http://en.wikipedia.org/wiki/Spherical_harmonics
The representation of the earth’s gravitational field you learned in high school, as being spherically symmetrical is not quite true.
In Newtonian mechanics the graviational field resulting from a spherical object of uniform density can be replaced for points outside the object by a point mass located at its center. You may realize, that the actual earth is not a homogenous sphere. If it were (and it didn’t rotate) a plumb bob would always point to the center of the earth. In fact, the earth, or the sea level surface, can be well approximated by an ellipsoid of rotation, with the polar radius about 56 kilometers less than the equatorial radius. (A cross section of the ellipsiod passing through the line joining the North and South pole, will be an ellipse.)
The actual direction that a plumb bob points, ignoring rotation, is always perpendicular to the equipotential surface. This means that away from the poles and the equator the plumb bob does not point at the center of the earth, in the northern hemisphere it points slightly to the south of the center, in southern hemisphere to the north. The effect is greatest at 45 degrees latitude, and the deflection is about a tenth of a degree.
If you were a satellite in a low orbit around the earth, and you happened to be at 45 degrees north latitude you would experience a gravititational acceleration, not towards the center of the earth, but about a tenth of a degree to the south of it. The effect can be resolved into two components, one toward the center of the earth, the J0 component and one at an angle, due to the earth’s equatorial bulge (that extra 56 kilometers around the equator). We call that the J2 component, or Second Zonal Harmonic. (There is no first Zonal Harmonic, the zeroth harmonic is due to the sphercial component of the field.) The effect of the zeroth harmonic decreases as R-squared, the J2 Harmonic as R-cubed. At the surface of the earth it is already only about one one-thousandth as strong as the zeroth component and for objects far enough away it can be ignored.
The latest earth gravity model, EMG96 has over 250,000 harmonics! The first good estimate of the the Earth’s J2 component was not available until we started tracking Sputnik in 1957. There are proposed models with over two million components. Good luck with that.
Thanks much.