In my book an equation is stated for the gravitational potential $V(r,\theta)$ of a nearly spherical body, such as the Earth. It says that the equation is derived using Laplace's equation though the derivation is not given and it contains the terms $J_0$, $J_2$, $J_3$, as well as the terms $P_0$, $P_1$, $P_2$: $$V(r,\theta)=-\frac{GM}{r}\Big(J_0P_0-J_1\frac{a}{r}P_1(\cos{\theta})-J_2\frac{a^2}{r^2}P_2(\cos{\theta}) ...\Big)$$ As the derivation is not given, it is not clear to me at all what the $J_i$ and $P_i$ terms stand for. It says that the $J_i$ terms represent the distribution of mass, but what does that mean? And what do the $P_i$ terms mean?
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4$\begingroup$ $P_i$'s are probably Legendre polynomials, and $J_i$'s are just model parameters (to be set/found for specific cases). $\endgroup$– AlexanderCommented Oct 25, 2016 at 21:23
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$\begingroup$ My guess is about the same as @Alexander but I think the $J_i$ terms might be Bessel functions, rather than model parameters, any thoughts? $\endgroup$– IntuitivePhysicsCommented Oct 25, 2016 at 21:38
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$\begingroup$ Apparently my guess was wrong. The $J_i$ terms are indeed measured. $\endgroup$– IntuitivePhysicsCommented Oct 25, 2016 at 22:04
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The $P_i$ terms are indeed Legender Polynomials. I am no expert on this system, but apparently the $J_0$ term is typically taken to be 1. The $J_1$ term is zero if the center of the coordinate system is the center of mass. The $J_2$ term describes the ellipticity of the object. I found quite a detailed discussion of the "nearly spherical mass potential" here:
http://www-gpsg.mit.edu/12.201_12.501/BOOK/chapter2.pdf
and the good news is the author does appear to be somewhat of an expert on the matter, and includes some derivations.
answered Oct 25, 2016 at 22:23