# Introduction

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:

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

and

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

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?

• Transform the field to Cartesian coordinates and make a shift. – Alex Trounev Mar 15 at 22:20
• @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? – jvriesem Mar 26 at 21:50
• You have a function $V(r,\theta , \phi )$. Transform this function to Cartesian coordinates $V(x,y,z)$. Make a shift $z=z'+\Delta z$. – Alex Trounev Mar 26 at 22:42
• @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. – jvriesem Apr 8 at 4:33