Geopotential model in spherical harmonics

Remarks

Example: the model with $N = 2$ and only $\bar C_{20} \ne 0$ describes an axially symmetric flattened Earth; with $J_2 = -\sqrt{5}\,\bar C_{20}$ it reads $V = \frac{GM}{r}\big(1 - J_2 (a/r)^2 P_2(\sin\phi)\big)$, where $P_2(t) = (3t^2 - 1)/2$; for the Earth, $J_2 \approx 1.083 \times 10^{-3}$. Non-example: the function $GM/r + c\,r^2$ with $c \ne 0$ is not of this form, since $r^2$ terms grow at infinity; it is not the potential of bounded masses. Each term $r^{-n-1}\bar P_{nm}(\sin\phi)\cos m\lambda$ is a solution of Laplace's equation, since $\bar P_{nm}(\sin\phi)\cos m\lambda$ is a spherical harmonic of degree $n$. The degree $1$ terms are absent because the origin is the centre of mass. The series for $N = \infty$ converges outside a sphere around the origin that contains all masses.