Generalized Kompaneets formalism for inelastic neutrino-nucleon scattering in supernova simulations

Based on the Kompaneets approximation, we develop a robust methodology to calculate spectral redistribution via inelastic neutrino-nucleon scattering in the context of core-collapse supernova simulations. The resulting equations conserve lepton number to machine precision and scale linearly, not quadratically, with number of energy groups. The formalism also provides an elegant means to derive the rate of energy transfer to matter which, as it must, automatically goes to zero when the neutrino radiation field is in thermal equilibrium. Furthermore, we derive the next higher order in $ϵ/m{c}^2$ correction to the neutrino Kompaneets equation. Unlike other Kompaneets schema, ours also generalizes to the case of anisotropic angular distributions, while retaining the conservative form that is a hallmark of the classical Kompaneets equation. Our formalism enables immediate incorporation into supernova codes that follow the spectral angular moments of the neutrino radiation fields.

Figure 1

The dynamical structure function versus the neutrino energy transfer for some representative parameters. The temperature here is $T=3\mathrm{MeV}$. Top: $\mu =-0.5$. Bottom: $\mu =0.5$, where $\mu$ is the cosine of the angle between the incoming and outgoing neutrinos. A mass density of ${10}^{10}\mathrm{g}{\mathrm{cm}}^{-3}$ is assumed. Each line is actually a superposition of the exact [Eq. (2)] and approximate [Eq. (4)] structure functions.

Figure 2

Comparison between particle number transfer and energy transfer calculated in different ways. Here, the nucleon temperature is $T=1\mathrm{MeV}$ and neutrino temperature is 5.5 MeV in the left two panels, while $T=3\mathrm{MeV}$ and $T_{\nu }=3.5\mathrm{MeV}$ in the right two panels. The top two panels show the particle number transfer calculated by the standard Kompaneets equation (black solid line), corrected Kompaneets equation (red solid line), and by doing the exact integration numerically (black dashed line). The bottom two panels show $I_{\nu }$ calculated with (red solid line) and without (black solid line) the correction factor. The exact energy deposition rate (black dashed line) is calculated by doing the integration numerically. The $y$ axes are arbitrarily scaled.

Figure 3

Comparison between particle number transfer and energy transfer calculated in different ways with only 12 energy bins. The nucleon temperature is $T=1\mathrm{MeV}$ and neutrino temperature is 5.5 MeV in left two panels, while $T=3\mathrm{MeV}$ and $T_{\nu }=3.5\mathrm{MeV}$ in the right two panels. The top two panels show the particle number transfer calculated by the corrected Kompaneets equation (red dots) and by doing the integration numerically (black dashed line). The bottom two panels depict $I_{\nu }$ calculated with the correction factor (red dots). The exact energy deposition rate (black dashed line) is calculated by doing the integration numerically. The y axes are arbitrarily scaled. Red dots in the upper panels are located at the centers of each logarithmic energy bin, while in the lower panels they are located at the edges.

Figure 4

Comparison between particle number transfer and energy transfer calculated in different ways. The neutrino distribution is $f=\frac{1}{2}\exp (-\frac{(ϵ-{ϵ}_0{)}^2}{2{\sigma }^2})$. This distribution is very different from the real case, but the formulas work well. Left panels: ${ϵ}_0=10\mathrm{MeV}$ and $\sigma =5\mathrm{MeV}$. Right panels: ${ϵ}_0=20\mathrm{MeV}$ and $\sigma =5\mathrm{MeV}$. Top panels: particle number transfer calculated by standard Kompaneets equation (black solid line), corrected Kompaneets equation (red solid line), and by doing the integration numerically (black dashed line). Bottom panels: $I_{\nu }$ calculated with (red solid line) and without (black solid line) the correction factor. The exact energy deposition rate (black dashed line) is calculated by doing the integration numerically. The y axes are arbitrarily scaled. Note that red lines fluctuate around the exact values. This is because we throw away the total derivative terms in the correction. Such fluctuations have little effect on ${\dot{Q}}_{\nu }$, and the relative errors of ${\dot{Q}}_{\nu }$ in both cases are below 2%.