A note on gravitational instability

Volume 3, number 7

PHYSICS

LETTERS

15 February 1963

p r o p o r t i o n a l (again b a r r i n g a c c i d e n t a l c a n c e l l a t i o n s ) feasibility of high-energy physics experiments with to a unique f u n c t i o n of 0, r j l ( e ) , which i s d e t e r m i n e d polarised photons. I would like to thank Professors by l , a s w e l l a s J. M e a s u r e m e n t of t h i s d i s t r i b u t i o n K. W. Ford and F. E. L o w for conversations. I a m w i l l . of c o u r s e , be f a c i l i t a t e d when the b a r y o n r e m u c h indebted to Professor Don Lichtenberg for s u , ~.g f r o m the e x c i t e d b a r y o n d e c a y i t s e l f d e c a y s correcting an earlier statement 5) that the differw e a k l y with p a r i t y n o n - c o n s e r v a t i o n , s e r v i n g a s an ence, eq. (5), depended upon I as well as J. a n a l y s e r of i t s p o l a r i s a t i o n in the s t a n d a r d m a n n e r . References The e d e p e n d e n c e of the f u n c t i o n s r j l ( 8 ) f o r J = ~, 1) R.Adair, Phys. Rev. 100 {1955) 1540. l =land2, and f o r J = ~ , l = 2 and 3 a r e g i v e n in 2) N. Cabibbo et al., Phys. Rev. Letters 9 (1962) 270. 3) Private communication of calculations performed by t a b l e 2. S. Berko. 4) G.Bardiellini et al., Phys. Rev. Letters 9 (1962) 396. The a u t h o r w a s s t i m u l a t e d to this i n v e s t i g a t i o n 5) Talk given at the Eastern Theoretical Physics Conf. b y P r o f e s s o r S. B e r k o ' s c o m m e n t s on the p o s s i b l e (Univ. of Va. 1962). * * * * *

A

NOTE

ON

GRAVITATIONAL

INSTABILITY

*

R. L. L I B O F F Courant Instituteof Mathematical Sciences, and Physics Department, University Heights **, New York University Received 19 January 1963

In 1902 J e a n s I) e x a m i n e d the s t a b i l i t y of an i n f i n i t e s y s t e m of m a s s p o i n t s i n t e r a c t i n g u n d e r a N e w t o n i a n force• The m a c r o s c o p i c a n a l y s i s i n d i cated t h a t the s y s t e m was u n s t a b l e and, f u r t h e r m o r e , the growth r a t e of the i n s t a b i l i t y i n c r e a s e d with the w a v e l e n g t h of the p e r t u r b a t i o n . I n t e r e s t in the m i c r o s c o p i c f o r m u l a t i o n of t h i s p r o b l e m h a s r e c e n t l y b e e n r e v i v e d 2 - 4 ) , and, in a d d i t i o n to the J e a n s r e s u l t one a l s o f i n d s that the d i s t u r b a n c e v a n i s h e s f o r w a v e l e n g t h s l e s s t h a n *** d, w h e r e d2 -

which, in turn, is realised in the limit of large wavelength perturbations. If the Vlasov-Boitzmann equation is linearised about a Maxwellian distribution and related N e w tonian force field, and solutions to the subsequent equation are sought, of the form, exp[-i~t+ik.x], then one obtains the dispersion equation 3,4) (for longitudinal fluctuations),

k2 = f -oo

C2 tvmtlo

v ~pdv -~ F(Z) , 13

-

Z

1

= (2~) -~ e-~ v 2

,

The equilibrium thermal speed is C, n o is the e q u i l i b r i u m n u m b e r d e n s i t y , m i s the m a s s of the e l e m e n t s , and e, i s the f o r c e c o n s t a n t , in t e r m s of which N e w t o n ' s law i s w r i t t e n ,

V . 0 = - otmn.

(I)

T h e f o r c e f i e l d i s O. In t h i s note we w i l l f o r m a l l y r e c a p t u r e , t h r o u g h a m i c r o s c o p i c a n a l y s i s , J e a n s ' r e s u l t that the m a x imum growth rate is ~o -1, where ~ o2 = ~mno ,

(2)

* The work presented in this paper was carried out at the Courant Institute of Mathematical Sciences, New York University, under contract AT(30-1)-1480 with the United States Atomic Energy Commission. ** Permanent address. *** d is the analogue of the Debye distance 4,5) in plasmas,

322

z = ~/~

(3) (4) (5)

The n o n d i m e n s i o n a l wave n u m b e r k i s w r i t t e n f o r

kd as the n o n d i m e n s i o n a l f r e q u e n c y ~ i s w r i t t e n f o r (cO/~o).

It is easily shown 4) that Im F > 0 in the first quadrant of the z-plane (including the real positive axis), Irn F < 0 in the second quadrant (including the negative real axis), and Im F = 0 along the imaginary axis. It follows that all of the unstable roots of (3) (i.e., roots lying in the upper half plane) lie along the imaginary axis, or equivalently for

z:i~,

~--[~[.

F o r t h e s e v a l u e s of z, F a s s u m e s the f o r m

(6)

Volume 3, number 7

PHYSICS #

f(i~) =

15 F e b r u a r y 1963 3

v2_~ dv v2 + #]2 "

-~

LETTERS

(7)

If k 2 i s r e w r i t t e n a s k 2 = ~2/#]2 ,

(8)

It f o l l o w s t h a t f o r l a r g e #] both k2 and t h e f i r s t d e r i v a t i v e of k2 a r e l a r g e r than t h e r e l a t e d v a l u e s of F ff ~2 > 1, w h i c h , in d i m e n s i o n a l f o r m , a p p e a r s a s

w h e r e v i s the g r o w t h r a t e 00 = i v , t h e n (3) a p p e a r s as

v2 ~ ~2/#]2 = -oo v 2 + dv#]2

(9)

"

Both s i d e s of (9) a r e p l o t t e d in fig. 1.

__

\

~

F(#)

\%, # Fig. 1. T h e i n e q u a l i t i e s w h i c h a r e d e p i c t e d in fig. 1 a r e o b t a i n e d b y e x a m i n i n g the a s y m p t o t i c b e h a v i o u r of F f o r l a r g e #], a c c o r d i n g to w h i c h

REGGE

V>

0~0 ,

w h e n c e t h e r e a r e no s o l u t i o n s f o r v > o~o and ~Oo r e p r e s e n t s the m a x i m u m g r o w t h r a t e of t h e s e i n s t a b i l i t i e s. A s a c o n c l u d i n g r e m a r k we note t h a t a l l of t h e remaining modes are damped. The exact forms are o b t a i n e d b y c o n t i n u i n g F a n a l y t i c a l l y into t h e l o w e r half z-plane. The only physically relevant fact a b o u t t h e s e d a m p e d m o d e s i s t h a t s o m e of t h e m p r o p a g a t e ( v i z . , the r o o t s t h a t l i e off t h e i m a g i n a r y axis). References 1) J . J e a n s , Phil. Trans. A 199 (1902) 49. 2) D. Lynden-Bell, Monthly Notices Roy. Astron. Soc. 124 (1962) 279. 3) R.Simon, Bull. Acad. Roy. Belg. s e r i e s 5, 47 (1962) 7. 4) R. L. Liboff, Gravitational instability and one-component plasma oscillations (New York University, NYO9754, November 1962). 5) P.Debye and E.Htlckel, Phys. Z. 24 (1923) 185.

POLE HYPOTHESIS AND POLARISATIONS IN yp AND Kp SCATTERING H. 0 B E R A L L * Harrison M. Randall Laboratory of Physics, University of Michigan, Ann Arbor, Michigan Received 2 January 1963

R e g g e p o l e h y p o t h e s i s 1) p e r m i t s one to o b t a i n the a s y m p t o t i c b e h a v i o u r of d i f f e r e n t i a l and t o t a l c r o s s s e c t i o n s in the s c h a n n e l f o r high e n e r g y , b y a s s u m i n g p a r t i c l e s c o r r e s p o n d i n g to the t r a j e c t o r i e s in the C h e w - F r a u t s c h i d i a g r a m 2) b e i n g e x c h a n g e d in the t c h a n n e l of the s a m e r e a c t i o n . T h e high e n e r g y b e h a v i o u r of v a r i o u s t o t a l c r o s s s e c t i o n s w a s o b t a i n e d in t h i s w a y b y U d g a o n k a r 3). G r i b o v a n d P o m e r a n chuk 4) have p o i n t e d out t h a t p o l a r i s a t i o n s d e p e n d on i n t e r f e r e n c e s b e t w e e n v a r i o u s e x c h a n g e d t r a j e c t o r i e s , t h u s p r o v i d i n g a d d i t i o n a l i n f o r m a t i o n on t h e t r a j e c t o r i e s and t h e i r c o u p l i n g s . P o l a r i s a t i o n s w e r e o b t a i n e d b y H a r a 5) f o r NN s c a t t e r i n g . In t h i s p a p e r , e x p r e s s i o n s f o r the p o l a r i s a t i o n s a r e p r e s e n t e d f o r ~N and KN, KN s c a t t e r i n g , t o g e t h e r with d i f f e r e n t i a l and t o t a l c r o s s s e c t i o n s . U s i n g the g e n e r a l f o r m of the s c a t t e r i n g m a t r i x f o r a z e r o - s p i n and a D i r a c p a r t i c l e 6), we o b t a i n f o r t h e d i f f e r e n t i a l c r o s s s e c t i o n , a s y m p t o t i c a l l y f o r s ~ oo a t ~ 4~-5

]A + ~

sl

-

]AI2

'

* Supported in part by the Office of Naval Research, U.S. Navy. 323