Influence of a magnetic field on beta-processes in supernova matter

An influence of a magnetic field on beta-processes is investigated under conditions of a core-collapse supernova. For realistic magnetic fields reachable in astrophysical objects we obtain simple analytical expressions for reaction rates of beta-processes as well as the energy and momentum transferred from neutrinos and antineutrinos to the matter. Based on the results of one-dimensional simulations of a supernova explosion, we found that, in the magnetic field with the strength $B\sim {10}^{15}\mathrm{G}$, the quantities considered are modified by a few percents only and, as a consequence, the magnetic-field effects can be safely neglected, considering neutrino interaction and propagation in a supernova matter. The analytical results can be also applied for accretion discs formed at a merger of compact objects in close binary systems.

Figure 1

Definitions of vectors and angles used in the analysis. Here, ${\mathbf{n}}_B=\mathbf{B}/B$ is the unit vector along the magnetic-field strength, ${\mathbf{n}}_R$ specifies the star radial direction, $\mathbf{q}$ is the neutrino momentum. The angles $\beta$, $\theta$, and $\vartheta$ between the vectors and the polar angle $\phi$ are also shown.

Figure 2

The protoneutron star (PNS) of radius $R_{\mathrm{PNS}}$ and momentum $\mathcal{F}$ transferred from neutrinos and antineutrinos to the matter at the point $M$, being at the distance $R$ from the PNS center. ${\mathbf{n}}_R$ and ${\mathbf{n}}_B$ are unit vectors in the radial direction from the PNS center and along the magnetic field strength, respectively.

Figure 3

The gain radius $R_{\mathrm{gain}}^{PV}$ from PROMETHEUS-VERTEX simulations [29].

Figure 4

The deviation of a gain radius $R_{\mathrm{gain}}(B=0)$ for a matter without the magnetic field, obtained by analytical calculations, from the gain radius $R_{\mathrm{gain}}^{PV}$ from PROMETHEUS-VERTEX simulations [29].

Figure 5

The deviation of a gain radius $R_{\mathrm{gain}}(B)$ in the presence of the magnetic field with $\cos \beta =1$ from an unmagnetized one. Magnetic field strengths are $B={10}^{15}\mathrm{G}$ (blue line) and $B={10}^{16}\mathrm{G}$ (green line).

Figure 6

The relative deviation of the proton-to-neutron reaction rate of beta-processes in the magnetic field from the unmagnetized case as a function of distance $R$ from the PNS center for several values of the time $t$ after a bounce and different directions of the magnetic field. Configurations of magnetic field are $\cos \beta =-1$ (top panel), $\cos \beta =0$ (middle panel) and $\cos \beta =1$ (bottom panel). Red lines: $t=0.1\sec$; orange: $t=0.5\sec$; yellow: $t=1.5\sec$; green: $t=4\sec$; cyan: $t=5.5\sec$; blue: $t=10\sec$; violet: $t=13\sec$.

Figure 7

The relative deviation of the total energy transferred from neutrinos and antineutrinos to the matter. The legend is the same as in Fig. 6.

Figure 8

The total momentum projection along the field, ${\mathcal{F}}_B(B)$, compared with the momentum projection ${\mathcal{F}}_r(0,0)$ on the radial direction. The legend is the same as in Fig. 6.

Figure 9

The relative deviation of the radial component, ${\mathcal{F}}_r(B,\cos \beta )$, of the total momentum transferred from neutrinos and antineutrinos to the matter from its representative value ${\mathcal{F}}_r(0,0)$, corresponding to the unmagnetized matter. The legend is the same as in Fig. 6.

Figure 10

The ratio ${\phi }_k(\eta )/{\phi }_k(\eta =0)$. Dotted line: $k=3$. Dashed line: $k=4$. Solid line: $k=5$.

Figure 11

The ratio ${\psi }_k(\eta )/{\phi }_k(\eta )$. Dotted line: $k=3$. Dashed line: $k=4$. Solid line: $k=5$.