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The gravitational instability of an infinite homogeneous medium when a coriolis acceleration is acting
344 GriNs, R . GOLD, T .
The gravitational instability of an infinite homogeneous m e d i u m . .
. .
GORTER, C. J .
. .
. .
.
. .
.
. .
.
. .
.
.
. . .
. . .
. .
. .
. .
.
.
.
.
1912 1952 1953 1947
Ann. Phys. (Leipzig), 37, 881. Nature (London), 169, 322. M.N., 112, 215. Paramagnetic Relaxation (Elsevier Publ.
1949 1950
Science, 109, 166. Publ. U.S. Naval Obs., IVashingt~n, 17
1949 1951 1949 1951 1954 1952
Science, 109, 165. Ap. J., 114, 241. Science, 109, 461. Ap. J., 114, 187. Ap. J., 119, 465. Bull. Harvard Coll. Obs., 921, 26.
Co., New York). HALL, J. S . . . . . . . . . HALL, J. S. and ~ I K I ~ S E L L , A. H . HILTNER, W. A .
.
.
.
.
.
. .
.
. .
.
. .
.
. .
.
.
.
SI'ITZER, L., J r . and TUKEY', J. W . . . . . . STRAXA~A:¢, G . ZIRIX, H . . .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
(Pt. I).
The Gravitational Instability of an Infinite Homogeneous Medium when a Coriolis Acceleration is Acting ~. C H A N D R A S E K H A R Yerkes O b s e r v a t o r y , U n i v e r s i t y of Chicago SUMMARY I t is s h o w n t h a t JEANS'S criterion for the gravitational instability of an infinite homogeneous m e d i u m remains unaffected even if the s y s t e m is p a r t a k i n g in r o t a t i o n a n d Coriolis forces are acting.
1. I~TTRODUCTION
O~E of the fundamental concepts in modern cosmogony is that of gravitational instability due to JEANS (1902). Its importance lies in the fact that it enables one to estimate the scale of the condensations which may take place in an extended gaseous medium ; and the possibility of estimating this will give precision to the ideas one may have regarding the formation of stars and of galaxies from an "original primeval nebula". To give only two examples of recent applications of gravitational instability and which at the same time illustrate the fruitfulness of JEANS'S concept, we may refer to SPITZER'S efforts (1951) to interpret and incorporate in a picture of the formation of stars, the highly irregular distribution of the interstellar matter and GAMOW'S speculations (1950) on the origin of the galaxies. The principle underlying JEAI~IS'Sestimate of the scale of the condensations which may take place in an extended gaseous medium is the following: starting from an initial state of homogeneity and rest, we consider the velocity of propagation of a wave of density fluctuation through the medium. I f the gravitational consequences of the density fluctuations are ignored, the problem is the classical one of the propagation of a sound wave; and, as is well known, the velocity of sound in a gaseous medium is independent of the wavelength and is given by c =
V(~'p/p)
.
.
.
.
.
(1)
where p denotes the gas pressure, p the density and ~ is the ratio of the specific heats. On the other hand, if the change 5 V, in the gravitational potential consequent to the
S. CHANDRASEKItAR
345
density fluctuation, 6p, is taken into account in the equations of motion in accordance with Poisson's equation, • V2~V ~ - - 47rG(~p .... (2) then the velocity of propagation of a wave is no longer independent of its wavelength, 2. It is given by Vg ~. ~/(c u -- Op2~/Tr) . . . . . (3) The velocity of wave-propagation, therefore, becomes imaginary when )" > ~ Ge =
~
....
= ~+ (say).
(4)
But an imaginary velocity of wave-propagation means, only, that the amplitude of the wave can increase exponentially with time. Accordingly, if an arbitrary initial perturbation in density is represented by a Fourier integral, then the amplitudes of the components in the Fourier representation which have wavelengths greater than AJ will increase exponentially with time; 2j is therefore a measure of the linear dimensions of a "condensation" which m a y take place in the medium. This is JEANS'S result. Now doubts have been expressed (cf. SPITZER) regarding the applicability of JEANS'S criterion (4) to a system partaking in rotation. And it has been stated that if ~:~ > ~Gp
. . . . (~)
where ~ denotes the angular velocity of rotation, then gravitational instability in the sense of JEANS cannot take place. I f this were true, the usefulness of the concept of gravitational instability would, indeed, be limited. Certainly, its applicability to the interstellar medium could be questioned. For, with the known angular velocity of galactic rotation ( ~ 10-15 per sec) the limit on the density imposed by (5) is p ~ 4 × 10-2a gm per cm ~.
.
.
.
.
(6)
Since the average density of interstellar space is appreciably less than this we might conclude t h a t gravitational instability offers or suggests no clue. However, an examination of the underlying dynamical problem shows t h a t no such inequality as (5) limits the applicability of JEA~s's criterion. It is the object of this paper to show this.
2. THE SOLUTION OF THE FROBLEM
Consider an extended gaseous medium which is partaking in a rotation with angular velocity ~ . The equations of motion and continuity governing the motions consequent to infinitesimal fluctuations in density (~p) and pressure (~p) are : 3v -P 3t and
grad 5p + p grad
(~ V +
3~ 5p = -- p div v .
2p v
.
×
~
.
....
.
.
(7) (8)
346
Tim gravitational instability of an infinite homogeneous medium
I f the changes in pressure and density are assumed to take place adiabatically, @ =
....
(9)
where c denotes the velocity of sound (equation (1)). In addition to the foregoing equations we have also Poisson's equation (2). We shall consider a solution of equations (2), (7), (8), a n d (9), which correspond to the propagation of a wave in the z-direction (say) ; for such solutions, 3 / 3 z is the only non-vanishing component of the gradient and the equations become 3~ + 2 ~ v ~ -- 2 ~ v ~ = o Ovu
3v~ 3t
c2 3
3
+
-
v
-
3
=
. . . . (10)
°
3v~
32
and
~~z 6 V + 47rGbp ~
O.
In writing these equations we have supposed t h a t the orientation of the co-ordinate axes has been so chosen t h a t ~ lies in the (y, z) plane and that, = (0, ~ ,
~z) .
.
.
.
.
(11)
For a solution which represents the propagation of a wave in the z-direction, 3 3 3-t = ice and 3 z - -
ik .
.
.
.
.
(12)
where co denotes the frequency and k the wave-number. Substituting (12) in equations (10), we obtain a system of linear homogeneous equations which can be written in matrix notation in the form : 2i~ z
co
-- 2i~
0
0
Vy
co
-- 2i~
0
0
0
Vx
0
÷ 2i~
co
- - c2k/p
k
Vz
(i
0
0
0
-- pk
o~
0
4~G
= 0.....
(13)
0 -- k 2
6V
The condition t h a t the foregoing system of equations has a non-trivial solution is t h a t the determinant of the m a t r i x on the left-hand side should vanish. On expanding the determinant, we find t h a t it can be reduced to the form . . . . (14)
CO4 __ ( 4 ~ 2 _~ ~'~j2)co2 _~_ 4 ~ z 2 ~ d 2 --_ 0
where (cf. equation (3)) =
-
4
cp) .
.
.
.
.
(15)
S. CHANDRASEKHAR
347
F r o m equation (14) it follows t h a t there are, in general, two modes in which a wave can be propagated t h r o u g h the medium. I f w1 and c% are the frequencies of these two modes, then (cf. equation {14)) 0)12 _~_ 0) 2 __~ 4~2 _{_ ~ j 2 and
0)1w2 -~- 2~z~'~ J .
. . . . (16) .
.
.
(17)
.
Hence i f ~ j is i m a g i n a r y , t h e n either eo1 or 0 2 m u s t be i m a g i n a r y . In either ease, there is a mode of wave-propagation which is unstable. Now the condition for ~ j to be imaginary is k s < 4~Gp/e~;
....
(1
s)
but this is precisely JEaNs's criterion. The condition for gravitational instability is, therefore, unaffected b y the presence of Coriolis forces. There is only one exception to this rule, namely, when the wave is propagated in a direction at right angles to the direction of ~ . In t h a t case ~= = 0, and equation (14) gives 0)5 = 4 ~ 2 -~
~-2j 2 =
4g2~ q- c2k ~ - - 4rrGp .
.
.
.
.
(19)
Accordingly, in this plane there is only one mode of wave-propagation ; a n d if the condition (5) should in addition be fulfilled, ms > 0 and gravitational instability cannot arise. This m a y explain how the inequality (5) came to be current. However, as we have already stated J E a x s ' s criterion for gravitational instability applies for wave-propagation in every other direction which is n o t at right angles to ~ .
REFERENCES GA•ow,
.
190"2 1929
Theory of Atomic Nucleus and Nuclear Energy. Sources, O x f o r d , p. 336. Phil. Trans. A., 199, 1 ; see p. 49. Astronomy and Cosmogony, C a m b r i d g e , see
.
1951
J. Washington Acad. S c i . , 41, 309.
G. a n d CRITCHFIELD, C . L .
JEANS, J . H .
.
.
.
.
.
.
.
.
1950
pp. 345-7. SPITZF.a¢, L .
.
.
.
.
.
.
.
.
The gravitational instability of an infinite homogeneous m e d i u m . .
. .
GORTER, C. J .
. .
. .
.
. .
.
. .
.
. .
.
.
. . .
. . .
. .
. .
. .
.
.
.
.
1912 1952 1953 1947
Ann. Phys. (Leipzig), 37, 881. Nature (London), 169, 322. M.N., 112, 215. Paramagnetic Relaxation (Elsevier Publ.
1949 1950
Science, 109, 166. Publ. U.S. Naval Obs., IVashingt~n, 17
1949 1951 1949 1951 1954 1952
Science, 109, 165. Ap. J., 114, 241. Science, 109, 461. Ap. J., 114, 187. Ap. J., 119, 465. Bull. Harvard Coll. Obs., 921, 26.
Co., New York). HALL, J. S . . . . . . . . . HALL, J. S. and ~ I K I ~ S E L L , A. H . HILTNER, W. A .
.
.
.
.
.
. .
.
. .
.
. .
.
. .
.
.
.
SI'ITZER, L., J r . and TUKEY', J. W . . . . . . STRAXA~A:¢, G . ZIRIX, H . . .
. .
. .
. .
. .
. .
. .
. .
. .
. .
. .
(Pt. I).
The Gravitational Instability of an Infinite Homogeneous Medium when a Coriolis Acceleration is Acting ~. C H A N D R A S E K H A R Yerkes O b s e r v a t o r y , U n i v e r s i t y of Chicago SUMMARY I t is s h o w n t h a t JEANS'S criterion for the gravitational instability of an infinite homogeneous m e d i u m remains unaffected even if the s y s t e m is p a r t a k i n g in r o t a t i o n a n d Coriolis forces are acting.
1. I~TTRODUCTION
O~E of the fundamental concepts in modern cosmogony is that of gravitational instability due to JEANS (1902). Its importance lies in the fact that it enables one to estimate the scale of the condensations which may take place in an extended gaseous medium ; and the possibility of estimating this will give precision to the ideas one may have regarding the formation of stars and of galaxies from an "original primeval nebula". To give only two examples of recent applications of gravitational instability and which at the same time illustrate the fruitfulness of JEANS'S concept, we may refer to SPITZER'S efforts (1951) to interpret and incorporate in a picture of the formation of stars, the highly irregular distribution of the interstellar matter and GAMOW'S speculations (1950) on the origin of the galaxies. The principle underlying JEAI~IS'Sestimate of the scale of the condensations which may take place in an extended gaseous medium is the following: starting from an initial state of homogeneity and rest, we consider the velocity of propagation of a wave of density fluctuation through the medium. I f the gravitational consequences of the density fluctuations are ignored, the problem is the classical one of the propagation of a sound wave; and, as is well known, the velocity of sound in a gaseous medium is independent of the wavelength and is given by c =
V(~'p/p)
.
.
.
.
.
(1)
where p denotes the gas pressure, p the density and ~ is the ratio of the specific heats. On the other hand, if the change 5 V, in the gravitational potential consequent to the
S. CHANDRASEKItAR
345
density fluctuation, 6p, is taken into account in the equations of motion in accordance with Poisson's equation, • V2~V ~ - - 47rG(~p .... (2) then the velocity of propagation of a wave is no longer independent of its wavelength, 2. It is given by Vg ~. ~/(c u -- Op2~/Tr) . . . . . (3) The velocity of wave-propagation, therefore, becomes imaginary when )" > ~ Ge =
~
....
= ~+ (say).
(4)
But an imaginary velocity of wave-propagation means, only, that the amplitude of the wave can increase exponentially with time. Accordingly, if an arbitrary initial perturbation in density is represented by a Fourier integral, then the amplitudes of the components in the Fourier representation which have wavelengths greater than AJ will increase exponentially with time; 2j is therefore a measure of the linear dimensions of a "condensation" which m a y take place in the medium. This is JEANS'S result. Now doubts have been expressed (cf. SPITZER) regarding the applicability of JEANS'S criterion (4) to a system partaking in rotation. And it has been stated that if ~:~ > ~Gp
. . . . (~)
where ~ denotes the angular velocity of rotation, then gravitational instability in the sense of JEANS cannot take place. I f this were true, the usefulness of the concept of gravitational instability would, indeed, be limited. Certainly, its applicability to the interstellar medium could be questioned. For, with the known angular velocity of galactic rotation ( ~ 10-15 per sec) the limit on the density imposed by (5) is p ~ 4 × 10-2a gm per cm ~.
.
.
.
.
(6)
Since the average density of interstellar space is appreciably less than this we might conclude t h a t gravitational instability offers or suggests no clue. However, an examination of the underlying dynamical problem shows t h a t no such inequality as (5) limits the applicability of JEA~s's criterion. It is the object of this paper to show this.
2. THE SOLUTION OF THE FROBLEM
Consider an extended gaseous medium which is partaking in a rotation with angular velocity ~ . The equations of motion and continuity governing the motions consequent to infinitesimal fluctuations in density (~p) and pressure (~p) are : 3v -P 3t and
grad 5p + p grad
(~ V +
3~ 5p = -- p div v .
2p v
.
×
~
.
....
.
.
(7) (8)
346
Tim gravitational instability of an infinite homogeneous medium
I f the changes in pressure and density are assumed to take place adiabatically, @ =
....
(9)
where c denotes the velocity of sound (equation (1)). In addition to the foregoing equations we have also Poisson's equation (2). We shall consider a solution of equations (2), (7), (8), a n d (9), which correspond to the propagation of a wave in the z-direction (say) ; for such solutions, 3 / 3 z is the only non-vanishing component of the gradient and the equations become 3~ + 2 ~ v ~ -- 2 ~ v ~ = o Ovu
3v~ 3t
c2 3
3
+
-
v
-
3
=
. . . . (10)
°
3v~
32
and
~~z 6 V + 47rGbp ~
O.
In writing these equations we have supposed t h a t the orientation of the co-ordinate axes has been so chosen t h a t ~ lies in the (y, z) plane and that, = (0, ~ ,
~z) .
.
.
.
.
(11)
For a solution which represents the propagation of a wave in the z-direction, 3 3 3-t = ice and 3 z - -
ik .
.
.
.
.
(12)
where co denotes the frequency and k the wave-number. Substituting (12) in equations (10), we obtain a system of linear homogeneous equations which can be written in matrix notation in the form : 2i~ z
co
-- 2i~
0
0
Vy
co
-- 2i~
0
0
0
Vx
0
÷ 2i~
co
- - c2k/p
k
Vz
(i
0
0
0
-- pk
o~
0
4~G
= 0.....
(13)
0 -- k 2
6V
The condition t h a t the foregoing system of equations has a non-trivial solution is t h a t the determinant of the m a t r i x on the left-hand side should vanish. On expanding the determinant, we find t h a t it can be reduced to the form . . . . (14)
CO4 __ ( 4 ~ 2 _~ ~'~j2)co2 _~_ 4 ~ z 2 ~ d 2 --_ 0
where (cf. equation (3)) =
-
4
cp) .
.
.
.
.
(15)
S. CHANDRASEKHAR
347
F r o m equation (14) it follows t h a t there are, in general, two modes in which a wave can be propagated t h r o u g h the medium. I f w1 and c% are the frequencies of these two modes, then (cf. equation {14)) 0)12 _~_ 0) 2 __~ 4~2 _{_ ~ j 2 and
0)1w2 -~- 2~z~'~ J .
. . . . (16) .
.
.
(17)
.
Hence i f ~ j is i m a g i n a r y , t h e n either eo1 or 0 2 m u s t be i m a g i n a r y . In either ease, there is a mode of wave-propagation which is unstable. Now the condition for ~ j to be imaginary is k s < 4~Gp/e~;
....
(1
s)
but this is precisely JEaNs's criterion. The condition for gravitational instability is, therefore, unaffected b y the presence of Coriolis forces. There is only one exception to this rule, namely, when the wave is propagated in a direction at right angles to the direction of ~ . In t h a t case ~= = 0, and equation (14) gives 0)5 = 4 ~ 2 -~
~-2j 2 =
4g2~ q- c2k ~ - - 4rrGp .
.
.
.
.
(19)
Accordingly, in this plane there is only one mode of wave-propagation ; a n d if the condition (5) should in addition be fulfilled, ms > 0 and gravitational instability cannot arise. This m a y explain how the inequality (5) came to be current. However, as we have already stated J E a x s ' s criterion for gravitational instability applies for wave-propagation in every other direction which is n o t at right angles to ~ .
REFERENCES GA•ow,
.
190"2 1929
Theory of Atomic Nucleus and Nuclear Energy. Sources, O x f o r d , p. 336. Phil. Trans. A., 199, 1 ; see p. 49. Astronomy and Cosmogony, C a m b r i d g e , see
.
1951
J. Washington Acad. S c i . , 41, 309.
G. a n d CRITCHFIELD, C . L .
JEANS, J . H .
.
.
.
.
.
.
.
.
1950
pp. 345-7. SPITZF.a¢, L .
.
.
.
.
.
.
.
.