Interaction of gravitational waves with strongly magnetized plasmas

We study the interaction of a gravitational wave (GW) with a plasma that is strongly magnetized. The GW is considered a small disturbance, and the plasma is modeled by the general relativistic analogue of the induction equation of ideal MHD and the single fluid equations. The equations are specified to two different cases, first to Cartesian coordinates and a constant background magnetic fields, and second to spherical coordinates together with a background magnetic field that decays with the inverse radial distance. The equations are derived without neglecting any of the nonlinear interaction terms, and the nonlinear equations are integrated numerically. We find that for strong magnetic fields of the order of ${10}^{15}\mathrm{G}$ the GW excites electromagnetic plasma waves very close to the magnetosonic mode. The magnetic and electric field oscillations have very high amplitude, and a large amount of energy is absorbed from the GW by the electromagnetic oscillations, of the order of ${10}^{23}\mathrm{erg}/{\mathrm{cm}}^3$ in the case presented here, which, when assuming a relatively small volume in a star’s magnetosphere as an interaction region, can yield a total energy of at least ${10}^{41}\mathrm{erg}$ and may be up to ${10}^{43}\mathrm{erg}$. The absorbed energy is proportional to $B_0^2$, with $B_0$ the background magnetic field. The energizing of the plasma takes place on fast time scales of the order of milliseconds. Our results imply that the GW-plasma interaction is an efficient and important mechanism in magnetar atmospheres, most prominently close to the star, and, under very favorable conditions though, it might even be the primary energizing mechanism behind giant flares.

Figure 1

Top: Average energy density ${ϵ}_{gw}(t)$ of the GW as a function of time. Bottom: Mean total energy density ${ϵ}_{\mathrm{pl}}(t)$ of the plasma as a function of time.

Figure 2

Spatial Fourier transform $|{\widehat{E}}_y(k_z,t){|}^2$ of $E_y(z,t)$ at a fixed time $t=0.00545\mathrm{s}$.

Figure 3

The energy density ${ϵ}_{\mathrm{pl}}$ absorbed by the plasma (at maximum absorption) as a function of the background magnetic field $B_0$ (diamonds), together with a reference line of logarithmic slope 2 (dotted line).

Figure 4

The energy density ${ϵ}_{\mathrm{pl}}$ absorbed by the plasma (at maximum absorption) as a function of the background matter density ${\rho }_0$ (diamonds, connected with a dotted line).

Figure 5

Top: Mean energy density ${ϵ}_{\mathrm{pl}}$ absorbed by the plasma as a function of box length $L$, with the inner edge of the simulation box at $r_{\mathrm{in}}=5r_{\star }$ (plus symbols, connected with solid line), together with a reference line of logarithmic slope $-0.7$ (dashed line). Bottom: Energy absorbed by the plasma in the conical volume of radial length $L$ and with inner edge at $r_{\mathrm{in}}=5r_{\star }$ as a function of $L$ (plus symbols, connected with solid line), together with a reference line of logarithmic slope 2.3 (dashed line).

Figure 6

Top: Mean energy density ${ϵ}_{\mathrm{pl}}$ absorbed by the plasma in the conical volume as a function of the position $r_{\mathrm{in}}$ of the inner edge of the simulation box, with fixed box length $L=540\mathrm{km}$ (plus symbols, connected with solid line), together with a reference line of logarithmic slope 2.3 (dashed line). Bottom: Energy absorbed by the plasma in the conical volume of radial length $L=540\mathrm{km}$, as a function of the position of the inner edge $r_{\mathrm{in}}$ of the simulation box (plus symbols, connected with solid line), together with a reference line of logarithmic slope 2.7 (dashed line).

Figure 7

Sketch: The GW interacts with a plasma volume in the equatorial plane (see text for details).

Figure 8

Top: Estimate of the energy $E_{\mathrm{pl}}$ absorbed by the plasma on the base of Eq. (45) as a function of the inner distance $r_{\mathrm{in}}$ from the star, for different interaction lengths $L$ (dots: $L=100\mathrm{km}$; short dashes: $L=250\mathrm{km}$; long dashes: $L=500\mathrm{km}$; and solid line: $L=1000\mathrm{km}$), and for $\alpha =10°$, $\phi =180°$. Bottom: Estimate of the energy $E_{\mathrm{pl}}$ absorbed by the plasma as a function of the inner distance $r_{\mathrm{in}}$ from the star, on the base of Eq. (61), i.e. for the case of a decaying background magnetic field, and for $L=5\times {10}^4\mathrm{km}$ (dash-dotted line), $L=1\times {10}^3\mathrm{km}$ (solid line), $L=5\times {10}^2\mathrm{km}$ (long dashes), $L=2.5\times {10}^2\mathrm{km}$ (short dashes), and for $L=1\times {10}^2\mathrm{km}$ (dotted line), and for $\alpha =10°$, $\phi =180°$.