Can there be planets with extremely strong magnetic fields?

Question

Would it be possible for a planet with surface conditions suitable for humans landing on it to have a very strong magnetic field, with field strengths on the planet's surface similar to the surface field strength of a typical kitchen magnet?

Answer 1

Wikipedia states that the strength of the magnetic field near a kitchen magnet is approximately 5 mT ($5 \times 10^{-3}$ Tesla), roughly 1,000 times as strong as Earth's magnetic field in that same kitchen. Note that this is measuring the $\mathbf{B}$ field of the magnet.

The magnitude of a magnetic field of a dipole at the planet's equator is $$B=\frac{\mu_0}{4\pi}\frac{p}{r^3}$$ where $p$ is the magnetic moment, $r$ is the radius, and $\mu_0$ is a constant. We also know that the magnitude if $\mu_0$ is $4\pi\times10^{-7}$. Therefore, setting $B=5\times10^{-3}$, we have $$5\times10^{-3}=1\times10^{-7}\frac{p}{r^3}\to p=r^3\times10^4$$ Substituting in Earth's radius, we find that the planet must have a magnetic moment of about $2.59\times10^{20}$ J/T. By contrast, if we use values for a magnetar, find find $p\approx8\times10^{19}$ J/T, at most.

What happens if we shrink the radius of the planet a bit while also changing the magnetic moment to preserve magnetic field strength? Even if we shrink it by a factor of about 2.5, to make it more like Mercury, we still have a magnetic moment about the same order of magnitude incomparable to that of normal planets. If we do the reverse, making the planet bigger, then we can afford to have a bigger magnetic moment.

The one thing I haven't taken into account yet is that the mechanisms for magnetic field formation are different in magnetars and planets. Flowers and Ruderman (1977) present a convincing argument that the magnetic fields in magnetars are residual, formed around the time of the supernovae that created them. Other theories posit that one of two other mechanisms (the battery model and the thermoelectric mechanism) are at work. The point is, these differ from the traditional dynamo theory that explains magnetic fields in planets and normal stars. So a planet generating a magnetic field through this method could not have as strong a magnetic field as a magnetar.¹

I suppose I'll end this on an uplifting note. Bodies called Thorne–Żytkow objects have been studied theoretically and are being searched for experimentally. They consist of an M-type red giant or red supergiant which has collided with a neutron star; the neutron star has come to rest at its center, forming a new core. We could - and this is a stretch, but still not entirely impossible - hypothesize that you could have a gas giant planet that collides with a magnetar, which then forms something similar to a TŻO.

Is it a stretch? Yes. The resulting body would differ from TŻOs in more ways than I can explain here. Would the resulting body we a planet you could land on? No. But it's as close as you can get to having a magnetic field that powerful. Oh, and on the equator, the field would be weak, because gas giants are normally quite large. But still, it's a start.

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^{Summarized from one of my answers on Physics Stack Exchange.}