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on the Moon’s orbit [Q&A]

Vinay

I will write up here brief answers to many astronomy related questions that were asked by students.  The first one is by Vibha of Poornaprajna College, who asks why is it that the Moon is under the gravitational influence of only the Earth and why is it not affected by the Sun, and why doesn’t it orbit the Sun independently?

The short answer is that it is affected by the Sun.  Gravitational forces are additive, i.e., the total gravitational force on the Moon is the sum of the forces due to the Earth, the Sun, the other planets, etc.  They can be calculated using Newton’s prescription,

FMoon-body = G MMoon Mbody / d2

where d is the distance between the Moon and the other body, G=6.67 10-8 dyne cm2 gm-2 is the Gravitational constant, MMoon=7.3459 1025 gm is the mass of the Moon.

In fact, we can directly compare the gravitational force on the Moon due to the Sun (FMoon-Sun) to that due to the Earth (FMoon-Earth). The ratio of the forces on the Moon due to the Sun and the Earth is

FMoon-Sun/FMoon-Earth = (MSun/MEarth) (dMoon-Earth/dMoon-Sun)2,

and putting in the appropriate values, MSun=2 1033 gm, MEarth=6 1027 gm, dMoon-Earth=1.284 light-seconds, and dMoon-Sun≈dEarth-Sun=508 light-seconds, we find that

FMoon-Sun/FMoon-Earth = 2.13

that is, the force of the Sun on the Moon is twice as large as the force of the Earth on the Moon!

What is more, when viewed from the Sun, the orbit of the Moon does not look like a spiral arranged around the Earth’s orbit, but is actually everywhere concave towards the Sun. There is a nice description of it in the wikipedia page on the orbit of the Moon and in references therein.

So this brings up the obvious question, why do we consider the Moon a satellite of the Earth and not an independent planet orbiting the Sun? The answer is that the Moon is still within the Earth’s sphere of influence (if it were not, it would have wandered away a long time ago) — the Sun is far enough away and the Earth and Moon are close enough together that the gravitational acceleration due to the Sun is very similar to both bodies, and it is mathematically convenient to treat the Earth-Moon system together when considering their orbit around the Sun.

The Moon is a very special satellite, nothing like it exists anywhere else in our solar system.