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Many people have been astounded by Swift's apparent knowledge of these Martian moons 150 years before they were discovered by Asaph Hall. It is particularly interesting why Swift should have placed the inner moon so close to the planet, with such an unusually fast period of revolution.
However, as amazing as this may seem at first glance, a detailed analysis shows that Swift's choices are quite in accord with the available scientific knowledge of the day.
In 1610, following Galileo's discovery of 4 moons around Jupiter, Kepler speculated that Mars would be found to have 2 moons. He based this on observations that in proceeding out from the Sun, Venus had no moons, Earth one, Mars two, the "missing" planet three, and Jupiter four. This belief would have been strengthened in Swift's time, as by then five moons had been discovered around Saturn.
If we look at the orbital radii of the moons of Jupiter, we might see why Swift chose the orbits he did for the Martian moons.
Once the orbital radius is given then Kepler's or Newton's laws may be called upon to compute the period. Two quantities are needed for this calculation - the diameter of Mars and its mass (or alternatively Kepler's constant for Mars). The first quantity was known (although it was believed to be 7700 km). It appears that Swift assumed an Earth mass, as this gives the figures he quoted. And so we see that maybe the "mystery" is not so mysterious after all. Some people have stated that Swift could not possibly have performed the required calculations. This may be so. But he undoubtedly had friends who could!
Originally posted by Hanslune
...................................Inner Moon............ Outer Moon
....................................(Swift) / (Actual) (Swift) / (Actual)
Radius (Mars Diameters).....3............1.38......5..........3.46
Revolution period (hrs).......10...........7.7......21.5......30.3