I N HIS C LAS SIC A L investigations on the dynamics of elliptical galaxies, Sir James Jean has said, "We have seen that the density of matter in the central lenticular masses of nebulae is of the order of 10-21 g/cm3• The free path in a gas of this density is 1014 cm., whereas the diameter of the central mass of Andromeda nebula is about 1.6 X 1021 cm .... It follows that the various nebular configurations may legitimately be interpreted as configurations of masses of rotating gas."1 He further calculates the free path of a star, allowing for its gravita-tional interactions with other stars of the cloud, and finds that for the same mean density it is equal to 10:.l9 cm.
(about 50,000,000 times the diameter of the nebula). He concludes that the concept of gas pressure cannot be legi-timately used in connection with nebular dynamics when the nebula is supposed to be a cloud of stars, and that clouds of stars should not assume the special shapes of observed nebulae.
Today we know that the elliptical galaxies and the cen-tral bodies of the spirals are made up entirely of stars.
This fact, first suggested by the stellar type of their spec-tra, was established beyond any doubt by the recent work of Baade2 who was able to resolve the celestial objects into a multitude of individual stars.
To -reconciliate Jeans' theoretical consideration with the observed fact, George Camow suggests that the regular shapes of elliptical galaxies were established in some past epoch when they were entirely gaseous, and are now re-tained as "dead skeletons" after all the original gaseous material was condensed into stars. This sttggestion agrees with the recent theory of the evolution of the expanding universe proposed by Dr. Camow,3 according to which the masses and sizes 0 f galaxies can be predicted on the
as-67
sumption that they originated as the result of "gravita-tional instability" of the expanding primordial gas. These gaseous clouds would tend to assume spherical or ellip-soidal forms depending upon the amount of angular mo-mentum, and their internal density distribution was pre-sumably that of the rotating isothermal gas-spheres. The linear velocities of gas-masses at each point were pro-portional to the distance from the rotation axis. As the star forming condensations took place under the forces of gravitation and radiation pressure, the net outward force due to the difference in gas (and radiation) pressure be-tween the surfaces toward and away from the center of the galaxy was reduced as a result of the reduction in the volume and surface area. Consequently, the stars were accelerated toward the center under the influence of un-balanced gravitational force, assuming that at no time dur-ing the history of the galaxy do the stars exchange any appreciable amount of energy with their immediate neigh-bors as a result of close encounters.
The only overall forces acting on a star will be: 1. The resistance of the gas to its motion, which will be a radial force as long as the motion is radial and which will dimin-ish in importance as the gas is consumed in the star forma-tion process. 2. The radial force exerted by the smoothed gravitational potential of the remainder of the stars. Then, the stars will tend to oscillate radially through the center of the galaxy. The amplitude of oscillation or the points of maximum elongation of the newly acquired elongated elliptical orbits of the stars should correspond to the dis-tances at which they were originally formed. As more and more stars were formed, the major axes of original ellip-tical orbits were gradually changing due to the gravita-tional action of other stars which have originated outside
68
of them, but are now penetrating during a certain fraction of their motion into the interior regions. As a result of these interactions, the stars acquired an altered distribu-tion. It is expected, however, that the stellar orbits were not shuffled; i.e" the stars which were formeda-t larger distances from the center and therefore with larger angu-lar moments' will also have their points of maximum elon-gation farther out from the center. In observing mean radial velocity of stars at different distances from the original rotation axis, with respect to an observer, we will essentially obtain the tangential velocities of those stars which pass through their point of maximum elongation at that distance. Since the stars whiCh move beyond that point will be much smaller in number and will also present us only with the projection of their actual velocity, it can be expected that the observed rotational velocities will in-crease with the distance from the center (linearly in the first approximation), giving the impression that the entire galaxy is rotating approximately as a solid body. This
"solid body" rotation is actually observed in elliptical galaxies and in the central bodies of spirals, and was con-sidered an unexplainable phenomena in view of the large free passes of individual stars.
In the present work we. plan to analyze the observed density distribution in the elliptical galaxies which was very carefully measured by Dr. Hubble.'
Computation of the initial density distribution in a spiral galaxy before formation of stars:
Let p (r) dr
=
the present mass in the shell between radii rand l'+
drf3(a)da = the initial mass between the shells of radii a and a
+
daper)
=
the present potential at r.A stellar mass starting at a and dropping inward t.o r will lose the potential energy
1'1'llP(a) -.; P(r)]
=
11tV2/2 and hence acquire the velocityv = y'2[P(a) - per)] .
The time which the star will spend in the shell of thick-ness dr will be dr
Iv
(on the inward trip), and the frac-tion of all its time which it spends in this shell will be dr/uT(a), where T(a) is the time required for the mass to drop from a to the center. The contribution to the mass observed now between rand r+
dr arising from the mass which started to drop from between a and a+
do. is,8(a) da dr T(a)v
SCIENTIFIC COM PUT A T I ON
Then the total mass between rand r
+
dr is ( )d _r
R p(a)da drp r r -
Jr
T(a)v .Therefore
_~
r
R pea) dapCr) = y'2Jr T(a) [pea) -- P(r)]I/Z'
dA 1
= l M ·
-da a
dQ 1
- = ..
~:f·--dr r
M
=
modulus=
.434 thenpea)
=2y'2. M T(a)
~[S~~(r) -
pea) dPd(g) dQJ. (1)7r a . dA A.
For integration, the Gregory formula is suggested, since the errors of integration are readily estimated:
IJ.a+nw (1 . 1)
W
af(x)dx= 2
fo+
fl+
f2+ ... +
fn-l+
2,fn .1 , 'f ) 1 (A iif . A iif ) - 12 (I::::.l!n_l - 1::::.1 0 - 24 W n-2 - W 0
19 (A
iiif
A iiif' ) _ - 720 W n-3 - W ,0 •••For obtaining derivatives near the end of the tabular se-quence we use the following:
'lV/'(a) = I::::.if(a) -
~
I::::.iif(a)+ j
I::::.iiif(a)-*
I::::. iv f (a)+ ...
Away from the ends of the tabular sequence the following central difference formula converges more rapidly:
wf'(a) =
~
(I::::.ia 112+ l::::.ia1/Z)1 ( A ". A i " )
- 12 W llla 1/~
+
W Ila1/2+
~O
(I::::. va_11:l+
I::::. va1/2) - 2!0 (I::::. vila_liz+
I::::. viia1/2 )+ ...
By definition
T(a) '"
f ~
=~2f
[Pea) ! rp(r)]lI'dr 1 r r
Q
=
log1or ;dQ=
dQ/dr=
M=
.434 T a _ _ 1 __r-a
rdQ( ) - My'2
JA
[Pea) - Per) ]1/2 (2)Near r = a, P (r) becomes nearly equal to P (a) and the integrand diverges. To overcome this a series approxima-tion in the neighborhood of r = a is suggested.
Let h = (a - r). Hence the first term is
~/-2-
-a'JMaX
yh
(3)For small a the first term approaches zero, and in the second we may set 3Ma
=
%-a3 p (0), and pea)=
p(O).Let h
=
a; then to the first approximation 1 ~I
1TCa)
= 4'J 6rrG p(O) . (4) The third term of the series should be calculated if greater accuracy is desired.The equations (1), (2), (3) and (4) with which we are here concerned, lend themselves readily and very con-veniently to solutions on standard IBM equipment. It is gratifying that a computing laboratory of this nature is available for these investigations. Although to date no con-clusive results are available, a more thorough investigation
of the problem is possible. It is the hope of the authors that in the near future they will be in a position to publish results of these investigations which will conclusively de-cide whether this theory is valid.
REFERENCES
1. SIR J. H. JEANS, Astronomy and Cosmogony (Cambridge, 1928), p. 335.
2. W. BAADE, "The Resolution of . . . the Central Region of the Andromeda Nebula," Astrophys. J., 100 (1944), p. 141.
3. G. GAMOW, "The Origin of Elements and the Separation of Galaxies," Phys. Rev., 74 (1948), pp. 505-6.
4. E. HUBBLE, "Distribution of Luminosity in Elliptical Nebulae,"
Astrophys. J., 71 (1930), pp. 231-76.
DISCUSSION
Mr. Hollander: Do we have any proof that stars are moving radially, either in or out?
Mr. B elzel': It would take very many years to notice any motion in the stars. I think that according to Dr. Gamow's theory of the evolution of the universe, in a period of about 200,000,000 years the stars might have made seven or eight oscillations.