NEUTRINO OSCILLATION AND STELLAR COLLAPSE

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NEUTRINO OSCILLATION AND STELLAR COLLAPSE

A. Pérez-Canyellas, J. Bernabéu

To cite this version:

A. Pérez-Canyellas, J. Bernabéu. NEUTRINO OSCILLATION AND STELLAR COLLAPSE. Journal de Physique Colloques, 1984, 45 (C3), pp.C3-143-C3-146. �10.1051/jphyscol:1984327�. �jpa-00224041�

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JOURNAL DE PHYSIQUE

Colloque C3, suppl6ment au n03, Tome 45, mars 1984 page C3-143

NEUTRINO OSCILLATION AND STELLAR COLLAPSE

A. P6rez-Canyellas and J. Bernab6u

Department o f Theoretical Physics, University o f Valencia, Spain

RESUME

Les effets des oscillations de neutrinos sur 116quation d16tat des supernovae sont 6tudi6s 2 l'aide d'arguments simples. Leur influence sur la dynamique de lleffrondrement est analysge. Les r6sul- tats les plus importants concernent le flux de neutrinos qui apparaxt presque proportionnel au nombre de g6n6rations participant aux oscillations.

ABSTRACT

The effects of neutrino oscillations on the equation of state of supernova matter are explored with simple arguments. Its influence on the dynamics of the collapse is analyzed. We find the most dra- matic results in the numerical flux of neutrinos, which appears almost proportional to the number of generations participating in the oscillation.

1. INTRODUCTION

One of the most important ingredients in the general scenario of the stellar collapse is the acceptance that neutrinos remain practically trapped during the collapse1, the so-called neutrino trapping..

The consequences of this trapping on the thermo9namical behaviour of the collapsing core are well k n 0 w n ~ 9 ~ . On the other hand, as long as the densities do not reach values of the order of nuclear matter densities, the pressure is determined mainly by the leptons, which are degenerate, whereas the nucleons remain essentially confined within the nuclei.

An important question is the neutrino flux towards the outer region of the core, on which the problem of neutrino oscillations could show some important modifications. It is not immediate to understand the implications of this phenomenon on the outer shells of the star, even if the flux of neutrinos changes dramatically as suggested in this work. The reason is that the energy spectrum of the scaping neutrinos has not been determined by us. This would require a more detailed study of the diffusion equation.

2.NEUTRINO OSCILLATIONS AND DISTRIBUTIONS

Neutrino oscillations appear as a possibility when neutrinos are massive4y5. In this case, the states with definite flavour, i.e., the ones with well defined properties under weak interactions, do not coincide with the states of definite mass. On the contrary, there will be a non trivial unitary transformation between both sets of states in the form

~ i i > -

-

^(JMi ^luL> ⁽¹⁾

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984327

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C3-144 JOURNAL DE PHYSIQUE

where I &) are the states of definite flavour ( b( 5 Q , p , t,.,J and I M) denotes the states of definite mass mi.

As a consequence, if at time say t = 0 a defirite flavcure state with momentum 3 is produced, after a time t the probability of having a behaviour as a flavour is given by

kP ct, = I &i

Gi

~ ~ ~ ' ' 1 ' ^{( 2 )}

Under the assumption of time reversal invariance

-

p((t) Our problem is the study of the influence of neutrino osci- , one has QP ^((L

i l

!

&

o n

s on their statistical distribution. To explore this, we add the corresponding term in Boltzmann equations

og =

^D,

g

^005,

g

^{( 3 )}

where 1 accounts for the evolution of ] ⁽³^, ^,^t^{) ,}the neutrino statistical distribution, as given by the collisions with mat,ter, essentially nuclei, and Oosc 1 . 1s the contribution of the oscillations

Let us consider an scenario of equilibrium of neutrinos with matter . This situation has sense considering that the characteristic time for equilibrium with matter, independent of the neutrino type, is much smaller than the characteristic time for electron capture, therefore for neutrino production

Under equilibrium, the oscillatory term in Boltzmann equations va- nishes and, if the mixings in a; are large enough and/cr the charac-- teristic times for oscillations are sma.11 enough against 7 4 % , ^the

equilibrium condition implies for all momenta 3 ^{and all}

airs

An immediate result of these conditions is that, for N neutrino generations satisfying these requirements, there will be equal number of neutrinos for all gener~rtions. _- Furthermore. the distributions will be of Fermi type-

€<$,

( 6 ) so that the chemical potential will be the same for all d

3. THE EQUATION OF STATE

The equation of state is given by the leptons, which are ex- t remely relativistic and statistically degenerate, in the form

where P is the pressure, <eu>the mean energy of neutrinos and k y their number per volume unit and, analogously, for the electrons

3 We use the equations for electron captures as given by Bethe et al. ,

modified by the fact of having N neutrino distributions in the form discussed above, The equations are solved numerically, from the border towards the inrter region of neutrino trapping, i.e., for densities &

10 12gr/cm . Only in tha.t region the above ccnsiderations on the equilibrium with matter have sense. We have used. a constant value of the lepton per baryon ratio Y, = C.311. The numerical integration gives the electron fraction per baryon in terms of the density Ye ( P ⁾

This function is monotically decreasing with increasing values of P

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Furthermore, for a given density, Ye is smaller for increasing values of N. The effect of the oscillatior~s has been the increase of electron captures, This can be understood as follows: electron captures are blocked by the presence of electron neutrinos in the Fermi sea; when they oscillate to other neutrino types, there is a partial deblocking and a corresponding increase of the captures.

Apart from these results which refer to the chemical comgo- sition of the core, we need a dynamical configuration at a given time in order to calculate the neutrino flux. This would provide also its influence on the collapse dynamics itself. We shall use the part'cular solution of homoiogous collapse given by Goldreich & Weber ', available in analytic form when the equation of state is

where K is a constant. The equation obtained by us can be written as

p = np"

where 1.31. Therefore, we can accept with enough confidence the value =: i/3.

Our results indicate that the effect of the oscillations on the collapse dynamics is rather minute, around 5% on the radios scale for the same value of the central density. Applying the results obtained for the dynamics, together with those for the chemical composition, we calculate the neutrino flux for different values of N. We obtain that this flux is practically proportional to N.

4. CONCLUSIONS

Neutrino oscillations can have a major role in the stellar collapse if the number N of neutrino generat.ions verifying the assumed equilibrium conditions is large enough. The model used is not probably realistic enough, but should describe the most relevant properties in the range of densities considered here. With the simplified equation of state, we have shown that the dynamics of the collapse is not strongly affected, although it could change the behaviour at larger densities. Independent of this, the important increase in electron captures leads automatically to the conclusion that the neutrino flux in- creases, almost linearly, with the value of N.

With thlS conclusion in mind, it seems that the consideration of this possible effect in the studies of stellar collapse, including larger densities, will have important implications on the later stages

~f the collapse.

REFERENCES

1) W.D. Arnett, Ap. J. 218 (1977) 815

2) D. . Lamb, J.M. Lattimer, C.J. Pethick and D.G.Ravenhal1, Phys.Rev.Lett 41 (1978) 1623

3) K.A. Bethe, G.E. Brown, J , Applegate and J.M. Lattimer, Nuclear Phys. (1979) 487

4) B. Pontecorvo. JETP (Sov. Fiz) 53 (1967) 1717 5) Nbcabibbo , Phys. Lett. 72B (1978) 333

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C3-146 JOURNAL DE PHYSIQUE

6) R.F. Sewyer, D.J. Scalapino and A. Soni; Proceedings of neutrino 79, BERGEN

7) P. Goldreich and S , V . Weber, Ap. J. 238 (1980) 993