Origin of the magnetic field of neutron stars

The strong magnetic fields in neutron stars are supposed to come from magnetic flux conservation. If we have:

$Phi_B = int B dS = const$

where $Phi_B$ is the magnetic field flux, $B$ is the magnetic field strength, and $dS$ is the elemental closed surface; then, this integral is constant through the surface.

If we consider the star surface over which take the integral, than

$S = 4pi R^2$

where $R$ is the star radius. This can be translated, altogether with the magnetic flux conservation law, as:

$B_f = B_i (frac{R_i}{R_f})^2$

where $i$ and $f$ are the indices for initial and final stages.
We know that the star implodes from a whatever star size to $sim10$ km. So the radii ratio is huge. You just need a starting magnetic field of $10-100$ G, to get a final magnetic field of the order of $10^{12}$ G, that is typical in neutron stars.

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