Formation and structure of neutrino astronomical objects

Pergamon Press. Printed in Great Britain 0275-1062/81/040377-04$07.50/O

Chin.Astron.Astrophys.2 (1981) 377-380 Act.Astron.Sin.

-22 (1981) 207-212

FORMATION

AND STRUCTURE

OF NEUTRINO

ASTRONOMICAL

OBJECTS

LU Tan f?an jirag University, LUO Liao-fu Neimenggu University, Xebei CoZlege of Engineering. YANG Gou-then Received 1980 October 9

ABSTRACT Neutrinos with non-zero mass could gather to form a new kind of astronomical bodies: the Neutrino Astronomical Objects (NAO). We have investigated the mechanism of their formation and the relation of this formationtothat of the galaxies, ascertained their e, p, He4 content, whose presence should produce a series of observable effects. NAOs are a peculiar kind of heavenly bodies with many new properties. They have a linear size of the order of 100 pc, a total neutrino content of the order of lo"& and an e , p, He4 content of the order of lO*M,.

The question of whether the mass of neutrino is zero or non-zero is important both from the particle physics and the astrophysics viewpoints. For example, one way of solving the wellknown riddle of the solar neutrino is to give the neutrinos a non-zero mass. Recently, two types of experiments on neutrino mass have aroused interest. One is the measurement of the 8-spectrum of light nuclei, where it was found that, /I/, ??zur - 34rt:4ev. (1) The other is the study of the phenomenon of neutrino oscillation, /Z/, which showed mrrm mPp = m,. (21 Despite their crude and preliminary nature, these results have stimulated considerable response. At least, people have begun to take seriously the possibility of a non-zero neutrino mass. It is highly probable, that the neutrino mass is not zero, but its exact value may still differ somewhat from that in (l), probably to the smaller side. Accordingly, our calculations will use mv= 34eV for the sake of concreteness, but we shall also consider the ease of a general mv, and particularly the case of mu< 34eV. There is a great abundance of neutrinos in the universe /3/. As long as the neutrino mass is not too small, many interesting astrophysical phenomenon must necessarily follow.

If neutrinos have a mass that is not too small, then this must lead to the existence of a new type of celestial bodies, analogous to neutron stars 141. We shall call these Neutrino Astronomical Objects or NAO for short. We reported on this question at the Symposium on Cosmic Ray and High-Energy Astrophysics, held in Qingdao, 1980 August 5-10. We can use a method similar to Oppenheimer-Volkoff'sto calculate the upper limit of the mass of an NAO and the corresponding radius. For a completely degenerate neutrino gas, we find Me) = K-'R(m./m,yM$$,

(3)

Rb"'p PR(m,/m,YRb":,

(4) where suffix OF refers to original Oppenheimer-Volkofflimits /5f, and K(=6) is the number of 100 pe. neutrino species fzis,i?a,Up, 3~~ 2%. iir!. For mu= 34 eV we have A4$')=2XlOl'M,,Rp)"'Hence we can re-write (3) and (4) as M6"b 2 X 10"(34/m.(cV)YMe, (3-l) (4-l) RP)L~ 100(34/m,(cV)Yp% We see that NAO's are big and massive bodies, with masses on the order of clusters of galaxies.

378

Neutrino Astronomical Objects

Obviously, with such huge masses, NAOs cannot be pure neutrinos, they must contain a quantity of other material gases (e, p, He&) and stars. Observations of their gases and stars should provide information on the NAOs. As an example, according to (3) and (4), the redshift of a spectral line emitted at the surface of an NAO is Z‘ -

- 1 = 0.13.

i

and this value is independant of the neutrino mass. Since the neutrino gas is transparent to photons, we should also see lines emitted in the interior, with redshifts greater than 2,. For the homogeneous model of NAO, the redshift from the centre of 2, will be greater than 2s by a factor of 1.5; for a non-homogeneousmodel, 2, will be greater still. Such non-uniform redshifts arising from various parts of an NAO are an important feature for NAO identification. The value at (5) refers to the most massive NAO. For the less massive ones, the redshifts will bs smaller, but the non-uniform character of the redshift will still be maintained. For - 0.85MI", we have R'") = 1.4Rp), Z,e 0.08, Z, & 0.12;for M’” - 0.43M!"', we have example, for M ('1 Z, > 0.04. R'") - 2.2Rb", Z ,* 0.025,

3. Because of its large mass and size, an NAO will radiate copiously following accretion. We can estimate the luminosity so radiated according to the following formula used in the model of free-falling particles /61, L - ,#,v&,,c( R"'Y(ZGM"'/R'~v:)S. (6) Here no and V, are the number density and velocity of particles (mainly protons) far from the NAO, @%O.l mPcz is the gravitational energy released by each proton accreted, and Clis the probability of a falling particle remaining inside the NAO and no longer escaping, For a rough estimate we take values pertaining to the space between clusters of galaxies, na~10e4 cm-', an effective temperature of ??,I=mpU~/k)~107K. Then for the heaviest NAO, we have L - 3 x 10'(34/m.(cV))'Q

(7)

The precise value of Q will depend on the content of material gases (e, p, He'...) inside the NAO. Obviously we have a very intense radiation.

4. Neutrinos are abundant in the universe. If we take ~"~3~eV,then, in terms of energy density, neutrinos will be about 100 times all the other material put together. In the course of evolution of the universe, the backgroung neutrinos may condense through Jeans' gravitational instability into celestial bodies. Note, however, that according to the standard model universe, after de-coupling,neutrinosand antineutrinos no longer took part in collisions and they would have kept separately their early form (but redshifted) of nondegenerate distribution A_'(ePyC'ltT. + l)_'dP#. (8) whileT.-T(4/ll)'",T being the background radiation temperature. Hence, NAOs formed in the course of evolution of the universe are likely to be not completely degenerate, According to Jeans' theory, the Jeans mass of the neutrino system can be written as MI” -

4%

-P.

(9) 3 where o"(am,) is the mass density of neutrinos, and vsv is the speed of sound in the neutrino gas, which can be calculated according to (8). Roughly speaking, we have

In order to form a body as large as in (3-l), we require a Jeans temperature of

379

Neutrino Astronomical Objects

T,- 2 X lO'(m.(eV)/34)K. (12) As the universe expanded and cooled to the temperature Tj, Jeans' instability began to appear in the neutrino gas, which then condensed into NAOs. According to the usual theory, galaxies were formed when Ts4000K. Therefore there is a critical neutrino mass mv(") such that NAOS were formed before or after the galaxies, according as the neutrino mass is greater or less The structure of the NAO formed in these two cases, including the content of than mycC). distribution of material gases, may be very different from each other because of the very different material background at the time of their formation.

5. We now proceed to study the abundance of e, p, He4... inside an NAO. Ifm.Zm?,then when NAOs began to form, there were still no galaxies and e, p, He4 etc. were still in the gaseous state. Since there is only the weak interaction operating between the neutrinos and the material gases, we can regard them as two fluids moving independently. Of course, there is gravitational coupling between them. The equation describing the motion of the two fluids coupled gravitationallyare $ + v * (P,Ui) - 0, (13) +(o,* V)Ui ---Lwi+g,

z

Pi

vxg-0,

(14) (15)

v f g--4nc(&%+

P,).

(16) Here suffix i represents either v (neutrino gas) or m (material gas), and g is the gravitational field. Using a method similar to Weinberg's 131, we take oi=const., Pi=const., vi=O, g=O to be the zeroth order (unperturbed)solution, then from (13)-(16) we derive the equations governing the behaviour of the first-orderedperturbed quantities PIi, Pti, VLi and get and gl, and then eliminate V’i B'plr - v:iv'pl, + 4&&I. + PI"). (17) 81' is the velocity of sound in the gas i. For (17), a plane wave in which V,$i=(Pli/p,i) solution exists:

which is a wave after the coupling. Substituting this in (17')gives two gravitationallycoupled dispersion relations, &.-

y&z.4=Gp,(a,+ %Il), ==yv~u,-4%Gp,(lt.+ a,>.

(18)

(19) fe%z;s;;,=i~~~uite high, there is strong coupling between photons and the ordinary matter, J, the speed of sound in the ordinary matter should be (20) 4P, & - -c= = 0.332. 3 (3& + 4P,1 meanwhile, according to (lo), we have a:.- a x 10-Y. (21) Noting the strong coupling between the ordinary matter and the photons, the pm and a, in (13)-(19) should be understood as pm+,,and CZ,+~. Eliminating w between (18) and (19), and noting that, at the onset of Jeans' instability, WQO approximately,or + %I+,) - 0. ytl:.a. - 4nCp.(u. we

have a,+, _ 3L _z a.

2

Pn+r Pr’

(22)

Thus, we see that the ordinary matter is condensed to a far smaller extent than the neutrinos. The fraction of ordinary matter in an NAO is even smaller than in the average background.

6. We now consider the effect of the expansion of the universe. We take the uniform expansion solution

Neutrino

380

pu = cd&/R

I

Astronomical

Objects

(t) )‘,

Pnr+r = ~mo(Ro/R

“u = V,

(t>‘)l, r(d/R),

-

g---r--

(23)

! 4;c 1 (P" + Pn+r)r

as the zeroth order approximation to the solution of (13)-(16). Here, R is the scale factor in the Robertson-Walker metric. Using the Newtonian approximation, we can find the coupled equations satisfied by the first-order perturbed quantities under the universal expansion. Let PI1- PI{(~) ==p{iq * r/R(t) 1, (24) with similar forms for vii andgI. Then the coupled equations will assume the form

Pli(Z) + ipv&)

’ PIi(t) + $

( &i(r) + $ vt,(t> = --i 2

-0,

(25)

Pli(r)q + g1(t),

(26)

I7 x &3(t) - 0, ’ iq. g,(t) - -4zGR(p,,(t) + ,~,+,(t)).

(27) (28) (29)

VU(~) - VIU + i+,(f)

Let

s{(t)- -iq: si(I)ld>,

(v,,**q-0,

(30)

F%(t) - tii(Ro/R)'7((t)- ~a;($). Substituting

in (25) and (26) gives qi3 P sir

(31)

R

31;1+ g.,i, - 0, l? o:i E{+ -- .si- - F7J,+ R

From

(32)

4&R

P.11. + Pnr+r%+r1.

y'(

(33)

(31) and (33) we have q,+2$i;+

Jg !

which may be written

>

9i -

47&P”%

approximately

+

Pm+,%+,)

-

0.

ij,+z~li.+~,~-44nGp,ll.~O, R’

ii,+,+ 2J-&+,

(34)

as (35)

+ $+r

(36)

= ~*GP,T,.

This shows that the ordinary matter will hardly influence the behaviour of the neutrino gas, Considering while the effect of the neutrino gas on the ordinary matter will be very great. that at the onset of
Lyubimor,

[21

Reinea. F., et al., UCI-IOpI9-144,

[31

Weinberg,

S., Gravitation and Cosmology:

Relativity,

1972.

V. A., et al., Preprint ITEP-62 1960.

(Moscow), 1980. Phus.Lett.94B (1.980)266. Phys.Rev.Latt. 45 (1980) 1307 Principlea and Applications

141 I5 7

Luo Liao-fu. i2’&rwnpqu Iiamc XU&Q~ 11 (19&o) Oppenheimer, J. R., Volkoff, Q. Mb., mg8. Rev., 55(1939), 374.

[ 6 I

Zeldotich,

Ya. B.. Novikov,

I. D., Relativistic

Adrophgsies,

Vol.

of the General Theory of

112.

1, 1971.