General-relativistic radiative cooling in neutron star magnetospheres

Abstract

Radiation reaction cooling plays an important role in describing the extreme plasma conditions found in the magnetospheres of astrophysical compact objects. Strong electromagnetic fields, characteristic of these environments, can trigger the development of anisotropic ring-shaped plasma distributions with inverted Landau populations in momentum space. In this work, we present the first systematic investigation of this mechanism in realistic astrophysical configurations, by accounting for how non-uniform electromagnetic field geometries and general-relativistic effects modify the phase-space dynamics of radiatively cooled plasmas. We demonstrate analytically that drift velocities favour the formation of spiral-shaped momentum distributions that still display inverted Landau populations, and estimate the minimum and maximum plasma injection distances required for inverted momentum distributions to be able to power the emission of coherent radiation through kinetic instabilities. From numerical simulations, we conclude that curved spacetime increases the gradient of the distribution function responsible for the development of kinetic instabilities, and prolongs the persistence of the inverted momentum structure relative to flat spacetime, confirming that realistic astrophysical conditions preserve and enhance the conditions necessary for synchrotron-powered emission of coherent radiation to occur.

Figures (5)

• Figure 1: Deformation of an initially gyrotropic momentum space volume (green) into a spiral structure (blue) due to the presence of a drift velocity and the combined action of the Lorentz and radiation reaction forces. The magnetic field points along $\bm{e}_\parallel = \bm{B}/B$ and the drift velocity along $\bm{e}_{\perp 2}$. In the laboratory frame (a), the guiding center of each individual particle shifts to the point $\gamma m \bm{v}_d$, which lies near the dashed line in plot (a2), implying that all particles should gyrate around this curve. Moreover, since particles gyrate with angular frequency $\omega_c/\gamma'$, particles with higher $p_\parallel$ gyrate at slower rates than particles with lower $p_\parallel$. Over time, this builds up a phase difference between particles that deforms the phase-space volume into the spiral shape observed in plot (a1). In the frame comoving with the drift velocity (b), the spiral is now centered around $p'_\perp = 0$, as shown in plot (b1). The compression of phase-space volume due to radiative cooling now happens in $\{ p_\parallel,p'_\perp \}$-space, shown in plot (b2), where we have depicted the streamlines of the trajectories of particles in the background, with the color representing the magnitude of the radiation reaction force. Similar to the case of a uniform B-field, the radiative cooling rate is still anisotropic and a non-linear function of particle momentum, but now it increases with the Lorentz-boosted perpendicular momentum, $p'_\perp$.
• Figure 2: Momentum distribution function at $t = 10\,t_R$ for an ensemble of $10^7$ electrons initialized at $r_i=1.3\,R_*$ and $\theta_i = 5{}^\circ$ from a Maxwell-Jüttner distribution with $\gamma_b = 2000$ and $p_{th} = 25\,mc$, subject to a magnetic dipole field with flat-spacetime polar surface intensity $B_* = 5\times10^{10} \:\mathrm{G}$. In the top panel, particles were launched in non-rotating Minkowski spacetime, while in the middle panel the metric is Schwarzschild with stellar compactness $r_s/R_* = 0.5$, and in the bottom panel the metric is Kerr-slow with $r_s/R_* = 0.5$ and stellar rotation period $P = 0.1 \:\mathrm{s}$. Inverted spiral-shaped distributions are found to emerge in different curved spacetime configurations as well. Ring-shaped momentum distributions emerge in Schwarzschild spacetime, while in Kerr-slow spacetime the distributions are spiral-shaped, due to the $\bm{E} \times \bm{B}$ drift.
• Figure 3: Time evolution of the ring radius and maximum perpendicular momentum gradient for an ensemble of $10^7$ electrons initialized at $r_i=1.3\,R_*$ and $\theta_i = 5{}^\circ$ from a Maxwell-Jüttner distribution with $\gamma_b = 2000$ and $p_{th} = 25\,mc$, subject to a magnetic dipole field in non-rotating Minkowski, Schwarzschild and Kerr-slow spacetimes with flat-spacetime polar surface intensity $B_* = 5\times10^{10} \:\mathrm{G}$. For the Kerr-slow metric, the stellar rotation period is $P = 0.1 \:\mathrm{s}$, while for Schwarzschild and Kerr-slow the stellar compactness is $r_s/R_*=0.5$. The dashed line represents the theoretical prediction, equation (\ref{['eq:pRMJ']}). When the stellar mass is taken into account, the ring radius diminishes and the perpendicular momentum gradient increases, while the stellar rotation has little influence due to its reduced value.
• Figure 4: Time evolution of the ring radius and maximum perpendicular momentum gradient for an ensemble of $10^7$ electrons initialized at $r_i=R_*$ and $\theta_i = 5{}^\circ$ from a Maxwell-Jüttner distribution with $\gamma_b = 100$ and $p_{th} = 10\,mc$, subject to a non-rotating magnetic dipole field in Schwarzschild spacetime with flat-spacetime polar surface intensity $B_*/B_{sc} = 1$. The simulations were performed for varying values of the stellar compactness $r_s/R_* = \{0, 0.1, \ 0.3, \ 0.5, \ 0.7\}$. Increasing the stellar compactness reduces the ring radius and augments the maximum perpendicular momentum gradient.
• Figure 5: Time evolution of the ring radius and maximum perpendicular momentum gradient for an ensemble of $10^7$ electrons initialized at $r_i=R_*$ and $\theta_i = 5{}^\circ$ from a Maxwell-Jüttner distribution with $\gamma_b = 100$ and $p_{th} = 10\,mc$, subject to a rotating magnetic dipole field in Kerr-slow spacetime with flat-spacetime polar surface intensity $B_*/B_{sc} = 1$. The simulations were performed for varying values of the angular velocity $\Omega_* = \{0, 0.002, \ 0.02, \ 0.05, \ 0.1\} \, c/R_*$. The ring radius increases with the stellar rotation due to the broadening of the distribution in $p_\perp$ caused by the frame-dragging effect, overcoming the effect of the gravitational acceleration, while the maximum perpendicular momentum gradient shows no dependence on the angular velocity at fixed compactness.