Astronomical Math for Dummies?

Question

How would I go about calculating the gravitational flux between Jupiter and Io, and between the Earth and the Moon?

I'd like to see if it is possible that the earth's mantle is partially heated by gravitational friction between the earth and the moon. The National Geographic channel seems to think it doesn't.

Answer 1

This is usually called tidal friction.



The gravitational flux between two bodies is simply the well-know gravitational equation. Calculating the heat absorbed/lost by a planet is another much more complicated matter. There are several mathematical models used, but since some constants aren't well-known, the models usually have a large amount of error. These are problems that people get Ph.D's just for doing a small amount of work on them, so they're not simple or easily solved systems.



The earth's oceans are currently assumed to absorb a lot of the energy, so it doesn't go directly into the Earth's crust.



See the (vague) reference below.

Answer 2

Not entirely sure, but the mass of the moon is about 1/6 the mass of the earth. The gravitational field falls off with the inverse square of the distance (so an object 3 times further away feels 1/9 the gravitational attraction). The moon is about 250,000 miles away, but the earth is only about 7000 miles in diameter. So if the mantle were 2,000 miles thick, the inner portion would be 248,000 miles from the moon, while the outer portion is 250,000 miles away. This amounts to 2 parts in 10,000 difference, at a great distance, from a relatively low mass object. This is probably why National Geographic channel thinks it doesn't. The effect is not completely negligible, though, or we wouldn't have tides...

Answer 3

If I were pushed for an answer I would say:

If the moon can raise the water of the earth by gravitational attraction. Then the raised water has gained potential energy, and when that energy becomes kinetic, heat will be released as a form of said energy.

It won't be a great deal, unlike the massive Jupiter's gravity, but it's real all the same.

Rather like a bird lands on a boat, the boat sinks deeper into the water by a fraction of a millimetre. Think about it.

I hope that helps your project.

Answer 4

I don't have the formula, but it's worth remembering that the Earth and Moon orbit a common centre of mass about 3000 miles from the centre of the Earth. The system is losing orbital momentum.



The main source of thermal heating of the Earth's mantle is radioactive decay as well as mass concentration and gravity.



Gut feeling tells me that orbital mechanics and gravitational flux do contribute to the heating of the mantle, but it is probably minute and hence NG says it doesn't.

Answer 5

Here's the formula you need to compute the gravitational attraction between two objects (like planets, stars, etc.,.)



F = G x M1 x M2 / r^2



'G' - gravitational constant (6.673^minus 11)

'M1' - mass of object one

'M2' - mass of object two

'r' - distance between M1 and M2 (center to center)



Keep in mind, though, that the gravitational attraction between solar system objects isn't quite as straight forward as two objects pretty much isolated from other masses.

Answer 6

It doesn't seem likely; any tidal heating of the Earth's interior would probably be so slight as to be completely swamped by the planet's "native" heat, from radioactive decay.