# Inclined Pulsar Magnetospheres in General Relativity: Polar Caps for the Dipole, Quadrudipole and Beyond

## TL;DR

This paper develops a general analytical and numerical method to determine the shape and charge distribution of pulsar polar caps, incorporating general relativity effects for various magnetic field configurations, including dipole and quadrudipole.

## Contribution

It introduces a novel combined analytical-numerical approach to model polar caps in relativistic pulsar magnetospheres with arbitrary magnetic fields.

## Key findings

- Inclined dipole polar caps are significantly affected by general relativity.
- Quadrudipole fields can produce thin annular polar caps.
- The method enables realistic modeling of pulsar emission regions.

## Abstract

In the canonical model of a pulsar, rotational energy is transmitted through the surrounding plasma via two electrical circuits, each connecting to the star over a small region known as a "polar cap." For a dipole-magnetized star, the polar caps coincide with the magnetic poles (hence the name), but in general, they can occur at any place and take any shape. In light of their crucial importance to most models of pulsar emission (from radio to X-ray to wind), we develop a general technique for determining polar cap properties. We consider a perfectly conducting star surrounded by a force-free magnetosphere and include the effects of general relativity. Using a combined numerical-analytical technique that leverages the rotation rate as a small parameter, we derive a general analytic formula for the polar cap shape and charge-current distribution as a function of the stellar mass, radius, rotation rate, moment of inertia, and magnetic field. We present results for dipole and quadrudipole fields (superposed dipole and quadrupole) inclined relative to the axis of rotation. The inclined dipole polar cap results are the first to include general relativity, and they confirm its essential role in the pulsar problem. The quadrudipole pulsar illustrates the phenomenon of thin annular polar caps. More generally, our method lays a foundation for detailed modeling of pulsar emission with realistic magnetic fields.