# 3.8: Summary

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- 3524

Let's review what we've learned about the many aspects of gravity so far.

## Orbital Mechanics

We fist covered orbital mechanics, which uses observations and theories to examine the Earth's elliptical orbit, its tilt, and how it spins. One of the most important part of orbital mechanics is Kepler's Third Law, in which the relationship \(T^2\propto r^3\) between the planet's orbital period T and its radius r can be represented in this equation:

\[\frac{T_1^2}{T_2^2}=\frac{R_1^3}{R_2^3}\]

This relationship is very useful for determining the orbital periods or radii of a variety of bodies including suns, moons orbiting planets, and satellites orbiting planets or the sun. Now relating gravity to mass, we found that

\[\frac{T^2}{r^3}=\frac{4\pi^2}{GM}\]

By observing the orbital radius and period of the orbiting object, we can then calculate the mass of the object being orbited.

## Moments of Inertia

We next discovered that Earth is not a perfect sphere and is in fact an ellipsoid. Understanding this can help us calculate the planet's moment of inertia and thus learn more about its internal structure. We can usually determine reasonable 2-layered planetary structures with additional information about the composition of density at the surface.

In determining the moment of inertia, several steps must be taken. First, we must assume that the planet behaves like a fluid over long periods of time, ie the hydrostatic assumption.

We can use the Darwin-Radau Approximation:

\[f_{hyd}\approx\frac{\frac{5}{2}\frac{a^3\omega^2}{GM}}{1+\frac{25}{4}(1-\frac{3}{2}\frac{C}{Ma^2})^2}\]

Where we can measure/observe everything except the moment of inertia \(\frac{C}{Ma^2}\)

To remedy this, the Darwin Radau equation should be solved for the moment of inertia:

\[\frac{C}{Ma^2}\approx\frac{2}{3}(1-\sqrt{\frac{2}{5}\frac{\frac{a^3\omega^2}{GM}}{f_{hyd}}-\frac{4}{25}})\]

## Geoid

We next dived into gravitational potential, U, which has the relationship to variables we discussed previously

\[U=\frac{Gm}{r}.\]

Our ultimate goal is to accurately represent the gravitational potential on the surface of a planet, and to do this we must use spherical harmonics. Summing these spherical harmonies together allows us to represent any surface on a sphere

\[U\approx-\frac{Gm}{r}+\frac{Gma^2}{2r^3}J_2[3\sin^2\phi-1]-\frac{1}{2}\omega^2r^2\cos^2\phi\]

## Isostasy and Satellite-derived Gravity and Geoid

One of the final topics we covered in this chapter is isostasy, which is topography supported by viscous stresses ie pressure, and can be compensated or uncompensated. Smaller feature such as seamounts or trenches are compensated and supported by the strength of the lithosphere thus they have a gravity anomaly. Larger features such as continents and mountain ranges are supported by viscous stresses, ie isostasy, and have no gravity anomaly.

The ultimate goal is to obtain isostatic balance and use known \(\rho_i\) and h_{i}'s to find an unknown \(\rho_i\) or h_{i}. Each real world example of isostasy such a continent-ocean or depth of an ocean basin can be written in its simplest form as

\[P=\sum_{i=1}^{n}\rho_ih_ig\;with\;n\;layers\]