Coupling between the radiation field and the fluid

The neutrino radiation field in the supernova is tightly coupled to the high density fluid. In order to obtain a well defined link between the evolution of radiation quantities and the evolution of hydrodynamical quantities, we recall that the solution of the microscopic Boltzmann equation updates the radiation moments according to Eqs. (28)-(30). We may now subtract these updates from the global evolution equations (6)-(12) to find the hydrodynamical part of the evolution equations

$$\frac{\partial }{\partial t}\left[ \frac{\Gamma }{\rho }\right] = \frac{\partial }{\partial a}\left[ 4\pi r^{2}\alpha u\right]$$ (88)

$$\frac{\partial }{\partial t}\left[ \Gamma \left( 1+e\right) \right] = -\frac{\partial }{\partial a}\left[ 4\pi r^{2}\alpha up\right]$$

$$- \alpha \Gamma \int \left( \frac{j}{\rho }-\chi F\right) E^{3}dEd\mu +\alpha u\int \chi FE^{3}dE\mu d\mu$$ (89)

$$\frac{\partial }{\partial t}\left[ u\left( 1+e\right) \right] = -4\pi r^{2}\frac{\partial }{\partial a}\left[ \alpha \Gamma p\right]$$

$$- \frac{\alpha }{r}\left[ \left( 1+e\right) \left( 1+\frac{6Vp}{m}\right) \frac{m}{r}+4\pi r^{2}\left( \left( 1+e\right) \rho K+pJ\right) \right.$$

$$+ \left. \left( 2u^{2}-\frac{m}{r}\right) \frac{2p}{\rho }\right]$$

$$+ \alpha \Gamma \int \chi FE^{3}dE\mu d\mu -\alpha u\int \left( \frac{j}{\rho }-\chi F\right) E^{3}dEd\mu$$ (90)

$$\frac{\partial }{\partial t}Y_{e} = \mp \alpha \int \left( \frac{j}{\rho }-\chi F\right) E^{2}dEd\mu$$ (91)

$$\frac{\partial V}{\partial a} = \frac{\Gamma }{\rho }$$ (92)

$$\frac{\partial m}{\partial a} = \Gamma \left( 1+e+J\right) +uH$$ (93)

$$\rho \left( 1+e\right) \frac{\partial \alpha }{\partial a} = -\frac{\partial }{\partial a}\left[ \alpha p\right] +\frac{\alpha }{4\pi r^{2}}\int \chi FE^{3}dE\mu d\mu$$ (94)

$$\frac{\partial e}{\partial t} = -p\frac{\partial }{\partial t}\left( \frac{1}{\rho }\right) -\alpha \int \left( \frac{j}{\rho }-\chi F\right) E^{3}dEd\mu .$$ (95)

The equations are written such that the presence of the radiation field only enters in terms of energy and momentum exchange. In the limit of a decoupled radiation and matter flow we therefore solve for ideal hydrodynamics and free streaming in an independent and numerically stable manner, no matter what the size of the radiation field is. The only remaining interactions between the radiation field and the fluid are of a gravitational nature, for example in the contribution of the radiation field to the common gravitational mass in Eq. (), or in the term in the momentum equation (). The detailed discretization of Eqs. ()-() has been described in Liebendoerfer_Rosswog_Thielemann_02. The interaction terms between the radiation field and the fluid are

$$e_{i'}^{\{\rm ext\}} = \sum _{j',k'}\left( \frac{j_{k'}(\rho _{i'},T_{i'}^{*},Y_{e,i'}^{... ...(\rho _{i'},T_{i'}^{*},Y_{e,i'}^{*})F_{i',j',k'}\right) E_{k'}^{3}dE_{k'}w_{j'}$$

$$S_{i+1}^{\{\rm ext\}} = -\sum _{j',k'}\chi _{k'}(\rho _{i'},T_{i'}^{*},Y_{e,i'}^{*})F_{i',j',k'}E_{k'}^{3}dE_{k'}\mu _{j'}w_{j'}$$

$$Y_{e,i'}^{\{\rm ext\}} = \mp \sum _{j',k'}\left( \frac{j_{k'}(\rho _{i'},T_{i'}^{*},Y_{e,i... ...\rho _{i'},T_{i'}^{*},Y_{e,i'}^{*})F_{i',j',k'}\right) E_{k'}^{2}dE_{k'}w_{j'},$$ (96)

where the minus sign in the -term is used for electron neutrinos and the plus sign for electron antineutrinos. Emission and absorption of and -neutrinos do not change the electron fraction. The star superscript for the temperature and electron fraction variables indicates again an evaluation of the cross sections for emission and absorption with a consistently updated thermodynamical state as described in the next section.