(adsbygoogle = window.adsbygoogle || []).push({}); Planets' Magnetic Moments ??

QUESTION:

C.T. Russelet alsay:

Is this correct?From dimensional considerations we would expect theMagnetic Momentof Venus to scale as the frequency of rotation and thefourth powerof the radiusof the core*.

For, theMagnetic Momentof a current distribution is:

[tex]\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \, dV[/tex]But, theCurrent Density(J), in the world's core, would seemingly be:

[tex]\vec{J} = \rho_{c} \, \vec{v} \approx \rho_{c} \, \vec{\omega} \times \vec{r}[/tex]So,

[tex]\vec{\mu} \approx \frac{1}{2} \int \rho_{c} \, \vec{r} \times \left( \vec{\omega} \times \vec{r} \right) dV[/tex]Then,Dimensional Analysisimplies that:

[tex]\mu \propto \rho_{c} \, R_{c} \, \omega \, R_{c} \, V_{c} \propto \rho_{c} \, R_{c}^{5} \, \omega[/tex]This analysis suggests thatPlanetary Magnetic Moments"should" scale as R_{c}^{5}. So, doPlanetary Magnetic Momentsactually scale as R_{c}^{4}?? What, then, is the otherLength Scalethat causes the units to work out ([tex]\left\| \mu \right\| = A \, m^{2}[/tex]) ??

ADDENDUM:

The resultingPlanetary Magnetic Field, at radial distance R, scales as:

[tex]B_{p} \propto \frac{\mu}{R^{3}}[/tex]Assuming that worlds'Core Radii(R_{c}) scale w/ those of the planets themselves (R_{p}), then near the surface (R = R_{p}) we have (??) that:

[tex]B_{p, surface} \propto \rho_{c} \, R_{p}^{2} \, \omega \approx \rho_{c} \, \frac{L_{p}}{M_{p}}[/tex]This suggests, that worlds' surface magnetic field strengths scale with theirSpecific Angular Momentum(L / M). For stars,Specific Angular Momentumincreases 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.

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# Planets' Magnetic Moments ?

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