Pulsar Polar Cap

We begin by considering the shape of magnetic field lines of a magnetic dipole. The components of the magnetic field are given by

$B_z = \frac{\mu}{r^3} \left(3 \cos^2 \theta - 1\right)$

$B_{\rho} = 3 \frac{\mu}{r^3} \cos \theta \sin \theta$

where $\mu$ is the magnitude of the magnetic dipole, $\theta$ is the angle relative to the axis of the dipole $\hat{z}$. The trajectory of a particle moving along the field lines is given by

$\frac{d z}{d \rho} = \frac{d}{d \theta} \left(r \cos \theta\right)/\frac{d}{d \theta} \left(r \sin \theta\right) = \frac{B_z}{B_{\rho}} = \frac{3 \cos^2 \theta - 1}{3 \cos \theta \sin \theta}$

Solving for $r$ yields

$r = K \sin^2 \theta$

where $K$ is a constant.

The polar cap of a pulsar is the location on the surface where the field lines close outside the light cylinder. The radius of the light cylinder is given by $r_l \approx c P$ where $c$ is the speed of light and $P$ is the spin period of the pulsar. Let consider the magnetic field line the passes through the intersection between the equatorial plane and the light cylinder

$r = c P \sin^2 \theta$

The same magnetic field line connects to the pulsar (whose radius we denote by $R$) at latitude

$R \approx c P \sin^2 \theta_p \Rightarrow \theta_p \approx \sqrt{\frac{R}{c P}}$

where we assume a small angle. The region on the pulsar above latitude $\theta_p$ is called the polar cap. Charged particles moving along field lines in this region can escape the pulsar. Other charged particles move along field lines that close within the light cylinder.