A planetary magnetic field $\vec{B}$ can be described outside of the planet using Gauss coefficients $g_n^m$ and $h_n^m$ and a spherical harmonic expansion:

$$\vec{B} \;=\; -\vec{\nabla} V(r,\theta,\phi)$$


$$V(r,\theta,\phi) \;=\; \sum_{n=1} \sum_{m=0}^l R_p \left(\frac{R_p}{r}\right)^{n+1} \big[g_n^m \cos(m\phi) + h_n^m \sin(m\phi) \big] P_n^m(\cos\theta) \text{,}$$

where $R_p$ is the planetary radius and $P_n^m(\cos\theta)$ are the associated Legendre polynomials.

If we assume the magnetic field is axially symmetric, then only the $m=0$ terms remain:

$$\begin{align} V(r,\theta,\phi) \;=&\; \sum_{n=1} R_p \left(\frac{R_p}{r}\right)^{n+1} g_n^0 P_n^0(\cos\theta) \\ =&\; \sum_{n=1} R_p \left(\frac{R_p}{r}\right)^{n+1} g_n^0 P_n(\cos\theta) \text{,}\end{align} $$

where $P_n^0(\cos\theta) = P_n(\cos\theta)$ are the Legendre polynomials.

My Question

Suppose I have the axisymmetric Gauss coefficients (say, $g_n^0$ for $1 \le n \le N$) for a given planet in a coordinate system centered on the planet. Let's suppose the magnetic equator is located northward from the planet's equatorial plane by a distance $\Delta z$, which implies the magnetic field is not symmetric about the planet's equator.

I want to treat the magnetic field as a dipole, probably in a coordinate system shifted northward along the polar axis by $\Delta z$. How could I calculate from those coefficients the "best-fit" dipole field in the shifted coordinate system?

  • $\begingroup$ Transform the field to Cartesian coordinates and make a shift. $\endgroup$ – Alex Trounev Mar 15 at 22:20
  • $\begingroup$ @AlexTrounev: How would I do that if I only have the Gauss coefficients for the unshifted field? Sure, I could transform the coordinates and vector components, but I'd still have the higher order terms. If I shifted only the dipole term, I'd lose the effect of the higher-order terms on the shifted dipole. How would I get the dipole term in the shifted coordinate system? $\endgroup$ – jvriesem Mar 26 at 21:50
  • $\begingroup$ You have a function $V(r,\theta , \phi )$. Transform this function to Cartesian coordinates $V(x,y,z)$. Make a shift $z=z'+\Delta z$. $\endgroup$ – Alex Trounev Mar 26 at 22:42
  • $\begingroup$ @AlexTrounev That’s what you said in your first comment. My reply asked a question. Shifting coordinate systems — by itself — doesn’t give a new value or expression for a coefficient in the unshifted frame. (I actually do as much already in a similar problem.) Shifting coordinate systems is trivial. I don’t know how that would affect the dipole term. Perhaps it would help if you expressed your intentions mathematically: something like $g_1^{0,’} = f(g_1^0)$, where the prime denotes the shifted frame. $\endgroup$ – jvriesem Apr 8 at 4:33

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