Notes on Stellar Statistics II The Mathematical Expression of the Law of Tangential Velocities

VOL. 11, 1925

87

ASTRONOMY: W. J. L UYTEN

rabbits. Obviously it may, under certain conditions, pass the kidney barrier from which it must be supposed to circulate with the blood. No one has, as far as I know, detected the virus in the urine in man. I am not aware that it has even been searched for there. Although certain Italian investigators report having demonstrated it in blood taken from persons suffering from febrile herpes, the experiments, as reported, arouse distrust. But the finding of the virus, if rarely, in the cerebrospinal fluid of man and also, if very exceptionally, in the rabbit, and in the urine of the rabbit may be taken to show an occasionaf, at least, general or blood distribution of herpes virus in both species. Although the rabbit reacts to the presence of the virus in the blood with symptoms of severe encephalitis, the human beingisnotknown to suffer in any degree from this source. Other rodentsguinea-pig, rat, mouse-respond with fatal encephalitis to intracranial but not to intravenous injection of the virus, while birds and other mammalsmonkey, dog, cat-are unaffected even by direct inoculation of the virus into the brain tissues.

NOTES ON STELLAR STATISTICS. H1: THE MATHEMATICAL EXPRESSION OF THE LAW OF TANGENTIAL VELOCITIES By WILLEM J. LUYTIN HARVARD COLLIGI OBSZRVATORY

Communicated December 3, 1924

Of the velocity laws used in recent years, the form proposed by Schwarzchild, i.e., a normal error function in the logarithms of the velocities, has met with considerable success. Subsequently it has become necessary to make the geometric mean velocity in this law a function of the absolute magnitude. In view of the difficulty of grasping the physical significance of such a logarithmic law, it is well to bear in mind that this law is only a rough statistical description of the celestial phenomena. If the approximation is not carried too far the logarithmic law suffices to represent space as well as tangential velocities and total as well as corrected velocities. With more numerous and more accurate data it has been thought necesary in some quarters to apply slight modifications to the logarithmic law. The advantage gained in representation, however, is not enough to outweigh the greater difficulty in physical interpretation. On the other hand the suggestions of a limiting velocity or a "velocity restriction" have empha-sized again the importance of the Maxwellian law. It appears advantageous to follow the original suggestion made by Kapteyn and others to assume a limited number of Maxwellian functions. In -

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ASTRONOMY: W. J. L UYTEN

PROC. N. A. S.

accordance with the views stated above the simplest assumption that can be made appears to be that,of a finite number of isotropic distributions on each of which a drift is superposed. The first problem is to obtain the frequency function of the observed tangential velocities and for this it will be enough to treat but one of these Maxwell functions. If h is the constant of this distribution, V the speed of the drift, and X the angular distance between a certain infinitesimal area in the sky and the apex of the drift, we have the following expressions for the distributions of velocities directed toward the apex and of velocities at right angles to this direction:

exp.[-h2(V - vo)2]dv,

f(v)dv =

f('r)dr =

h\gexp.[-J2r2]dT.

(1)

Introducing the total tangential velocity T by: v = TsinO,

r =

TcosO

(2)

the frequency function of linear tangential velocities for all stars in the sky follows: /'r/2 2k2 dO A dX i(T)dT = TdT exp. [-h2(T2 + V2 sin2X-2TV sin X sin 0) ]sin X (3) or:

*(T)TdT = TdT X

exp.[-h2(T2 + V2 sin2X)] sinl^J o T

/~~~~~~~~~~~~2

exp. [2h2TVsinXsinOI.dO

The second integral is of the form:

F(x)= f

r/2

exp.(xsinO).dO

(4)

Comparing this with the definite integral expressing the Bessel function, or, still easier, differentiating equation (4) twice and writing a2F o 6X2= F (x) --1 -6F ~x 6x we see that the solution of (4) is given by F(x) = 7rJo(ix) = 7rJo(2h2VT sine NF=i) We then have:

¶(T)dT

=

TdT f 2h2 exp.[-h2(T2 + V2 sin2X)].

Jo(2h2VT sin V/ 1).sinXdX.

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ASTRONOMY: W. J. LUYTEN

89

= TdT/ 2h2 exp. [- 2(T2 + V2 sin2X) ].

(h2VT)2"

E

Writing cos X = u, this transforms into:

*I(T)dT = TdT j

2h2 exp. [-h2(T2 + V2 -V2u2)]. co

i

j (hj VT)2(-2) -j) .(1 U2)'du. -

In performing the integration it appears advantageous to put:

a = h2V2,

z = hT

Repeatedly integrating by parts and expanding into power series we obtain for the final answer:

*1(T)dT = 2k2e- h2(T2+ VI) TdT. 2

Ajz2j.

(5)

0

Aj

=

=

2 J aj

1+

L1.3.5.7...(2j +

a

1).O!

3.5...(2j+3).1! 57 .. (2+5).2!

I

These final expressions contain T and V only in the combinations hT and hV, accordingly, since h is connected with the mean peculiar velocity w by the equation h = 1/wV/r, 'I(T) gives us in explicit form the frequency function of the ratio between the tangential and the mean peculiar velocity, and contains only one parameter, the ratio between the drift velocity and the mean peculiar velocity. Using z and a exclusively in the right hand side of equation (5) we obtain: eaaiCiz2 2ezdzE' Aje az2j or: *I(T)dT = 2e-'2 In most instances we have to deal with valuesof a less than 1.27 (corresponding to V = 2w) in which case the series expressions derived for the coefficients Aj converge very rapidly. The term aJ makes the Aj very sensitive to small changes in a, accordingly it is desirable to leave this factor out, and to tabulate, not Aj, but Cj = Aj,/a, as a function of a, a procedure which will facilitate interpolation considerably. Table I gives the values of the first seven coefficients Cj for values of V/w between 0 and 2. The value V/w = Vir = 1.7724, is inserted, since for this particular value, a = 1, and the power series take on their simplest form. The tenth column gives the values of e 6; with these, and the val-

zdzE

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MA THEMA TICS: G. C. E VA NS

PROC. N. A. S.

ues for a shown in the second column, the coefficients of zi may be easily computed for any specific value of V/w. The coefficients A- satisfy the condition i j! = ea, necessary, in

order that I (T) may be a true frequency function, and f =

I(T)dT

1.

Having once obtained I (T) we can now calculate the arithmetic mean tangential velocity Ta, which will be purely a function of a = V2/rW2. We have:

Ta = f

T.A(T)dT =

f

a2

a3

2e"2z2dz. Aje -azJ.

Integration gives:

'rj /iFi

a

a4

2h a L1+ 1.3 3.5.2! +5.7.3! 7.9.4! + . J The values of Ta, expressed in terms of the mean value T,, when a = 0 (To = Vi/r/2h), are entered in the last column of Table I for different values in the argument a. In a subsequent paper we shall derive a similar expression for the uncorrected radial velocity Ra; and we shall then be able to make an immediate comparison between crude proper motions and crude radial velocities. TABIZ I CG.10 C1.103

CG.104

C..1016

6-a

Ta/To

0.705 0.711 0.717 0.727 0.739 0.756 0.776 0.778 0.801

0.256 0.258 0.260 0.263 0.267 0.272 0.278 0.278 0.285

0.658 0.661 0.665 0.672 0.680 0.692 0.705 0.706 0.721

1.0000 0.9235 0.8361 0.7274 0.6082 0.4886 0.3773 0.3682 0.2799

1.000 1.026 1.055 1.10 1.15 1.22 1.30 1.31 1.38

V/w

a

Co

Cl

C2

0.000 0.50 0.75 1.00 1.25 1.50 1.75 1.772 2.00

0.0000 0.0796 0.1791 0.318.3 0.4974 0.7164 0.9748 1.0000 1.273

1.000 1.027 1.058 1.117 1.195 1.308 1.454 1.463 1.653

0.667 0.678 0.684 0.712 0.745 0.784 0.832 0.835 0.896

0.133 0.135 0.136 0.138

0.127 0.128 0.129 0.132

0.143 0.148 0.156 0.156 0.164

0.135 0.138

0()143 0.143

0.149

ECONOMICS AND THE CALCULUS OF VARIATIONS BY G. C. EVANS Tun RPic INSTITUTT Communicated November 5, 1924

1. The writer is not the first to venture to state a general theory in mathematical terms of asubjectwhichisnotunfairlyregardedascompounded somewhat indefinitely of psychology, ethics and chance. Being more than amere mixture, however, it is equally fair to say that a separate analysis may apply; indeed, in economics we are interested in the body of laws or de-