# Origin of the magnetic field of neutron stars

It seems a little counter intuitive that neutron stars possess such strong magnetic fields. Its electric charge is presumably zero, so however fast it spins, it shouldn't generate any magnetic field. Or is it due to the electric charges of quarks or their intrinsic spins?

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

$$\Phi_B = \int B\ \mathrm{d}S = \text{const}$$

where $\Phi_B$ is the magnetic field flux, $B$ is the magnetic field strength, and $\mathrm{d}S$ 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

$$S = 4\pi R^2$$

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

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

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

• You might like to add that this is unlikely to be the whole solution for the very strong magnetic fields found in magnetars and similar objects, where some sort of dynamo during the core collapse may be required. – Rob Jeffries Oct 8 '15 at 10:44
• Why is there a conservation of magnetic flux in a star (a collapsing one, to boot!) but apparently not in a planet like earth (whose magnetic field changes direction occasionally)? I would expect any "order" (parallel spins, or any ordered flows of charged matter which produce a magnetic field) to get weaker over time, for very general reasons like increasing entropy. (Why) is that not the case? – Peter - Reinstate Monica May 24 '16 at 15:46
• @PeterA.Schneider, the magnetic flux conservation is referred to the magnitude of the magnetic field (it is an integral). This is what gets conserved during collapse. For the rest of the comment: I am sorry but I do not get what you mean. Maybe you can create an entirely new question? – Py-ser May 31 '16 at 13:20
• Thanks for your answer. It is simply that I (being an interested layman) have never heard of a conservation of magnetic flux which seems self-evident to you, and apparently to the other readers. In general, in particular at larger time scales, there does not seem to be such a conservation: Earth's magnetic flux, for example, changes considerably over (geological) time. Why can the conservation of magnetic flux be assumed for a collapsing star? – Peter - Reinstate Monica May 31 '16 at 17:26
• As I suggested, this deserve an entirely new question. Especially time scales must be specified for a coherent answer. – Py-ser Jun 1 '16 at 12:16