What counts as a body when calculating the gravitational force between two objects?

Question

My question when we calculate the gravitational force of attraction between two objects say the earth and a satellite, can we add the mass of the atmosphere to the mass of the earth and consider it as a single body (the atmosphere also attracts the satellite)? If yes can this also be extended to non connected bodies,for example can we consider a sphere of space consisting of many planets and stars to be a single body ?

Answer 1

You can always define any collection of objects as a single "body". I think what you want to ask is when we can apply $-GMm/r^2$. The law holds for two particles only. So the question is when you can consider a "body" as a point particle. It will usually be a good approximation when the distance between the two bodies is much larger than their sizes. Another case is when the above is not true but the bodies have spherical symmetry. Then by shell theorem they can be considered as if they were point particles at the center.

Answer 2

Newton's formula for universal gravitation

$$ F_G = -G \frac{m_1 \, m_2}{r^2} $$

is defined for particles (effective objects small enough for their dimension to be neglected), but it also applies (as written) to an important category of macroscopic objects.

The shell theorem tells us that external to any spherically symmetric object the gravitational effect of that object is the same as if the entire mass were concentrated in a point particle at the geometric center.

This is very useful because the major objects in the solar system (the sun, the planets (major and dwarf), and many of the large moons are approximately spherical, which makes treating them with Newton's simple formula acceptable for the purposes of many calculations.

For non-spherical objects one has to divide the objects into many small pieces, determine the effect of each pair (one from each body) separately, and then add up those effects. Take to the proper mathematical limit that is expressed as a double integration over both the volumes something like

$$ F_G = -G \iiint_{V_1} \mathrm{d}\vec{x}_1 \iiint_{V_2} \mathrm{d}\vec{x}_2 \frac{\rho_1(\vec{x}_1) \, \rho_2(\vec{x}_2)}{\left( \vec{x}_1 - \vec{x}_2 \right)^2} \,,$$

where the $V$ are the volumes of the two objects, and the $\rho$s are their densities. This is obviously a much less convenient thing to evaluate.