Could a binary planet system have a shared magnetic field?

We can treat a planet as a magnetic dipole. In this case, the strength of the field scales as an inverse cube law (rather than the more familiar inverse square law), with some angular dependence. At the magnetic equator, we can write this as $$B(r)=B_S{\left(\frac{r}{R_p}\right)}^{-3}$$ where $R_p$ is the radius of the planet and $B_S$ is the magnetic field at the planet's surface. Therefore, if the other planet orbits, say, $10R_p$ away, the magnetic field coming from the other planet will be 0.1% of the surface field on the source planet. Even if we cut that in half, the field strength rises to only 1%, which is arguably not enough to provide the second planet with any of the benefits of the first.

For comparison, the Moon orbits Earth at 60 Earth radii, and is only about 1% the mass of Earth. Bringing the other planet close enough to experience a strong magnetic field would cause massive tidal effects, which would probably offset the benefits of a magnetic field!

The upside of a close orbit, of course, is that it reduces the time needed for the planets to become tidally locked to one another. That timescale goes as $\tau \propto r^6$, where $r$ is the distance between the centers of the planets. If $r$ is large, tidal locking may take a long time, but if $r$ is small, tidal locking can happen on reasonable timescales.

Some numbers

You seem to want the gravitational force from one planet to a person on the other planet's surface to be half the surface gravity of the other planet. That means that the distance from the center of one planet to the surface of the other is $\sqrt{2}{R}_p$. This means you'd have a surface magnetic field on the second planet of $B_S(\sqrt{2}{)}^{-3}\approx 0.35B_S$ - not bad.

On the other hand, you'd have some extremely strong tidal forces. The distance between the planet's cores is about $2.5R_p$. Tidal forces scale as $F_T\propto m/r^3$, where $m$ is the mass of the perturbing planet, and so you've got tidal forces about 1.3 million times as strong as the Earth experiences from the Moon.

This is . . . not a setup conducive to life.

posted over 5 years ago

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