There's a fun theorem in orbital mechanics. If you can reproduce the entire solar system, all the planets, and the sun, at any scale you like and with the exact densities of the original, even a billionth scale, the orbital periods will be the same as they are now, at least considered as two body problems. So the whole thing, theoretically, should work the same way to a first approximation.
The sun would be a challenge there, as its center is far denser than any normal terrestrial material. But you could use its average density instead: 1.41 tons per cubic meter (water is 1 ton per cubic meter).
But you'd need to do it in free space, probably far from any planet also, as tidal effects would disrupt it.
On Earth, well the Earth's gravity would cause problems. Perhaps theoretically if you could suspend it in a frictionless vacuum chamber through magnetic levitation adjusted to exactly counteract the Earth's gravity and automatic feedback to counteract any other external forces?
But the forces between the planets would be so tiny it would be hard to keep it stable I think. Also either the vacuum chamber would be huge, or the planets just tiny like specks of dust. And the lunar and sun tides would be a problem too.
Here are instructions on how to make a 1: 6,336,000,000 scale model of our solar system: How Big is the Solar System?
See also Could Another (small) Satellite Orbit The International Space Station?
I wonder if this is also true more generally for the multi-body problem? Anyone know? Do comment if you have the answer (or do your own answer to the question of course). For the moment anyway I am not sure how one could see it, if it is true.
The force law
doesn't scale down in a simple way if you reduce all the dimensions by a scale factor and use same densities. With scale factor 10 say, then the masses reduce by 1000 each, radius reduces by 100, so the force decreases by 10,000.
The sun would be a challenge there, as its center is far denser than any normal terrestrial material. But you could use its average density instead: 1.41 tons per cubic meter (water is 1 ton per cubic meter).
In the formula for the orbital period of a satellite,
See Orbital period as a function of central body's density
Here, notice there's an a cubed on the top - that is half the maximum distance between the planets in their orbit (semi-major axis). On the bottom, the masses M1 and M2 depend on the cube of their radii also.
So if you reduce the semi major axis a by a scale factor and reduce the radii of the planets by the same scale factor, in a scale model of the same density, the period remains the same.
But you'd need to do it in free space, probably far from any planet also, as tidal effects would disrupt it.
On Earth, well the Earth's gravity would cause problems. Perhaps theoretically if you could suspend it in a frictionless vacuum chamber through magnetic levitation adjusted to exactly counteract the Earth's gravity and automatic feedback to counteract any other external forces?
But the forces between the planets would be so tiny it would be hard to keep it stable I think. Also either the vacuum chamber would be huge, or the planets just tiny like specks of dust. And the lunar and sun tides would be a problem too.
Here are instructions on how to make a 1: 6,336,000,000 scale model of our solar system: How Big is the Solar System?
See also Could Another (small) Satellite Orbit The International Space Station?
I wonder if this is also true more generally for the multi-body problem? Anyone know? Do comment if you have the answer (or do your own answer to the question of course). For the moment anyway I am not sure how one could see it, if it is true.
The force law
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Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
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