Pulsar spin-down luminosity: Simulations in general relativity

Adopting our new method for matching general relativistic, ideal magnetohydrodynamics to its force-free limit, we perform the first systematic simulations of force-free pulsar magnetospheres in general relativity. We endow the neutron star with a general relativistic dipole magnetic field, model the interior with ideal magnetohydrodynamics, and adopt force-free electrodynamics in the exterior. Comparing the spin-down luminosity to its corresponding Minkowski value, we find that general relativistic effects give rise to a modest enhancement: the maximum enhancement for $n=1$ polytropes is $\sim 23\%$. Evolving a rapidly rotating $n=0.5$ polytrope we find an even greater enhancement of $\sim 35\%$. Using our simulation data, we derive fitting formulas for the pulsar spin-down luminosity as a function of the neutron star compaction, angular speed, and dipole magnetic moment. We expect stiffer equations of state and more rapidly spinning neutron stars to lead to even larger enhancements in the spin-down luminosity.

Figure 1

Poloidal magnetic field lines for a nonspinning TOV star with compaction $C=0.174$ and endowed with a GR dipole magnetic field. The lines at $t/M=0$ (black dashed curves) overlap the field lines at $t/M=442$ (blue solid curves). The time $t/M=442$ corresponds to $\sim 2/3$ Alfvén crossing time. Left panel: Far zone magnetic field lines. Right panel: Near zone magnetic field lines.

Figure 2

Angular frequency of the magnetic field lines ${\mathrm{\Omega }}_F$ normalized by the angular velocity of the star $\mathrm{\Omega }$ as a function of the polar angle in the $x-z$ plane at $r=0.2R_{LC}$ and $r=0.5R_{LC}$ for a TOV star with $C=0.33$. Two different resolutions are shown (low and high). As expected, the magnetosphere within the light cylinder $R_{LC}$ corotates with star and the higher the resolution the higher is the degree of corotation.

Figure 3

Poloidal magnetic field lines on the $x-z$ plane for a rotating neutron star with eccentricity $e=0.799$ and angular velocity $\overline{\mathrm{\Omega }}=0.20$ (see Table 2) in flat spacetime after the system has relaxed at $t\approx 6\pi /\mathrm{\Omega }$. The shaded areas designate the stellar interior. Left panel: Magnetic field lines in the far zone. The dashed vertical line indicates the light cylinder. Right panel: Magnetic field lines in the near zone.

Figure 4

Pulsar spin-down luminosity $L$ normalized by $L_0=1.02{\mu }^2{\mathrm{\Omega }}^4$, our flat-spacetime result, for models listed in Tables 1 and 2. Left panel: $L/L_0$ vs compaction $C$. Right panel: $L/L_0$ vs polar redshift $Z_p$, where the points are connected by the fitting functions defined in Eqs. (14)–(17). The parameter space for rotating stars is contained between the left dashed line (the mass-shedding limit) and the right dashed line (maximum compaction). The top point (triangle) corresponds to the supramassive neutron star limit for $n=1$. The lower shaded zone is the area of the parameter space that cannot be reached, unless we assume flat spacetime.

Figure 5

Poloidal magnetic field lines at the relaxed state of the rapidly rotating $n=0.5$ NS model (see Table 2). The shaded area designates the stellar interior. Left panel: Far-zone solution. The vertical dashed (dotted) line indicates the GR (flat spacetime) prediction for the location of the light cylinder. Right panel: Near-zone solution. The $Y$ point coincides with the GR prediction for the location of the light cylinder.

Figure 6

Radial dependence of the Poynting luminosity $L/L_0$ for a uniformly rotating star with compaction $C=0.183$ and angular velocity $\overline{\mathrm{\Omega }}=0.15$ (see Table 2). Notice that beyond a radius of 2 light-cylinder radii and out to 10 light-cylinder radii, the luminosity drops only by $\sim 4\%$ which indicates that the dissipation must be small.