3 Basic equations and assumptions

The hydrodynamic equations are considered in Eulerian form for spherical symmetry with source terms for Newtonian gravity and neutrino energy and momentum exchange with the stellar medium. The equations of continuity, momentum, and energy are:

$${\partial \rho \over \partial t}+{1\over r^2}{\partial \over \partial r} (r^2\rho v)= 0 \ ,$$ (2)

$${\partial (\rho v)\over \partial t}+{1\over r^2}{\partial \ov... ...al P\over\partial r} - \rho{\partial \Phi \over \partial r}\ ,$$ (3)

$${\partial e\over \partial t}+{1\over r^2}{\partial \over \par... ...\rbrack= -\rho v {\partial \Phi \over \partial r} + Q_{\nu}\ .$$ (4)

Here r, v, $\rho$, P, t are radius, fluid velocity, density, pressure, and time, respectively, and e is defined as the sum of internal energy density, $\varepsilon$, and kinetic energy density of the gas:

$$e\,=\, {1\over 2}\rho v^2 + \varepsilon \ .$$ (5)

The term $Q_{\nu}$ denotes the rate of energy gain or loss per unit volume by neutrino heating and cooling. $\Phi(r)$ is an effective potential which contains contributions from the gravitational potential and from the momentum transfer to the stellar gas by neutrinos. Neglecting self-gravity of the gas in the region between neutrinosphere and supernova shock, it can be written as

$$\Phi\,=\,-\,{G{\widetilde M}\over r}\,=\,-\,{G\over r}\left( ... ...ppa_{\rm {t}} \rangle L_{\nu}\over 4\pi Gc\rho} \right)\ \cdot$$ (6)

Here G is the gravitational constant, c the speed of light, M the mass inside $R_{\nu }$ and ${\widetilde M}$ means an effective mass that includes the momentum transfer term and is defined by the term in brackets on the right side of Eq. (6). When self-gravity is disregarded, the mass of the gas between $R_{\nu }$ and $R_{\rm {s}}$ must be negligible compared to the neutron star mass M, i.e.,

$$\Delta M\,=\,\int_{R_{\nu}}^{R_{\rm {s}}}{\rm {d}}r\,4\pi r^2\rho(r) \ll M\ .$$ (7)

In Eq. (6) $L_{\nu} = \sum_{\nu_i}L_{\nu_i}$ is the total neutrino luminosity and $\langle \kappa_{\rm {t}} \rangle$ the mean total opacity calculated as an average of the total opacities of neutrinos $\nu_i$ and antineutrinos $\bar\nu_i$ of all flavors according to

$$\langle \kappa_{\rm {t}} \rangle L_{\nu}\,\equiv\, \sum_{\nu_... ...+ \sum_{\bar\nu_i}\kappa_{{\rm {t}},\bar\nu_i}L_{\bar\nu_i}\ .$$ (8)

The total opacity $\kappa_{{\rm t},\nu_i}$ of neutrino $\nu_i$ is considered to be averaged over the spectrum of the corresponding energy flux. Note that in this paper, opacities are defined as inverse mean free paths and are thus measured in units of 1/cm. Equations (3), (4), and (6) imply that the momentum transfer rate from neutrinos to the stellar gas is written as $\langle \kappa_{\rm {t}} \rangle L_{\nu}/(4\pi c\,r^2)$ with $L_\nu$ and $\langle \kappa_{\rm {t}} \rangle/\rho$not depending on r. This is approximately fulfilled in the optically thin regime for neutrinos, i.e., exterior to the neutrinosphere where the neutrino luminosities and spectra are roughly constant. Yet it is not exactly true, because the concept of "the'' neutrinosphere is fuzzy and neutrino emission and absorption continue even outside the neutrinosphere. In addition, the opacity depends on the composition which varies with the radius. During all of the post-bounce evolution, however, the typical total neutrino luminosity is only a few per cent of the Eddington luminosity,

$$L_{{\rm {Edd}},\nu}\,\equiv\,{4\pi GMc\rho \over \langle \kappa_{\rm {t}} \rangle}\ \cdot$$ (9)

Therefore the neutrino source terms for momentum in Eq. (3) and for kinetic energy in Eq. (4), which are carried by the potential $\Phi$, are always small and the approximate treatment following below is justified.

Neutrinos transfer momentum to the stellar medium by neutral-current scatterings off neutrons and protons. The corresponding transport opacity for these scattering processes is

$$\kappa_{\rm sc}\,\approx\,{5\alpha^2+1\over 24} {\sigma_0 \la... ...rm e}c^2)^2}\,{\rho\over m_{\rm u}}\, (Y_{\rm n}+Y_{\rm p})\ .$$ (10)

Here $m_{\rm u}\approx 1.66~10^{-24}\,$g is the atomic mass unit, $m_{\rm e}c^2 = 0.511\,$MeV the rest-mass energy of the electron, $\sigma_0 = 1.76~10^{-44}~$cm², and $Y_{\rm n} = n_{\rm n}/n_{\rm b}$ and $Y_{\rm p} = n_{\rm p}/n_{\rm b}$ are the number fractions of free neutrons and protons, i.e., their particle densities normalized to the number density $n_{\rm b}$ of nucleons. A minor difference between the neutrino-proton and the neutrino-neutron scattering cross section due to different vector coupling constants is ignored, and also the axial-vector couplings are assumed to be the same and to be equal to the charged-current axial-vector coupling constant in vacuum, $\alpha = -1.26$. Additional scattering reactions with electrons and positrons can be neglected because of their much smaller cross sections, and neutrino scattering off nuclei is unimportant because the post-bounce medium exterior to the neutrinosphere is nearly completely disintegrated into free nucleons.

In case of $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ also the charged-current absorptions on neutrons and protons, respectively, need to be taken into account due to their large cross sections. The absorption opacity is

$$\kappa_{\rm {a}}\,\approx\,{3\alpha^2+1\over 4} {\sigma_0\lan... ...}\, \left\{\matrix{Y_{\rm n} \cr Y_{\rm p} \cr}\right\}\ \cdot$$ (11)

In Eqs. (10) and (11) the recoil of the nucleon and phase space blocking effects for the fermions are neglected, which is very good at the conditions considered in this paper. In both neutral-current and charged-current processes only the leading terms depending on the squared neutrino energy, $\epsilon_{\nu}^2$, are taken into account. Averaging over the spectrum of the neutrino energy flux yields the factor $\langle \epsilon_{\nu}^2 \rangle$, for which Eq. (25) provides a suitable definition, if minor differences between the spectra of neutrino energy density and flux density are disregarded.

The total opacity includes the contributions from scattering and absorption and is given as $\kappa_{{\rm t},\nu_i} = \kappa_{{\rm sc},\nu_i}+ \kappa_{{\rm a},\nu_i}$. With typical values $L_{\nu_{\rm e}} = L_{\bar\nu_{\rm e}} = L_{\nu_x} = {1\over 6}L_{\nu}$ (with $\nu_x \in \lbrace \nu_{\mu},\,\bar\nu_{\mu},\,\nu_{\tau},\,\bar\nu_{\tau}\rbrace$and $L_{\nu_x}$ being the luminosity of each individual type of $\nu_x$), $\langle \epsilon_{\bar\nu_{\rm e}}^2 \rangle\approx 2 \langle \epsilon_{\nu_{\rm e}}^2 \rangle$and $\langle \epsilon_{\bar\nu_x}^2 \rangle\approx 4 \langle \epsilon_{\nu_{\rm e}}^2 \rangle$, the total opacity averaged for all neutrinos and antineutrinos can be estimated from Eq. (8) as

 

$$\langle \kappa_{\rm {t}} \rangle \approx {113\alpha^2+25\over 144} {\sigma_0\langle \epsilon^2_{\nu_{\rm e}} \rangle\over (m_{\rm e}c^2)^2}\,{\rho\over m_{\rm u}}\, (Y_{\rm n}+Y_{\rm p})$$

$$\approx 1.9~10^{-7}\rho_{10}\, \left( {kT_{\nu_{\rm e}}\over 4\,{\rm MeV}} \right)^{\! 2}\ \ \left\lbrack {1\over {\rm cm}} \right\rbrack \ \cdot$$ (12)

For deriving the first expression, the factor $Y_{\rm n}+2Y_{\rm p}$ in the absorption term was replaced by $Y_{\rm n}+Y_{\rm p}$. This is a reasonably good approximation because between the neutrinosphere and the shock, and the corresponding change of the absorption opacity of electron antineutrinos implies only a minor error in the total opacity. In the second equation use was made of $Y_{\rm n}+Y_{\rm p} \approx 1$. If the neutrino flux spectrum has Fermi-Dirac shape with vanishing degeneracy, the neutrino temperature $T_{\nu}$ is related to the mean squared neutrino energy by $\langle \epsilon^2_{\nu} \rangle\approx 21\,(kT_{\nu})^2$. k is the Boltzmann constant and $\rho_{10}$ the density measured in 10$^{10}\,$g/cm³. For a total neutrino luminosity $L_{\nu} = 10^{53}\,$erg/s, neutrino momentum transfer reduces ${\widetilde M}$ in Eq. (6) relative to M by about 3.8 10^-2 $M_{\odot }$, which is indeed a small correction.