APH N.S., Heavy Ion Physics 13 (2001) 259–265
HEAVY ION PHYSICS
c Akad´ emiai Kiad´ o
Interaction between Black Holes and Neutrino Stars G. Kupi Department of Atomic Physics, R. E¨ otv¨ os University H-1117 Budapest, Hungary Received 23 August 2000; revised version 16 November 2000 Abstract. If dark matter consists of neutrinos and antineutrinos then they may annihilate and radiate X-rays. The neutrino matter in galactic nuclei may be very dense. Black holes assumed at the center of such neutrino stars can swallow the matter permanently. Observable signs and time evolution of such a system are studied in this paper. Keywords: neutrino, quasar PACS: 95.35+d
1. Introduction There are more and more observations concerning the centers of the galaxies. The stellar-kinematical data indicate there dark compact objects of 106.5 to 109.5 solar masses. There is further evidence that the enormous energy radiation of the quasistellar objects are powered by such objects. From the time variability of the energy output few light days or light hours can be obtained for the size of these objects. These observations are mostly explained by supermassive black holes. However, the size of these objects can be larger than their Schwarzschild radius. Viollier and Trautmann [1] suggested an alternative of black holes in galactic nuclei. They stated that these objects could be neutrino stars. The relic neutrinos (or neutralinos) may have been collected into the centers of the galaxies. If this selfgravitating degenerate neutrino matter consisted of neutrinos with about 17 keV/c 2 rest mass then it could mimic phenomena that are expected around the supermassive black holes. According to recent empirical data mνµ ≤ 19 keV/c 2 and mντ ≤ 18.2 MeV/c 2 [2]. These allow the 17 keV/c 2 rest mass. The neutrino matter can be dense enough, if we take into consideration the gravitational phase transition of fermionic matter [3]. The gaseous matter extends over the whole galaxy (and much beyond) [4] but the condensed phase only over a few light days.
1219-7580/01/ $ 5.00 c 2001 Akad´emiai Kiad´ o, Budapest
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If we assume a neutrino star at the center of a galaxy then it is evident to study the interaction of such a “pseudo” black hole and a real black hole. A simple model of this system is studied in the present paper.
2. The Simple Model of the System A very simple case is studied: a black hole in a spherically symmetric degenerate neutrino cloud. Further it is supposed that the flow of the infalling matter is stationary. We adopt Shapiro’s notations, who has described the relativistic behaviour of the baryonic matter around a black hole with similar conditions [5]. A black hole of mass M is represented by the Schwarzschild line element, ds2 = (1 − 2m/r)c2 dt2 −
dr2 − r2 (dθ2 + sin2 θdφ2 ), 1 − 2m/r
where m = GM/c2 . To describe the behaviour of the system we need the following equations (Landau and Lifshitz [6]). The equation of continuity is (nU i );i = 0, where U i = dxi /ds is the 4-velocity of the fluid and n is the scalar number density of neutrinos measured in the frame in which the fluid element is at rest. The relativistic generalizations of Euler’s equations are: ωU k Ui;k = −
∂P ∂P − Ui U k k , i ∂x ∂x
where P is the pressure, ω = + P is the internal enthalpy per unit proper volume, and is the proper internal energy density. If we apply the above equations to our model we get from the equation of continuity 4πnur2 = A = constant, where |U 1 | = u/c. From Euler’s equations: dP u2 1 d u2 2m 1 m − 1 + = − − 2. 2 2 2 dr c (P + ) dr c r r We need further an equation for from the 1st theorem of thermodynamics: d dn P dn Λ − − = , dr n dr n dr u where Λ is the energy loss by annihilation of neutrino–antineutrino pair: Λ = vσn, where v is the average velocity of particles. For v we use the velocity obtained from the Fermi momentum. σ is the cross section of the ν + ν → 2γ process: 3 α2 2 2 mν g mν σ= β −1 , 144π me
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where α is the electric fine structure constant, g is the Fermi coupling constant, mν is the rest mass of neutrinos, me is the rest mass of electrons, β = v/c [7]. We need further P = P (n). We use the equation of state of the degenerate gas. For β 1 it is approximately P =
h2 5mν
for β ∼ 1 it is P =
3 8π
2/3
n5/3 ,
ch 3 n4/3 . 4 π
In order to integrate the system of equation we need the value of A, A=
π(GM )2 n0 , c3s0
according to Shapiro’s and Bondi’s paper [5, 8], where n0 is the number density at infinity and cs0 is the sound velocity at infinity. It is supposed that at infinity the infall velocity is 0. We need to know only two parameters. It is taken mν = 17 keV and n0 = 2.5 · 1030 m−3 for one of Viollier’s neutrino stars. If the mass of this star is specified then the spectrum from the annihilation can be calculated. Let Mtot = M + Mν = 109 M .
3. The Spectrum The isotropic emissivity of the neutrino gas jν is obtained from the assumption that the energy states are filled completely up to the Fermi energy. So the annihilation radiation is subject to the Doppler effect. In order to obtain the spectrum Lν0 at infinity we have to transform firstly the jν and ν. (The quantities measured at infinity are denoted by 0.) jν = jν
(1 − v 2 /c2 ) , (1 − (v/c) cos Θ )2
where v(r) =
ν = ν
(1 − v 2 /c2 )1/2 , (1 − (v/c) cos Θ )
dr 1 u(r) . = 2 dt 1 − 2m/r [u (r)/c2 + 1 − 2m/r]1/2
The angle Θ is the angle between the velocity v, which points radially inward, and the line of sight. The comma denotes the quantity measured by a stationary observer in the Schwarzschild frame. This is a local special relativistic transformation. One may write for the energy radiation δLν0 δt0 δν0 = jν δV δΩ δt (1 − 2m/r)1/2 ,
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where
δt0 = δt /(1 − 2m/r)1/2 , δν0 = δν (1 − 2m/r)1/2
and
δV = 4πr2 δr/(1 − 2m/r)1/2 .
We obtain
δLν0 = jν 4πr2 δrδΩ .
Integrating the result over r and Ω we have Lν0 = 8π
2
r∗
2
cos Θ∗
r dr 2m
where ν0 =
−1
jν
(1 − v 2 /c2 ) d(cos Θ ), (1 − (v/c) cos Θ )2
ν[(1 − v 2 /c2 )(1 − 2m/r)]1/2 [1 − (v/c) cos Θ ]
and r∗ the radius of the neutrino star. Rays that emerge at an angle less than Θ∗ (measured in the stationary frame) are captured by the black hole. 1/2 2 2m 27 2m −1 +1 | cos Θ | = . 4 r r
∗
All these remarks on the spectrum can be found in Shapiro’s paper [5].
4. Time Evolution of the System One can estimate the life-time of this system. The accretion rate is approximately dM = 4πρ0 G2 M 2 /c3s0 = KM 2 , dt where M is the mass of the black hole [9]. After integration we get M (t) =
1 . 1/M (0) − Kt
It takes some ten million years to swallow the whole cloud of mass ∼ 106 M by the black hole. It is a faster process for more massive clouds. Figures 1 and 2 show the number density and the infall velocities of neutrinos. Figures 3 and 4 show the spectrum of the system. The numbers beside the curves denote the mass of the central black hole: M/M . The neutrino star radiates for a long time permanently because the black hole grows hyperbolically. But at the end of the life of the neutrino star the radiation decreases fast and stops in some years.
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30
5⋅10
30
4⋅10
8
9⋅10 30
n [m-3 ]
3⋅10
6
10 7
10
7
5⋅10
30
2⋅10
8
10
8
3⋅10
5⋅10
30
1⋅10
0 0.0
12
13
13
1.0⋅10
5.0⋅10
8
1.5⋅10
r [m]
Fig. 1. The number density of neutrinos 1.0 8
9.9⋅10 0.8
0.6 v/c
8
9⋅10 0.4
8
4⋅10 8
2⋅10
8
8
7⋅10
10
0.2
7
10
0.0 0.0
12
5.0⋅10
13
1.0⋅10
13
1.5⋅10
r [m]
Fig. 2. The infall velocity of neutrinos
13
2.0⋅10
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G. Kupi E[keV] 11 0 2.0⋅10
10
20
30
40
6
10 11
1.5⋅10
8
I[W/Hz]
10
11
1.0⋅10
8
3⋅10
8
10
5⋅10
5.0⋅10
8
7⋅10 8
0.0
9⋅10 0
18
18
18
4⋅10
2⋅10
6⋅10
18
8⋅10
ν [Hz]
Fig. 3. The X-ray spectrum of the system
E [keV] 90 2.5⋅10
10
20
30
8
9.2⋅10
9
2.0⋅10
I [W/Hz]
9
1.5⋅10
9
8
1.0⋅10
9.5⋅10
8
5.0⋅10
8
0.0
9.8⋅10 8 9.9⋅10 0
18
2⋅10
18
4⋅10
18
6⋅10
ν [Hz]
Fig. 4. The X-ray spectrum of the system at the end of its evolution
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5. Conclusion Now we can see the main properties of the spectrum: the Doppler broadening and the gravitational redshift of the line corresponding to the rest mass of neutrinos. If we accept this simple model (as basic case) one may become able to observe either hardly varying or fast decreasing and redshifting radiation from the neutrino star because of the hyperbolic growth of the black hole. This decreasing ends in a short time. The (neutrino) quasars work mostly in the early Universe and we see that the big neutrino clouds have very little chance to exist now in the galactic nuclei. Black holes in galactic nuclei are derived usually from baryonic hyperstars. These objects radiate only for some decades. The heavy nuclei (70 ≤ A ≤ 209) are synthesized by r process [10]. There are no conditions of r process in normal stars but only in hyperstars at the end of their life. So these heavy nuclei can be found in bulk only in such galaxies which contained hyperstars. Galaxies comprising neutrino quasars contain fewer heavy nuclei now.
Acknowledgement The author is grateful to G. Marx for his valuable comments.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
R.D. Viollier and D. Trautmann, Phys. Lett. B 306 (1993) 79. The European Physical Journal C 15 1–4 (2000) 350. N. Bili´c, R.D. Viollier, Phys. Lett. B 408 (1997) 75. G. Marx and G. Kupi, WIN’99, Proceedings of the Workshop on Weak Interactions and Neutrinos, Cape-Town, 1999, p. 269. S.L. Shapiro, The Astrophys. Journal 180 (1973) 531. L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Reading, Mass., Addison-Wesley Publishing Co., 1965. J. Balog and G. Marx, Acta Phys. Hung. 58 1-2 (1985) 35. H. Bondi, Monthly Notices of Royal Astronomical Society 112 (1952) 195. A. Treves, L. Maraschi and M. Abramowicz, Publications of the Astronomical Society of the Pacific 100 (1988) 427. E.M. Burbidge, G.R. Burbidge, W.A. Fowler and F. Hoyle, Rev. Mod. Phys. 29 (1957) 547.
