A Pedagogical Review of the Vacuum Retarded Dipole Model of Pulsar Spin Down

TL;DR

This paper reviews the vacuum retarded dipole model of pulsar spin-down, providing detailed derivations, corrections to historical equations, and discussing its relevance as a simplified approximation despite more advanced simulations.

Contribution

It offers a comprehensive pedagogical review of the VRD model, including corrected derivations and clarifications of its assumptions and limitations.

Findings

Provides detailed derivations of the VRD spin-down equations

Corrects typographical errors in Deutsch (1955) model equations

Highlights the model's usefulness as a qualitative approximation

Abstract

Pulsars are rapidly spinning highly magnetised neutron stars. Their spin period is observed to decrease with time. An early analytical model for this process was the vacuum retarded dipole (VRD) by Deutsch (1955). This model assumes an idealised star and it finds that the rotational energy is radiated away by the electromagnetic fields. This model has been superseded by more realistic numerical simulations that account for the non-vacuum like surroundings of the neutron star. However, the VRD still provides a reasonable approximation and is a useful limiting case that can provide some qualitative understanding. We provide detailed derivations of the spin down and related electromagnetic field equations of the VRD solution. We also correct typographical errors in the general field equations and boundary conditions used by Deutsch (1955).