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The Earth does not have a defined geometric shape. Then how do space scientists who want to put a satellite in orbit model the gravitational field of Earth? Do they assume a single object with minute variations from sphericity, or as a combination of multiple objects of defined shapes? Or, are there different ways of doing it right?

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Quick answer for now; the Earth's field has been more and more carefully measured by precise tracking of satellites over time. It stared with very careful visual tracking/timing and interferometrically tracking and recording the Doppler shift of the radio signals from late 1950's satellites like Sputnik and Vanguard and then many many others over the years, then on to pairs of satellites that track each others' distance GRACE and GRACE-FO in order to build an Earth Gravitational Model.

It's expressed as a bunch of coefficients of spherical harmonics. The monopole term is $GM$ the standard gravitational parameter, there is no dipole moment since the center is defined as the Earth's center of mass, and the quadrupole moment is expressed using $J_2$ and $J_{22}$, the first of which represents Earth's oblateness which is roughly a 0.1 % effect in low Earth orbit and is what makes Sun-synchronous and Molniya orbits possible.

The spherical harmonics of a given published model only work outside of (essentially) all of Earth's mass. Very near the Earth's surface (as opposed to in orbit) you would have to correct them for the shell of the mass of the atmosphere above you which is no longer affecting you as well as nearby mountain tops that rise above you.

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