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Neutrino transport in stellar collapse
PHYSICS REPORTS (Review Section of Physics Letters) 163, Nos. 1-3 (1988) 127-136. North-Holland, Amsterdam
NEUTRINO TRANSPORT IN STELLAR COLLAPSE Eric S. MYRA Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA
Abstract: A multigroup flux-limited diffusion approximation to the equations of neutrino transport is outlined. Because the approximation requires no assumptions about the functional form of the neutrino distribution, it enables detailed tracing of the evolution of neutrinos when applied, as it is here, as part of a description of a collapsing stellar core. Presented are some results of such collapse calculations made using this method. They show that during the crucial period of lepton loss from the system, the neutrinos are not accurately modelled by a thermal distribution.
1. Introduction
It could perhaps be argued that progress in research on supernovae is cyclic in character. Such an observation would be based on the way certain effects seem to resurface periodically as the important contributions that will explain the phenomenon. This state of affairs, an indication of the longevity of the problem, results from the tantalizing nature of candidate models. Those models that manage to succeed usually obtain explosions that are marginal (marginal here meaning that a slight variation in some uncertain or disputed parameter will lead to failure of an otherwise successful model). This can be understood when it is realized that only a small fraction (-1%) of the gravitational potential energy released during stellar core collapse is imparted to the ejected material. An important feature that accompanies the release of the gravitational potential is the production of neutrino radiation. Over the last few years, neutrino physics has been handled with increasing sophistication in collapse codes; it has become recognized that in order to treat neutrinos in a satisfactory manner, calculations must somehow account for a neutrino spectrum that is frequently nonthermal in some region of the core. At certain stages of the collapse and subsequent rebound, it appears that the form of this neutrino spectrum may be critical in determining the evolution of the nonthermalized regions. To obtain neutrino spectra in full generality, solution of the Boltzmann transport equation is required. Unfortunately, the already massive nature of most stellar collapse codes makes such a solution impractical. As an alternative, one can proceed by deriving a finite number of angular moment equations and then solve these by multigroup methods. While truncation at a finite number of equations requires some sort of "fixup" to handle the free-streaming regime correctly, reasonably accurate solutions are possible with a moderate expenditure of computer time. Summarized here is a multigroup flux-limited scheme and the results of some calculations in which it is employed. It will be seen that the neutrino spectrum is nonthermal during the crucial periods of neutrino production and loss from the system. The multigroup methods outlined should be especially appropriate for tracing this evolution. 0 370-1573/88/$3.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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128
2. The neutrino transport equations
The present scheme is based on work of Mihalas and Weaver [1] who have derived the photon equation of transfer in comoving co-ordinates. When there is nonuniform flow in the radiating fluid, this co-ordinate system gives rise to velocity dependent terms that cause spectral changes in the radiation. Upon dropping terms in the fluid velocity of higher order than v/c and performing an angular integration, Mihalas and Weaver obtain an equation that describes evolution of the radiation. For neutrinos of energy e , it takes the form
D (uff)
P -Dtt
8
c~ [v (3p _u.)+l Dp ] (Du.]
+ 4 zrp -f-mm ( rZl'~) + e~ -~e~ r
p -D-f P~ =\--D-t/M'
(1)
where 1
uv =
-
2. f dis I(e., -
¢
#)
(2)
-1
is the spectral neutrino energy density [the zeroth angular moment of the specific intensity I(e~,/x)], 1
27r f dlz lzI(ev, L=c -1
tz)
(3)
is the spectral energy flux density, and I
p . = 2 zr f d ~ tzZl(e~,' tz )
(4)
-1
is the spectral neutrino pressure. (Here, the term "spectral" indicates that the quantities have not been integrated over the neutrino energy.) The right-hand side of eq. (1) accounts for processes that transfer energy from the matter to the neutrinos and vice versa. Included are such processes as neutrino emission-absorption and nonconservative scattering. Conservative scattering makes no contribution to the (DuJDt)M term and affects only terms containing j~ and Pv. The 8/8m term on the left-hand side of eq. (1) describes the spatial transfer of neutrinos. The terms in 8/ae~ are velocity dependent and cause spectral rearrangement of neutrinos. To eliminate the higher moments of I(e v, Iz) present in eq. (1), closure relations are required. To eliminate j~, a form of Fick's law diffusion is assumed,
L-
cd or
Vuv,
(5)
where or is the reciprocal mean free path, c is the speed of light, and d is a dimensionless diffusion coefficient. The choice of d must insure that large, unphysical fluxes do not occur in regions of large concentration gradients. This restriction can be expressed as IJl -< cuv, with the equality holding in free-streaming regions. In the diffusion limit, d must be selected such that IJl = (c/3cr)lVuvl. One such d
E.S. Myra, Neutrino transport in stellar collapse
129
that has these desired properties is the diffusion coefficient of Levermore and Pomeraning [14],
1 ( d=~-~
1) cothR-R
1 2+R ---~ 6 + 3 R + R E'
(6)
which is adopted here in the second form. The dimensionless gradient R is given by
R = -IVuJ/ tou.,
(7)
the effective albedo to is tra*b + o'su, to -
,
(8)
oru v
and b is the Fermi-Dirac energy distribution function. The scattering and absorption coefficients are denoted by o"s and aa, respectively, with tr = o"s + tr*,. The asterisk on tra indicates that the effects of Fermi-stimulated absorption are included. The second moment of the specific intensity Pv can be eliminated through the relation (9)
pv=xu,,
where the present definition of d then yields an Eddington factor X of the form ( X=cothR
cothR-~
1)
2+R ~6+3R+R
( (2+R)RE ] 2 1+6+~/~-2].
(10)
In the present scheme, eq. (1) is rewritten using the quantity (11)
y=-uJ(evPNA),
the number of neutrinos per baryon per energy interval. With this definition and use of the equation of continuity, Dy 0Z a [ -e,, D----t-+ a--mm+ a-~e~
1 2
+ ar
x 11 ~
(12)
p Dt y = D-[ M'
is obtained, where the spatial neutrino current Z is given by Z = -4err 2 cd a(yp) or ar
(13)
Because terms on the right-hand side of this equation cannot change the number of leptons in the system and the second and third terms on the left-hand side are divergences, this form of the transport equation explicitly conserves lepton number. In a collapsing stellar core, the important neutrino-matter interactions are limited to three processes: (i) neutrino emission and absorption by free protons and nuclei through weak charged currents, (if) inelastic neutrino-electron scattering through both weak charged and weak neutral
Theory of supernovae
130
currents, and (iii) elastic scattering of neutrinos from nuclei and free nucleons through weak neutral currents. The present scheme accounts for only these processes. The details of the microphysics are given in Myra et al. [2]. For neutrino production by electron capture, the neutrino emissivity can be expressed as dA H
dAp
S= -d-~-e~Y H + ~ e~ Yp,
(14)
where dAx is
(G)x
dAx = In 2 ~
e2~(e~+
Wx)2fede..
(15)
Here, X represents electron capture on heavy nuclei (H) or on free protons (p) and the heavy nucleus to baryon and proton to baryon number ratios are given by YH and Yp, respectively. The Coulomb barrier penetration factor ( G ) x and the ft values for the two reactions considered were taken from Fuller [3]. Once the neutron number of the mean nucleus reaches 40, however, (ft)H is set to infinity. This corresponds to the complete suppression of forbidden transitions and the neglect of thermal unblocking. The Q-values of the reactions are denoted by Wx, and are as given in Myra et al. [2]. The electron phase space occupancy is denoted by re. To account for the effects of neutrino-electron scattering (NES), the present model replaces collision integrals by a Fokker-Planck approximation, closely following the scheme described in Bowers and Wilson [4] (see also Myra et al. [2]). The momentum space current in the Fokker-Planck approximation can be expressed as ZFP =
KNEsev[_y(l_f~) - e ~2T ~--~e 0~ (Y)]
,
(16)
where KNEs is the positive function proportional to the equilibration rate of the neutrinos as a function of energy and to the fractional energy change of a neutrino per collision. The assumptions under which the Fokker-Planck equation is derived will generally lead to a neutrino thermalization rate that is too slow. This shortcoming of the scheme can be remedied somewhat by setting a minimum value for the fractional energy exchange. This procedure gives satisfactory results, at least for the infall stages. The equilibration rates used are those of Tubbs and Schramm [5]. For the purposes of determining the scattering coefficient for spatial diffusion, the Bowers and Wilson fit to the tabular values of Tubbs and Schramm has been employed. With the calculation of the emission-absorption and the nonconservative scattering terms, the right-hand side of eq. (12) can now be given explicitly, 0ZFP
(17)
where L is the neutrino phase space occupancy and F is the neutrino absorption rate. Since the matter, though not necessarily the neutrino radiation, is in local thermodynamic equilibrium, F can be readily calculated from S using the Kirchhoff-Fermi law [6, 2]. Except for the neutrino-electron scattering discussed above, the scattering of neutrinos from matter
E.S. Myra, Neutrino transport in stellar collapse
131
is assumed to be conservative. In this scheme, conservative scattering of neutrinos via weak neutral currents is calculated for scattering from heavy nuclei and free nucleons based on the calculation of Lamb and Pethick [7, 2]. The numerical scheme for solving eq. (12) has been presented elsewhere [2]. It is solved in an implicit manner over the neutrino energy groups and in a semi-implicit manner over spatial zones following the n-precursor method developed by Noam Sack (here in its simplest form with n = 0). Typically, the solution is carried out over 60 spatial zones and requires about 1500 timesteps to reach core bounce and several thousand more to follow the fate of the shock. At no point in the calculation are the neutrinos assumed to be in equilibrium with the matter. Thus it is necessary to have fine grouping at energies where the neutrino Fermi surface is developing in order to closely track the exchange of energy between the matter and the neutrinos. Typically, about 30 groups are used, with 13 geometrically spaced groups centred on energies between 5 and 366 MeV. Additional groups are then inserted between 30 and 250 MeV. (The 5 MeV group which accounts for neutrinos between 0 and 6.0 MeV is sufficiently fine to account for low-energy neutrinos since the phase space occupied by such neutrinos is small relative to the important neutrinos in the problem.) In calculations with this scheme, it has been observed that the approach of the neutrinos to chemical and thermal equilibrium is very smooth and that once established, it is well maintained. Some saving in computation time should then be realized if this observation is used to reduce the amount of transport that needs to be done in full detail.
3. Results
This multigroup transport scheme has been used in a number of stellar collapse calculations. In this article, discussion is limited to the effects of transport on a number of Newtonian hydrodynamic calculations and will centre chiefly on the effects that are expected to be illuminated by use of a multigroup code. There are no explosions among the results reported here; this can largely be attributed to the neutrino production and transport in the models. Two aspects of the neutrino physics will be examined: the dependence on heavy capture rates and the importance of neutrino-electron scattering. Calculations were performed with a 15 M e star evolved by Weaver, Zimmerman and Woosley [8] (WZW 1978) and one of 13 M o by Weaver, Woosley, and Fuller [9] (WWF 1983). Table 1 shows a comparison between these two models for several important quantities, both prior to collapse and at the
Table 1 Effect of initial configurations
pre-collapse
WZW
WWF
15 1.6 1.0
13 1.4 0.8
Yoo
0.44
0.42
lie YL S
0.29 0.35 1.4 0.75
0.29 0.36 1.2 0.77
MMs/ M 0 MFe[Me sc
at bounce
M,onlc/M 0
132
Theory of supernovae
moment of bounce. It should be noted that the differences in the character of the pre-collapse iron cores are not caused by the difference in main-sequence mass, but rather by differences in reaction rates used to evolve the two models. (Particularly, revisions in the weak interaction rates used in the more recent model have led to greater neutrino production, loss, and cooling in the period just prior to collapse.) It can be seen that the small differences between the two initial conditions are preserved by the collapse. The slightly lower entropy of the WWF model results in less electron capture during collapse and hence a slightly larger unshocked core, as given by the position of the sonic point at bounce. Calculations were performed using two ft values for electron capture on heavy nuclei. As mentioned earlier, this capture was completely suppressed once the mean neutron number reached 40. With complete shell blockage, it can be seen (fig. 1) that the ft values change the shape of the deleptonization trajectories, but that the overall amount of electron capture by heavy nuclei is not affected. In this work, no calculations were made in an attempt to account for nonzero rates of forbidden transitions or thermal unblocking, though progress in calculating rates for these effects is being made [10]. Use of these more recent ft values should not change the present results to any great extent. Table 2 shows results for collapses using the WWF model with three treatments of neutrino-electron scattering: NES rates set to zero, "realistic" rates as given in Myra et al. [2], and the realistic rates multiplied by a factor of ten. It can be seen that with this medium entropy core, the effects of NES are measurable and deleterious to the lepton fraction. The contribution of neutrino-electron scattering to core deleptonization occurs through the process of neutrino downscatter, which is a result of the electron degeneracy and a nonthermal neutrino distribution in the collapsing core. (The electron chemical potential/z e is typically about 20 MeV while .,.,..,
0.50
0.5C
J
0.45
...i...,...,...,...,..b:
0.45 Ye
Ye 0.40
0.40
0.35
0.35
0.30
0.30
S
6
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.2 0.4 0.6 0.8 1.0 1.2 1.4 mOSS (M o)
mass (M o)
Fig. 1. Profiles of Ye at six stages of the collapse of the WWF model for (a) log(ft)H = 4.3 and (b) log(ft)n = 2.58. The profiles are shown at six instants when the central zone density has reached: (1) 10 TM, (2) 1011, (3) 1012, (4) 1013, (5) 10"gcm 3, and (6) at core bounce (Pc--4 x 10TMg cm-3). Table 2 Effect of neutrino-electron scattering Initial
no NES
NES
i0 x NES
Ye YL s
0.42 0.42 0.8
M,o,,ol M o
-
0.31 0.38 1.1 0.81
0.29 0.36 1.2 0.77
0.28 0.34 1.3 0.72
E.S. Myra, Neutrino transport in stellar collapse
133
the matter temperature is about 1 MeV. The neutrino distribution has almost zero occupation in states of low energy.) Energy exchange in a scattering event must then necessarily favour neutrino downscatter as the electrons have few or no available states into which they can downscatter. Since the typical cross sections scale as the square of the neutrino energy, downscattered neutrinos tend to escape from the system (which is the reason for the small occupation in low-energy states). As well, NES increases the matter entropy, leading to a higher proton fraction and higher neutrino production rates. Downscatter also unblocks neutrino states at high energies, the energies at which neutrinos are most favourably produced. These results are in qualitative (though not strictly quantitative) agreement with those of Bruenn [11-13]. The stage of the collapse where NES can be important is governed by the requirement that there be a significant number of neutrinos present to participate in scattering, but that the material be sufficiently diffuse to enable escape in great numbers from the core. Figure 2 shows that these requirements are met in the central portion of the core between densities of about 2 x 101] and 4 × 10 t2 gcm -3. Figure 3 compares neutrino spectra and local conditions at two densities, 1 x 10n and 1 × 10t2 g cm -3. The conditions at the lower density are virtually identical. However, by the time the density reaches 1 x 1012gcm -3, differences in the evolution have become apparent. Of particular importance is the downshifted nature of the neutrino spectrum in the calculation using NES and the resulting deleptonization and entropy generation. These sorts of differences are present not only in the centre of the core, but further out as well, though the magnitudes decrease somewhat as a result of spatial transport. It is observed that in calculations using NES there is more zone-to-zone transport from inner zones as a result of there being more low-energy neutrinos. For this reason, in outer zones, there are more occupied low-energy states and less downscatter. This NES-induced transport results in cores that are slightly less uniform in composition than those collapsed without NES (fig. 4).
1o°
OA5
YL
i0-~
0.40 !
0.35
ld 2 '
"
L2\
..... 10+1 10Im 1013 lOTM density (g/crn 3) Fig. 2. Evolution with density of the lepton fraction YLin the WWF model for the central mass zone in calculations with (1) NES modelled by the Fokker-Planck approximation and (2) NES neglected.
\\
-"\ \\ ......
10
100 ~
(MeV}
Fig. 3. Neutrino occupation fractions in the central zone for two stages of the collapse of the WWF model for (1) NES included and (2) NES neglected. The dashed line is the occupation at pc = 1 x 10Ugcm -3, at which time the conditions in the core are nearly identical in calculations (1) and (2): Yo=0.40, Y, =0.001, T= I.IMeV, and /~, = 17MeV. The solid lines indicate occupation at po=1x1012gcm-3, by which time in (1), Y+=0.35, Y,=0.0"21, T=I.6MeV, and /z,=35MeV, and in (2), Y~=0.37, Y~=0.014, T= 1.5MeV, and /z, = 36MeV. The downshifted spectrum in the calculation that includes NES is evident.
Theory of supernovae
134
0.50 0.45 0.40 0.35
1
0.30 0.2 0.4 0.6 0.8 1.0 1.2 mess (M e )
1.4
Fig. 4. Profiles of lie at the instant of core bounce of the ~,~/F mode] for (I) NES included and (2) NES neglected. The inclusion of NES, in addition to the depletion of the lepton number, also produces more transport within the core, resulting in more nonuniformity in composition relative to the calculation in which it is neglected.
4. Conclusions It can be seen from the results of the detailed treatment of neutrino transport, that there are stages in the collapse when it is necessary to model the neutrino physics carefully. The present multigroup calculations show that tracing the detailed evolution of the neutrino spectra illustrates important consequences of various physical processes. Here, the role of neutrino-electron scattering has been emphasized, though other processes should also be illuminated by use of this scheme. NES has been found to have a measurable effect on deleptonization. It is responsible for reducing by 0.02 the trapped lepton fraction and increasing the inner core entropy by about 0.1 (in units of Boltzmann's constant per baryon). As well, it leads to shock formation at a point 0.04 M o closer to the centre of the core at bounce. In present models, however, since shocks fail even in the absence of NES, its presence is insufficient to differentiate between shock success and failure.
Acknowledgement This research was supported in part by the US Department of Energy under contract EY-76-C-023071 at the University of Pennsylvania.
References [1] [2] [3] [4] [5] [6] [7]
D. Mihalas and R.P. Weaver, Los Alamos Report No. LA-UR-82-743 (1982). E.S. Myra, S.A. Bludman, Y. Hoffman, I. Lichtenstadt, N. Sack and K.A. Van Riper, Astrophys. J. 318 (1987) 744. G.M. Fuller, Astrophys. J. 252 (1982) 741. R.L. Bowers and J.R. Wilson, Astrophys. J. Suppl. 50 (1982) 115. D.L. Tubbs and D.N. Schramm, Astrophys. J. 201 (1975) 467. S.A. Bludman, in: Proc. Intern. Neutrino Conf. (Aachen, 1976), eds H. Faissner et al. (Vieweg, Braunschweig, 1977). D.Q. Lamb and C.J. Pethick, Astrophys. J. (Lett.) 209 (1976) L77.
E.S. Myra, Neutrino transport in stellar collapse
[8] T.A. Weaver, G.B. Zimmerman and S.'E. Woosley, Astrophys. J. 225 (1978) 1021. [9] T.A. Weaver, S.E. Woosley and G.M. Fuller, private communication (1983). [10] G.M. Fuller, W.A. Fowler and M.J. Newman, Astrophys. J. 293 (1985) 1. [11] S.W. Bruenn, Astrophys. J. Suppl. 58 (1985) 771. [12] S.W. Bruenn, Astrophys. J. Suppl. 62 (1986) 331. [13] S.W. Bruenn, preprint (1986). [14] C.D. Levermore and G.C. Pomraning, Astrophys. J. 248 (1981) 321.
135
NEUTRINO TRANSPORT IN STELLAR COLLAPSE Eric S. MYRA Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA
Abstract: A multigroup flux-limited diffusion approximation to the equations of neutrino transport is outlined. Because the approximation requires no assumptions about the functional form of the neutrino distribution, it enables detailed tracing of the evolution of neutrinos when applied, as it is here, as part of a description of a collapsing stellar core. Presented are some results of such collapse calculations made using this method. They show that during the crucial period of lepton loss from the system, the neutrinos are not accurately modelled by a thermal distribution.
1. Introduction
It could perhaps be argued that progress in research on supernovae is cyclic in character. Such an observation would be based on the way certain effects seem to resurface periodically as the important contributions that will explain the phenomenon. This state of affairs, an indication of the longevity of the problem, results from the tantalizing nature of candidate models. Those models that manage to succeed usually obtain explosions that are marginal (marginal here meaning that a slight variation in some uncertain or disputed parameter will lead to failure of an otherwise successful model). This can be understood when it is realized that only a small fraction (-1%) of the gravitational potential energy released during stellar core collapse is imparted to the ejected material. An important feature that accompanies the release of the gravitational potential is the production of neutrino radiation. Over the last few years, neutrino physics has been handled with increasing sophistication in collapse codes; it has become recognized that in order to treat neutrinos in a satisfactory manner, calculations must somehow account for a neutrino spectrum that is frequently nonthermal in some region of the core. At certain stages of the collapse and subsequent rebound, it appears that the form of this neutrino spectrum may be critical in determining the evolution of the nonthermalized regions. To obtain neutrino spectra in full generality, solution of the Boltzmann transport equation is required. Unfortunately, the already massive nature of most stellar collapse codes makes such a solution impractical. As an alternative, one can proceed by deriving a finite number of angular moment equations and then solve these by multigroup methods. While truncation at a finite number of equations requires some sort of "fixup" to handle the free-streaming regime correctly, reasonably accurate solutions are possible with a moderate expenditure of computer time. Summarized here is a multigroup flux-limited scheme and the results of some calculations in which it is employed. It will be seen that the neutrino spectrum is nonthermal during the crucial periods of neutrino production and loss from the system. The multigroup methods outlined should be especially appropriate for tracing this evolution. 0 370-1573/88/$3.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Theory of supernovae
128
2. The neutrino transport equations
The present scheme is based on work of Mihalas and Weaver [1] who have derived the photon equation of transfer in comoving co-ordinates. When there is nonuniform flow in the radiating fluid, this co-ordinate system gives rise to velocity dependent terms that cause spectral changes in the radiation. Upon dropping terms in the fluid velocity of higher order than v/c and performing an angular integration, Mihalas and Weaver obtain an equation that describes evolution of the radiation. For neutrinos of energy e , it takes the form
D (uff)
P -Dtt
8
c~ [v (3p _u.)+l Dp ] (Du.]
+ 4 zrp -f-mm ( rZl'~) + e~ -~e~ r
p -D-f P~ =\--D-t/M'
(1)
where 1
uv =
-
2. f dis I(e., -
¢
#)
(2)
-1
is the spectral neutrino energy density [the zeroth angular moment of the specific intensity I(e~,/x)], 1
27r f dlz lzI(ev, L=c -1
tz)
(3)
is the spectral energy flux density, and I
p . = 2 zr f d ~ tzZl(e~,' tz )
(4)
-1
is the spectral neutrino pressure. (Here, the term "spectral" indicates that the quantities have not been integrated over the neutrino energy.) The right-hand side of eq. (1) accounts for processes that transfer energy from the matter to the neutrinos and vice versa. Included are such processes as neutrino emission-absorption and nonconservative scattering. Conservative scattering makes no contribution to the (DuJDt)M term and affects only terms containing j~ and Pv. The 8/8m term on the left-hand side of eq. (1) describes the spatial transfer of neutrinos. The terms in 8/ae~ are velocity dependent and cause spectral rearrangement of neutrinos. To eliminate the higher moments of I(e v, Iz) present in eq. (1), closure relations are required. To eliminate j~, a form of Fick's law diffusion is assumed,
L-
cd or
Vuv,
(5)
where or is the reciprocal mean free path, c is the speed of light, and d is a dimensionless diffusion coefficient. The choice of d must insure that large, unphysical fluxes do not occur in regions of large concentration gradients. This restriction can be expressed as IJl -< cuv, with the equality holding in free-streaming regions. In the diffusion limit, d must be selected such that IJl = (c/3cr)lVuvl. One such d
E.S. Myra, Neutrino transport in stellar collapse
129
that has these desired properties is the diffusion coefficient of Levermore and Pomeraning [14],
1 ( d=~-~
1) cothR-R
1 2+R ---~ 6 + 3 R + R E'
(6)
which is adopted here in the second form. The dimensionless gradient R is given by
R = -IVuJ/ tou.,
(7)
the effective albedo to is tra*b + o'su, to -
,
(8)
oru v
and b is the Fermi-Dirac energy distribution function. The scattering and absorption coefficients are denoted by o"s and aa, respectively, with tr = o"s + tr*,. The asterisk on tra indicates that the effects of Fermi-stimulated absorption are included. The second moment of the specific intensity Pv can be eliminated through the relation (9)
pv=xu,,
where the present definition of d then yields an Eddington factor X of the form ( X=cothR
cothR-~
1)
2+R ~6+3R+R
( (2+R)RE ] 2 1+6+~/~-2].
(10)
In the present scheme, eq. (1) is rewritten using the quantity (11)
y=-uJ(evPNA),
the number of neutrinos per baryon per energy interval. With this definition and use of the equation of continuity, Dy 0Z a [ -e,, D----t-+ a--mm+ a-~e~
1 2
+ ar
x 11 ~
(12)
p Dt y = D-[ M'
is obtained, where the spatial neutrino current Z is given by Z = -4err 2 cd a(yp) or ar
(13)
Because terms on the right-hand side of this equation cannot change the number of leptons in the system and the second and third terms on the left-hand side are divergences, this form of the transport equation explicitly conserves lepton number. In a collapsing stellar core, the important neutrino-matter interactions are limited to three processes: (i) neutrino emission and absorption by free protons and nuclei through weak charged currents, (if) inelastic neutrino-electron scattering through both weak charged and weak neutral
Theory of supernovae
130
currents, and (iii) elastic scattering of neutrinos from nuclei and free nucleons through weak neutral currents. The present scheme accounts for only these processes. The details of the microphysics are given in Myra et al. [2]. For neutrino production by electron capture, the neutrino emissivity can be expressed as dA H
dAp
S= -d-~-e~Y H + ~ e~ Yp,
(14)
where dAx is
(G)x
dAx = In 2 ~
e2~(e~+
Wx)2fede..
(15)
Here, X represents electron capture on heavy nuclei (H) or on free protons (p) and the heavy nucleus to baryon and proton to baryon number ratios are given by YH and Yp, respectively. The Coulomb barrier penetration factor ( G ) x and the ft values for the two reactions considered were taken from Fuller [3]. Once the neutron number of the mean nucleus reaches 40, however, (ft)H is set to infinity. This corresponds to the complete suppression of forbidden transitions and the neglect of thermal unblocking. The Q-values of the reactions are denoted by Wx, and are as given in Myra et al. [2]. The electron phase space occupancy is denoted by re. To account for the effects of neutrino-electron scattering (NES), the present model replaces collision integrals by a Fokker-Planck approximation, closely following the scheme described in Bowers and Wilson [4] (see also Myra et al. [2]). The momentum space current in the Fokker-Planck approximation can be expressed as ZFP =
KNEsev[_y(l_f~) - e ~2T ~--~e 0~ (Y)]
,
(16)
where KNEs is the positive function proportional to the equilibration rate of the neutrinos as a function of energy and to the fractional energy change of a neutrino per collision. The assumptions under which the Fokker-Planck equation is derived will generally lead to a neutrino thermalization rate that is too slow. This shortcoming of the scheme can be remedied somewhat by setting a minimum value for the fractional energy exchange. This procedure gives satisfactory results, at least for the infall stages. The equilibration rates used are those of Tubbs and Schramm [5]. For the purposes of determining the scattering coefficient for spatial diffusion, the Bowers and Wilson fit to the tabular values of Tubbs and Schramm has been employed. With the calculation of the emission-absorption and the nonconservative scattering terms, the right-hand side of eq. (12) can now be given explicitly, 0ZFP
(17)
where L is the neutrino phase space occupancy and F is the neutrino absorption rate. Since the matter, though not necessarily the neutrino radiation, is in local thermodynamic equilibrium, F can be readily calculated from S using the Kirchhoff-Fermi law [6, 2]. Except for the neutrino-electron scattering discussed above, the scattering of neutrinos from matter
E.S. Myra, Neutrino transport in stellar collapse
131
is assumed to be conservative. In this scheme, conservative scattering of neutrinos via weak neutral currents is calculated for scattering from heavy nuclei and free nucleons based on the calculation of Lamb and Pethick [7, 2]. The numerical scheme for solving eq. (12) has been presented elsewhere [2]. It is solved in an implicit manner over the neutrino energy groups and in a semi-implicit manner over spatial zones following the n-precursor method developed by Noam Sack (here in its simplest form with n = 0). Typically, the solution is carried out over 60 spatial zones and requires about 1500 timesteps to reach core bounce and several thousand more to follow the fate of the shock. At no point in the calculation are the neutrinos assumed to be in equilibrium with the matter. Thus it is necessary to have fine grouping at energies where the neutrino Fermi surface is developing in order to closely track the exchange of energy between the matter and the neutrinos. Typically, about 30 groups are used, with 13 geometrically spaced groups centred on energies between 5 and 366 MeV. Additional groups are then inserted between 30 and 250 MeV. (The 5 MeV group which accounts for neutrinos between 0 and 6.0 MeV is sufficiently fine to account for low-energy neutrinos since the phase space occupied by such neutrinos is small relative to the important neutrinos in the problem.) In calculations with this scheme, it has been observed that the approach of the neutrinos to chemical and thermal equilibrium is very smooth and that once established, it is well maintained. Some saving in computation time should then be realized if this observation is used to reduce the amount of transport that needs to be done in full detail.
3. Results
This multigroup transport scheme has been used in a number of stellar collapse calculations. In this article, discussion is limited to the effects of transport on a number of Newtonian hydrodynamic calculations and will centre chiefly on the effects that are expected to be illuminated by use of a multigroup code. There are no explosions among the results reported here; this can largely be attributed to the neutrino production and transport in the models. Two aspects of the neutrino physics will be examined: the dependence on heavy capture rates and the importance of neutrino-electron scattering. Calculations were performed with a 15 M e star evolved by Weaver, Zimmerman and Woosley [8] (WZW 1978) and one of 13 M o by Weaver, Woosley, and Fuller [9] (WWF 1983). Table 1 shows a comparison between these two models for several important quantities, both prior to collapse and at the
Table 1 Effect of initial configurations
pre-collapse
WZW
WWF
15 1.6 1.0
13 1.4 0.8
Yoo
0.44
0.42
lie YL S
0.29 0.35 1.4 0.75
0.29 0.36 1.2 0.77
MMs/ M 0 MFe[Me sc
at bounce
M,onlc/M 0
132
Theory of supernovae
moment of bounce. It should be noted that the differences in the character of the pre-collapse iron cores are not caused by the difference in main-sequence mass, but rather by differences in reaction rates used to evolve the two models. (Particularly, revisions in the weak interaction rates used in the more recent model have led to greater neutrino production, loss, and cooling in the period just prior to collapse.) It can be seen that the small differences between the two initial conditions are preserved by the collapse. The slightly lower entropy of the WWF model results in less electron capture during collapse and hence a slightly larger unshocked core, as given by the position of the sonic point at bounce. Calculations were performed using two ft values for electron capture on heavy nuclei. As mentioned earlier, this capture was completely suppressed once the mean neutron number reached 40. With complete shell blockage, it can be seen (fig. 1) that the ft values change the shape of the deleptonization trajectories, but that the overall amount of electron capture by heavy nuclei is not affected. In this work, no calculations were made in an attempt to account for nonzero rates of forbidden transitions or thermal unblocking, though progress in calculating rates for these effects is being made [10]. Use of these more recent ft values should not change the present results to any great extent. Table 2 shows results for collapses using the WWF model with three treatments of neutrino-electron scattering: NES rates set to zero, "realistic" rates as given in Myra et al. [2], and the realistic rates multiplied by a factor of ten. It can be seen that with this medium entropy core, the effects of NES are measurable and deleterious to the lepton fraction. The contribution of neutrino-electron scattering to core deleptonization occurs through the process of neutrino downscatter, which is a result of the electron degeneracy and a nonthermal neutrino distribution in the collapsing core. (The electron chemical potential/z e is typically about 20 MeV while .,.,..,
0.50
0.5C
J
0.45
...i...,...,...,...,..b:
0.45 Ye
Ye 0.40
0.40
0.35
0.35
0.30
0.30
S
6
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.2 0.4 0.6 0.8 1.0 1.2 1.4 mOSS (M o)
mass (M o)
Fig. 1. Profiles of Ye at six stages of the collapse of the WWF model for (a) log(ft)H = 4.3 and (b) log(ft)n = 2.58. The profiles are shown at six instants when the central zone density has reached: (1) 10 TM, (2) 1011, (3) 1012, (4) 1013, (5) 10"gcm 3, and (6) at core bounce (Pc--4 x 10TMg cm-3). Table 2 Effect of neutrino-electron scattering Initial
no NES
NES
i0 x NES
Ye YL s
0.42 0.42 0.8
M,o,,ol M o
-
0.31 0.38 1.1 0.81
0.29 0.36 1.2 0.77
0.28 0.34 1.3 0.72
E.S. Myra, Neutrino transport in stellar collapse
133
the matter temperature is about 1 MeV. The neutrino distribution has almost zero occupation in states of low energy.) Energy exchange in a scattering event must then necessarily favour neutrino downscatter as the electrons have few or no available states into which they can downscatter. Since the typical cross sections scale as the square of the neutrino energy, downscattered neutrinos tend to escape from the system (which is the reason for the small occupation in low-energy states). As well, NES increases the matter entropy, leading to a higher proton fraction and higher neutrino production rates. Downscatter also unblocks neutrino states at high energies, the energies at which neutrinos are most favourably produced. These results are in qualitative (though not strictly quantitative) agreement with those of Bruenn [11-13]. The stage of the collapse where NES can be important is governed by the requirement that there be a significant number of neutrinos present to participate in scattering, but that the material be sufficiently diffuse to enable escape in great numbers from the core. Figure 2 shows that these requirements are met in the central portion of the core between densities of about 2 x 101] and 4 × 10 t2 gcm -3. Figure 3 compares neutrino spectra and local conditions at two densities, 1 x 10n and 1 × 10t2 g cm -3. The conditions at the lower density are virtually identical. However, by the time the density reaches 1 x 1012gcm -3, differences in the evolution have become apparent. Of particular importance is the downshifted nature of the neutrino spectrum in the calculation using NES and the resulting deleptonization and entropy generation. These sorts of differences are present not only in the centre of the core, but further out as well, though the magnitudes decrease somewhat as a result of spatial transport. It is observed that in calculations using NES there is more zone-to-zone transport from inner zones as a result of there being more low-energy neutrinos. For this reason, in outer zones, there are more occupied low-energy states and less downscatter. This NES-induced transport results in cores that are slightly less uniform in composition than those collapsed without NES (fig. 4).
1o°
OA5
YL
i0-~
0.40 !
0.35
ld 2 '
"
L2\
..... 10+1 10Im 1013 lOTM density (g/crn 3) Fig. 2. Evolution with density of the lepton fraction YLin the WWF model for the central mass zone in calculations with (1) NES modelled by the Fokker-Planck approximation and (2) NES neglected.
\\
-"\ \\ ......
10
100 ~
(MeV}
Fig. 3. Neutrino occupation fractions in the central zone for two stages of the collapse of the WWF model for (1) NES included and (2) NES neglected. The dashed line is the occupation at pc = 1 x 10Ugcm -3, at which time the conditions in the core are nearly identical in calculations (1) and (2): Yo=0.40, Y, =0.001, T= I.IMeV, and /~, = 17MeV. The solid lines indicate occupation at po=1x1012gcm-3, by which time in (1), Y+=0.35, Y,=0.0"21, T=I.6MeV, and /z,=35MeV, and in (2), Y~=0.37, Y~=0.014, T= 1.5MeV, and /z, = 36MeV. The downshifted spectrum in the calculation that includes NES is evident.
Theory of supernovae
134
0.50 0.45 0.40 0.35
1
0.30 0.2 0.4 0.6 0.8 1.0 1.2 mess (M e )
1.4
Fig. 4. Profiles of lie at the instant of core bounce of the ~,~/F mode] for (I) NES included and (2) NES neglected. The inclusion of NES, in addition to the depletion of the lepton number, also produces more transport within the core, resulting in more nonuniformity in composition relative to the calculation in which it is neglected.
4. Conclusions It can be seen from the results of the detailed treatment of neutrino transport, that there are stages in the collapse when it is necessary to model the neutrino physics carefully. The present multigroup calculations show that tracing the detailed evolution of the neutrino spectra illustrates important consequences of various physical processes. Here, the role of neutrino-electron scattering has been emphasized, though other processes should also be illuminated by use of this scheme. NES has been found to have a measurable effect on deleptonization. It is responsible for reducing by 0.02 the trapped lepton fraction and increasing the inner core entropy by about 0.1 (in units of Boltzmann's constant per baryon). As well, it leads to shock formation at a point 0.04 M o closer to the centre of the core at bounce. In present models, however, since shocks fail even in the absence of NES, its presence is insufficient to differentiate between shock success and failure.
Acknowledgement This research was supported in part by the US Department of Energy under contract EY-76-C-023071 at the University of Pennsylvania.
References [1] [2] [3] [4] [5] [6] [7]
D. Mihalas and R.P. Weaver, Los Alamos Report No. LA-UR-82-743 (1982). E.S. Myra, S.A. Bludman, Y. Hoffman, I. Lichtenstadt, N. Sack and K.A. Van Riper, Astrophys. J. 318 (1987) 744. G.M. Fuller, Astrophys. J. 252 (1982) 741. R.L. Bowers and J.R. Wilson, Astrophys. J. Suppl. 50 (1982) 115. D.L. Tubbs and D.N. Schramm, Astrophys. J. 201 (1975) 467. S.A. Bludman, in: Proc. Intern. Neutrino Conf. (Aachen, 1976), eds H. Faissner et al. (Vieweg, Braunschweig, 1977). D.Q. Lamb and C.J. Pethick, Astrophys. J. (Lett.) 209 (1976) L77.
E.S. Myra, Neutrino transport in stellar collapse
[8] T.A. Weaver, G.B. Zimmerman and S.'E. Woosley, Astrophys. J. 225 (1978) 1021. [9] T.A. Weaver, S.E. Woosley and G.M. Fuller, private communication (1983). [10] G.M. Fuller, W.A. Fowler and M.J. Newman, Astrophys. J. 293 (1985) 1. [11] S.W. Bruenn, Astrophys. J. Suppl. 58 (1985) 771. [12] S.W. Bruenn, Astrophys. J. Suppl. 62 (1986) 331. [13] S.W. Bruenn, preprint (1986). [14] C.D. Levermore and G.C. Pomraning, Astrophys. J. 248 (1981) 321.
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