The 3D pulsar magnetosphere with machine learning: first results

TL;DR

This work tackles the lack of a reference steady-state solution for the 3D ideal force-free pulsar magnetosphere by developing a domain-decomposed, current-sheet-aware solver implemented with meshless Physics-Informed Neural Networks (PINNs) in the corotating frame. By solving two independent open/closed region problems and iteratively shaping the interim separatrix to satisfy pressure balance and $B^2-E^2$ continuity, the authors obtain a dissipationless steady-state solution for an inclined dipole with a fixed polar-cap shape. The results reveal a Y-point inside the light cylinder, a $B_p\to0$ condition at the Y-point with $B_\phi\neq0$, and a topologically T-shaped Y-point, along with an undulating equatorial current sheet and automatic $\alpha$-adjustment along open field lines crossing the light cylinder. The total Poynting flux is nearly conserved in the open region, yielding $L \approx 1.2L_0(1+\sin^2\lambda)$ for the test case, demonstrating the potential of PINNs to generate new ideal 3D pulsar magnetosphere solutions. While the approach ignores microphysical reconnection and current-sheet dissipation, it provides a controllable framework to explore magnetospheric topology across parameter space and offers a path toward more comprehensive models that can be linked to observations and other astrophysical systems.

Abstract

All numerical solutions of the pulsar magnetosphere over the past 25 years show closed-line regions that end a significant distance inside the light cylinder, and manifest thick strongly dissipative separatrix surfaces instead of thin current sheets, with a tip that has a distinct pointed Y shape instead of a T shape. We need to understand the origin of these results which were not predicted by our early theories of the pulsar magnetosphere. In order to gain new intuition on this problem, we set out to obtain the theoretical steady-state solution of the 3D ideal force-free magnetosphere with zero dissipation along the separatrix and equatorial current sheets. In order to achieve our goal, we needed to develop a novel numerical method. We solve two independent magnetospheric problems without current sheet discontinuities in the domains of open and closed field lines, and adjust the shape of their interface (the separatrix) to satisfy pressure balance between the two regions. The solution is obtained with meshless Physics Informed Neural Networks (PINNs). In this paper we present our first results for an inclined dipole rotator using the new methodology. We are able to zoom-in around the Y-point and inside the closed-line region, and we observe new interesting features. This is the first time the steady-state 3D problem is addressed directly, and not through a time-dependent simulation that eventually relaxes to a steady-state. We have trained a Neural Network that instantaneously yields the three components of the magnetic field and their spatial derivatives at any given point. Our results demonstrate the potential of the new method to generate new solutions of the ideal pulsar magnetosphere.

Figures (8)

• Figure 1: Evolution of the total NN losses for the solution with $\lambda=20^\circ$ and $\theta_{\rm pc}=36^\circ$ in the closed-line (blue) and open-line (orange) regions with respect to the number of training epochs (the difference in the number of epochs has to do with the different number of steps the training algorithm takes during its second-order optimization). We started the training with a first order optimizer and switched to a second order one after $10^4$ training epochs. Beyond that, we adjusted the shape of the interim separatrix every $3\times 10^4$ epochs (blue and orange high spikes). The achieved total training loss between $10^{-5}$ and $10^{-6}$ is deemed satisfactory.
• Figure 2: Cross section of steady-state solution representing an inclined rotator with $\lambda=20^\circ$ and $\theta_{\rm pc}=36^\circ$. In this simulation $r_*=0.25 R_{\rm lc}$. Rotation axis along $z$. The inclined magnetic axis lies along the corotating $xz$ plane shown. This particular cross section represents phase $0$ (or $\pi$) of the corresponding rotating time-dependent solution. Closed thick black lines: separatrix between open and closed field lines. Open thick black lines: separatrix between open field lines that originate from the north and south magnetic poles. This is where the equatorial current sheet lies. Red lines: initial dipolar shapes of the interim separatrices before readjustment. For this particular choice of the polar cap, the dipole is significantly stretched outwards closer to the light cylinder (represented by the two dashed lines at $x/R_{\rm lc}=\pm 1$). The equatorial current sheet lies where lines from the north and south polar cap rim meet. Color scale: ratio $B_p/B$. This represents the development of the azimuthal magnetic field $B_\phi$ accross the magnetosphere. Notice that at the magnetospheric Y-point where the equatorial current sheet connects to the separatrix current sheet, $B_p=0$ and $B_\phi\neq 0$ as expected 2003ApJ...598..446U. Notice also that in this solution the Y-point is still significantly inside the light cylinder.
• Figure 3: Same as Fig. \ref{['fig:magnetic_solution0']} but along the corotating $yz$ plane. This particular cross section represents phase $\pi/2$ (or $3\pi/2$) of the corresponding rotating time-dependent solution.
• Figure 4: The steady-state solution of Fig. \ref{['fig:magnetic_solution0']} in 3D as seen from above. Yellow cylinder: light cylinder. Pale undulating surface: equatorial current sheet. We plot only open magnetic field lines just inside the rim of the northern polar cap. The Y-point in this solution is significantly inside the light cylinder. We see the clear azimuthal break of open field lines expected very close to the Y-point where $B_p\rightarrow 0$ and $B_\phi \neq 0$.
• Figure 5: The steady-state solution of Fig. \ref{['fig:magnetic_solution1']} as seen from the side. We plot open magnetic field lines just inside the rims of both polar caps.The undulating shape of the equatorial current sheet between them is clearly seen. Notice that it does not develop the physical instabilities seen in all previous numerical solutions because this physics is missing in our work (we obtain the solution without a current sheet and then reverse the direction of the field in the southern magnetic hemisphere).