On the possibility of two-bang collapse

IL NUOVO CIMENTO VOL. 11 C, N. 3 Maggio-Giugno 1988

On the Possibility of a Two-Bang Supernova Collapse.

V. S. BEREZINSKY (*), C. CASTAGNOLI, V. I. DOKUCHAEV (**) and P. GALEOTTI

Istituto di Cosmogeofisica del CNR - Torino Istituto di Fisica Generale and Sezione INFN, Universit4 di Torino - Torino

(ricevuto il 15 Gennaio 1988)

Summary. - - The possibility of a two-bang stellar collapse originating SN 1987a, and having the characteristics of the events recorded in Mont Blanc and Kamiokande, is discussed here. According to the ,,standard,, collapse models of nonrotating stars, which predict the formation of a neutrino- sphere with a nondegenerate neutrino gas inside the star, the Mont Blanc and Kamiokande data for the first burst give a too large stellar mass. On the contrary, a degenerate neutrino gas with low temperature T ~ 0.5 MeV, and chemical potential ~-~(12+15), predicts a relatively low total energy outflow W~ ~ (2 + 6)- 1054 erg, and a small number of expected interactions in Kamiokande. A possible scenario is suggested: a massive (M~20M| rotating star is fragmented into two pieces, one light and the other heavy, at the onset of the collapse. The massive component collapses to a black hole, and produces the first burst. Neutrinos are trapped inside the collapsing star because of elastic scattering in the outer core off heavy nuclei, with A ~> 300. It is shown that neutrinos fill up the quantum sl~ates, producing a degenerate neutrino gas. The second burst is explained by coalescence of the light fragment (M= (1 +3)M| onto the massive black hole. The time delay between the two observed bursts (4.7h) is mostly connected with gravitational braking, when the light fragment falls down onto the black hole, with an accompanying emission of gravitational waves for times of order of hours.

PACS 94.40 - Cosmic rays.

1. - Introduction.

Two burs ts have been detected in underground neutrino observatories during the occurence of SN 1987a, separated in t ime by 4.7 h, which, assuming they are

(*) On leave of absence from Institute for Nuclear Research of the Academy of Sciences of USSR, Moscow, USSR. (**) Institute for Nuclear Research of the Academy of Sciences of USSR, Moscow, USSR.

287

288 V.S. BEREZINSKY, C. CASTAGNOLI, V. I. DOKUCHAEV and P. GALEOTTI

due to neutrino interactions, impose serious difficulties in understanding the stellar collapse from which this supernova originated. In fact, since the pioneer works (1,2), no previous model predicted a two-bang collapse, as the detection of two neutrino bursts suggests. The first burst was recorded (8.,) on real time at the occurence in the Mont Blanc LSD experiment, and marginally detected (5) also in the Kamiokande experiment; the second burst was recorded in the Kamiokande (6), IMB (7) and Baksan (8) detectors. From the experimental point of view, it has been shown (9) that the two bursts are not contradictory, because of the different energy thresholds and experimental techniques used in the different detectors.

However, both bursts are difficult to explain within the framework of the current models of stellar collapses. Indeed, the first one involves a high luminosity outflow (Wv e > 10 ~ erg) of low-energy neutrinos (Eve ~ (7 + 9) MeV), while the second one shows a considerable anisotropy of the recorded pulses,

(1) YA. B. ZELDOVICH and O. KIt. GUSEINOV: Soy. Phys. Dokl., 10, 524 (1965). (2) G. V. DOMOGATSKY and G. T. ZATSEPIN: in Proceedings of the XIX International Cosmic Ray Conference (London, 1966), p. 1030. (3) M. AGLIETTA, G. BADINO, G. BOLOGNA, C. CASTAGNOLI, A. CASTELLINA, V. L. DADYKIN, W. FULGIONE, P. GALEOTTI, F. F. KALCHUKOV, V. B. KORTCHAGUIN, P. V. KORTCHAGUIN, A. S. MALGUIN, V. G. RYASSNY, O. G. RYAZSKAYA, O. SAAVEDRA, V. P. TALOCHKIN, G. TRINCHERO, S. VERNETTO, G. T. ZATSEPIN and V. F. YAKUSHEV: Europhys. Lett., 3, 1315 (1987). (4) V. L. DADYKIN, G. T. ZATSEPIN, V. B. KORCHAGUIN, P. V. KORCHAGUIN, A. S. MAL'GIN, O. G. RYAZSKAYA, V. G. RYASSNYI, V. P. TALOCKIN, F. F. KAL'CHUKOV, V. F. YAKUSHEV, M. AGLIETTA, G. BADINO, G. BOLOGNA, C. CASTAGNOLI, A. CASTELLINA, W. FULGIONE, P. GALEOTTI, O. SAAVEDRA, G. TRINCHERO and S. VERNETTO: JETP Left., 45, 593 (1987). (5) A. DE RUJULA: Phys. Lett. B, 193, 514 (1987). (e) K. HIRATA, T. KAJITA, M. KOSHIBA, M. NAKAHATA, Y. OYAMA, N. SATO, A. SUZUKY, M. TAKITA, Y. TOTSUKA, T. KIFUNE, T. SUDA, K. TAKAHASHI, T. TANIMORI, K. MYIANO, M. YAMADA, E. W. BEIER, L. R. FELDSCHER, S. B. KIM, A. K. MANN, F. M. I NEWCOMER, R. VAN BERG, W. ZHANG and B. G. CORTEZ: Phys. Rev. Lett., 58, 1490 (1987). (~) R. M. BIONTA, G. BLEWITT, C. B. BRATTON, D. CASPER, A. CIOCIO, R. CLAUS, B. CORTEZ, M. CROUCH, S. W. DYE, S. ERREDE, G. W. FOSTER, W. GAJEWSKI, K. S. GANEZER, M. GOLDHABER, T. J. HAINES, T. W. JONES, D. KIELCZEWSKA, W. R. KROPP, J. G. LEARNED, J. M. Lo SECCO, J. MATTHEWS, R. MILLER, M. S. MUDAN, H. S. PARK, L. R. PRICE, F. REINES, J. SCHULTZ, S. SEIDEL, E. SHUMARD, D. SINCLAIR, H. W. SOBEL, J. L. STONE, L. R. SULAK, R. SVOBODA, G. THORNTON, J. C. VAN DER VELDE and C. WUEST: Phys. Rev. Lett., 58, 1494 (1987). (s) E. N. ALEKSEEV, L. N. ALEKSEEVA, V. I. VOLCHENKO and I. V. KRIVOSHEINA: JETP Left., 45, 589 (1987). (9) M. AGLIETTA, G. BADINO, G. BOLOGNA, C. CASTAGNOLI, A. CASTELLINA, V'. L. DADYKIN, W. FULGIONE, P. GALEOTTI, F. F. KALCHUKOV, V. B. KORTCHAGUIN, P. V. KORTCHAGUIN, A. S. MALGUIN, V. G. RYASSNY, O. G. RYAZSKAYA, O. SAAVEDRA, V. P. TALOCHKIN, G. TRINCHERO, S. VERNETTO, G. W. ZATSEPIN and V. F. YAKUSHEV: Europhys. Lett., 3, 1321 (1987).

ON THE POSSIBILITY OF A TWO-BANG SUPERNOVA COLLAPSE 289

which cannot be explained as due to (~e-e) scattering. In any case, a reliable interpretation of the two bursts, and a comparison between them, is difficult because of the poor statistics.

When comparing the observational data with the standard theoretical predictions (which have the difficulty to turn the core collapse into the envelope explosion), one should remember that the latter do not include the effects of rotation and magnetic field. If one takes into account rotation, the characteristics of the neutrino emission change drastically(l~ and rotation was indeed suggested (~1) to explain the two-bang scenario of SN 1987a as an alternative explanation of the collapse on two stages (~,~2): first into a neutron star and subsequently into a black hole. Here we suggest a possible scenario to explain the two-bang collapse, with particular emphasis on the first neutrino burst.

A brief review of the basic properties of the Mont Blanc and Kamiokande experiments is needed before comparing the data recorded in these detectors. In any experiment, the main signal induced by supernova neutrinos is given by positron pulses, produced through the capture reaction (~p, ne § with free protons, with a cross-section

(1) z(E~) = 9.6- 10-~(E~ - 1.29)[(E~ - 1.29) 2 - 0.51~] 1~ cm~/MeV 2

and an energy threshold E~ ~-1.8 MeV. A comparison of the signals expected in the Mont Blanc and Kamiokande

experiments, based only on the mass ratio NKH/N~ B = 17 of the two detectors, is misleading because positrons are detected in a different way in the two experiments. In water (Kamiokande) the positron is detected through the emission of Cerenkov light, and the total visible energy released by a positron with kinetic energy Te is ECs = Te - To, where Tc ~ 0.25 MeV is the threshold for ~erenkov light production by positrons in water. Therefore, in water the visible energy is E c = E v . - 2.1 MeV. In liquid scintillator (Mont Blanc) the visible energy is given by the total positron energy (kinetic and rest-mass): ESs = Te + 2mec 2 = Eve - 0.8 MeV, because the positron annihilates into 2 gammas and produces an extra energy release of about 1 MeV, detectable in the scintillator but not in water. Therefore, for the same neutrino energy, the visible energy in the scintillator is higher than in water by the amount AE = E s - ECs ~ 1.3 MeV.

In addition, the detection efficiency of electrons with total energy Ee is alSO higher in scintillator than in water at the energies of interest, as can be seen from fig. 1 where the efficiency curves of the Mont Blanc and Kamiokande

(10) V. S. IMSHENNIK and D. K. NADJOZHIN: Astrophys. Space Phys. Rev., edited by R. A. SYUNYAEV, Vol. 2 (Harwood Acad. Publ., 1983), p. 75. (11) L. STELLA and L. TREVES: Astron. Astrophys., 185, L5 (1987). (12) W. HILLEBRANDT, P. HOEFLICH, P. KAFKA, E. MUELLER, H. V. SCHMIDT and J. W. TRURAN: Astron. Astrophys., 180, L20 (1987).

290 V . S . BEREZINSKY, C. CASTAGNOLI, V. I. DOKUCHAEV and P. GALEOTTI

100

�9 ,~ 50

0

energy (MeV)

680 t LSD / ~.~. .....................................

f Kamiokancle ~ jB ~

,.. / t 216.0 t / / / / /.." , ..." /

/ / - /

: / .."/

" / - / I/

// -/

-/ -!

-/

-'~, , ~ ~ I ~ ~ J ~ I ~ ~ ~ J I

10 15 20

Fig. 1. - Efficiencies of LSD and Kamiokande detectors.

experiments are reported. For example, from fig. i one deduces a 50% ~ffic~hcy at ~ 6 MeV in Mont Blanc, and at = 9 MeV in Kamiokande.

The total energy output of ~-e's in the first burst is higher than the predicted value. Taking into account only the 5 detected neutrinos, and adopting a distance r = 52 kpc for SN 1987a, the Mont Blanc collaboration reported (~8) an energy outflow

(2) W ~ - 4=r2 ~ Ei N MB ~1 a(Ei) V~B(Ei) ~ 6.3. l0 w erg,

where VMB(Ei) is the Mont Blanc detection efficiency, and a corresponding number of events in Kamiokande, given by

N~H ~, ~(Ei) (3) NK= N~B i=, ~M,(Ei)

(13) M. AGLIETTA, G. BADINO, G. BOLOGNA, C. CASTAGNOLI, A. CASTELLINA, W. FULGIONE, P. GALEOTTI, O. SAAVEDRA, G. TRINCHERO, S. VERNETTO, V. L. DADYKIN, V. B. KORTCHAGUIN, P. V. KORTCHAGUIN, A. S. MALGUIN, V. G. RYASSNY, O. G. RYAZSKAYA, V. P. TALOCKIN, G. T. ZATSEPIN and V. F. YAKOUSHEV: ESO Workshop on SN 1987a, edited by J. DANZIGER (Garching bei Munchen, 1987).


of order of 10, because the visible energies detected in the Mont Blanc scintillator in the range (5.8 + 7.8) MeV correspond to visible energies in water in the range (4.5+6.5)MeV, for which the Kamiokande detection efficiency is ~K~ 0.1.

Since, during the first burst, Kamiokande detected (~) 2 events, there is no great conflict between the two experiments, and the difference in the predicted and observed number of events in Kamiokande can be explained by the poor statistics and by systematic errors, mainly in the efficiency curves of both experiments.

2. - ,<Standard~ m o d e l pred ic t ions for the f irst n e u t r i n o burst .

In the ,,standard,, calculations (1~17) of neutrino emission, a nonrotating stellar core is considered, and the effect of a strong magnetic field is also neglected. The basic conclusion is the prediction of a Kelvin cooling stage during the collapse (after the core becomes nontrasparent to neutrinos), where the volume emissivity of neutrinos is replaced by the surface emissivity. The internal thermal pressure halts the collapse even in the case of very massive cores (14), until the neutrino emission from the core surface cools the matter, a phase which lasts for about 10 s. Subsequently, the collapse enters into the relativistic stage of contr:action, with the final formation of a black hole. For smaller core masses, a hot neutron star is formed, and also in this case the cooling from the neutrino- sphere lasts for about (10 + 20)s; thereafter, the core becomes an usual cold neutron star.

During this stage, matter consists mainly of neutrons, with a small admixture of protons and electrons (he ~ np ~ 0. lnn). The equilibrium is maintained by the usual reactions: (e+e-o2~) , (e+no~-ep), ( e - p ~ e n ) . Also muon and tau neutrinos, generated in the reaction (e+e---.~V), are expected to be in thermal equilibrium with matter inside the core, and trapped because of elastic scattering with nucleons. In the scattering reactions with electrons neutrinos are thermalized, and they diffuse out of the core. As a consequence, in the standard models all the 3 neutrino species are in thermal equilibrium inside the neutrino sphere.

The total flux of neutrinos with energy E , emitted per second from the neutrino sphere, is given by

(4) 3 2=c E 2

~=1F~(E) = N~ 4rcR~ (hc) 3 exp [E/T] + 1'

(14) D. K. NADJOZHIN: Astrophys. Space Sci., 53, 131 (1978). (15) D. K. NADJOZHIN and I. V. OSTROSCENKO: Soy. Astron., 24, 47 (1980). (16) j. R. WILSON: Astrophys. J., 163, 209 (1971). (17) R. MAYLE, J. R. WILSON and D. N. SCHRAMM: Astrophys. J., 318, 288 (1987).

292 V. S. BEREZINSKY, C. CASTAGNOLI, V. I. DOKUCHAEV and P. GALEOTTI

where N. = 3 is the number of flavours, R~ is the radius and T~ is the temperature of the neutrino sphere.

This picture, however, must be considered as a first approximation of the real collapse. In particular, opposite results have been obtained on the high-energy tail of the neutrino spectrum: there could be an enhancement (16.~7) because of accretion in the process of collapse, or a cut-off (~4,~) by a factor exp [ - a.E~/T 2] because of re-absorption of the high-energy ~e neutrinos in the stellar envelope outside the neutrino sphere. In any case, whether or not high-energy neutrinos are produced by accretion, their absorption outside the neutrino sphere is an unavoidable physical process, since the neutrino sphere is defined as the layer where neutrinos with the average energy are absorbed, and, since the neutrino cross-section increases as E~, those with higher energy are stronger absorbed outside the neutrino sphere.

Generally speaking, the radius R~ of the neutrino sphere depends on the neutrino energy, because higher the energy stronger is the coupling with matter; therefore, R~ is larger for high-energy neutrinos than for the low-energy ones. Since the temperature decreases with the stellar radius, this implies a high-energy cut-off in the neutrino spectrum, and the effect is strong if the temperature gradient is large in the region near the neutrino sphere.

Using the basic features of the ,,standard~, models, namely: 1) equilibrium of the 3 neutrino species, 2) neutrino sphere with average radius R , and 3) energy spectrum of neutrinos at the Earth given by

(5) j~(E) = K 1 + exp [EJT] exp [ - aE~/Te],

where a = 0 corresponds to the black-body spectrum (4), one can evaluate from the Mont Blanc data the total energy output of SN 1987a. Assuming the detection of NMB = 5 neutrinos in the Mont Blanc experiment, the normalization factor K in eq. (5) can be computed from

(6) (6) NMB = K N ~ B dE~z(E.)~(E.) exp[EJT] + 1 exp[ - xE~/T2], o

where N~ B = 8.4.10 ~~ is the number of hydrogen atoms in LSD, ~(Ev) is the (TeP, ne § cross-section given by eq. (1), and VMB(E~) is the Mont Blanc detection efficiency, shown in fig. 1. Thus one obtains

(7) E~ exp [ - ~E~/T2], W~ e = 4~2K dE~ exp [E~IT] + 1

where r = 52 kpc is the distance of SN 1987a from the Earth. The corresponding


number of expected interactions in Kamiokande is given by

By (8) N K = K N K dE~z(E~)vK(E~) e x p [ E J T ] + l exp[-aE~/T2]'

o

where N K = 1.4.10 8~ is the number of hydrogen atoms, and vK(E~) the detection efficiency in Kamiokande, shown in fig. 1. The results of these calculations are

1.2

1.0

0.8 o "~'~ 0.6

i

0.4

f f / f 1 1 I,.......

I / f f ~

1:7 1:9 2'1 2:3 2'.s z:7 2:s T (MeV)

70

60

50

~0

30

2O

Fig. 2. - Total energy output released in ~-e-neutrinos (full curve) and expected number of events (dashed curve) in Kamiokande for different temperatures, calculated on the basis of the 5 Mont Blanc events and on the assumption of Planck spectrum of neutrinos (a = 0).

shown in fig. 2 for different temperature values of the Planck spectrum, with a = 0. The main discrepancy here is the large number of events expected in Kamiokande on the basis of the Mont Blanc signal, which comes because of the large number of hydrogen atoms sensitive to detect the high-energy tail of the neutrino spectrum, which is not empty for a = 0.

Turning to the neutrino spectra with a high-energy cut-off (a ~ 0), the results of our calculations are shown in fig. 3 for different ~ values. One can note that the number of predicted events in Kamiokande decreases with increasing the values, down to about 10 interactions, which we regard acceptable, but the neutrino energy output simultaneously increases.

We wish to discuss briefly the physical reason of the previous discrePancy. A low temperature is needed to predict a reasonable number of events in Kamiokande, but this implies a large energy outflow, which requires also a large radius R~ of the neutrino sphere, according to the relation

7 2 4 (9) L~ = ]-~ zSB 47:R~ T ,

294 V. S. BEREZINSKY, C. CASTAGNOLI, V. I. DOKUCHAEV and 1 ~. GALEOTTI

L,O

30

,e o 20

10

At~ / / i / / -

/ / / /

/ / / 1

30

a = 0 . 1 0 J = 0 . 1 5 " / / / /

/ / / / / /

/ / / / / / /

/ / / / /

/ / / / / / / : 0 . 2 0

I / / / / /

/ / / / /

/ / / / /

/ / / / J

/ / I / / / / / /

/ s~sll /

/ / / ~ / / / / / / / /

/ / / / / / f / / / * *

/

0

, , , ,. ,. - - 7 - - 1.9 2.1 2.3 2 5 2 7 2.9

E ( M e V )

2o~

10

.5 117 0

Fig. 3. - Total energy output released in re-neutrinos (full curves) and expected number of events (dashed curves) in Kamiokande for the truncated spectrum of neutrinos (a r 0). The numbers on the curves show the values of a.

where L~ = W,/~:v is the neutrino luminosity (z, ~ 10 s being the duration of the burst), and zss is the Stefan-Boltzmann constant.

By definition, neutrinos are in thermal equilibrium with matter at the surface o f the neutrino sphere; this condition requires a strong coupling between matter and neutrinos at the neutrino sphere, and hence some minimal density of matter. Then, because of the large radius R , given by eq. (9) as

(10) Rv = Ro(LJIO ~ erg/s) 1 : - 3 TMeV,

with

(11) R0 = 1.3. l0 s cm,

a too large mass M > (4/3)=poR~ must be inside the neutrino sphere. Firstly, let us estimate the matter density Po at the neutrino sphere. The

thermal equilibrium in the neutrino sphere is reached through reactions changing the number density of neutrinos (i.e. for ~-e, this is given by ~%p, ne+).


Using the cross-section z(T) ~ zo T2~v for ~,-neutrinos with energy E., = T (which gives the larger cross-section for small T, and hence smaller p0 in comparison with the true values) the condition of thermal equilibrium at the surface of the neutrino sphere, assuming the 3-neutrino scattering within the layer give A R ~ ( 1 / 3 ) R , is given by

(12) o0 T~v ~nx AR~ ~ 3,

where n~ is the nucleon number density at the surface of the neutrino sphere, and is the fraction of protons to all nucleons. From eqs. (10) and (12) we have

(13) 10mr 1 (_10~ erg/s) 1/2 ~ (_1053 erg/s) 1~

P~ zoRo ~ ~ = 1.4.1013 ~ g/am 3,

where m~ is the nucleon mass. One can see from these calculations that the density P0 is a rather universal

value: it does not depend on the temperature and it is of the same order of magnitude as in the model calculations. From eq. (13), one can obtain immediately the lower limit of the mass inside the neutrino sphere as

(14) M >-~7:~oR~ = 6.4

which is unacceptably large for the small temperature values (T< 1.7MeV) suggested by the observations.

3. - A nonstandard possibility.

As shown in the previous section, the ,, standard ~ models of collapsing stellar cores are unable to fit the experimental data. However, the present predictions for the neutrino radiation from a gravitational collapse will probably change drastically in the future, when magnetic field and especially rotation will be taken into account. Two basic differences with nonrotating models were found in some preliminary calculations (1o): a lowering of the temperature of the neutrino sphere, and of the neutrino flux.

Here, we discuss a nonstandard possibility for which rotation can play a crucial role: namely, we assume that neutrinos have a low-temperature (T < 1 MeV) degenerate spectrum. In this case, both the troublesome quantities before mentioned (the total energy output and the predicted number of events in Kamiokande) will be reduced, as we shall show. Let us consider the ~.-neutrino

29@ V. S. BEREZINSKY~ C. CASTAGNOLI, V. I. DOKUCHAEV and P. GALEOTTI

flux at the Earth as

(15) j~(E~) = KE 2 1 1 + exp [(EJT) - ~]'

being the chemical potential, and K computed from the number of events recorded in the Mont Blanc detector

f (16) NMB = KN~ B dE~z(EJ V~B(E~) 1 + exp [(EJT) - ~]" 0

For this spectrum, the total energy flux and the number of events predicted in Kamiokande are, respectively,

F E~ (17) W~ = 4=reK J dE~

o 1 + exp [(EJT) - ~]'

(18) N K = K N ~ dE~z(E~)vK(EJ 1+ e x p [ ( E J T ) - ~ ] " 0

Figure 4 shows the results of these calculations, which axe quite reasonable especially if statistical errors are taken into account, for 1 2 < ~ < 15, and 0 . 4 M e V < T < 0 . 5 M e V . For example, for ~ = 1 2 . 5 and T = 0 . 5 M e V , one gets

o

15 ," , , ,, , , 3 0 , / / /

I / i i / i / / i /

1 5 / 1 4 / 13/ / 12/" / / /

/ / / / / / / /

/ / /

1 / / /

- / / "" 2 0 / / / / /

l / J / / ' / / / / / / / / /

/ / / / / / / / / /

/ / / / / / / / / / �9

/ ] / / / / / / / / /

10

- ~ 14 12 13 J

J i I 0 0.4 0.5 0.6 0.7

T ( M e V )

Fig. 4. - Total energy output released in Te-neutrinos (full curves) and expected number of events (dashed curves) in Kamiokande for a degenerate neutrino spectrum with dimensionless chemical potentials ~ = ~/T, indicated on each curve.

ON T H E PO SSIB ILITY OF A TWO-BANG S UP ERNOVA COLLAPSE 297

W~ = (2.3 + 1.1). 10 ~ erg and NK = 12.2 _ 5.5. The low counting rate predicted for Kamiokande is due to the fact that the spectrum decreases sharply with energy when E~ >FT-

In addition, the low temperature, T=0 .5MeV, and the high chemical potential result in a low density of equilibrium positrons, and consequently in a small flux of muon and tau neutrinos, which are produced only through the (e+e---*v?) channel. This reduces drastically the total (all flavours) energy neutrino output.

4. - A possible scenario.

We discuss here a possible scenario in which a burst of degenerate neutrinos is emitted by a low-temperature neutrino sphere. However, we are aware that this is only a suggestion for the physical processes producing such a burst of neutrinos, and that detailed numerical calculations are needed in order to confirm (or reject) this possibility.

he~vynucLei neutron co(e __ i v :&e~:k hole .v v

~contre~ction~ contro~~qntra.~

s~o~ge I stooge I I s~r~ge I I I s~;c~ge IV

Fig. 5.

At the onset of the collapse (see fig. 5, stage I), we assume the existence of a massive core (M ~- (10 + 20) Mo), made of heavy nuclei with A ~-(300+ 1000)(~8,19), collapsing into a black hole on dynamical time scale (~= 10s) which is governed by rotation, namely by transfer of angular momentum outside. The neutrino emission starts when a neutron inner core is formed (see fig. 5, stage II), with temperature and density higher than those in the outer core, but still low enough to avoid production of ~ and ~ neutrinos.

The characteristics of the neutrino emission from the star are defined by the properties of the outer core. In fact, mainly because of coherent scattering

(is) H. A. BETHE, J. E. BROWN, J. COOPERSTEIN and J. R. WILSON: Nucl. Phys. A, 403, 625 (1983). (19) D. Q. LAMB and C. J. PETHICK: Phys. Rev. Left., 28, 948 (1972).

20 - I1 Nuovo Cimento C.

298 v . s . BEREZINSKY, C. CASTAGNOLI, V. I. DOKUCHAEV and P. GALEOTTI

vee--->~ee, neutrinos are trapped inside the outer O~re, their temperature is somewhat lowered, and ~ degeneracy is produced because they accumulate in the outer core during its contraction phase (see fig. 5, stage III).

The neutrino emission from the outer core occurs after the diffusion time td, which is considerably reduced when the outer core turns into a thin envelope (see fig. 5, stage IV) which traps neutrinos less effectively. At this stage, one should expect that the temperature of the neutrino gas approaches the temperature of the inner core, and the formation of a black hole begins.

Let us consider now the neutrino emission (stages II to IV, fig. 5). To estimate the radius Ro of the outer core and its average density Po, only one neutrino type should be considered. The energy density of Ve'S inside the outer core is given by

E s 1 (19) p;.(E0 = (hc) s 1 + exp [ (EJT) - ~r

Because of v + A--* ~ + A scattering, an isotropic-neutrino angular distribu- tion can be assumed near the boundary of the outer core; thus the neutrino luminosity, calculated as p~c=4=R~, is

(20) Lie= 1 ,,44 ,~ ~ S B 1 7~-~0x ,

where the values of • defined as

15f x dx (21) x = - ~ exp [x - ~] + 1 '

are reported in table I. For a chemical potential ~ = 14, eq. (20) gives

(22)

and

R0 17107( L,e , 2 .10~erg / s ] \0 .5MeV cm

(22) 1012 M ' T �9 10~erg/s] \0 .5MeV

TABLE I.

0 3 8 10 12 14

x 7/8 11.4 2.07. 102 4.64.10 2 9.09. 102 1.63. 102


From (22) and (23) one deduces a reasonable estimate of the mass inside the neutrino sphere.

For consistency of the estimates (22) and (23), we have to prove that elastic scattering off heavy nuclei is efficient in trapping neutrinos inside the outer core and to provide their degeneracy. The neutrino mean free path in matter dominated by heavy nuclei with A ~ 300 is much smaller (19) than R0; indeed, for sin20w = 0.25, one has

(24) ;(~ = 2.104 p~ (E J10 MeV) -2 (N/A) -2 cm,

where N / A is the fraction of neutrons in the nucleus. The neutrino diffusion outward is governed by the diffusion coefficient D = )~c/3; thus, the trapping time td ~ R~/D is

(25) ( L~. ~(. T )-4 {. E~ ]2{N]2

t d~ l . 5 2 .10~erg/s j \0 .5MeV. P12~10--~-~eV] ~ ] s.

Let us now estimate the characteristic time needed for neutrino degeneracy when propagating through the outer core. If electrons are degenerated in the outer core, then the processes ~ + ~--) e § + e- and ~ + ~--. e § + e- are suppressed, and thermal equilibrium with chemical potential t~ = 0 for neutrinos is not reached during the propagation time. Coherent-scattering alone off heavy nuclei (A---300 + 1000) can however provide neutrino degeneracy(~); in this process, ~ + A - - * ~ + A , the fraction of neutrino energy loss is given by A E / E ~ E / ( A m w c2). The degeneracy characteristic time can be estimated as the time during which N ~ E / ~ collisions occur:

N td (n~AC)

10_2[ E ~-3/'1012\ 2 . s

The emission of ~ and ~ neutrinos, produced through e+e - annihilation, is negligible at the temperatures and densities here considered. The neutrino emissivity from the outer core is strongly suppressed by the factor exp [ - E F / K T ] , where EF~>70MeV is the Fermi energy of the degenerate

ielectrons. For the inner core, even if degeneracy is absent, the neutrino luminosity is less than (21)

L~ 4-10 u ( . ^ T /s = M ~< 10 ~ erg/s. P

(2~ T. J. MAZUREK: Astrophys. J., 207, L87 (1976). (21) K. R. LANG: Astrophys. Formulae (Springer Verlag, Berlin, 1974), p. 409.


As far as the ~/;~ ratio is concerned, it is clear that a value larger than 1 can be due only to deoelectronization processes e- + (A, Z) --. (A, Z ~- 1) + ~e during the neutronization stage of the outer core. We anticipate here that in our model this process proceeds gradually as the outer core contracts. An upper limit for the energy output W~ from de-electronization processes is given by Er Ne, where N~ is the total number of electrons in the outer core, which can be readily computed from eq. (23). As a result we have

W~ < 4.9. IO~((Z/A)/O.3)r erg.

Since in our model we have W; ~-(1+2). 1054erg, the neutrino flux is characterized by a ratio ~ / ;~ -1 , and the total energy output in ~e and ;~ neutrinos corresponds to ~ 10% of the mass of the collapsing core

(26) W~ + W~ ~ 3.6. IO~(M/2OMo) erg.

5. - T i m e de lay o f t h e s e c o n d n e u t r i n o burst .

We discuss here a possible explanation, again based on rotation and similar to the one already suggested (19, of the observed time delay (4.7 hours) between the two bursts.

During the collapse, the star passes through the stage of a flattened spheroid (three-axis ellipsoid, and bearlike body) and then breaks into two fragments with masses M~-2OMo and m ~ (1 +2)Mo. The massive component continues to collapse and produces the first neutrino burst during the black-hole formation, on time scale of order (10 + 100) s depending on the evolutionary stage at which the break-up of the star occurs. Meanwhile, the low-mass star approaches the massive component, orbiting around it, because of gravitational radiation losses, until it is disrupted and its matter is accreted by the massive block hole, thus producing the second neutrino burst.

The orbital period P of the binary and its lifetime, i.e. the characteristic time z~ of gravitational energy losses (~), are given by

(27) C 4~'2R3 ~1/2 ~ R ~3/2~20M| 1/2

P=~G(M+m)] 1"1~4.3.108cm] ~M--+--mm] s,

_ 5 c0 R4 ( R 41M (20 o 2 (28) z~ 256 G 8 ,M e 4.7h ~t.3.10Scm] ---~--~----M-~] '

(~) L. D. LANDAU and E. M. LIFSttITZ: Theory of Fields (Pergamon Press, Oxford, 1958).


where ~ = Mm/(M + m), R is the separation of the binary system (approximately equal to the initial radius of the massive component), and G is the gravitational constant.

For consistency of the picture given above, it is required that R >> R0, where R0 is the radius of the outer core, given by eq. (22) as R0 ~ 107 cm, and a characteristic time of gravitational braking ~g considerably larger (of order of hours) than all the other time scales, namely: the duration of the collapse of the massive component and the time of disruption of the low-mass component and its matter accretion onto the black hole. As the latter is concerned, an appropriate estimate was given (11) of order or less than 1 s. Also the former time scale is very short in the absence of rotation, and defined by the hydrodynamical time

[ 7~2 R3 ~ 1/2 vh ~ [ 4 - ~ ) - 0,3 (4 ,3 , R .~3/'z/. 2 0 / ~ ~ 1/2

i-0'cm s.

In fact, as discussed in the previous section, the time scale of the collapse is driven by angular-momentum losses, and it is of the order of 10 s. We do not discuss here the characteristics of the second neutrino burst, since it is fitted by the ((standard,) picture of accretion of matter on a black hole (10).

6 . - C o n c l u s i o n s .

Neutrinos produced during a stellar collapse are the deepest probe to study the physical processes occurring in a collapsing core. The calculations of a stellar collapse are technically one of the most difficult problems in modern physics, and the present results are not the final word in this problem: the effects of rotation and strong magnetic fields will probably change drastically the present view of the collapse. The results of the neutrino underground experiments will be the judge for this complicated problem, as for the internal structure of the Sun the results of the solar-neutrino experiments will do. It is interesting to remind once again the prophetic words of Schwartzshild in his book Evolution of the stars: although the things we study could be perfect, the human mind is far from perfection. Left alone, it has a property to wear the mistakes, which we witnessed hundred times, and only new, found with difficulties, experimental data, brought it back to the right way.

In this paper we addressed two questions. Is it possible to explain the first neutrino burst within the framework of the ~(standard~) models, predicting the existence of a neutrino sphere during the Kelvin stage of the core cooling, and with black-body emission of all the three types of neutrinos? We arrived at a negative answer to this question. The reason is that, to explain the weak signal in Kamiokande, one should assume a low temperature of the neutrino spectrum; then, from the luminosity found in the Mont Blanc signal, a too large radius


(Rv ~ 108 cm) of the neutrino sphere is deduced from the Stefan-Boltzmann law. In addition, the matter density at the neutrino sphere must be large enough to provide coupling of neutrinos with matter, which implies a minimum mass inside the neutrino sphere of order of 10SMo.

The second question we have discussed here is what kind of models can explain both the observed neutrino bursts. The suggested scenario is the following: because of rotation a massive progenitor star (M~20Mo) is fragmented into two pieces at the onset of the collapse, and the massive component collapses in a black hole producing the first neutrino burst, the time scale of the collapse being driven by angular momentum losses. Neutrinos are trapped inside the star because of elastic-scattering processes in the outer core off the heavy nuclei with A ~> 300; the result is neutrino emission during their trapping time, from a low-temperature (T~0.5MeV), degenerate (EF ~- (6 + 8) MeV) gas made of ~e and Ve neutrinos. The other neutrino flavours are produced with very low efficiency through electron pairs annihilation in the low-temperature, degenerate electron gas. The neutrino spectrum has a sharp cut-off at energy E~ > EF, because of the factor exp [ - EJT], which explains the low counting rate in Kamiokande, and requires a relatively small total energy output, Wv ~ (2 + 6). 10 ~ erg.

The second neutrino burst is explained by accretion of matter of the disrupted light star (M ~ (1 - 3)Mo) onto the massive black hole. The time delay between the two neutrino bursts is mostly due to gravitational braking, as the ligth fragment spiralls towar~ls the black hole, producing a large amount of gravitational radiation for times of order of hours.

Some decrease in luminosity can be obtained if one considers the most probable, flattened (disklike) configuration of the collapsed object, with the accidental position of the observer in the direction of the maximum flux. The suggested possibility to explain the first neutrino burst involves the formation of a massive black hole (M~ 10Mo); therefore, the whole story will have its epilogue when the supernova shell becomes transparent.

Finally, we want to emphasize that the scenario here outlined is a suggestion which needs more detailed calculations, to confirm or reject the discussed possibility.

* * *

The authors would like to thank many people with whom they discussed the problem of a two-bang collapse, for both support and criticism. First of all, we wish to thank G. T. Zatsepin, for stimulating this work and many useful conversations. We benefited much understanding from conversations with V. M. Chechetkin, A. E. Chudakov, D. K. Nadyozhin, O. G. Ryazskaya, V. G. Ryassny, O. Saavedra, S. Vernetto. A great help in calculations and in understanding the problem was given to us by G. C. Trinchero.

One of us (VSB) thanks the Istituto di Cosmogeofisica of CNR for the hospitality.


�9 R I A S S U N T O

Si discute la possibilit~ c h e l a supernova 1987a abbia avuto origine da un collasso a due stadi, avente le caratteristiche degli eventi registrati nelrOsservatorio Neutri- nico del Monte Bianco e nell 'esperimento di Kamioka. Secondo i modelli ,,standard,, di collasso di stelle senza rotazione, che prevedono la formazione di una neutrino- sfera con un gas non degenere di neutrini al suo interno, i dati del Monte Bianco e di Kamioka per il primo burst comportano un valore troppo alto della massa stella- re. A1 contrario, un gas degenere di neutrini a bassa temperatura T - 0 . 5 M e V e con potenziale chimico ~ ~ ( 1 2 - 15) comporta un'emissione totale in neutrini relativa- mente bassa W . ~ (2 + 6). 10~erg ed un piccolo numero di interazioni in Kamiokan- de. Si suggerisce un possibile scenario: una stella massiva (M ~ 20M~) in rotazione si frammenta in due oggetti, uno leggero e l'altro pesante, all'inizio del collasso. L'oggetto massivo collassa in un buco nero e produce il primo burst. I neutrini vengono intrappolati entro la stella collassante a causa deUo scattering elastico nel core esterno dovuto a nuclei pesanti, con A ~> 300. Si mostra che i neutrini riempio- no gli stati quantici, producendo un gas degenere di neutrini. I1 secondo burst spiegato dalla caduta del frammento leggero M ~ (1 + 3)M~ sul black hole massivo.

O BO3MO~OlOCTH Koa~anca ceepxnoeofi, xapagTep~ymmeroca ~yMa n3p~maMn.

Pe3mMe (*). - - B pa6ore o6cy~aeTcH BO3MO~a~ocr~ KO~Uianca cBepxHoBo~ SN 1987a, xapaKxeprt3yIomerocH asyM~ BapbISaMH a HMeIomero oco6eHaocrrt, 3aperacTpHpoBanabm Ha MOH BHane a B KaMnOKaHIIe. Cor~acno ~CTa~dlapTHbIM~ MO~eHHM Ko~anca HeBpaII1aIOIIg~CH 3Be3~, KOTOpI,Ie npe1IcKa3bIBaiOT 06pa30BaHlle HeI~TpI4HHOI~I cqbep~I c HeBBIpOTK/IeHtlbIMH HeI~ITpHItHblM ra3oM BHyTpI4 3Be3/lhl, ~arnabm Ha MOH BHaRe rt B KaM~OKaHae aTm nepBo~ BCnbtmrai aamT CnnmKOM 60nbmym Maccy 3Be3JIbI. HanpoTrm, Bbtpomaenn~n~ He~Tpmarmn~ ran c Hn3KO~ TeMnepaTypofi T ~ 0 . 5 M a B n XnMrrqecKnM rIoTeat~a.rlOM /z ~ 12 + 15 npe~icKa31,maeT BI,Ixo~ OTrlocrrreHi, no MRTIOI~I HOYIHOIVI ~Heprrn4 W ~ ( 2 + 6 ) . 10 s4 ~pr n Ma,rioe ~mcsio O~KrtaaeMi,LX s3a~Moae~cTsm~I B KasmoKarxae. IIpeaHaraeTcH SO3MOgm, n~ caeHapm~: MaecvmHa~ (M~2OMo) spataammaacH 3ee3aa pa3zezaereH Ha abe qacTn, O~Ha zerxa~ H apyra~ Tageza~, s Haqaze xoamanca. MaccnBHa~ KOMnoaeHra KoHHancapyeT B aepmT~o a~xpy a npon3soaaT nepBym BCn~,~tuKy. He~TpaHO 3axBarbmalOTeH SHyTpn KoHzancapymme~ 3Be3a~ sczeaeTsne ynpyroro pacce~Hrm SO Bnettmefi O6OHO~IKe Ha T~)KeZ~,tX a;tpax c A>~300. IIoKa3~,maeTcH, qTO He~Tprmo 3aIIO~lI-IgIOT KBaHTOBbIe COCTONHH~I, o6pa3ya 8~ipo~ewa~,n~ HefiTprnmh~fi ra3. BTopa~ scnbttUKa o6~,acrmeTcH Koazee~emme~ smrxoro qbparMenTa (M = (1 + 3) M| c MaCCaBHOfi qepnofi abIpOfi. B p e ~ 3anaaabmaHrm MeT, qly ~IByM~ na6~io~leHrmtMH BCIlbIIIIKa/vn, I (4.7 qaca) B OCHOanOM caHasiBaeTcH c rpaBnTataiOHn~IM TOpMO)KeHneM, Korea HerKa~ qbparMeHT naaaeT Ha qepnyIO a~,tpy, qTO conpoao~aeTcg a3Hy~eaaeM rpaBnTatmOHn~IX BOJIH B TeqenHe BpeMenlI, nop~jIKa HeCKOHt,KHX qaCOB.

(*) Hepeaec)eno peOata~uegt.

On the possibility of two-bang collapse

Documents

Transcript of On the possibility of two-bang collapse