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I have read that Galileo was able to start observing the four large satellites of Jupiter in 1610. Did he ever attempt to estimate the relative sizes of the four orbits, and their periods?

I made a plot of the period of the four satellites versus their orbit size, as a ratio to the diameter of Jupiter at the equator. That would have been easier to sketch/estimate than absolute size.

Is there any record of a plot or graph somewhat like this - to visualize even qualitatively what kind of relationship there was?

Today one would make a log-log plot and immediately "discover" the 3/2 slope, but logarithms weren't available yet.

About eight years later, Kepler "articulated" what we now call Kepler's third law, the period squared being proportional to the diameter (semi-major axis) cubed. Did that realization come from these moons, or from looking at planetary motion instead?

One reason I ask about the Galilean moons - the eccentricity of orbits of these four moons is very low - their motion would be easily interpreted using just circles, while several planets have significant eccentricity and their positions would require more sophisticated math.

Since the relationship of the planets and that of the Jovian moons would have a different constant - was it immediately interpreted as a measure of the ratio of masses of Jupiter and the Sun?

This is my first question here - the wording may seem colloquial, but my question is serious.

Galilean Moons of Jupiter Kepler's Third Law

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The Wikipedia article on Harmonices Mundi states that Kepler gave only the conclusion.

Since he had taken all of Brahe's observations, the presumption is that he used this data, for he was very familiar with it, and it was more than adequate for the task. His published result describes the relationship in terms of the sun and planets, but not planets and moons.

See also Was Jupiter's mass “guessed at” by Kepler or Galileo?

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  • $\begingroup$ Thank you - that's a pretty clear answer! Also thanks for the linked question. At first I thought mine was a duplicate, but actually I'm primarily interested in just the "3/2 power law" behavior. As I mentioned in my comment here I'm surprised that Kepler didn't see the relationship much sooner by watching Jupiter's Galilean satellites! $\endgroup$
    – uhoh
    Commented Mar 25, 2016 at 23:59
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Kepler's third law was discovered on the basis of comparison of periods and distances of the planets. This was in 1619. Only in 1621 Kepler noticed that Galileo moons of Jupiter also satisfy this law. This fact was later used by Galileo as an argument in favor of Copernican system.

By the way, Kepler was one of the first astronomers who used logarithms. (Napier's discovery was communicated to Ticho Brahe in 1590, and Kepler was Ticho's assistant in 1600-1601.)

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  • $\begingroup$ Having spent many hours admiring the motion of those satellites through an eyepiece, I am really curious why the "3/2 power law" wasn't deduced in a matter of months or a year at most from their display. The periods are short and the orbits are essentially circular. It seems to me that anybody with mathematical skills and interest, after seeing those four dots dancing would have been compelled to make a plot like the one in the question. I wonder if Kepler noticed it but didn't want to mention it until it was shown to hold for much more "substantial and important" bodies (e.g. the planets) $\endgroup$
    – uhoh
    Commented Mar 25, 2016 at 23:38
  • $\begingroup$ I hadn't realized logarithms were known at all at that time! Thank you for that. Speeding up arithmetic calculations is one use, drawing straight line fits to log-log plots to demonstrate power law behavior is another. I just searched "log-log" here and found this answer which is interesting reading. I have just asked this question as a follow-up. $\endgroup$
    – uhoh
    Commented Mar 25, 2016 at 23:49
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    $\begingroup$ I suppose they were not generally "known at that time". But I suppose they were known to Brahe and his assistants, including Kepler, because we have direct evidence. Information was spread slowly in pre-Internet times:-) $\endgroup$
    – Alexandre Eremenko
    Commented Mar 28, 2016 at 4:32

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