A Necessary Condition for the Submergence of Proto-Neutron Star Magnetic Fields by Supernova Fallback

Abstract

Central compact objects (CCOs) are a subclass of neutron stars with a dipole magnetic field strength considerably weaker than those of radio pulsars and magnetars. One possible explanation for such weak magnetic fields in the CCOs is the hidden magnetic field scenario, in which supernova fallback submerges the magnetosphere of a proto-neutron star beneath a newly formed crust. However, the fallback mass and timescale required for this submergence process remain uncertain. We perform one-dimensional general relativistic magnetohydrodynamic simulations of the supernova fallback onto a magnetized proto-neutron star, while considering neutrino cooling. In our simulations, the infalling material compresses the magnetic field and drives a strong shock. The shock initially expands outward, but eventually stalls and recedes as neutrino cooling becomes significant. After the shock stalls, the gas density above the magnetosphere increases rapidly, potentially leading to the formation of a new crust. To understand the shock dynamics, we develop semi-analytic models that describe the resulting magnetospheric and shock radii when the shock stalls. By comparing the fallback time scale with the shock stalling time scale, corresponding to the waiting time for the new crust formation, we derive a necessary condition for the submergence of the PNS magnetic field. Our results will provide guidance for investigating the diversity of young isolated neutron stars through multidimensional simulations.

Figures (9)

• Figure 1: A schematic view of the supernova fallback. The white circle in the left figure denotes the PNS with a radius of $r_{\rm PNS}$. The gray arrows stand for the velocity vector of the free-falling gas, and black wavy lines with arrows indicate the neutrinos propagating outward. We denote the magnetospheric and shock radii as $r_{\rm M}$ and $r_{\rm sh}$, respectively, in terms of the spherical radius $r$. The overall structure of the supernova fallback can be divided into free-fall region ($r>r_{\rm sh}$), post-shock region ($r_{\rm M}<r<r_{\rm sh}$), and the PNS magnetosphere ($r_{\rm PNS}<r<r_{\rm M}$). The shadow in the post-shock region represents the gas density and temperature. The right two panels are enlarged views around $r=r_{\rm M}$. A new crust forms above the PNS magnetosphere when neutrino cooling is efficient (panel a), whereas no crust forms when neutrino cooling is inefficient (panel b).
• Figure 2: Left panel: cooling rate due to the neutrino emission against the gas temperature. We employ $\rho=10^{6},~10^{8},~10^{10}$, and $10^{12}~{\rm cm^{3}~s^{-1}}$. Right panel: steady solutions of the shock radius as a function of fallback accretion rate. Here, we adopt $r_{\rm PNS}=10~{\rm km}$ and $M_{\rm PNS}=1.4M_\odot$ (see Equations \ref{['eq:pair']} and \ref{['eq:URCA']}). The vertical dashed line denotes the mass accretion rate when $r_{\rm sh,pair}=r_{\rm sh,URCA}$ ($\dot{M}_{\rm fb,a}$).
• Figure 3: Left: radial profiles of three different pressures: the magnetic pressure $p_{\rm mag}$ (blue), the radiation pressure $p$, and the ram pressure $p_{\rm ram}$. Dashed lines indicate the magnetospheric radius $r_{\rm M}$ and the shock radius $r_{\rm sh}$. The thin line represents the radial dependence of $p$ in Equation (\ref{['eq:shock_pres']}). Right: various time scales against radius. We express the dynamical time scale as $t_{\rm dyn}=r/v^{(r)}$ and the cooling time scales due to the pair process as $t_{\rm pair}=e/\dot{q}_{\rm pair}$ and the URCA process as $t_{\rm URCA}=e/\dot{q}_{\rm URCA}$. The shaded region stands for a region of $r<r_{\rm M}$. Both panels are results in the fiducial model, B14Mm5R10, at $t=0.118~{\rm s}$.
• Figure 4: The time-sequenced images of $p_{\rm ram}$ (left), $p$ (middle), and $p_{\rm mag}$ (right) in the fiducial model, B14Mm5R10. Dashed lines in the left panel represent the shock formation time $t_{\rm shock}$ and the shock stalling time $t_{\rm stall}$. For $t_{\rm shock}<t<t_{\rm stall}$, the shock expands over time. However, it stalls at $t=t_{\rm stall}$ and recedes for $t>t_{\rm stall}$ due to the neutrino cooling. The white line in the right panel indicates the radius of $\beta=1$. The region on the lower and upper sides of this line correspond to $\beta<1$ and $\beta>1$, respectively.
• Figure 5: Top and middle: plots for the magnetospheric (top) and shock radii (middle) as a function of the fallback accretion rate. All plots are the results when $t=t_{\rm stall}$. The solid lines indicate the semi-analytical solutions calculated from Equations (\ref{['eq:semi_ana1']}), (\ref{['eq:pair2']}), and (\ref{['eq:URCA2']}). The shaded regions in the top and middle panels denote where the deviation from the semi-analytic solution for $B_{\rm PNS}=10^{15}~{\rm G}$ is within 20% and 50%, respectively. We also plot the analytical solution of $r_{\rm sh}$ for $B_{\rm PNS}=0$ (Equation \ref{['eq:pair']}). Bottom: shock stalling time scale ($t'_{\rm stall}=t_{\rm stall}-t_{\rm shock}$) against the fallback accretion rate. Solid lines represent the fitting results. All panels are the results of $r_{\rm PNS}=10~{\rm km}$.