# On the magnetic field evolution timescale in superconducting neutron star cores

## TL;DR

This paper clarifies the correct form of the induction equation and the relevant magnetic field evolution timescale in superconducting neutron star cores, emphasizing the importance of solenoidal velocity fields and irrotational terms.

## Contribution

It resolves a controversy by analyzing the roles of irrotational and solenoidal components in the induction equation for neutron star cores.

## Key findings

- Irrotational terms dominate the ambiguity in timescale estimation.
- Only the solenoidal electric field contributes to magnetic field evolution.
- Stationary velocity fields are essential for accurate long-term evolution modeling.

## Abstract

We revisit the various approximations employed to study the long-term evolution of the magnetic field in neutron star cores and discuss their limitations and possible improvements. A recent controversy on the correct form of the induction equation and the relevant evolution timescale in superconducting neutron star cores is addressed and clarified. We show that this ambiguity in the estimation of timescales arises as a consequence of nominally large terms that appear in the induction equation, but which are, in fact, mostly irrotational. This subtlety leads to a discrepancy by many orders of magnitude when velocity fields are absent or ignored. Even when internal velocity fields are accounted for, only the solenoidal part of the electric field contributes to the induction equation, which can be substantially smaller than the irrotational part. We also argue that stationary velocity fields must be incorporated in the slow evolution of the magnetic field as the next level of approximation.