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When say, calculating the gravitational force between the Earth and the Moon, we use Newton's law of universal gravitation to show:

$$F_{(gravity)} = G \ \frac{m_E \ m_M}{r^2}$$

Where:

  • $m_E$ = mass of the Earth
  • $m_M$ = mass of the Moon
  • $r$ = distance between the Earth and the Moon

The above is how my textbooks describe it. However, considering that we treat all the mass of the Earth and the Moon as being concentrated at a point source (their centre of mass), shouldn't we also account for the radii of the Earth and the Moon to find the true distance between the centre of mass of the Earth and the centre of mass of the Moon, so that the equation is instead:

$$F_{(gravity)} = G \ \frac{m_E \ m_M}{(r + r_E + r_M)^2}$$

Where:

  • $r_E$ = radius of the Earth
  • $r_M$ = radius of the Moon

Is this what is actually done and my textbooks are just glossing over the finer details, or is there some reason why we don't do this? Are the radii of the Earth and the Moon (or any other two bodies) considered to be negligible compared to the distance between them?

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    $\begingroup$ Take a look at the shell theorem $\endgroup$ – valerio Jul 18 '17 at 11:45
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    $\begingroup$ The distance considered ($r$) SHOULD be the distance between the two pint masses! The formula you use depends on whether you are considering $r$ the distance between the center of the Earth and the center of the Moon or the distance between the surfaces, but nobody would use the latter..! $\endgroup$ – JalfredP Jul 18 '17 at 11:46
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    $\begingroup$ So then the distance is actually the distance between the two centres of mass. I always suspected that, but the textbooks didn't specify it. Thanks. If you could write that up as an answer I'll accept it. Thanks also to @valerio92 for the link to shell theorem. It's explained something else that I was wondering about. $\endgroup$ – Pancake_Senpai Jul 18 '17 at 12:05
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As described in the comment, your assumption about the variable $r$ was not correct:

$r$ in the given formula is the distance of the centers of mass of the two bodies and not the shortest distance of the surfaces of the two bodies (you made another minor error: you defined $r$ as the distance between the Sun and the Moon and not Earth and Moon).

Contrary, if you want to stay with your own definition or $r$, your adjusted formula where you added the radius of Earth and Moon would also be fine.

Finally, as described also in the comment, it is also important to understand that for the given problem (spherical objects with homogeneous mass distribution), the size of the objects can be completely ignored and the whole masses can be thought as centered at the centers of mass. This can be shown mathematically by integration techniques over the whole volume (this was not your question but often asked in the same context).

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First of all $r$ refers the distance from the center of earth to the center of the other body (say moon). So, you don't need the radius of either the moon or earth, and the formula will be $$F=\frac{GMm}{r^2}$$

If $r$ is the distance between the surface of the earth and the surface of the other object (say moon), then, the formula will be

$$ F=\frac{GMm}{(r+r_e+r_m)^2}$$

But, in this case also $r_m\ll r$ and $r_e\ll r$. So, even if we neglect $r_e$ and $r_m$ in the above formula, and measure the distance from surface to surface, it will not make much difference in the value of $F$.

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