris orbit motion. It’s obvious that every debris needs to be calculated for predicting its orbital character according to this algorithm. When the number of debris is not so large, it is convenient and effective. However, the number of debris from spacecraft collision breakup is very large. Generally speaking, the number of debris over 10 cm size is thousands, and that debris of centimeters is tens thousand, and smaller debris is more. So this algorithm is difficult because of huge data quantity and long calculation time, and it is necessary to find a faster and effective algorithm. 3. Faster algorithm of debris cloud orbital character NASA standard breakup model presents size distribution function of debris from spacecraft collision breakup. A power law relation exists between the number and size of debris. As numerical calculation, the size of debris can be divided into multi-section, Lci [Lc(i), Lc(i + 1)) (i = 1, 2, . . .). So the number of debris in every size section, N(Lci), can be calculated. N ðLci Þ ¼ N ðLc ðiÞÞ N ðLc ði þ 1ÞÞ ¼ 0:1ðMÞ0:75 Lc ðiÞ1:71 Lc ði þ 1Þ1:71

Fig. 1. The spacecraft collision breakup algorithm.

ð1Þ

where the characteristic length Lc = (x + y + z)/3 (in m); x is the longest linear body dimension, y is the longest body dimension orthogonal to x, and z completes the triad and is measured normal to the x–y plane. The value of M is defined as the mass (in kg) of both objects in a catastrophic collision. In the case of a non-catastrophic collision, the


value of M is defined as the mass (in kg) of the projectile multiplied by the collision velocity (in km/s). Then, area-to-mass ratio of debris is divided into multisection, A/Mj [A/M(j), A/M(j + 1)) (j = 1, 2, . . .). According to area-to-mass ratio distribution function, distribution probability of debris with A/Mj in the section of Lci, PA=M (Lci, A/Mj), can be calculated.

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Finally, orbital lifetime of representative debris in LEO, Lt (A/Mj, DVk, dirr), can be calculated by taking into account air drag (Vallado, 2004), and orbital character of debris can be analyzed. At the time of post-breakup t, in the size range of [Lc(i), Lc(i + 1)), the ratio of orbital debris to breakup debris is: X P ðt; Lci Þ ¼ P A=M ðLci ; A=M j ÞP DV ðA=M j ; DV k ÞP dir ðDV k ; dirr Þ

P A=M ðLci ; A=M j Þ ¼ F A=M ðkci ; vj Þ

j;k;r

ð5Þ

S=C

DA=M ðkci ; vj Þ jvjþ1 vj j ¼P S=C j DA=M ðkci ; vj Þ jvjþ1 vj j

ð2Þ

where DS=C A=M ðkc ; vÞ is the area-to-mass distribution function for spacecraft collision fragments based on NASA’s standard breakup model, kc ¼ log10 ðLc Þ, and v = log10(A/ M) is the variable in the area-to-mass distribution. Similarly, delta velocity of debris can be divided into multi-section, DVk [DV(k), DV(k + 1)) (k = 1, 2, . . .). According to delta velocity distribution function, distribution probability of debris with DVk in the section of A/Mj, PDV(A/Mj, DVk), can be calculated. P DV ðA=M j ; DV k Þ ¼ F DV ðvj ; mk Þ DCOLL ðvj ; mk Þðmkþ1 mk Þ ¼ P DVCOLL ðvj ; mk Þðmkþ1 mk Þ k DDV

ð3Þ

where, DCOLL ðv; mÞ is the delta velocity distribution funcDV tion for spacecraft collision fragments based on NASA’s standard breakup model, and m = log10(DV) is the variable in the delta velocity distribution. In the same way, an equal-angle subdivided polyhedron grid, such as Icosahedron, can be adopted for debris delta vp is velocity velocity directions (see Fig. 2). In Fig. 2, ~ vt ;~ vector of target and projectile, respectively, and ~ vcm is the equivalent velocity vector of system in the case of conservation of momentum. According to delta velocity direction equal-angle distribution, distribution probability of debris with dirr (r = 1, 2, . . .) in the section of DVk, Pdir(DVk, dirr), can be calculated. P dir ðDV k ; dirr Þ ¼

1 maxðrÞ

ð4Þ

where j, k and r should meet the case, Lt (A/Mj, DVk, dirr) > t. Obviously, in this size range, the number of orbital debris is N ðt; Lci Þ ¼ N ðLci ÞP ðt; Lci Þ

ð6Þ

4. A case calculation and analysis USA 193 was an American military satellite launched on December 14, 2006, and its precise function and purpose were classified. The satellite malfunctioned shortly after deployment. On February 14, 2008, U.S. officials announced the plan to destroy USA 193 before atmospheric reentry, stating that the intention was ‘‘saving or reducing injury to human life”. On February 21, 03:29 GMT U.S. navy employed a SM-3 missile intercepting the USA 193 satellite successfully. The satellite weight is 2270 kg, circle orbit, and height 247 km. The interceptor weight is about 30 kg, flight velocity 2.6667 km/s. Their collision velocity reached 9.8333 km/s (Wikipedia, 2008). According to size distribution function, the number of debris in various size sections is calculated (see Table 1). It is shown that the number of debris rises rapidly as size decreases. As calculating debris cloud by the faster algorithm, we subdivide area-to-mass ratio into 103–102.5, 102.5– 102, 102–101.5, 101.5–101, 101–100.5, 100.5–100, 100–100.5, and 100.5–101, eight sections (include almost all debris area-to-mass ratio). At the same time, delta velocity is subdivided into 0–100.5, 100.5–100, 100–100.5, 100.5–101, 101–101.5, 101.5–102, 102–102.5, 102.5–103, 103–103.5, and 103.5–104, 10 sections (include almost all debris delta velocity). An equal-angle subdivided Icosahedron grid will be adopted for debris delta velocity directions. Supposing that survival height of debris is 100 km, atmosphere density refers to USSA76 model, drag coefficient CD = 2.2. At the time of post-breakup t, in various size ranges, the ratio of orbital debris to breakup debris and the number of orbital Table 1 The number of debris in various size sections from USA 193 test.

Fig. 2. Sketch map of debris delta velocity direction.

Debris size (m)

Debris number (piece)

Accumulative number (piece)

P1.0 0.5–1.0 0.1–0.5 0.05–0.1 0.01–0.05

33 76 1594 3870 81,784

33 109 1703 5573 87,357

1530


Table 2 Space debris ratio (unit: %) and number (unit: piece) from faster algorithm. Post-breakup time

Debris size (m) 0.01–0.05

0h 2h 1 Day 7 Days 30 Days 60 Days 180 Days

0.05–0.1

0.1–0.5

0.5–1.0

P1.0

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

100 16.30 9.18 4.05 0.51 0.12 0.00

81,784 13,331 7508 3314 417 95 1

100 10.25 4.49 2.46 0.36 0.13 0.00

3870 397 174 95 14 5 0

100 11.32 4.46 2.55 0.30 0.15 0.00

1594 180 71 41 5 2 0

100 8.97 3.27 2.10 0.28 0.14 0.00

76 7 2 2 0 0 0

100 11.26 3.96 2.71 0.37 0.18 0.00

33 4 1 1 0 0 0

Table 3 Space debris ratio (unit: %) and number (unit: piece) from general algorithm. Post-breakup time

Debris size (m) 0.01–0.05

0h 2h 1 Day 7 Days 30 Days 60 Days 180 Days

0.05–0.1

0.1–0.5

0.5–1.0

P1.0

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

Ratio

Number

100 17.66 9.55 4.08 0.94 0.56 0.45

81,784 14,443 7813 3335 766 455 364

100 10.10 5.09 2.33 0.44 0.23 0.16

3870 391 197 90 17 9 6

100 11.04 5.46 2.63 0.38 0.06 0.00

1594 176 87 42 6 1 0

100 9.21 3.95 1.32 0.00 0.00 0.00

76 7 3 1 0 0 0

100 12.12 3.03 0.00 0.00 0.00 0.00

33 4 1 0 0 0 0

debris are calculated by the faster algorithm. The results see Table 2. It can be seen from the result that 81,784 pieces of debris with 0.01–0.05 m size are induced by the collision breakup, and they will fall into atmosphere rapidly, orbital debris ratio and number are 16.30% and 13,331 pieces, respectively, in post-breakup 2 h, 9.18% and 7508 pieces in 1 day, 4.05% and 3314 pieces in 7 days, 0.51% and 417 pieces in 30 days, and almost no one lives in half year. Orbital debris with bigger than 0.05 m size can be generally surveyed by people nowadays, and 5573 pieces of debris with this size are from this breakup. Orbital debris ratio and number are 10.55% and 588 pieces, respectively, in post-breakup 2 h, 4.45% and 248 pieces in 1 day, 2.49% and 139 pieces in 7 days, 0.34% and 19 pieces in 30 days, and no one lives in half year. For validating the faster algorithm, we simulate the USA 193 satellite breakup event by general algorithm and compare their results. At the time of post-breakup t, in various size ranges, the ratio of orbital debris to breakup debris and the number of orbital debris are calculated by general algorithm. The results see Table 3. It can be seen from the result that 81,784 pieces of debris with 0.01– 0.05 m size are induced by the collision breakup, and orbital debris ratio and number are 17.66% and 14,443 pieces, respectively, in post-breakup 2 h, 9.55% and 7813 pieces in 1 day, 4.08% and 3,335 pieces in 7 days, 0.94% and 766 pieces in 30 days, and very little part lives in half year. 5573 pieces of debris with bigger than 0.05 m size are from this breakup. Orbital debris ratio and number are 10.37%

and 578 pieces, respectively, in post-breakup 2 h, 5.17% and 288 pieces in 1 day, 2.39% and 133 pieces in 7 days, 0.41% and 23 pieces in 30 days, and almost no one lives in half year. By comparing data, we can find that calculation results of two algorithms are very close, and difference between corresponding orbital debris ratios from two tables is less than 1% generally. So the presented faster algorithm is correct and effective. We write the calculation programs of faster algorithm and general algorithm by MATLAB, and perform calculation on a Pentium 4 computer. Table 4 shows calculation times of two algorithms in every size range. It is obvious that calculation time of general algorithm is effected by the number of debris, and rises accordingly. Whereas calculation time of our faster algorithm is independent of debris quantity, and is about 17 s. When debris size is in 0.1– 0.5 m, calculation times of two algorithms are pretty much the same thing, in 0.05–0.1 m, calculation time of faster algorithm is equal to 60.71% of general algorithm, but in Table 4 Calculation times of two algorithms. Debris size (m)

Debris number (piece)

General algorithm (s)

Faster algorithm (s)

P1.0 0.5–1.0 0.1–0.5 0.05–0.1 0.01–0.05

33 76 1594 3870 81,784

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Advances in Space Research Journal ~ Volume 43, Issue 10, Pages 1471-1594, 15 May 2009

Feminist Review: Journal, Issue 43

Advances in Cancer Research Volume 43

Advances in Botanical Research, Volume 43

Advances in Botanical Research, Volume 43

Manual Therapy Journal - Volume 10, Issue 2, Pages 93-174 (May 2005)

Advances in Genetics Volume 43

Advances in Food Research, Volume 15

Advances in Computers, Volume 43

Advances in Accounting Behavioral Research, Volume 10

Advances in Ecological Research, Volume 10

Advances in Cancer Research, Volume 10

Advances in Immunology Volume 43

ADVANCES IN CATALYSIS VOLUME 10, Volume 10

International Journal of Powder Metallurgy Volume 43 Issue 6

International Journal of Powder Metallurgy Volume 43 Issue 5

International Journal of Powder Metallurgy Volume 43 Issue 4

Advances in Schizophrenia Research 2009

Manual Therapy Journal - Volume 12, Issue 2, Pages 85-198 (May 2007)

Manual Therapy Journal - Volume 8, Issue 2, Pages 63-128 (May 2003)

Advances in Computers, Volume 15

Advances in Agronomy, Volume 15

Advances in Geophysics, Volume 15

Advances in Parasitology Volume 15

Manual Therapy Journal - Volume 9, Issue 2, Pages 59-120 (May 2004)

Manual Therapy Journal - Volume 11, Issue 2, Pages 91-166 (May 2006)

Advances in Immunology Volume 15

Advances in Parasitology Volume 15

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