Gravitational collapse, neutrinos and supernovae

ANNALS

OF

PHYSICS:

Gravitational

43, 42-73 (1967)

Collapse, ROBERT

Department

of Physics,

Brown

Neutrinos

and

Supernovae*

A. SCHWARTZt University,

Providence,

Rhode

Island

02912

The collapse and explosion of a highly evolved stellar core is treated in the framework of general relativistic hydrodynamics. The hydrodynamic equations for spherically symmetric systems, including radiative transfer by neutrinos, are formulated, and numerical methods for their solution are developed. These methods are then applied to the calculation of the dynamics of an unst.able model stellar core which has been constructed by Chiu. It is shown that the heating of the imploding material by neutrinos from a hot. neutron core is sufficient to reverse the implosion. The subsequent explosion of the envelope is identified with a supernova. I. INTRODUCTION

This paper discussessome new results on the theory of supernova explosions, using the complete general relativistic hydrodynamic equations. Newtonian calculations have suggested a mechanism for the explosion of a supernova, and and have indicated that a relativistic calculation may be necessary (I), (2). In addition argument#son the “issue of principle” of gravitational collapse have suggested that any massive objects, unless numerical coincidences are such that it is able to eject most of its mass, will eventually undergo dynamical relativistic collapse. With this motivation, it was decided to undertake a study of the relativistic collapse and possible explosion as a supernova of a massive star. The observations of supernovae, which have recently been summarized by Zwicky (S), indicate that there exist at least five distinct types of supernovae. The star whose dynamics are discussed in detail here probably corresponds to Zwicky’s type I, although it is not certain, becauseno attempt is made to compute the optical properties of the supernova. Therefore, this paper will make no attempt to identify the resulting supernova explosion with any of the types which have been observed; such identification will depend on a study of the properties of the envelopes of the star, which has not been included in our model. It has long been known that no star of greater than 1.4 solar massescan support itself by electron degeneracy (4), (5), but only recently has the nature of * Based on a thesis submitted by the author in partial for the Ph.D. in Physics, Columbia University, 1966. t Supported in part by NASA grant, NGR-40-002-009. 42

fulfillment

of t,he requirements

SUPERNOVAE

FIG.

1. Initial

models

for pre-supernova

st,ars

t,he instability which occurs when a star exceeds this limit been investigated. The first suggestion of how this collapse was initiated was that of Burbidge, Burbidge, I~owler and Hoyle (5) (hereafter referred to as BBFH) that, at t,emperature of !I’9 = 6, FeS6 was photodissociated into alpha part,icles and neutrons, with a corresponding decrease in the thermal pressure. [See also Fowler and Hoyle (7).] This was enough to make the star collapse t’o densit’ies sufficiently high that elect’rons were captured onto the nuclei, and the collapse to nuclear ckwsities was on. Another viewpoint on the supernova instability has recently been advanced by Chiu (8). He has emphasized the fact t’hat a highly evolved star, in the stage of carbon-burning or later, emits most of its luminosity in the form of neutrinos. The neutrino star has a very different structure from an ordinary phot,on star, as he has shown in detailed numerical integrations. The models which result, are more centrally condensed and cooler (and thus more degenerate) than the models proposed by BBFH. The run of temperature and density in t,wo such models, for stellar cores of mass 10 and 2.5 solar masses. is shown in Fig. 1. The evolution of the 2.5 solar mass model is the subject of this paper; the comput,at,ion of the evolution of the 10 solar-mass core is in progress and will be the subject of a future publication. After the suggestion of BBFH that the iron-helium transition would serve to trigger the supernova collapse, Colgate and White (I) performed a numerical calculation of the hydrodynamics of a supernova explosion, and demonstrated

44

SCHWARTZ

that it was possible for energy to be transferred by neutrinos in sufficient amounts to eject most of the mass of the star. Later calculations by Arnett (2) have confirmed the conclusions of Colgate and White for less massive supernovae (less than 5. although there are indications that the energy transfer is less effective for more massive stars (9). These calculations have indicated that the remnant of a supernova explosion probably is sufficiently massive and dense for general relativity to have important effects on its dynamics. I therefore have developed a numerical method for treating hydrodynamical problems in general relativity, and applied it to the supernova problem. Although many discussions have been published of hydrostatic equilibrium calculations (IO)-(12) and of small oscillations about equilibrium (13)-(15), but the only attempts at dynamical calculations have been either special solutions (13) or studies of physically unrealistic models (15). In order to study realistic examples of gravitational collapse to relativistic densities, it is first necessary to formulate a fully relativistic version of the equations of hydrodynamics. II.

RELATIVISTIC

HYDRODYNAMICS

The relativistic hydrodynamic equations, including dissipative processes such as heat conduction or radiation, have been discussed by Landau and Lifshitz (17). They are obtained from the special relativity equations by replacing the ordinary derivatives in those equations with covariant derivatives. If TwY is the stress-energy tensor of the fluid, and u’ its four-velocity, then the equations of Landau and Lifshitz are T,“; y = 0 CPU”); P = 0

( “Euler

Equation”),

( “Equation

of Continuity”).

(1) (2)

Here, p is the proper density of particles, or, strictly speaking, baryons, which we know to be conserved in all processes. Since mass is not conserved in special or general relativity, the equation of continuity must involve only the particle number. The units adopted in this section are G = c = 1. It is easily verified that these equations in fact reduce to the correct nonrelativistic limits. In order to put these equations in a form which is suitable for numerical integration, we must first choose a convenient coordinate system in which to express them. In order to exploit the assumed spherical symmetry of the problem. we will choose a metric (the most general spherically symmetric one) cl.? = -e”

dt2 + (R’/y)’

dp’ + R2( de2 + sin” 0 dqi2).

(3)

The coordinate p is a Lagrangean coordinate, or comoving coordinate as it is usually called in relativity. The four-velocity in this coordinate system is thus u = (1, 0, 0, 0) by spherical symmetry. The prime (‘) on any quantity stands for alap, and a dot over the quantity means a/at. In the metric above, if cp = 0

x5

SUPERNOVAE

and y = 1, the Lorentz metric results. In the general case, the function R( t, p j is not a radius, but a “circumferential” variable; t,he circumference of a circle is 21rR. Since the fluid is at rest’ in this frame, the stress-energy tensor may be writt,en as T” = t ” + ATVY where 1.-------!;----;I t’l” zz

(41

0

I /O

I

P

/ I’J

is the isotropic stress-energy tensor of the fluid, and AT”” is t)he anisotropic portion which represents the energy transfer. In the above equations, p is the number density of particles in the fluid, multiplied by the proton mass to put it in t#he correct units, c is the total internal energy, excluding rest mass, per gram, and P is the pressure. The term includes the isotropic portion of the neutrino pressure, as well as all other sources of pressure (which are, of course, isot’ropic). The form of the anisotropic part of T’” must be the same as t,he special rel:ltivity form, since we are working in the rest frame of the fluid, and it is alway:: possible to instantaneously transform away the gravitational field. WC may write f RAE i

9 _---I-_---------_--

AT”” =

; 1

/ AI’,,

q / I

1 0

API

0i

(.-,I

AP,]

in this case, AC is the energy density of the anisot,ropic radiation, (1 is the flux of energy, and AP,, and API are respectively the components of the anisotropic pressure parallel and perpendicular to the vector q. If we assume that the cnergJ transfer is purely radial, then g,, = (0, p, 0, 0). The form of 4 is uniquely determined by the second law of thermodynamics, i.e., by the requirement that the entropy produced by the dissipat#ive term be positive. If we assume in addition that the heat flux is proport,ional only to the temperature gradient, and to no higher derivatives, it was demonstrated hy Eckart (18) that 4 must be given by b” = -x(g’”

+ u”u”)(T,v

+ Tu,;aua),

((ii

where T is the temperature. The term T,, is just the classical term in Fourier’s law of heat conduction. The second term is a relativistic one, and is due to the fact that heat too has inertia. Its interpretation becomes clear if one considers the

46

SCHWARTZ

following situation: In a constant gravitational field, let there be two bodies at different gravitational potentials, which are in thermal equilibrium with each other, and are held stationary. If they exchange energy by means of photons, it is easy to see that, the condition for equilibrium must be T(~oo)“* = constant, and not merely T = constant, because of the gravitational red-shift of the photons. It is easily verified that the second term in the relativistic heat-diffusion law (6) leads to just this result. If we define the luminosity, L, to be L = 4rR2q, then the first law of thermodynamics, whose relativistic form is u,( T”; “) = 0,

(7)

becomes

(8) where we define t,he comoving proper time derivative

Dt = e-‘d/dt.

(9)

In order to determine L, we must look at the classical limit, which gives q = -XVT. Since we have already decided to keep only first derivatives of T, this means using the ordinary diffusion approximation from the theory of stellar interiors, which gives x = $$acT3/Kp, where 1 -= K

m 1 dB, dv s0 --KY aT

dB, aT Ob

(10)

This is the usual Rosseland mean opacity, and a is the Stefan-Boltzmann constant in the case of photon transport. In this case, however, we are concerned with transport by neutrinos, not photons, so there must be a few minor modifications made in this equation. The factor a must be changed to Qa, since the energy density of a neutrino gas is not, aT4, but 3gaT4. The equation for the Rosseland mean opacity is still correct,, provided one uses, instead of the Plan& distribution, the FermikDirac distribution for BY( T). Since the opacities for neutrinos and antineutrinos are, in general different, the Rosseland mean must be taken as the average of that for the neutrinos and the antineutrinos. These changes are discussed further in Appendix B. The hydrodynamic equations may now be obtained from the “Euler equations” (1) the equation of continuity, (2) and the Einstein equations, RF’ = SaT”“. The derivation has been given by Misner and Sharp (19), (90). The discussion below follows theirs. If we define the Lagrangean coordinate /L to be proportional to the number

SUPERNOVAE

of baryons

enclosed,

then the law

of conservation

of baryons,

T‘ = -hrpR2R’.

r=lrl With the additional

47

( 2) beconws ( 11)

definitions 111’ = (1-t

tjr

u = DtR

(121

w = 1 + E + P;‘p the 220sEinstein equation symmetry j becomes

(which

is the same as the R,+ cquat,ion by the assunwd

In this equation, t#he anisotropic stresses AP have neglected. If we then use the R: Einstein equation to evnluatc D/r, ( 11) the result8 is

In the numerical integration use, instead of p, a variable

of this equation,

and put this int,c)

it was found mow

a = pR2

(aonvenicnt

to

( 1.-I1

In t’erms of LY,( 14) becomes

One of the Euler equations has already nontrivial one because of the symmet’ry.

given us ( S) ; t#hcrc is only one otlw It now read::

cp’ = -I’/plV The luminosit,y,

(17

given by ( 6 j is z

i

=

142a

3

R”

Ii

e-4’

( T4dy )’

(ISI

Given an appropriate equation of &ate, expressing P, and t as a funct~iw (Jf and T, and the correct boundary conditions, these equations ( 8), (11‘)~( 13 i 1 nrd ( 16)-( 1s) are a complete system, suitable for numeriral integration.

p

48

SCHWARTZ

There is still some freedom, however, in defining the time coordinate. This is associated with the fact that we have not yet specified the boundary conditions at the outside of the star. Of course, if one chose a Schwarzschild like time coordinate for this problem, it would be impossible to treat the problem of continued collapse, just as in the P = 0 case, Since we are allowing radiation to take place, it is necessary to match the geometry at the boundary to the Vaidya solution [see Lindquist, Schwartz, and Misner (21) this paper is referred to as LSM]. It would be possible to use the coordinate system of LSM, which is similar to the Kruskal coordinates for the ordinary Schwarzschild metric, to do this fitting, but we would obtain no new information about the interior behavior from this procedure. The geometry is already completely specified by our equations; the only thing that the fitting at this boundary can tell us is what that geometry looks like to an observer who uses a different coordinate system. Since the boundary remains in a classical region (i.e., Gm/c’R << 1) it will suffice for our purposes to pick a time coordinate which is just that of an observer moving along with the matter at the edge of the star. This corresponds to the boundary condition a(po , t) = 1, where ~0 is the total baryon number of the st,ar. The other boundary condition that is needed at the edge of t’he star is that 011 the luminosity. Actually, this boundary condition is not applied at the physical boundary of the star, but rather at the point at which the star becomes optically thin to the neutrinos. Since it is difficult to identify this point, several methods of defining it, and several t,ypes of boundary conditions were tried. These are further discussed in the next section. The boundary conditions at the center of the star are obvious. They must be ,112= 0, L = 0, !C = 1, R = 0, and U = 0. Th e condition r = 1 is a consequence of the ident’ity I’ = 1 + U2 - %1/R, which is just the expression of the Roe Einstein equat’ion in these coordinates. In addition, since there was no attempt to make a detailed treatment of the envelope, the boundary conditions P = 0 and p = 0 were used at the surface. The equations above are the ones used in the numerical integration scheme described in Appendix A. Given the proper initial conditions, they determine completely the evolution of a spherically symmetric body, subject’ only to the condition that the mean free path of the neutrinos be short in the regions in which energy transfer is import’ant. A numerical scheme for the integration of these equations, neglecting heat transfer, has been constructed by May and White (.26). They have applied the method to the calculation of the collapse of a uniform sphere of gas with a polytropic equation of state, in order to investigate the problems of relativistic collapse. The numerical method used in this work is a generalization of that employed by May and White in the adiabatic problem.

49

SUPERNOVAE

III.

THE

SUPERNOVA

MODEL

The equations discussedabove may now be applied to the computation of the evolution of a star at the point of instability. In this work, no attempt is made to construct such a pre-supernova model; instead the results of Chiu on presupernova evolution (8), (22) are used as the starting point’ for the calculation. These models were constructed by an accurate integration of t’he equations of hydrostatic equilibrium. A method has been developed for the integration of the general relativistic hydrodynamic equations by a Lagrangean finite difference method, patt’ernecl after that which has long been used in nonrelativistic problems [Richtmyer, (23)]. The code treat,s shocks by the Von Neumann-Richtmyer artificial viscosit? method, which automatically satisfies the Rankine-Hugoniot conditions, and ensures a constant shock thickness. The mathematical details of the method are discussedin Appendix A. There is one further difficulty which must be overcome before the Chiu models can be used to begin t’he calculation. In the computations of Chiu, an exact equation of st’ate was used for the electrons, t,aking partial degeneracy into account. The method used for Chiu for the evaluat,ion of this equation of statme cannot be used for the dynamical calculation; it’ is so slow that it would increase t,he running time by a factor of 100 or more. It was therefore decided to approximate the electron pressure by taking only the zero-temperature cont)ribution (see Appendix B). Thus, the pressure calculated from this equation of state, at a given temperature and density, is lessthan the true pressure used by Chiu. Thus, t’he models which are in hydrostatic equilibrium according to the true equation of state are not in equilibrium according to the approximate equation of state used in these calculat,ions. Since it is absolutely essential to the correct treat*ment of the dynamics that the original model be in hydrostatic equilibrium, it n-as decided t’o buy this equilibrium at t’he price of getting the temperatures wrong. Before the computation was begun, the temperature of every zone was adjusted by a numerical method so that the pressure of that zone, computed according to the approximate equation of state, was equal to the t#rue pressure, as comput,ed by Chiu. In this procedure, it was found that the error in the temperature t,hat was made was always lessthan 50 %. However, as the collapse goes on, t,hc error is compounded, because the absence of the electron thermal energy term makes the temperature rise more steeply than it should in an adiabatic compression. For radiative or shock heating, the situation is even more serious. It is therefore the opinion of the author that no temperature computed in this work should be trusted to ,&thin a factor of two. In the following, it is very important to keep this in mind, in order not to receive a false impression of the accuracy (Jf the numbers presented.

50

SCHWARTZ

With this caveat, we begin the discussion of the dynamics of the supernova with the treatment of the hydrostatic approach to the supernova instability. In agreement with the ideas of Chiu discussedin Section I, the assumption was made that the pre-supernova core evolves on such a low adiabat (high p, low 2’) that it never reaches the region of the iron-helium phase transition. The collapse is induced by neut’rino emission, which makes the star contract to a density high enough so that the electrons are captured, and the equilibrium composition shifts into the neutron-rich region. The rates of neutrino emission in this stage are discussedin Appendix C. In order to treat this quasi-static evolution with the dynamic equations used here, it is necessary to artificially speed up the evolution, which would otherwise require a prohibitive amount of computing. This was done by the simple artifice of multiplying the neutrino emission rate by a factor of IO’, thus effectively speeding up the evolution rate by this factor. Such a method is justified as long as the evolution remains quasistatic, since the rate of quasistatic evolution is always greater than the rate of energy transfer from one part of the star to another. Once the inertial terms become appreciable, the method becomesinapplicable. It was therefore decided to cut the neutrino emission back from the artificially enhanced value to the correct one as soon as the kinetic energy of any zone became 10% of its gravitational energy. The results of this phase of the calculation are shown in Fig. 2. This represents the state of the star in its last moments of hydrostatic life. The central density has evolved to a lit’tle over lOlo g/cm3, and electron captures are beginning to becomeimportant, pushing the star over into dynamical instability. The inversion of the temperature gradient near the center of the star seemsto be characteristic of the evolution by neutrino loss. It has been found by Chiu (unpublished) in some of his highly evolved hydrostatic models, and seemsto be a real effect. It will be discussedin a fortbcoming publication by Chiu, but has not been investigated in this work. When the star has evolved to the point shown in Fig. 2, it becomesunstable due to electron captures. As soon as the central zone has reached bhe density at which it starts capturing electrons, the neutrino sink is cut back to the normal value. In practice, this is a more stringent control of the 10% kinet’ic energy criterion. At the point of electron capture, the star was still found to be very nearly quasistatic. The effects of the method of initiating this instability were studied by repeating the calculation with varying densities for electron capture (seeAppendix B). Three calculations were made, with electron captures initiated at 6 X lo”, 1 X lOlo, and 6 X lOlo g/cm3. In each case, the number of electrons per cm3 was held constant as the density increased beyond the electron capture point, until the number per gram had fallen to l/1000 of its initial value. Thereafter, the number per gram was held constant. This procedure is discussedfurther in

SUPERNOVAE

100

I

I

IO

FIG.

Y. Configuration

of 2.5. stellar

core

on the verge

of gravitational

collapse

Appendix I3 on the equation of state. It was found that there were no significant differences in the mass, density or temperature of the resulting neutron core in the three cases cited above. In all of the complete calculations, the electron capture threshold n-as assumed to be lOlo g/cm3. From this point on, several assumptions were made about the equation of state and the neutrino processes, and the complete dynamical history of t’he supernova was then computed. All of the models described in the following started from the initial state shown in Fig. 2. III the following, we will give a detailed discussion of the dynamical history of one particular model, that employing the nuclear equation of state developed by- Skyrme and allowing energy transfer by electron-type neutrinos in t~he diffusion approximation, and losses due to muon-type neutrinos, computed according to the formula (C17). Other models which have been computed will then be discussed more briefly, in terms of comparisons with this model. When nuclei in the central zone of the star begin capturing electrons, the pressure defect is so severe that this zone goes essentially into free fall. This leads to a rarefactjion wave which propagates out,ward through the star with the speed of sound. As the rarefaction reaches a given zone, its support is removed, and it too begins moving inward with very nearly the free-fall velocity. The progress of this instability is shown in Figs. 3 and 4. Figure 3 represents a “snapshot” of the

52

SCHWARTZ

5X10S

0

z G -5x108 2 9 -IO9

-1.5X10s

-2 x IO9

0

IO

20

30

40

50

ZONE FIG. 3. Velocity the center of the

profile of star after star to zone 12 (24y0

start of collapse. A rarefaction of the total mass).

has progressed

from

star taken just after the central zone has started to collapse. The rarefaction has reached zone 12 (of 50 zones) ; the velocities of the outer zones represent fluctuations about the equilibrium configurations before the start of the collapse. The inner zones are falling in with very high velocities, because they have been in free fall for a longer time. Figures 5 and 6 are snapshots taken about 3 msec later showing different properties of the star. Figure 5 shows the central zone of the star collapsed to a density of 4 X 1Ol4 g/cm3, which is slightly higher than normal nuclear density. The temperature, in units of mec2, has risen to 90, or nearly 45 MeV, in the center, and slightly higher nearby. At this point, the effect of the degenerate neutron pressure is making itself felt. Figure 6 shows that this pressure has been sufficient to halt the collapse of the first five zones. They are standing almost still, while the outer material rains down upon them, forming a very strong standing shock at the edge of the core. The discontinuity in temperature and density across this shock is seen in Fig. 7. At this point (and this remains a general feature of the problem) the inner 6 zones have become optically thick to neutrinos, while the total optical depth of the remaining outer portion of the star is less than 0.1. The region interior to the core shock is optically thick because of the high temperature and density dependence of the neutrino opacity. As discussed in Appendix C, the cross section for neutrino interaction with matter increases approximately with the square of

SUPERNOVAE

10-l

FIG.

1 IO6

I

I

I

IO6

IO’O

IO

4. Temperature-density

I IO6

configuration

I 12

IO

at the same

14

lOiS

instant

as Fig.

I

I

I

I

I

IO6

IO'O

lOI2

lOI

lOI

3

log p FIG.

5. Temperature-density

configuration,

3 msec

after

Figs.

3 and

4

54

SCHWARTZ

5X10S

0 > k is -5x10* -I 2 -IO9

-l.5xlos

-2x10S 7

0 ZONE

6. Velocity

FIG.

lOI

profile

at the same

I

I

instant.

as Fig.

I

I

30

40

5

lOI

lOI Q H

~ IO'O

IO6

IO" -

u

IO

r

20

50

ZONE FIG.

7. Density

profile,

10 msec

after

Figs.

5 and

6

SUPERNOVAE

.h)

t’he momentum t,ransfer. Thus, for interaction with degenerate ekC~r(Jlls, tdhcl opacity is proportional to the temperature and the Fermi energy; for nondcgenerate matter it is proportional to the square of the temperature. This strong variation of t#he opacity will tend to retain the thermal energy in the shocketl material, and to make the shock front very narrow. Indeed, as seen in Fig. 7, the shock front is only about two zones thick. It’s thickness is being determinctl by the artificial viscosity, which does not allow shocks to be less that a few zonc~ in thickness. The radiative shock width is small by comparison. Nonetheless, there is considerable radiation of neutrinos across the shock. although t,he effect, mentioned above keeps the time scale for neutrino cooling much longer t,han the hydrodynamical time scale. In this calculation, t#he IICI~trino luminosity of t’he core is found to be 1O54ergs/set, or /1,2a solar rest mass per second I ! These neutrinos are emitted with roughly the energy corresponding to blackbody radiation at the t,emperature of the emitting surface, which is just behind the shock front. This temperature is about 10 meV (see Figure 5) at this time, so the mean energy of the neutrinos is about 30 meV. These neut’rinos, even t,hough they are passing t,hrough relat’ively cold matter, have a compar:ttively short mean free path, considerably less than the radius of the star. This means that, they will be absorbed in the cooler outer layers, and deposit most 01 the energy that has been radiated by the shock front. This is the mechanisni b!y which the supernova explodes. The outer layers of the star are not ejected by :1 hydrodynamical shock which propagates out t’hrough the star from the cor(’ bounce. The situation envisaged is then as follows. The core shock acts as RII emit’ting surface, producing high energy neutrinos. These neutrinos are absorbed in t,hr envelope of t#he star, and event’ually heat it up sufficiently so it may expand outward, convert its internal energy into mass motion, and be blown off. ‘I’h(h kinetic energy of the ejected envelope, in this model, comes entirely from t,hr energy deposited by the neutrinos. This is exactly the process which was suggested by Colgate and Whit’e. However, Colgate and White argued that, half the thermal energy of the core was deposited in the envelope; this calculatlion actually computes the transport process, and demonstrat,es thatt it’ works almost exactly as Colgate and White suggest,ed. In order to solve the transport problem in the optically thin region CJutsick the core, an approximation was used which was the opposite of the diffusion approximation. In the diffusion approximation, it is assumed that the source function for the radiation is given by t’he equilibrium value. In the optically thin region, it is assumed that the source function is zero. This is equivalent to saying that, there is no emission at all, and that once a neutrino is scattered or absorved, it is thereafter thrown out of the problem. Thus, the transport problem is reduced to a simple exponential attenuation of a beam. The mean free pat,h

56

SCHWARTZ

I

5x10S -

0 z 3 sw -5x10* >

--

-

-

-IO9 -

-l.5x10S -

-2x10g

0

I IO

I

!

I

20

30

40

-.I

50

ZONE

FIG. 8. Velocity

profile, at the same instant

as Fig. 7

used in the exponential is that characteristic of the high energy neutrinos from the emitting surface. This method is similar to the Schuster-Schwarzschild approximation in the theory of stellar atmospheres. This cavalier treatment of a subtle and difficult problem is far from a rigorous solution, but it is hoped that it will suffice to show the plausibility of the supernova mechanism being investigated. The progress of this neutrino heating is shown in Figs. 7-9, which are snapshots of the star taken 10 msec later than Figs. 5 and 6. In this seriesof pictures, one may observe that the neutrino heating has had a sufficient effect on the outer layers to begin to propel1 them outward (zones 13-25 in Fig. 8). The core shock, which has now progressed to zone 9, is stiI1 quite strong, furnishing a neutrino luminosity of 9 X 1O32g/set or 8 X 1O53erg/set. Figure 7, which gives the density distribution of the star, shows the situation. The innermost ten zones have a density distribution which is characteristic of a neutron star. The central density is 6.0 X 1OL4g/cm3, and the mass of the now almost static neutron core is 0.5 solar masses.Another 0.1 solar mass of material is still in free fall, and will eventually join the neutron star. The remainder of the star has enough internal energy to give itself escape velocity, and will be ejected in the supernova explosion. Shortly after this point, the computation was halted. No attempt was made to follow the actual explosion of the envelope, for reasons explained below. In summary, then, we may say that the model discussedhere gives some hope

57

SUPERNOVAE

log p FIG.

9. Temperature-density

configuration,

at the same

instant

as Figs.

7 and 8

that a type I supernova may actually be exploded by neutrinos in the manner described. The calculation above shows a star with an evolved core of 2.5 solar masses, which collapses, explodes, and leaves a neutron star remnant of about 0.6 solar masses. The mass of the remnant is well under the Oppenheimer-Volkoff limit, so it is expected to remain as a stable neutron star. The model discussed here is only one of several whose behavior has been investigated is detail. A calculation exactly like that described above was also performed using the same initial model, but with an ideal Fermi gas equation of state for the nuclear matter. The results were almost the same as for the case described, and differed only slightly in numerical detail. It is not worthwhile to present the results of this calculation here, since the numerical differences b(btween the two models are less than the uncertainties in the calculation. In order to assess the effects of mu-neutrino emission on the calculation, the two models above were also recomputed without any losses due to mu-neutrinos. Again, the results were qualitatively similar, but, as expected, the mass of the neutron star remnant was smaller and its temperature higher in the latter case. IV.

RESULTS

AND

CONCLUSIONS

The calculation described above has omitted many effects which are of great interest in the problem of supernovae. The most important of these omissions is, of course, the neglect of nuclear physics. which has important effects on t*he dy-

58

SCHWARTZ

namics of the exploding envelope and on the equation of state in the electroncapture region. However, the complexity of the nuclear physics makes it impossible to treat as being coupled to the dynamics; one must calculate the dynamics as well as possible, get the temperature-density history of each zone, and then attempt to treat the nuclear problem. Although this type of calculation is of enormous importance, the temperatures obtained in t’his work are far too inaccurate to be useful in such a problem. Realistic calculations of the r-process in supernovae will have to wait until a more accurate treatment of the equation of state and the neutrino transfer problem is avaiIable. The results obtained here are, however, of sufficient interest to justify further attempts in this direction. In the preceding discussion, there has been no mention at all of general relativity. This is not an oversight; it is simply that the calculation never showed any relativistic effects to be important. Although the method used was one which took full account of all relativistic effects from the outset, the anticipation that this model would enter the relativistic regime was not borne out. This does not mean that the notion of relativistic collapse is completely irrelevant to supernovae; it merely indicates that the numerical peculiarities of the model chosen were such that it remained nonrelativistic. In an attempt to treat a relativistic problem, some experimental computations were made on a stellar core of 10 solar masses, the model of which had also been constructed by Chiu. The computation was begun in the same fashion as the 2.5 solar-mass case, and the star was evolved by neutrino emission until the onset of collapse. Since the collapse of this star also began in the center, a neutron core of approximately the same size as that found in the lighter star was expected. This is the point at which the computation ran into trouble. In the small star, the core initially covered four or five mass zones (Figure 6). In the large star, the same amount of mass was enclosed by the first zone alone, since the star was also divided into 50 mass zones. Thus, in order to follow the dynamics of such a star, it would be necessary to use much finer zoning in the core of the star. However, this is just the region of highest sound speed, so that the restriction on the time step placed by the Courant-Friedrichs-Lewy stability requirement would increase the computing time prohibitively. The problem was therefore abandoned, pending the availability of more powerful computers. The work contained in this paper is of a preliminary nature, and is obviously far from presenting a complete theory of the supernova phenomenon. The conclusions that can be drawn from this work are in the nature of encouragement to investigate the mechanism further, rather than definitive results. It is felt that further investigation of the model of supernovae discussed here will lead to an eventual understanding of this most important phenomenon. A practical method for the calculation of spherically symmetric hydrodynamical problems in General Relativity has been demonstrated, and there is hope that its further application will lead to a better understanding of the supernovae phenomenon.

*5!)

SUPERNOVAE

ACKNOWLEDGMENT

The author would like to thank all of those without whose aid and encouragement this work would not have been possible. In particular, my advisor, Hong-Yee Chiu, has been a source of invaluable information and insight. Stirling Colgate, who suggested this problem, also deserves special thanks. I am indebted to W. D. Arnett, A. G. W. Cameron, R. W. Lindquist, M. M. May, C. W. Misner, E. Schatzman, S. Tsuruta, J. A. Wheeler, R. H. White and many others too numerous to mention for valuable conversations. I am especially grateful for the assistance of Harry Pollack, who did much of the programming, and for the hospitality of the Goddard Institute for Space Studies. APPENDIX

A. NUMERICXL

METHODS

ln this appendix, we will first discuss the finite-difference methods used for the hydrodynamic equations, and then indicate the i comparitively simple) modifications made to include neutrino diffusion. The differential equations, discussedin Section II, are

DtR Dt(ln

= u,

a)

=

d/R’,

D, T = -(P

+ ac/a(l/p))ut

(f)/-$,

,,I’ = (1 + c)r, $9’ = -P’/pW, I‘ = &rpR’R’, where

D, = eCa/dt

’ = a/&,

a E e’

In addition to these equations, an equation of state is required to supply P, t, &/a( l/p) and &/aT as functions of p and T. The finite-difference scheme used for these equations is patterned after that which Colgate and White (1) and Arnett (2) have found useful in the nonrelativistic supernova problem. The star is divided into J zones; the first, zone represents the center of the star, and the Jth the outer boundary. Certain of the yuantities are defined in the centers of zones, and others at the zone boundaries, so in reality there are 2 J zones. The present method, however, only requires the storage of J values of each quantity in the computer’s memory. The centering of the principal quantities is shown in Fig. 10. The “mechanical” quantities, R and U are associated with zone boundaries. The “thermodynamic” quantities, p, P, E and T are to be associated with zone centers. If we pick t,hc

GO

SCHWARTZ

FIG. 10. Schematic view of zoning method for finite-difference equations. The star is divided into shells, each with radius Rj" and velocity U/-“2. Between the jth and j + ls” shells, the thermodynamic variables are defined for the gas between these shells.

space centering of these quantities then the centering of all the quantities derived from them is also fixed. The code also treats shocks automatically by the Von Neumann-Richtmyer artificial viscosity method [Von Neumann (,@), Richtmyer (%?)I. This term was added to the finite-difference equations after experience with the unmodified equations showed that oscillations of two zones in wavelength and large, but bounded amplitude developed. Since a problem with J zones is really equivalent, to approximating the gas by one with J molecules, Von Neumann and Richtmyer interpreted these oscillations as representing heat. Thus they introduced an artificial pressure, Q, into the equations, which has the function of feeding the energy of these oscillations into heat. If the Q-term has the form

Q = UO~P@U)*,

(as>

the thickness of the shock is independent of the shock strength. In practice, 2 = 2 was chosen, and the shockswere -34 zones thick. In the hydrodynamical ziuations, P is replaced everywhere by (P + Q) . S’mce the velocity is centered in spaceat zone boundaries, andin time at half time-steps, the momentum equation,

(iI

SUPERNOVAE

(Al ) , must, be replaced by n+1/2

ui

-

Uj

n-112

ajnfin

(A9)

-n+l/2U~+l!e CLj 3

(AlO)

=

At”

The posit,ion equation

(AZ),

then must be Rlfl

_

R,j =

Atn+l/2

Then, for proper fj”

_

centering ‘;;/$jn

of (A9),

da(Rj”)2

‘;++liz

we must have

HZ.‘~ 3 + 4dRi7’):’ (@)”

A;jp;-1/2

3 Here, a j~..!pk,~ = aj In the (A

p,,, .’

(All

i

Glde (-) over any quantity denotes a space average; e.g. p17L= + p~--l,2). A bar over a quantity denot,es a time extrapolation, e.g. ajn + $$(At,/At,-l)(ajn - aj”-‘). same way, the difference equations for (A3 ) and (A4) become

7,+1/z 111 a),;+1/2

=

zn+l/? aj+l/2

n+1/4 Uj+l ,;$I2

n+1/2

-

Uj

_

~gfllz

At”,

(

x

{ llP$l$

-

Al”)

l,‘P;+l/z).

We notice that Q is not properly centered in (A14), but we will neglect this, because Q is not a physical quantity, and its definition is somewhat arbitrary. After this sweep through the net, from center to outside, it is necessary to sweep through the net in the opposite direcbion in order to ir1t’egrat.e (A(i) ; (Al51 and to compute rl”&

=

;

p;:;,2[(R;32

+

Rj”=1’R:+’

+ (RS+‘?](Rjn++:.

(AlA)

- R;+‘)

and n+1 712j

=

7?1;2 + (1 + &:3)ri;+lliq

The procedure is now finished; it now suffices to choose a new begin again, computing the next time st,ep.

(‘4171 At,

and to

62

SCHWARTZ

The choice of time step is determined by the criteria of both accuracy and stability. The first is somewhat arbitrary, but, in line with usual practice, the time steps were set by not allowing the density or internal energy of any zone change by more than 2% in any step. The rigorous analysis of the stability of a highly nonlinear system such as this is beyond present mathematical capabilities, but a criterion can be derived from the heuristic Van Neumann approach. Following the method used by Richtmyer in the nonrelativistic case, we arrive at (AlS) for a region in which there are no shocks. recognized as the CourantFriedrichs-Lewy c, = (dP/dpy2

In the nonrelativistic condition, where

case, this is

is the sound speed.

In a region of strong shock, the equations become hyperbolic, and the stability criterion will be different from the above. In practice, the “Courant” time step ( AlS) proved satisfactory in shock calculations. In real cases, the time step was the minimum of 0.2 times the step ( AlS) and those determined by the density and internal energy controls. The internal energy condition was found to assure stability in problems where strong shocks were present. The accuracy of the computational scheme was checked by testing it on three different types of problems. The first was the free-fall problem, or the Friedmann universe. An initially stationary homogeneous model with zero pressure was used, and its collapse was calculated. This served principally as a check on the difference equation (Ala) for the density; if the density did not remain uniform throughout the collapse, this equation was at fault. In practice, many different equations for advancing the density were tried, and the method (Al2) was selected because it gave the best performance in this test. In order to serve as a check on the equation (A9) for the velocity, hydrostatic equilibrium models were used. This served the purpose of demonstrating that the code was able to reproduce small oscillations about equilibrium without the introduction of any numerical instability, as well as that of initiating t,he dynamical calculation (see Section III). Finally, the method was tested for its ability to reproduce several analytic shock-wave solutions. A similarity solution has been given by Taylor [see Courant and Friedrichs (!X)] for the problem of a spherically symmetric strong shock in a polytropic medium. If the shock is assumed to start at the origin at t = 0, then the position of the shock front is given by Y, = const X t2’jp-“j. Numerical computations indicate that this condition is satisfied to better than 5%.

63

SUPEHNOVAE

The code was also tested on a generalization of the Taylor blast-wave solution for shocks in a density gradient. The solution, given by Sedov (,26) is for a . ’ medium of density p = pOrVw. Th e position of the shock front is then f-, = con&. X t2’(5--o). With zones of equal mass, it was found that five zones pei scale height sufficed to reproduce the solution with w = 3 and 4. The equations now must be modified to include radiative diffusion. The equations of mot’ion, including the diffusion equation, are now Dt(ln DIT

=

-((P

+

aj = U’/R’ & +

- L/R,

d~/dp-')Il~(p-~)

+

eP(

Le")'),

i X3

j

( A4'

)

The remaining equations, (Al)-(AS) are the same. In order to write a difference equation for (A19), we note that t,he derivative (T”e”‘) is naturally centered at time n + 1 and space-step j. The luminosity will therefore be centered this way, and the difference equation becomes

where we have used (A15) (A-I’) is then

t,v eva1uat.e ‘p’. The difference

where ATo is the expression for ATni’12 ,+I,2 appearing Con, (A3’) must therefore be

equation

to replace

in i&414). The density

equa-

The method for solving these equations is as follows. Start,ing from the cent,ei of Dhe star, evaluate the expressions (A9), (AlO), ( h22), (A21) and (,A20) in that order, for each zone. This procedure advances all of the relevant quantities from time step n to 12 + 1. Then, the initial-value equations, (A15), (Al6 1, and (A17) must be evaluated. Finally, with the choice of a new time step, the procedure is finished, and we are ready to progress to step n + 2. The presence of the diffusion equation in the system, however, introduces another time st,ep control in addition to those discussed above. Since we arc

64

SCHWARTZ

using an explicit difference method for the diffusion equation, the time step is also limited by the stability requirement for it as well as for the hydrodynamic equations, It is impossible to perform a stability analysis for the coupled system, even in the linear approximation. However, for the linearized diffusion equation, aT/dt = D( a’T/a?), the stability criterion is [Richtmyer, (2S)] 2DAt/(Ar)’ Thus, we are led to the requirement,

This time control Courant condition,

proved, (AN).

for our system,

in practice,

APPENDIX

5 1.

A23 that

to be much less restrictive

than the

B. EQUATION OF STATE

The task of this appendix is to derive a crude analytic form for the equation of state of hot, dense matter. The most important demand of the equation of state is that it be expressed in a form which can be rapidly evaluated by an electronic computer, since this must be done over a million times during the calculation of each model. It is thus impossible to have any numerical integrations, searches through tables, or any other time-consuming procedures in the final equation of state. The reader is therefore asked to pardon the travesties on physics which will be committed below in the name of computational necessity. We will discussthe three regions of the equation of state separately, and indicate the approximations made in each. The first region is the low density regime, in which the electron Fermi level is so low as to have no effect on the nuclear physics. The second is the region of electron captures, in which the nuclei are becoming more and more neutron-rich as the density increases,finally forming an almost pure neutron gas. The third domain is that of the extreme densities that are usually characterized by the term “nuclear matter.” The low-density regime is the only one that is susceptible to exact treatment without any detailed knowledge of nuclear properties. The pressure here is given by three terms. The first two are trivial to evaluate; they are the ideal gas pressure of the nuclei and the radiation pressure. The electrons, however, are partially degenerate in the region of interest. In addition, the temperature is of the order of the electron rest mass, so pair creation must be taken into account. Chiu has developed a numerical method for evaluating the electron contribution under these conditions, but it was agreed that the use of his method would increase the computing time necessary for each problem a hundredfold, and thus be completely impractical. The situation is saved by the fact that the initial model starts out on a very

65

SUPERNOVAE

low adiabat, with the Fermi energy of the electrons at the center of the star being about 1OkT. Thus, one might hope that the approximation of perfectl! degenerate electrons would not be too bad. The degenerate elect)ron pressure is calculated by Chandrasekhar (5) to be

(2(2x2while the energy density

3)(x2 + 1Y”

+ 3 sinh--’ n:]

of the degenerate electrons

is

where .r = pfrm,i!jtzC. The total pressure in this region is thus P = PO + ,!iaT’ and the energy density

(per gram)

f RgpT//p,

133

is

e = UO + aT4/p + 3jR,l’/c(,.

134

In order to treat the equation of state in the region of electron captures prop erly, it would be necessary to make a calculation far longer t,han t*his entire paper! It would involve knowledge of not only the ground states, but the excited st$att?s of all the nuclei, and all of the nonequilibrium subtleties of t.he entire ,*-process. However, following the completely degenerate treatment of the previous scction, these difficulties too will be neglected. The dominant source of pressure in this region is still the electrons, so for our purposes the nuclear composition is irrelevant. The only thing that counts is the total number of elect,rons which are still around, and, since we have assumed they are at absolute zero anyway, t,he number density will be calculated by the same assumption. This problem has already been solved by Tsuruta (11) who calculated the composition of zero temperature matter from a density of lo’-lo’* g/cma. The results of these cdculations are shown in Fig. 11. (I am specially indebted t,o Dr. S. Tsuruta for permission to reproduce this figure.) At point (a) on Figure 11, the electron Fermi level has risen so high that it is equal to the neutron-proton mass difference in 4He, the nucleus most resistant to electron capture. At this point, a further increase in density leads only to thr capture of more electrons; the Fermi level remains the same. The electron captures start at, 10” g/cm3 and continue over approximately three decades of density. At 1014 g/cmY, the electrons make a negligible contribution t.o the total pressure, and may safely be neglected. The radiation pressure and nuclear pressure were also retained in this region. In this region, in addition to the three terms already discussed, the pressure of the degenerate neutrons makes a significant contribution There has been

66

SCH

COMPOSITION COMPOSITE

6

8

IO LOG

WAKTZ

DISTRIBUTION EQUATION OF

12 DENSITY

I4 (gm/cm3

FOR STATE

I6

18 )

11. Composition for an equation of state for non-interacting perature. Electrons are captured onto nuclei at point (a); Electron at (b), at which point all nuclei are dissociated into free protons these points, the nllmber density of electrons is almost constant. FIG.

20

particles at zero temcaptures are completed and electrons. Between

much discussion in the literature about the equation of state at and beyond nuclear density [see Chiu (22), Salpeter (27), Tsuruta (ll), Cameron (12) and HTWW). On the basis of different assumptions about nuclear forces, many authors have derived equations of state which have the same type of qualitative behavior, but differ in quantitative detail. The effect of the nuclear forces is to lower the pressure below that of a noninteracting Fermi gas at densities near normal nuclear density (a.9 X 1014g/cm3)and, because of the nucleon repulsion at very short distances, to increase t’he pressure above the noninteracting value at densities somewhat above nuclear density. The author believes that it is im-

tii

SUPERNOVAE

possible to choose any of these equations of state present ignorance of both the strong interactions As a representative of such equations of state, the for no better reason than t,hat it is expressed in a for rapid computation. According t)o Skyrme (28) the internal energy en = 7.98 x l&Y and the corresponding p,

as the correct one, given our and of many-body theory. Skyrme equation was chosen, simple analytic form suitable per gram of a neutron

+ 9.79 x 10-“p”:8 -

I.:#

gas is

x 1o”p

pressure is thus

= ;i.:sa x 109pa’3 + 1.63 x lom--“p8:” -

1.&S x lojp’?.

13;

These forms were used in the equation of state, both in the electron-captu1.e region, and in the region of high ( 2 1014) densities, where the electron number was assumed to be lo-” (Jf its value at the onset of electron capture. In an effort to estimate the effect, of the nuclear potential, computjations were also made using an ideal Fermi-gas equation of st,ate for the neutrons. This corresponds to retaining only the first term in (B.5) and ( BC,). Thus, in the electron capture region and in t’he nuclear matter region, the J)WSsure is given by I’ = PO + l’, , where POis the expression ( BZ) and I-‘, is the neutron pressure (BS). In the electron-capture region, the E’ermi level of the electrons is assumedt,o remain at 21 MeV over a range of three decadesin density. Thereafter, it, increaseswith the $$$power of the density. APPENDIX

C.

NEUTRINO PROCESSES

The effect of neutrinos on presupernova evolution has been discussedby Chin [(YE%‘),and to be published] and by P’owler and Hoyle ( 28). These aut,hors COW elude that the pair annihilation neutrino process ( Chiu (29 )] and the plasmorl process [Bdams, Ruderman and Woo (30)] are the most, import8ant processesin dissipating the star’s energy. The former is most important in t#hehot)ter region while the plasmon process is larger in regimes of high degeneracy. Chiu ( ‘I(i) has given analytic forms which approximate the nrut,rino emission due to thc>se processesin the pre-supernova phase. The plasmon rate is Qglasmon

?z

1.1 ( 2’,)“p ergs:/g!sec*.

j (‘I 1

while the energy loss due to the pair annihilat~ion neutrinos is Q Epair 1.3 X lo”( T!,j’/p ergs,‘g:‘sec*.

(C’2j

These rates were used in the initial stages of the c*ompntation, bcforc the initic tion of the dynamical collapse. In order t)o treat the problem of radiative transfer by neutrinos, we must POIIxider separately t,he transport equat8ionsfor neutrinos and antincutrinos, sincac

68

SCHWARTZ

their interactions with matter are different. This is slightly more complicated than the problem for photons, in which the polarization can be neglected in most stellar problems. The equation of transfer is (31)

--i ar,’ c

at

+ w*VIwh = - Ko’p( 1 + PXT)lof -N,I.=o,i[l-SKcos~d~]+J,’

(c3)

where the symbols have the usual meanings, and the labels + and - refer to neutrinos and antineutrinos respectively. The only difference from the corresponding equation for photons is that the expression (1 - e-hw’M) which appears in the photon equation becomes (1 + e--hw’kT) for the neutrinos. This is merely due to the change from Bose to Fermi statistics. For a derivation of this equation, see Chiu (31). The first term on the right-hand side of this equation represents absorbtion and emission of neutrinos, the second scattering, principally by free electrons. The phase function for this scattering is just (1 + cos’ {), so that the integral J K( ,t) cos E dfi will vanish in the cases considered. We will make the usual assumptions of the theory of stellar interiors in order to solve this equationapproximately. They are time independence (al,/at = 0), local thermodynamic equilibrium (Jw has its equilibrium value) and the diffusion approximation (keeping only first order departures from the equilibrium values). With these assumptions, plus the assumption that the temperature gradient is purely radial, the transport equations become

aI( w aT

aT - = (KLf + KY,*) pp*

)

aR

cc41

where fsc KU

=

Nscs*/p,

K:*

=

1 $ emhwlkT).

K~*(

(C5)

The flux of energy F+ = / I,’

cos ( dO dw

((35)

is then just ((27) where m

l/K+

= s 0

K,,+ ;

Kcsd)-i.!-$

dw

/s

0

oII arZP’ dw. aT

CC81

69

SUPERNOVAE

Here, we may effect an important simplification by noticing that, in thermodynamic equilibrium, we must have

Since we are interested in computing the total flux, and do not, particularly rare how much is carried by neutrinos and how much by ant,ineutrinos, we may \vrite

where 1 -2 =z+-” K

1 K

Interchanging the order of integration and differentiation, that [Landau and Lifshitz ($)]

T” w dw= 167 ac s I(O) 4~ ’

(Cl11 and remembering

(Cl”)

we obbain L = 4s~R’F = ; f

4~R2T” g,

Except for the factor x, which comes from the difference in energy density between a black-body neutrino gas and a photon gas, this is exactly the same as the equation of transfer commonly used in the theory of stellar interiors. Only the opacity is defined differently, but we knew this would be the case; all of the physics of the problem is always contained in the opacit)y. We 11ow proceed to the calculation of the relevant opacities. KEUTRINO

OPACITY

The interaction of neutrinos with dense matter has been discussedby Bahcall (32), by Bahcall and Frautschi (33) and Euwema (34). The processesconsidered were elastic scattering by electrons and inverse bet,a-decays m nucleons and nuclei. It was shown by Bahcall and Fraut.schi (33) that, 011 the basis of present laboratory experiments, the contributions to neutrino opacit,y due to any resonances in the neutrino-electron or neutrino-nucleon system are negligible. The elastic scattering of neutrinos and nucleons R-asalso shown to be negligible iu the same paper. I. Inverse beta decays Inverse beta-decays are the dominant source of neutrino opacity for ordinary (i.e., cool, nondegenerate) matter. For neutrinos, the reaction is

70

SCHWARTZ

(1)

v+n+e-+P,

while the corresponding (2)

absorbtion

is

v + p + e+ + n.

In the limiting (2) is (3)

antineutrino

case of interest to us, qP >> fac2, the cross section for both (1) and

u z uo( qv/nlc2)2,

where qv is the neutrino energy! and uo is the characteristic UO

cross section,

= 1.7 X 1O-44 cm”

However, as has been pointed out by Bahcall and Euwema, the nucleons in the regions of interest for the supernova problem are far from free. The regions of the star which were found in this calculation to be the crucial ones for determining the rate of energy transport by neutrinos were at densities of 10” grams/cm3 or greater. At this density, the Fermi energy of the electrons is already more than 10 MeV. Thus, the mean free path of neutrinos whose energy is less than the electron Fermi energy will be much longer than in nondegenerate matter. Euwema (34) has calculated the effect of degeneracy on the cross section for reaction (1) and finds that, for zero temperature, neutrinos with energies even slightly above + , the electron Fermi energy, are absorbed with an average cross section nearly equal to the free-nucleon value. The presence of a finite temperature will also t,end to make the cross section for even the lower energy neutrinos approach this value. Of more importance to the energy transfer, however, is the process (2). When the neutrons begin to become degenerate, as in the core of the imploding supernova, (2) becomes strongly inhibited. Furthermore, at such high densities, the number of protons is much less than the number of neutrons, further reducing the neutrino opacity due to (2). Euwema has calculated the opacity for this process under the assumption of perfect degeneracy, and with no inhibition due to degeneracy. The cross-sections are typically l/100 to l/1000 of those for (1) in both cases. As we shall see, this means that both of these processes are negligible compared to the neutrino opacity due to the scattering on electrons. The scattering of neutrinos by electrons at rest was discussed in the original paper of Feynman and Gell-Mann (55). The cross section for this process is u = uo( w2/1 + 2wJ

(G14)

This cross section is negligibly small compared with the inverse beta-decay cross section, and thus was neglected initially. However, it was pointed out by Bahcall

SUPERNOVAE

71

that, the cross section may be many times larger than this for a gas whic+h is very hot, very degenerate, or both. It was shown by Bahcall (32) that), for high-energy neutrinos and elect,rona, the cross section is approximately (r = aow2. One may understand this incrcasc as follows: It is well known that the cross-sections for all of the weak int,eractions increase approximately with the square of t’he energy in t’he center of mass system. III the scattering from electrons at rest, the center of mass energy is (w2/1 + 2w 1’ “? while, in a. head-on collision of an electron and neut8rino, bot,h of energy w the cm. energy is 2w. (All energies are in units of mc’. j For antinellbrine-electron scattering, the cross section is exactly UO of that for nelit,rinc )P electron scattering. The results of Bahcall’s computations are that the average cross se&m for :I neutrino of energy w in a nondegenerate gas of temperature kT >> WC* is u E 3.%$(~w)

(?’ in unit,s of IW?)

For a degenerat,e gas, the result is

These should be multiplied by fd for antineutrinos. In the transport calculations, the Rosseland mean opacity for the neut,rinos is required, since the diffusion approximation is being used. Since 6he mat,ter in the regions of importance in this problem is fairly degenerate, the form ( 10‘1 for the opacity is used. The Rosseland mean is given by (Cll). Using t,he (aross section ( C15), the Rosseland mean is thus K = 5.1 X 10?!‘2)/~L, cm”/g where 1’ is measured in units of llzc2 and pe is the mean molecular weight per eMron.

It is plain that the emission of large numbers of muon neutrinos can have a significant effect on the cooling of the neutron core in the early stages of its formation. The mu neutrinos can only interact with ordinary matter by producing muons, which means that the threshold for any interactions whatsoever is 105 MeV in the center of mass. Therefore, almost all of the mu neutrinos produced will leave the star wit,hout being scattered or absorbed, and Tvill act as :I sink for the int’ernal energy. The muon cooling rates have been computed under the assumption that the dominant sources of mu neutrinos are muon and pion decay. Since the mean collision time even of a pion is very much less than its lifetime, the number densities of muons and pions may be computed according to statistical equilibrium. The number densities are then [Landau and Lifshitz (d)]

72

SCHWARTZ

This expression is valid for kT << mc2, a condition which remains satisfied in the actual case. An evaluation of the exact expression for the pair density is out of the question, as it would involve too much computing time. The energy loss rate due to the decay of each type of particle is then Q = nAEC/r where E is the average fraction of the decay energy carried by the mu neutrino. For the muon, E= $5, for the pion R ,C ?i. RECEIVED:

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