MadSci Network: Astronomy |
You are correct that if there were some way to increase Jupiter's mass steadily, its moons would move closer continuously. To make this question more concrete, let's consider what it would take for the innermost Galilean moon Io to impact Jupiter.
Currently Io orbits at a distance of 422,000 km from Jupiter or at about 6 Jovian radii. What would it take to reduce its orbit to 1 Jovian radius so that it would spiral into Jupiter's outer atmosphere? In answering this question, I assumed that conservation of energy was almost valid. Because we are considering adding mass to Jupiter, conservation of energy is not strictly valid. However, if we add the mass slowly and in small amounts, then conservation of energy is almost valid. (If a small amount of mass is added, Io moves in just a small amount, so conservation of energy remains almost valid.) I combined this assumption with Kepler's Laws, specifically the third law.
Assuming that I've done the algebra correctly, I find that the mass of
Jupiter needed is Mf =
Mi(a/RJ)
, where Mi
is
Jupiter's current mass, RJ
is Jupiter's radius,
and a
is Io's current orbital distance. Above we said that
a/RJ ~ 6
, meaning that one would have to increase
Jupiter's mass by 6 times in order to get Io to spiral into it. Given that
Jupiter is 2 times more massive than all of the other planets in the solar
system combined, finding 6 times the mass of the solar system in meteorites
would be tough! Of course even larger amounts of mass would be required if
one wanted to consider any of the other Galilean satellites.
(The astute reader will note that I've made another assumption: Jupiter's radius does not change as mass is added to it. It turns out that this is a good assumption. In general one would think that adding mass to an object would increase its radius. That's not true of some objects, such as white dwarfs, for which adding more mass tends to compress them making them smaller. Jupiter lies just on the boundary between these two cases: Adding mass to Jupiter doesn't change its size very much.)
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