LETTER~. AL NUOV0 CIMENTO

VOL. 39, N. 9

3 Marzo 1984

On the Gravitational Red-Shlft of Galactic Halos. A. MALAGOLI and R. RUFFI.NI Department o] Physics (~G. M a r c o n i , Universith , L a Sapienza ~ - l~oma

(rieevuto il 30 Settembre 1983) PACS. 98.50. - The Galaxy, extragalactic objects and systems.

S u m m a r y . - Limits are established for the gravitational red-shift to be expected for galactic halos. The possibility that collapsed galactic halos have relevance in astrophysical systems is also advanced.

A variety of authors have discussed the fundamental role t h a t galactic halos of bosonic or fermionic particles can play in the process of the formation (1) and in the morphology and stability of galaxies (2). Direct experimental evidence for the existence of such halos comes mainly from the rotation curves of galaxies (3), while we can deduce potentially important information on the detailes of the density distribution profile from the dynamic of globular clusters and dwarf galaxies (4). I n this letter, we would like to explore an additional and independent piece of evidence for the existence of these halos, namely the effect which the gravitational field of the massive halo has on the frequency of an electromagnetic signal going into (blueshift) or leaving (red-shift) a galaxy. I t is well known that limits can be set on the extreme values of the gravitational reM-shift to be expected at the surface of a selfgravitating field. I n a classical paper BOND1 (5) has shown that the m a x i m u m red-shift at the surface of a self-gravitating perfect fluid with equation of state p = p(~) is z = 0.615, as long as the following conditions are fulfilled: [SP\ 89

c, = ~ - ~ ) < c

and

P

--~

1

where c, is the sound velocity and c is the light velocity.

(1) GAG GIAN GONG, t~. RUFFXNX: P h y s . Lett. B , 97, 388 (1980); 1~}, 47 (1981); R . FABBRI, R . JANTZEN and R . RUFFINI: ~4stron. ~ p p l . , 144, 219 (1982). (2) :M. P~. BALDESCHI, G. B. GELMII~I a n d R . RLTFINI: P h y s . Left. B , 122, 221 (1983). See also ref. (*). (') See e . g . V . R U m N : S p i r a l galaxies and the masses o] the Universe, in Gamow Cosmolgy, Proceedings of the L X X X V I E. F e r m i S u m m e r School, e d i t e d b y F. MEI,CHIORrtI a n d R . RUFFINI (1982). (~) R . FABBRI a n d R . RUFFINI: ~Islrophys. S p a c e S c i . , 82, 2~9 (1982) R . RL'FFINI (1983}, s u b m i t t e d t o P h y s . Lelt. (1) H . ]3ONDI-" Pr0C. R . Soe. Lo/l~on, S e t . A , 281, 39 (1964).

177

178

i.

,SIALAGOLI

and

R. RUFFINI

V~Pnat makes t h e analysis of a galactic halo p a r t i c u l a r l y interesting, h o w e v e r , is t h e fact t h a t the optically significant p a r t of a galaxy, t h a t e m i t t i n g or receiving e l e c t r o m a g n e t i c signals, is e m b e d d e d well w i t h i n a galactic halo. The radius of the c o m p o n e n t of visible m a t t e r of a g:flaxy is e x p e c t e d to h a v e a radius about o n e - t e n t h t h e radius of the (galactic) halo (see for e x a m p l e the model discussed by CROLLALANZA et al. (G)). U n d e r these conditions we arc in a situation quite similar to t h e one considered by IIoYLE and FOWLER (7). In this case it has been p u r p o r t e d t h a t any v a l u e of z could be a t t a i n e d , but it will become clear t h a t the existence of a critical mass against gravitatim~al collapse can impose additional constraints on the g r a v i t a t i o n a l shift from the inside of a self-gravitating configuration. The ratio between the frequency rob, m e a s u r e d by an observer m o v i n g in a g r a v i t a tional field w i t h four-velocity u,,%, and the frequency v,.~. m e a s u r e d by an e m i t t e r m o v i n g with four-velocity u,~.~, is simply given by

.......

(1)

v. . . .

7-

- ,

(uaKa)t.~a.

where K '~ is the p a r a l l e l - t r a n s p o r t e d w a v e v e c t o r e v a l u a t e d at the p o i n t of emission and observation. In order to e v a l u a t e the c o n t r i b u t i o n due only to t h e g r a v i t a t i o n a l field, we assume in (1) both t h e e m i t t e r and t h e observer at rest w i t h u a = 0, a - - - 1 , 2 , 3 , u ~ 1. T h e n eq. (1) becomes (2)

1 -t- z

=

v~176 v ....

--

(goo).t.m. (goo)~. "

In ordcr to give a q u a n t i t a t i v e e s t i m a t e of eq. (2), wc idealize the s t r u c t u r e of the halo and t h a t of t h e visible p a r t of the galaxy by means of two spherically s y m m e t r i c d i s t r i b u t i o n s of m a t t e r of c o n s t a n t d e n s i t y 01 and 0~- W e have the density profile

I

~l,

i~ ~ R I ,

q(R) = ~ q2,

.RI < .R < R2 ,

[0,

R >/~.

The gee c o m p o n e n t of the m e t r i c for such a configuration is given by 3

(a)

goo(1r =

[C~(/Cl}

, for O < ~ < R ~ ,

C~O(R1)z(R)

~

2

P

R~
(t) A. CROLLALANZA,GAO G]AN GONG and R. ]:~UFFI.~I: Left. Nuovo Cimenlo, 32, 411 (1981). (D F. t{ovLl~ and %V. A. FOWL~:Ir Nature (London), 213, 373 (1967).

179

ON THE GRAVITATIONAL RED-SItIFT OF GALACTIC HALOS

where

2GM

4/$

M = -~- [Ot(Ba)s + q,(R~ - - / ~ ) ] ,

C2

B 9"2

~(1~)=

1

R

01 = C, + ( e l - - o , ) B s ,

TI

'

it--rVr'--r"l(r.)~]t'

e,O(R,),

5'z=

(r~)~]

'

30(R,) z(R,~]. Bz= ~t(R1) 1- 2 (rs)2

If we assume t h a t for a g a l a x y similar to our own t h e visible c o m p o n e n t has a mass M ~ . -~ 1011 M o and a r a d i u s R,~. ~_ 10 kpc and t h a t t h e halo c o m p o n e n t has a mass MH.LO --~ 10 • Mvi. and a r a d i u s RHALO~-- 10R,M., t h e n t h e light e m i t t e d from t h e surface of t h e visible m a t t e r d i s t r i b u t i o n will suffer a g r a v i t a t i o n a l red-sfhit of a b o u t z = = A~/), ~_ 10 -6. Conversely a blue-shift will be i n t r o d u c e d by t h e halo of t h e g a l a x y in t h e m e a s u r e d frequencies of l i g h t f r o m e x t r a g a l a c t i c sources. I n d e e d t h e effect m a y be considerably larger if t h e halo is m o r e c o m p a c t , b u t in t h i s case n a t u r a l l i m i t s are i m p o s e d by t h e existence of a critical mass for t h e onset of t h e g r a v i t a t i o n a l collapse of a system of self-gravitating bosonie or fermionic particles. T h e v a l u e of t h i s mass for a halo of f e r m i o n s can be deduced f r o m t h e general r e l a t i v i s t i c t r e a t m e n t c a r r i e d o u t by ()Pe~.mt~IM~.R and VOT,KOFF (6) for a s y s t e m of fully degenerate p a r t i c l e s and is g i v e n b y

itCireRrr = ( r

'mT

where mp = (~c/G)t is t h e P l a n e k mass and ms t h e mass of t h e particle. I n a similar w a y RUFFX~I and BONAZZOLA (~) find, for a s y s t e m of bosons,

MERIT ~ 0.6

ma.

I n b o t h cases we see t h a t t h e m a x i m u m r e d - s h i f t at t h e centre is z ~ 0.31 and at t h e surface is z ~ 0.13. We can, therefore, conclude t h a t noneollapsed halos of such selfg r a v i t a t i n g particles can account for t h e a n o m a h m s red-shifts observed in some galaxy clusters (lo), w h e r e t h e difference in t h e red-shifts of some spectral lines can be a b o u t A z ~ 10-2 (n). L e t us t u r n now to t h e possibility of collapsed halos.

(8) (') (ta) (1,)

J'. R. G. A.

R. OPPENItEIVI'ER, G. M. VOLEOI;'F: Phys. /~v., 55, 374 (1939). RU~TINI and S. BONAZZOLA: Phys. Rev., 187, 1767 (1969). I4. FIELD, H. ARP and J. N. BAIICALL: The Red.shift Controversy (Reading, Mass., 1973), nOmSENBEaG: Phys. Scr., 17, 205 (1978).

].80

A. MA.LAGOLI

and

R. R U F F I N I

It is clear that at any z the systems of massive self-gravitating fermions or bosons originating in the eosmological expansion c'tn form bound systems if their mass is larger ~h'~n the corresponding value of the Jeans' Mass (~). These bound systems can undergo an indefinite gravitational collapse if no equilibrium configuration exists. This situation can be relevant if massive leptons of the kind considered by L ~ and W~I.~B~RG (~2) exist in nature with masses ~nL ~ 2 GeV (see rcf. (13)). V~'ith all duc reserve, therefore, we advance the view t h a t collapsed objects with masses larger than the above critical values and endowed with angular velocity may be the energy source of quasars, via the usual energy formula (14). Their formation would occur by the Jeans instability process. The gravitational red-shift distribution around such a collapsed halo will not bc subject to the above constraints and will be given by the well.known formula for the region outside the horizon of a black hole (~5). Details are going to appear elsewhere (~s).

(~z) B. W. LEE a n d S. ~,VElh'BERG: P h y s . Rev. Lelt., 39, 165 (1977). (~a) ]~.. RUFFII~'I: in Proceedings o! the F i r s t Equalortal School on Belativistlc Astrophysics, Bogof~, Colombia, 1984. (3,) D. CHR1STODOITLOU a n d R. RUFleINI : P h y s . Bey. D, 4, 3552 (1971). (~*) :R. }~.ur~'I,~1: P h y s i c s Outside the Horizon o/ a B l a c k Hole. in P h y s i c s and Astrophysics o] N e u t r o n Stars and Black Holes, Proceedings o! ihe L X ~ z E . F e r m i S u m m e r School, e d i t e d b y R. RUFFINI a n d R. GIACCO.~I (1978). (~o) A. MALAGOLI a n d R. R~TF[h'X: (1983) in p r e p a r a t i o n .