Online Material

  
Appendix A: Neutrino opacities

In this appendix we shall describe the neutrino-matter interactions that are included in the current version of our neutrino transport code for supernova simulations. We shall focus on aspects which are specific and important for their numerical handling and will present final rate expressions in the form used in our code and with a consistent notation.

   

Table A.1: Overview of all neutrino-matter interactions currently implemented in the code. For each process we list the sections where we summarize fundamental aspects of the calculation of the corresponding rate and give details of its numerical implementation. The references point to papers where more information can be found about the approximations employed in the rate calculations. In the first column the symbol $\nu$ represents any of the neutrinos ${\nu_{\rm e}},{\bar\nu_{\rm e}},\nu_\mu,\bar\nu_\mu,\nu_\tau,\bar\nu_\tau$, the symbols ${\rm e}^-$, ${\rm e}^+$, ${\rm n}$, ${\rm p}$ and A denote electrons, positrons, free neutrons and protons, and heavy nuclei, respectively, the symbol ${\rm N}$ means ${\rm n}$ or ${\rm p}$.

Reaction			Rate described in	Reference
$\nu {\rm e}^{\pm}$	$\rightleftharpoons$	$\nu {\rm e}^{\pm}$	Sects. A.1.2, A.2.4	Mezzacappa & Bruenn (1993b); Cernohorsky (1994)
$\nu {\rm A}$	$\rightleftharpoons$	$\nu {\rm A}$	Sects. A.1.2, A.2.3	Horowitz (1997); Bruenn & Mezzacappa (1997)
$\nu {\rm N}$	$\rightleftharpoons$	$\nu {\rm N}$	Sects. A.1.2, A.2.1	Bruenn (1985); Mezzacappa & Bruenn (1993c)
${\nu_{\rm e}}{\rm n}$	$\rightleftharpoons$	${\rm e}^- {\rm p}$	Sects. A.1.1, A.2.1	Bruenn (1985); Mezzacappa & Bruenn (1993c)
${\bar\nu_{\rm e}}{\rm p}$	$\rightleftharpoons$	${\rm e}^+ {\rm n}$	Sects. A.1.1, A.2.1	Bruenn (1985); Mezzacappa & Bruenn (1993c)
${\nu_{\rm e}}{\rm A}'$	$\rightleftharpoons$	${\rm e}^- {\rm A}$	Sects. A.1.1, A.2.2	Bruenn (1985); Mezzacappa & Bruenn (1993c)
$\nu\bar\nu$	$\rightleftharpoons$	${\rm e}^- {\rm e}^+$	Sects. A.1.3, A.2.5	Bruenn (1985); Pons et al. (1998)
$\nu\bar\nu ~{\rm N}{\rm N}$	$\rightleftharpoons$	${\rm N}{\rm N}$	Sects. A.1.3, A.2.6	Hannestad & Raffelt (1998)

A.1 Basic considerations

In the following we discuss the different neutrino-matter interaction processes which contribute to the source term ("collision integral'') on the rhs of the Boltzmann equation:

$$\frac{1}{c}\frac{\partial}{\partial t}~f + \mu\frac{\partia... ...mu^2}{r}\frac{\partial}{\partial \mu}~f = \sum_{\{I\}} B_I ~.$$ (A.1)

The sum on the rhs of this equation runs over all considered interaction processes that can change the distribution function of a particular type of neutrino. For simplicity we have written down the Boltzmann equation for static media, here. In the general case of a moving stellar fluid, velocity-dependent terms have to appear on the lhs of Eq. (A.1) when the neutrino quantities and the neutrino-matter interaction rates are evaluated in the local rest frame of the fluid. The phase-space distribution function f is related to the specific intensity ${\cal I}$ used in the main body of this paper by ${\cal I}=c/(2\pi\hbar c)^3\cdot\epsilon^3~f$. Accordingly, the source term on the rhs of the Boltzmann equation, Eq. (6), and the corresponding moment equations for neutrino number and neutrino energy or momentum (Eqs. (30), (31), (7), (8)) can be calculated from B_I as

 

$${C}(\epsilon,\mu) \frac{c}{(2\pi\hbar c)^3}\epsilon^3 \cdot \sum_{\{I\}} B_{I}(\epsilon,\mu) ~,$$ :=

$${C^{(k)}}(\epsilon) \frac{c}{(2\pi\hbar c)^3}\epsilon^3 \cdot \sum_{\{I\}} B^{(k)}_{I}(\epsilon) ~,$$ := (A.2)

$${\frak{C}^{(k)}}(\epsilon) \frac{c}{(2\pi\hbar c)^3} \epsilon^2 \cdot \sum_{\{I\}} B^{(k)}_{I}(\epsilon) ~,$$ :=

where $B^{(k)}_{I}(\epsilon):= 1/2\int_{-1}^{+1}{\rm d}\mu~\mu^k B_I(\epsilon,\mu)$. Here and in the following, the dependence of quantities on the space-time coordinates (t,r) is suppressed in the notation. Throughout Appendix A the temperature T is measured in units of energy.

  
A.1.1 Neutrino absorption and emission

The rate of change (modulo a factor of 1/c) of the neutrino distribution function due to absorption and emission processes is given by (see Bruenn 1985)

$$B_{\rm AE}(\epsilon,\mu)= j(\epsilon)[1-f(\epsilon,\mu)]-f(\epsilon,\mu)/\lambda(\epsilon) ~,$$ (A.3)

where j denotes the emissivity and $\lambda$ is the mean free path for neutrino absorption. The factor (1-f) accounts for fermion phase space blocking effects of neutrinos. Using the Kirchhoff-Planck relation ("detailed balance''), and introducing the absorption opacity corrected for stimulated absorption,

$$\kappa_{\rm a}^*:=\frac{1}{1-f^{\rm eq}}\cdot \frac{1}{\lambda}= j+\frac{1}{\lambda} ~,$$ (A.4)

Eq. (A.3) can be rewritten as

$$B_{\rm AE}(\epsilon,\mu)= \kappa_{\rm a}^*(\epsilon)[f^{\rm eq}(\epsilon)-f(\epsilon,\mu)] ~.$$ (A.5)

In Eq. (A.5) it becomes evident that the source term drives the neutrino distribution function $f(\epsilon,\mu)$towards its equilibrium value $f^{\rm eq}(\epsilon)=(1+\exp{[(\epsilon-\mu_\nu^{\rm eq})/T]})^{-1}$, where $\mu_\nu^{\rm eq}$ denotes the chemical potential for neutrinos in thermodynamic equilibrium with the stellar medium. In equilibrium, the source term vanishes in accordance with the requirement of detailed balance.

  
A.1.2 Scattering

Reduction of the collision integral:

The rate of change of the neutrino distribution function due to scattering of neutrinos off some target particles is given by the collision integral (cf. Cernohorsky 1994, Eqs. (2.1), (2.3))

 

$$B_{\rm S}(\vec{q})= \int \frac{{\rm d}^3\vec{\vec{q'}}~}{(2\pi)^3... ...},\vec{q'}) -f_\nu(1-f'_\nu)\widetilde{R}^{\rm out}(\vec{q},\vec{q'})\right] ~,$$ (A.6)

where $f'_\nu$ depends on $\vec{q'}$ and the scattering kernels $\widetilde{R}^{\rm in}$ and $\widetilde{R}^{\rm out}$ are defined as the following phase space integrals over products of the transition rate ${\cal R}$ and the Fermi distribution functions $F_{\rm T}$ of the target particles:

 

$$\widetilde{R}^{\rm in}(\vec{q},\vec{q'}) \frac{2}{c\cdot(2\pi \hbar c)^3}\cdot \int \frac{{\rm d}^3\vec{p'... ...\pi)^3} (1-F_{\rm T})F'_{\rm T}~ {\cal R}(\vec{p'},\vec{q'};\vec{p},\vec{q}) ~,$$ =

$$\widetilde{R}^{\rm out}(\vec{q},\vec{q'}) \frac{2}{c\cdot(2\pi \hbar c)^3}\cdot \int \frac{{\rm d}^3\vec{p'... ...\pi)^3} (1-F'_{\rm T})F_{\rm T}~ {\cal R}(\vec{p},\vec{q};\vec{p'},\vec{q'}) ~.$$ = (A.7)

The vectors $\vec{q}$ and $\vec{q'}$ are the momenta of the ingoing and outgoing neutrinos, respectively, and $\vec{p}$ and $\vec{p'}$ those of their scattering targets.

The scattering kernels are usually given as functions of the energies $\epsilon$ and $\epsilon'$ of the ingoing and outgoing neutrino and the cosine $\omega=\mu\mu'+\sqrt{(1-\mu^2)(1-{\mu'}^2)}\cos(\varphi-\varphi')$ of the scattering angle, where $(\mu,\varphi)$ and $(\mu',\varphi')$ are the corresponding momentum space coordinates. Expanding the scattering kernels ( $R^{\rm in/out}:=1/(2\pi c)^3\cdot\widetilde{R}^{\rm in/out}$) in a Legendre series

$$R^{\rm in/out}(\epsilon,\epsilon',\omega)= \sum_{l=0}^{\inft... ...2l+1}{2}~\phi_l^{\rm in/out}(\epsilon,\epsilon')P_l(\omega) ~,$$ (A.8)

with the Legendre coefficients

$$\phi_l^{\rm in/out}(\epsilon,\epsilon')= \int_{-1}^{+1}{\rm ... ...ega~ P_l(\omega){R}^{\rm in/out}(\epsilon,\epsilon',\omega) ~,$$ (A.9)

and applying the addition theorem for Legendre polynomials $P_l(\omega)=P_l(\mu)P_l(\mu')+2\sum_{m=1}^{l}\frac{(l-m)!}{(l+m)!}P_l^m(\mu)P_l^m(\mu')\cos[m(\varphi-\varphi')]$ (e.g., Bronstein & Semendjajew 1991, P_l^m are the associated Legendre polynomials), the integral over $\varphi'$ in the collision integral can be performed analytically (Yueh & Buchler 1977; Schinder & Shapiro 1982). For use in our Boltzmann transport code, it is necessary to truncate the Legendre expansion at some level. Note that the orthogonality relation $\int_{-1}^{+1}{\rm d}\omega~P_l(\omega)P_{l'}(\omega)=\frac{2}{2l+1}\delta_{ll'}$ (e.g., Bronstein & Semendjajew 1991) implies that any truncation of the Legendre series $R^{\rm in/out}(\epsilon,\epsilon',\omega)$ still gives the exact integral value $\int_{-1}^{+1}{\rm d}\omega~{R}^{\rm in/out}(\epsilon,\epsilon',\omega)$, independent of the level of truncation $l_{\rm max}>0$.

The collision integral and its first two angular moments finally read:

 

$$B_{\rm S}(\epsilon,\mu) 2\pi~\int_0^\infty {\rm d}\epsilon'~ {\epsilon'}^2$$ =

$$\Big\{(1-f_\nu) \cdot \sum_{l=0}^{\infty}(2l+1)P_l(\mu) \phi_l^{\rm in}(\epsilon,\epsilon')L'_l(\epsilon')$$

$$-f_\nu \cdot \sum_{l=0}^{\infty}(2l+1)P_l(\mu) \phi_l^{\rm out}(\epsilon,\epsilon')(\delta_{l0}-L'_l(\epsilon'))\Big\} ~,$$ (A.10)

 

$$B^{(0)}_{\rm S}(\epsilon) = 2\pi~\int_0^\infty {\rm d}\epsilon'~ ... ...lon) \phi_l^{\rm out}(\epsilon,\epsilon')(\delta_{l0}-L'_l(\epsilon'))\Big\} ~,$$ (A.11)

 

$$B^{(1)}_{\rm S}(\epsilon) = 2\pi~\int_0^\infty {\rm d}\epsilon'~ ... ...on)] \phi_l^{\rm out}(\epsilon,\epsilon')(\delta_{l0}-L'_l(\epsilon'))\Big\} ~,$$ (A.12)

where the "Legendre moment'' $L_l:=\frac{1}{2}\int_{-1}^{+1}{\rm d}\mu~~P_l(\mu) f(\mu)$ of order l of the distribution function has been introduced. These Legendre moments can obviously be written as linear combinations of the angular moments $M_m:=\frac{1}{2}\int_{-1}^{+1}{\rm d}\mu~\mu^m f(\mu)$, which occur in the formulation of the transport equations used in our code. Detailed balance requires

$$\phi_l^{\rm in}(\epsilon,\epsilon')= {\rm e}^{-(\epsilon-\epsilon')/T}~\phi_l^{\rm out}(\epsilon,\epsilon') ~.$$ (A.13)

Note that exploiting the in-out invariance of the transition rate (and therefore $\phi_l^{\rm out}(\epsilon,\epsilon')=\phi_l^{\rm in}(\epsilon',\epsilon)$; e.g. Cernohorsky 1994) leads to

$$\int_0^\infty{\rm d}\epsilon~\epsilon^2 B^{(0)}_{\rm S}(\eps... ...ty{\rm d}\epsilon~\epsilon^3 B^{(0)}_{\rm S}(\epsilon)\ne 0 ~,$$ (A.14)

which means that neutrino number is conserved in the scattering process, whereas there is a nonvanishing energy exchange between neutrinos and matter due to scatterings.

Isoenergetic scattering:

If for a particular scattering process the energy transfer between neutrinos and target particles can be neglected, we have $\phi_l^{\rm out}(\epsilon,\epsilon')= \phi_l^{\rm in} (\epsilon,\epsilon')=: \phi_l(\epsilon)\cdot\delta(\epsilon-\epsilon')$, and the source terms given by Eqs. (A.10), (A.11), (A.12) simplify to

   

$$B_{\rm IS}(\epsilon,\mu)= 2\pi\sum_{l=0}^{\infty}(2l+1)P_l(\mu)~ \epsilon^2\phi_l(\epsilon)L_l(\epsilon) -2\pi \epsilon^2\phi_0(\epsilon)~f_\nu ~,$$ (A.15)

$$B^{(0)}_{\rm IS}(\epsilon) = 0 ~,$$ (A.16)

$$B^{(1)}_{\rm IS}(\epsilon) = 2\pi ~\epsilon^2 L_1(\epsilon)\cdot\big(\phi_1(\epsilon)-\phi_0(\epsilon)\big)~.$$ (A.17)

  
A.1.3 Pair processes

Applying the procedures outlined in Sect. A.1.2 to the collision integral for the process of thermal emission and absorption of neutrino-antineutrino pairs by electron positron pairs and related reactions (e.g., nucleon-nucleon bremsstrahlung), the corresponding source terms read (see Bruenn 1985; Pons et al. 1998):

 

$$B_{\rm TP}(\epsilon,\mu) 2\pi\int_0^\infty {\rm d}\epsilon'~ {\epsilon'}^2$$ =

$$\Big\{ (1-f_\nu)\cdot \phi_0^{\rm p}(\epsilon,\epsilon') -\sum_{l=0}^{\infty}(2l+1)P_l(\mu) \phi_l^{\rm p}(\epsilon,\epsilon')\bar{L}_l(\epsilon')$$

$$+f_\nu \cdot\sum_{l=0}^{\infty}(2l+1)P_l(\mu) \phi_l^{\rm a}(\epsilon,\epsilon')\bar{L}_l(\epsilon')) \Big\} ~,$$ (A.18)

 

$$B^{(0)}_{\rm TP}(\epsilon) = 2\pi\int_0^\infty {\rm d}\epsilon'~ ... ...)\phi_l^{\rm a}(\epsilon,\epsilon') L_l(\epsilon)\bar{L}_l(\epsilon') \Big\} ~,$$ (A.19)

 

$$B^{(1)}_{\rm TP}(\epsilon) = 2\pi\int_0^\infty {\rm d}\epsilon'~ ... ...n') [ (l+1)L_{l+1}(\epsilon)+l L_{l-1}(\epsilon) ]\bar{L}_l(\epsilon') \Big\}~.$$ (A.20)

Here bars indicate quantities for antineutrinos. The absorption and production kernels are related by

$$\phi_l^{\rm a}(\epsilon,\epsilon')= \left(1-{\rm e}^{(\epsilon+\epsilon')/T}\right)\cdot\phi_l^{\rm p}(\epsilon,\epsilon')$$ (A.21)

in accordance with the requirement of detailed balance.

   
A.2 Numerical implementation of various interaction processes

In the following we describe the implementation of the various neutrino-matter interactions into our transport scheme. The expressions are mainly taken from the works of Tubbs & Schramm (1975), Schinder & Shapiro (1982), Bruenn (1985), and Mezzacappa & Bruenn (1993b,c).

All average values of the source terms within individual energy bins (cf. Eq. (20)) are approximated to zeroth order by assuming the integrand to be a piecewise constant function of the neutrino energy $\epsilon$, and hence:

$$B^{(k)}_{j+1/2}:= B^{(k)}(\epsilon_{j+1/2}) ~.$$ (A.22)

With the rest-mass energy of the electron, $m_{\rm e}c^2=0.511~{{\rm MeV}}$, and the characteristic cross section of weak interactions $\sigma_0:= 4(m_{\rm e}c^2 G_{\rm F})^2/(\pi(\hbar c)^4)=1.761\times 10^{-44}~{\rm cm}^2$($G_{\rm F}$ is Fermi's constant), we define:

$${\cal G}:=\frac{\sigma_0}{4~{m_{\rm e}}^2c^4} ~\cdot$$ (A.23)

  
A.2.1 Neutrino absorption, emission and scattering
by free nucleons

In the standard description of neutrino-nucleon interactions (see Tubbs & Schramm 1975; Bruenn 1985; Mezzacappa & Bruenn 1993c), many-body effects for the nucleons in the dense medium, energy transfer between leptons and nucleons as well as nucleon thermal motions are ignored. The corresponding simplifications allow for a straightforward implementation of the processes, using Eq. (A.5) for the charged-current reactions and Eqs. (A.15)-(A.17) for the neutral-current scatterings, respectively. Expressions for $\kappa_{\rm a}^*$ and the Legendre coefficients $\phi_0(\epsilon)$, $\phi_1(\epsilon)$ can be computed from the rates given by Bruenn (1985) and Mezzacappa & Bruenn (1993c).

For absorption of electron neutrinos by free neutrons ( ${\nu_{\rm e}}+ {\rm n} \rightarrow {\rm e}^- + {\rm p}$) the final result for the opacity is:

 

$$\kappa_{\rm a}^*(\epsilon)= {\cal G}\cdot(g_{\rm V}^2+3g_{\rm A}^... ...lon)} \;\eta_{\rm np} \cdot (\epsilon+Q)\sqrt{(\epsilon+Q)^2-m_{\rm e}^2c^4} ~.$$ (A.24)

The opacity of the absorption of electron antineutrinos by free protons ( ${\bar\nu_{\rm e}}+ {\rm p} \rightarrow {\rm e}^+ + {\rm n}$) is

 

$$\kappa_{\rm a}^*(\epsilon)= {\cal G}\cdot(g_{\rm V}^2+3g_{\rm A}^... ...silon)} \;\eta_{\rm pn} \cdot(\epsilon-Q)\sqrt{(\epsilon-Q)^2-m_{\rm e}^2c^4}~,$$ (A.25)

if the neutrino energy $\epsilon\ge m_{\rm e}c^2+Q$, and $\kappa_{\rm a}^*(\epsilon)=0$, else. Here, $Q:=m_{\rm n}c^2-m_{\rm p}c^2$ denotes the difference of the rest-mass energies of the neutron and the proton, and $F_{{\rm e}^\mp}$ are the Fermi distribution functions of electrons or positrons. The constants $g_{\rm V}=1$ and $g_{\rm A}=1.254$ are the nucleon form factors for the vector and axial vector currents, respectively. The quantities $\eta_{\rm np}$ and $\eta_{\rm pn}$ take into account the effects of fermion blocking of the nucleons. Making the approximations mentioned above, in particular ignoring the recoil of the nucleon, Bruenn (1985, Eq. (C14); see also Reddy et al. 1998) derived:

 

$$\eta_{\rm np} 2\int\frac{{\rm d}^3\vec{p}~}{(2\pi\hbar c)^3}F_{\rm n}(\epsilon)[1-F_{\rm p}(\epsilon)]$$ :=

$$\frac{n_{\rm p}-n_{\rm n}}{\exp{\left[\psi_{\rm p}-\psi_{\rm n}\right]}-1} ~,$$ = (A.26)

where $n_{\rm n}$, $n_{\rm p}$ are the number densities of neutrons and protons, respectively. The corresponding degeneracy parameters $\psi_{\rm n}$ and $\psi_{\rm p}$are calculated by inverting the relation (${\rm N}$ stands for ${\rm n}$ or ${\rm p}$)

$$n_{\rm N}=\frac{1}{2\pi^2}\left(\frac{2~m_{\rm N} c^2~T}{(\hbar c)^2}\right)^{3/2} F_{1/2}(\psi_{\rm N}) ~,$$ (A.27)

where F_1/2 is the Fermi integral of index 1/2 (see, e.g., Lattimer & Swesty 1991, Sect. 2.2). The quantity $\eta_{\rm pn}$ in Eq. (A.25) is defined accordingly by exchanging the subscripts n and p in Eq. (A.26). In the nondegenerate regime, where blocking is unimportant, one verifies the familiar limits $\eta_{\rm pn}=n_{\rm p}$ and $\eta_{\rm np}=n_{\rm n}$.

Scattering of neutrinos off free nucleons (${\rm n}$ or ${\rm p}$) is mediated by neutral currents only, which makes the distinction between neutrinos and antineutrinos unneccessary. Neglecting nucleon recoil and nucleon thermal motions, the isoenergetic kernel obeys ${R}_{\rm IS}(\epsilon,\omega)\propto (1+\omega)$ (e.g. Bruenn 1985; Mezzacappa & Bruenn 1993c), which implies $\phi_{l>1}(\epsilon)\equiv 0$. The non-vanishing Legendre coefficients read

  

$$\phi_0(\epsilon) \frac{{\cal G}}{8\pi}\cdot \left\{ \begin{array}{ll} \eta_{\rm nn... ..._{\rm V}-1)^2+\frac{3}{4}g_{\rm A}^2\right] &({\rm for\ p})~,\end{array}\right.$$ = (A.28)

$$\phi_1(\epsilon) \frac{{\cal G}}{24\pi}\cdot \left\{ \begin{array}{ll} \eta_{\rm n... ..._{\rm V}-1)^2-\frac{1}{4}g_{\rm A}^2\right] &({\rm for\ p})~.\end{array}\right.$$ = (A.29)

The factors

$$\eta_{\rm nn/pp}:=T\frac{\partial n_{\rm n/p}}{\partial \mu_{\rm n/p}}$$ (A.30)

approximately account for nucleon final state blocking (Bruenn 1985), and $C_{\rm V}=1/2+2\sin^2\theta_{\rm W}$ with $\sin^2\theta_{\rm W}=0.23$for the Weinberg angle. In order to accurately reproduce the limits $\eta_{\rm nn/pp}=n_{\rm n/p}$ for nondegenerate (and nonrelativistic), and $\eta_{\rm nn/pp}=n_{\rm n/p}\cdot 3T/(2 E^{\rm F}_{\rm n/p})$for degenerate (and nonrelativistic) nucleons ( $E^{\rm F}_{\rm n/p}$ is the Fermi energy) we prefer using the analytic interpolation formula proposed by Mezzacappa & Bruenn (1993c)

$$\eta_{\rm nn/pp}= n_{\rm n/p}\frac{\xi_{\rm n/p}}{\sqrt{1+\x... ...with}\quad \xi_{\rm n/p}:= \frac{3T}{2 E^{\rm F}_{\rm n/p}} ~,$$ (A.31)

instead of a direct numerical evaluation of Eq. (A.30).

  
A.2.2 Neutrino absorption and emission by heavy nuclei

As in the case of charged-current reactions with free nucleons, neutrino absorption and emission by heavy nuclei ( ${\nu_{\rm e}}+ A' \rightleftharpoons A + {\rm e}^-$) is implemented by using Eq. (A.5). For calculating the opacity of this process we adopt the description employed by Bruenn (1985) and Mezzacappa & Bruenn (1993c):

$$\kappa_{\rm a}^*(\epsilon)= {\cal G}\cdot g_{\rm A}^2~ \frac{2}{7... ...}(\epsilon)} ~ n_{A} \cdot(\epsilon+Q') \sqrt{(\epsilon+Q')^2-m_{\rm e}^2c^4}~,$$ (A.32)

if $\epsilon\ge m_{\rm e}c^2-Q'$, and $\kappa_{\rm a}^*(\epsilon)=0$ else. Here $Q':=m_{A'}c^2-m_{A}c^2\approx \mu_{\rm n}c^2-\mu_{\rm p}c^2 +\Delta$ denotes the energy difference between the excited state A'=(N+1,Z-1) and the ground state A=(N,Z). Following Bruenn (1985), we set $\Delta=3$ MeV for all nuclei. The internal structure of the nucleus with charge Z and neutron number N=A-Z is represented by the term $2/7~N_{\rm p}(Z)N_{\rm h}(N)$, with the functions $N_{\rm p}(Z)$, $N_{\rm h}(N)$ as given by Mezzacappa & Bruenn (1993c, Eqs. (31), (32)).

  
A.2.3 Coherent scattering of neutrinos off nuclei

We use reaction rates for coherent neutrino scatterings off nuclei which include corrections due to the "nuclear form factor'' (following an approximation by Mezzacappa & Bruenn 1993c; Bruenn & Mezzacappa 1997), and ion-ion correlations (as described in Horowitz 1997). For a detailed discussion of the numerical handling of both corrections, see the appendix of Bruenn & Mezzacappa (1997). The result of the latter reference can readily be used to calculate $B^{(1)}_{\rm IS,ions}$, the contribution of coherent scattering to the rhs of the first order moment equation. For the Boltzmann equation, we have to make a minor additional approximation: The correction factor $\left\langle\mathcal{S}(\epsilon)\right\rangle_{\rm ion}$ as provided by Horowitz (1997) applies for the transport opacity and thus for the combination $\phi_0-\phi_1$ (which is given by Bruenn & Mezzacappa 1997) but not necessarily for the Legendre coefficients $\phi_0$ and $\phi_1$ individually, which are needed for calculating the collision integral $B_{\rm IS,ions}$ for the Boltzmann equation. We nevertheless assume here that $\phi_{0/1}(\epsilon)\propto \left\langle{\cal S}(\epsilon)\right\rangle_{\rm ion}$. This implies

$$\phi_{0/1}(\epsilon)\approx \frac{1}{16\pi}~{\cal G}\cdot A^... ...y)\left\langle\mathcal{S}(\epsilon)\right\rangle_{\rm ion} ~,$$ (A.33)

with

$${\cal F}_0(y) \frac{2y-1+{\rm e}^{-2y}}{y^2} ~,$$ :=

$${\cal F}_1(y) \frac{2-3y+2y^2-(2+y){\rm e}^{-2y}}{y^3} ~,$$ := (A.34)

$$\frac{2}{5}\frac{(1.07A)^{2/3}\epsilon^2(10^{-13}{\rm cm})^2}{(\hbar c)^2} ~,$$ y :=

containing the effects of the nuclear form factor, and

$$\mathcal{Z}(A,Z):= \left[(C_{\rm A}-C_{\rm V})+(2-C_{\rm A}-C_{\rm V})\frac{2Z-A}{A}\right]^2 ~\cdot$$ (A.35)

$C_{\rm V}$ is defined as before (see Eqs. (A.28), (A.29)) and $C_{\rm A}=1/2$. In effect, the angular dependence of the ion-ion correlation correction is not exactly accounted for in the Boltzmann equation. The approximation, however, only influences the closure relation for the moment equations at a higher level of accuracy. It neither affects the Legendre coefficients at larger neutrino energies (where $\left\langle\mathcal{S}(\epsilon)\right\rangle_{\rm ion}$ is close to unity) nor does it destroy the consistency of the Boltzmann equation and its moment equations. Moreover, it has been shown that the overall effect of ion-ion correlations in stellar core collapse is only marginally relevant (Bruenn & Mezzacappa 1997; Rampp 2000).

  
A.2.4 Inelastic scattering of neutrinos off charged leptons

For the scattering of neutrinos off charged leptons, we approximate the dependence of the scattering kernel on the scattering angle by truncating the Legendre expansions (Eqs. (A.10)-(A.12)) at $l_{\rm max}=1$. For our purposes, this has been shown to provide a sufficiently accurate approximation (see Smit & Cernohorsky 1996; Mezzacappa & Bruenn 1993b), in particular also because the total scattering rate is correctly represented. Expressions for the Legendre coefficients are taken from the works of Yueh & Buchler (1977), Mezzacappa & Bruenn (1993b), Cernohorsky (1994), Smit & Cernohorsky (1996):

$$\phi_l^{\rm out}(\epsilon,\epsilon')= \alpha_{\rm I} A_l^{\... ...psilon') +\alpha_{\rm II} A_l^{\rm II}(\epsilon,\epsilon') ~,$$ (A.36)

with $k\in \{{\rm I},{\rm II}\}$ and

 

$$A_l^k(\epsilon,\epsilon')= \frac{{\cal G}}{\epsilon^2~\epsilon^{\... ...\rm e}(E_{\rm e}+\epsilon-\epsilon')\big)H^k_l(\epsilon,\epsilon',E_{\rm e}) ~.$$ (A.37)

The constants $\alpha_{\rm I/II}$, which depend on the neutrino species and also the type of charged lepton (i.e. electron or positron) involved in the scattering process, are combinations of the weak coupling constants (see, e.g., Cernohorsky 1994). For the $\mu$ and $\tau$ neutrinos and antineutrinos, which are treated as a single kind of neutrino in our code, we take the arithmetic mean of the corresponding coupling constants. For l=0,1 the functions $H^{\rm I/II}_l(\epsilon,\epsilon',E_{\rm e})$(in units of ${{\rm MeV}}^5$) are given by Yueh & Buchler (1977 with the corrections noted by Bruenn 1985).

Calculating the Legendre coefficients $\phi^{\rm in/out}_l(\epsilon_j,\epsilon_{j'})$ is a computationally expensive task, since for all combinations $(\epsilon_j,\epsilon_{j'})$and all radial grid points numerical integrals over the energy of the charged leptons (Eq. (A.37)) have to be carried out. It is, however, sufficient to explicitly compute $\phi^{\rm in/out}_l(\epsilon_j,\epsilon_{j'})$ for $\epsilon_j \le \epsilon_{j'}$. The missing coefficients can be obtained by exploiting a number of symmetry properties (see Cernohorsky 1994). In doing so, not only the computational work is reduced by almost a factor of two, but even more importantly, detailed balance ( $B_{\rm NES}\equiv 0$, if $f\equiv f^{\rm eq}$) can be verified for the employed approximation and discretization of the collision integral and its angular moments to within the roundoff error of the machine, provided that all integrals over neutrino energies are approximated by simple zeroth-order quadrature formulae (cf. (Eq. A.22)). Similarly, one can also verify the conservation of particle number (Eq. (A.14)) in the corresponding finite difference representation.

In our neutrino transport code the dependence of the source terms on the neutrino distribution and its angular moments is treated fully implicitly in time, i.e. the time index of the corresponding quantities in Eqs. (A.10)-(A.12) reads fⁿ⁺¹ and Lⁿ⁺¹_l. The thermodynamic quantities and the composition of the stellar matter which are needed to calculate the Legendre coefficients for inelastic scattering of neutrinos, however, are computed from $\varepsilon^{*}$ and ${Y_{\rm e}}^*$, representing the specific internal energy and electron fraction at the time level after the hydrodynamic substeps (cf. Sect. 3.6.1). This allows one to save computer time since the Legendre coefficients must be computed only once at the beginning of each transport time step. Comparing to results obtained with a completely time-implicit implementation of the source terms (i.e. using $\varepsilon^{n+1}$ and ${Y_{\rm e}}^{n+1}$ for determining the stellar conditions that enter the computation of the Legendre coefficients) we have found no differences in the solutions during the core-collapse phase, where neutrino-electron scattering plays an important role in redistributing neutrinos in energy space and thus couples the neutrino transport to the evolution of the stellar fluid.

  
A.2.5 Neutrino pair production/annihilation by thermal ${\rm e}^+~{\rm e}^-$ pairs

Following the suggestion of Pons et al. (1998) we truncate the Legendre series (A.18)-(A.20) for the pair-production kernels at $l_{\rm max}=2$. The Legendre coefficients are given by (Bruenn 1985; Pons et al. 1998)

 

$$\phi^{\rm p}_l(\epsilon_j,\epsilon_{j'})={\cal G}\cdot \frac{T^2}... ...epsilon_j,\epsilon_{j'}) +\alpha_2^2\Psi_l(\epsilon_{j'},\epsilon_j) \right) ~,$$ (A.38)

with $x:=(\epsilon+\epsilon')/T$ and ${\cal G}$ defined by Eq. (A.23). Explicit expressions for the (dimensionless) functions $\Psi_l(\epsilon_j,\epsilon_{j'})$ and details about their efficient calculation are given by Pons et al. (1998). Also the coefficients $\alpha_1,\alpha_2$ for the different neutrino flavours can be found there (Pons et al. 1998, Table 1).

  
A.2.6 Nucleon-nucleon bremsstrahlung

Legendre coefficients for the nucleon-nucleon bremsstrahlung process, assuming nonrelativistic nucleons, are calculated by using the rates given in Hannestad & Raffelt (1998). The non-vanishing coefficients read:

  

$$\phi^{\rm p}_0(\epsilon_j,\epsilon_{j'}) {\cal G}\cdot\frac{3~C_{\rm A}^2}{(2\pi)^2}~ \big({\rm e}^{-x}-1\... ...{\Gamma_\sigma}{(\epsilon+\epsilon')^2 + 0.25~(\Gamma_\sigma~g)^2}\cdot s(x) ~,$$ = (A.39)

$$\phi^{\rm p}_1(\epsilon_j,\epsilon_{j'}) -\frac{1}{9}\phi^{\rm p}_0(\epsilon_j,\epsilon_{j'})~,$$ = (A.40)

with

$$\Gamma_\sigma:=\frac{8\sqrt{2\pi}\alpha_\pi^2}{3\pi^2} \eta_... ... \quad \eta_*:=\frac{(p_{\rm F}~c)^2}{2 m_{\rm N} c^2 ~T}\cdot$$ (A.41)

In Eq. (A.40), $p_{\rm F}=\hbar\cdot(3\pi^2~n_{\rm N})^{1/3}$denotes the Fermi momentum for the nucleons, $\alpha_\pi=15$ is the pion-nucleon "fine-structure constant'' (see Hannestad & Raffelt 1998), and $m_{\rm N}$ is the nucleon mass. The analytic form of the dimensionless fit functions g and s(x), which both depend on the nucleon density $n_{\rm N}$ and temperature T of the stellar medium can be found in Hannestad & Raffelt (1998). The sum in Eq. (A.39) runs over the individual processes $\nu\bar\nu~{\rm nn}\rightleftharpoons {\rm nn}; ({\rm N}={\rm n})$, $\nu\bar\nu~{\rm pp}\rightleftharpoons {\rm pp}; ({\rm N}={\rm p})$, and $\nu\bar\nu~{\rm np}\rightleftharpoons {\rm np}; ({\rm N}={\rm np})$. The latter process is approximately taken into account by using the particle density $n_{\rm np}:=\sqrt{n_{\rm n} n_{\rm p}}$ and multiplying the corresponding contribution to the sum in Eq. (A.39) by a factor of 28/3 (cf. Thompson et al. 2000).

The production processes of neutrino-antineutrino pairs both by the annihilation of ${\rm e}^+~{\rm e}^-$ pairs and by the nucleon-nucleon bremsstrahlung are implemented into the discretized equations (Eqs. (21), (22), (28), (29)) by applying the techniques described for the inelastic scattering reactions (see Sect. A.2.4).