Fast magnetic field evolution in neutron stars: The key role of magnetically induced fluid motions in the core

In [Gusakov et al. Phys. Rev. D 96, 103012 (2017)], we proposed a self-consistent method to study the quasistationary evolution of the magnetic field in neutron-star cores. Here, we apply it to calculate the instantaneous particle velocities and other parameters of interest, which are fixed by specifying the magnetic field configuration. Interestingly, we found that the magnetic field can lead to generation of a macroscopic fluid motion with the velocity significantly exceeding the diffusion particle velocities. This result calls into question the standard view on the magnetic field evolution in neutron stars and suggests a new, shorter time scale for such evolution.

Figure 1

Meridional cross sections of an NS. Straight vertical lines show the symmetry axis of the star. Hatched regions show the NS crust, white regions show the core. The poloidal magnetic field lines are depicted by the thick black lines (lie on the surface of constant $\mathrm{\Psi }$). A direction of the field is shown by arrows and coincides with the direction of integration in Eqs. (45), (46), and (47). (a) The field line with a given $\mathrm{\Psi }$ is open in the core. The initial point ${\chi }_{0\mathrm{\Psi }}$ coincides with the point, where the field line enters the core. A point where the field line leaves the core is denoted as ${\chi }_{1\mathrm{\Psi }}$. (b) The field line with a given $\mathrm{\Psi }$ is closed in the core. The starting point ${\chi }_{0\mathrm{\Psi }}$ is defined as the nearest point to the crust.

Figure 2

A meridional cross section of an NS core and “minimal” configuration of the magnetic field considered in the text. Field lines are shown by white curves. The strength of the magnetic field is denoted by different colors.

Figure 3

The toroidal component of the baryon velocity $U_{b\phi }$ for the minimal magnetic field model. Colors denote the value of $U_{b\phi }$ in the logarithmic scale (in cm/s). Crossed circles represent the direction of $U_{b\phi }$ [cf. Fig. 9].

Figure 4

Radial dependence of the zeroth-order Legendre components of (rescaled) chemical potential perturbations, $100\times \delta {\mu }_n/(B_{\max }^2/n_0)$ and $10\times \mathrm{\Delta }\mu /(B_{\max }^2/n_0)$, for the minimal magnetic field model from Sec. 4a. Colored lines show the exact solution to Eqs. (30) with the boundary conditions (31) and (33b) for a set of redshifted temperatures indicated in the figure. Dashed lines are plotted in the low-temperature limit (almost coincide with the black lines); dot-dashed lines illustrate the high-temperature limit. Hatched regions correspond to the crust, where our approach is invalid.

Figure 5

The same as in Fig. 4, but for the second-order Legendre components of $\delta {\mu }_n$ and $\mathrm{\Delta }\mu$.

Figure 6

Two-dimensional density plot showing the ratio of $\delta {\mu }_n/(k\tilde{T})$ in the star for the minimal model of the magnetic field with $B_{\max }=5\times {10}^{15}\mathrm{G}$. The left, middle, and right panels are plotted for $\tilde{T}=2\times {10}^8\mathrm{K}$, $4\times {10}^8\mathrm{K}$, and $5\times {10}^8\mathrm{K}$, respectively.

Figure 7

The same as in Fig. 6 but for the ratio $\mathrm{\Delta }\mu /(k\tilde{T})$ and $\tilde{T}=2\times {10}^8\mathrm{K}$ (left panel), $\tilde{T}=4\times {10}^8\mathrm{K}$ (middle panel), and $\tilde{T}=5\times {10}^8\mathrm{K}$ (right panel); see the text.

Figure 8

Two dimensional density plots showing the (logarithm of) poloidal component of the baryon velocity ${\mathit{U}}_b^{(p)}$ (left panel), the proton diffusion velocity ${\mathit{w}}_p$ (middle panel), and the neutron diffusion velocity ${\mathit{w}}_n$ (right panel) for the minimal magnetic field model with $B_{\max }=5\times {10}^{15}\mathrm{G}$ and $\tilde{T}=2\times {10}^8\mathrm{K}$. Arrows indicate the velocity direction.

Figure 9

(a) Comparison of the dimensionless functions $f$ for the minimal magnetic field model from Sec. 4a (black line) and for the model of Sec. 4d (green line). (b) Magnetic field model from Sec. 4d (cf. Fig. 2). (c) Density plot for the toroidal component of the baryon velocity for the magnetic field model from Sec. 4d (cf. Fig. 3). Circles with crosses and dots indicate direction of the toroidal component.

Figure 10

The same as in Fig. 8, but for the magnetic field model from Sec. 4d.

Figure 11

Density plot showing the logarithm of the time scale ${\tau }_U$ (in yr) for the minimal model of the magnetic field from Sec. 4a (left panel) and for the model with no particle flows through the crust-core interface from Sec. 4d (right panel). Arrows indicate direction of the derivative $\partial \mathit{B}/\partial t$ given by Eq. (56). To plot the figure we assumed $B_{\max }=5\times {10}^{15}\mathrm{G}$ and $\tilde{T}=2\times {10}^8\mathrm{K}$.