Planets' Magnetic Moments ?

1. Jan 7, 2009

Widdekind

Planets' Magnetic Moments ??

QUESTION:

C.T. Russel et al say:
From dimensional considerations we would expect the Magnetic Moment of Venus to scale as the frequency of rotation and the fourth power of the radius of the core*.​
Is this correct?
For, the Magnetic Moment of a current distribution is:
$$\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \, dV$$​
But, the Current Density (J), in the world's core, would seemingly be:
$$\vec{J} = \rho_{c} \, \vec{v} \approx \rho_{c} \, \vec{\omega} \times \vec{r}$$​
So,
$$\vec{\mu} \approx \frac{1}{2} \int \rho_{c} \, \vec{r} \times \left( \vec{\omega} \times \vec{r} \right) dV$$​
Then, Dimensional Analysis implies that:
$$\mu \propto \rho_{c} \, R_{c} \, \omega \, R_{c} \, V_{c} \propto \rho_{c} \, R_{c}^{5} \, \omega$$​
This analysis suggests that Planetary Magnetic Moments "should" scale as Rc5. So, do Planetary Magnetic Moments actually scale as Rc4 ?? What, then, is the other Length Scale that causes the units to work out ($$\left\| \mu \right\| = A \, m^{2}$$) ??

The resulting Planetary Magnetic Field , at radial distance R, scales as:
$$B_{p} \propto \frac{\mu}{R^{3}}$$​
Assuming that worlds' Core Radii (Rc) scale w/ those of the planets themselves (Rp), then near the surface (R = Rp) we have (??) that:
$$B_{p, surface} \propto \rho_{c} \, R_{p}^{2} \, \omega \approx \rho_{c} \, \frac{L_{p}}{M_{p}}$$​
This suggests, that worlds' surface magnetic field strengths scale with their Specific Angular Momentum (L / M). For stars, Specific Angular Momentum increases strongly w/ mass*. If something similar is true for worlds too, then bigger planets will tend to have mightier magnetic fields.
* Carroll & Ostlie. Introduction to Modern Astrophysics, pg. ~893.

Last edited: Jan 7, 2009