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

Following Kepler's publication of his 3rd law of planetary motion1,

$$p^2 / r^3 = 1$$

in 1619, it would have been possible to use telescopic observations to arrive at an estimate of the orbital radii the Jovian moons observed by Galileo 1610, expressed in Earth orbital radii.2 Along with the observed periods of the moons, these radii could have been used in a Jovian version Kepler's formula to discover that (at least for Io, Europa, and Callisto3)

$$p^2 / r^3 \approx 1.053 \times 10^3$$

or as close as telescopic technology of the day would allow.

Did anyone at the time take this, or a similar approach, to arrive at the conclusion that there is an attribute the Sun and Jupiter in which they differ by a factor of approximately a thousand? That attribute, of course, turns out to be mass.

Did Kepler, Galileo, or any of their contemporaries perform this calculation in the early 1600s? If not, why?

1. For $p$ in Earth years and $r$ in Earth orbital radii (today's AU).
2. Applying trigonometry to the observed angular extent of the moons' orbits and the distance to Jupiter.
3. Ganymede produces a slightly different value.

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A corresponding question could have been asked about the Earth's Moon (the answer would also be interesting to know), but since the Earth has only one (then visible, regular) satellite the critical, suggestive observation, that $p^2/r^3$ is a (different) constant for each central body, could not have been made in that case. So, presumably, this would have had to come first. – raxacoricofallapatorius Sep 6 '12 at 22:14
Was your literature search successful? – Lupercus Sep 12 '12 at 0:18
"Galileo" isn't the proper name, is it? – NikolajK Nov 9 '12 at 12:47

No. Johannes Kepler published what is now known as his third law of planetary motion in 1619 (in his treatise Harmonices Mundi), but discovered it already on May 15, 1618. He simply related mean distance of a planet from the Sun to its mean angular motion, without a word about a mass, I think. He did write on gravity and mass (not the precise physical term) in a foreword to his earlier book Astronomia Nova.

Thanks to people (Rafael Gil Brand, Roger Ceragioli and R. H. van Gent) from H-ASTRO discussion forum I have the following update #1:

1, The original form of the third law (formulated for planets), freely traslated to English reads approximately:

"...it is absolutely certain and perfectly correct, that the ratio which exists between the periodic times of any two planets is precisely 3/2 of the ratio of the mean distances, i.e. of the spheres themselves, bearing in mind, however, that the arithmetic mean between both diameters of the elliptic orbit is slightly less than the longer diameter"

2, Although (as far as I know from my own experience with early observations of double stars by Galileo) it is virtually impossible to prove that an earlier observation/idea didn't exist, it seems that the first application of Kepler's Third Law to the Jovian satellite system, is found in Newton's Philosophiae Naturalis Principia Mathematica (2nd ed. of 1713), lib. III, prop. 8, resulting in 1/1033 solar mass.

It is possible that Riccioli had something about the topic in one of his monumental treatises published around the middle of the 17th century.

Update #2

Riccioli seem to discuss relation between elongation of Galilean satellites of Jupiter and their orbital periods both in his Almagestum Novum and Astronomia Reformata, and cites Vendelinus (Godefroy_Wendelin). The Wikipedia entry for him states:

"In 1643 he recognized that Kepler's third law applied to the satellites of Jupiter."

without further details.

I repost here the final answer by Christopher Linton from H-ASTRO:

"Kepler, in the Epitome of Copernican Astronomy (1618-1621), did apply his third law to the Jovian satellites (in Art. 553). He got the data from Simon Mayr's World of Jupiter (Mundus Jovialis, 1614). He establishes that $T^2/a^3$ is roughly constant and concludes that the physical mechanism which causes the planets to move as they do is the same as that which causes the Jovian satellites to rotate around Jupiter."

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Note that the question is not whether he used a concept similar to "mass" as we conceive of it today, but how early (a) the Jupiter-moons pattern was noted, and when (b) any connection between it and the Sun-planets pattern was commented on. Certainly this sort of thinking must have been quite widespread by the 1700s for Swift to discussed the corresponding Mars-moons pattern so casually. – raxacoricofallapatorius Nov 6 '12 at 13:21
@raxacoricofallapatorius I see... I will try to find out. – Leos Ondra Nov 6 '12 at 15:22
Excellent, thank you. The more I read of Kepler, the less it surprises me that he didn't miss extending the period-radios relationship to Jupiter. Very little escaped his imaginative mind. (And thanks for the pointer to Marius and Mundus Iovialis, about which I new nothing.) – raxacoricofallapatorius Nov 9 '12 at 16:26

The origin of Kepler's law can be very easily grasped by a modern readership (Landau & Lifsitz, Vol.1 Mechanics). Considering Newton's inverse square law applied to a planet orbiting the Sun (omitting here the vector notations) $$\frac{d^2r}{dt^2}=-G \frac{M_{Sun}}{r^2}$$ we can easily see that new similar orbits (valid solutions characteristic for further possible planets) are obtained by scaling a given solution $r(t)$ as follows $$r'(t)=\lambda \ r(t/\tau)$$ with the condition (Kepler's law) $$\frac{\lambda^3}{\tau^2}=1\ .$$ Should this scaling relation be not respected, the scaled solution $r'(t)$ is still valid if the mass of the central object is changed according to $M'= M_{Sun}\ \lambda^3/\tau^2$. Should then $M'$ represent the mass of Jupiter, we can obtain the "Jupiterian" Kepler's law $$\frac{\tau^2}{\lambda^3}=\frac{M_{Sun}}{M_{Jup.}}\approx 1000 \ .$$

This could hardly deserve to be called a "law" by today's standards, after 300 years of dynamic education of the general public. I suppose that in the early 1600s, when these brilliant people derived constant relationships from observational data, they were only thinking in purely kinematic terms, perhaps they even believed they discovered laws of (heavenly) kinematics as such. Long time after Kepler the concept of mass was not clear at all, neither for Newton, nor even for Mach 200 years later. Considering these plausible arguments I would answer your question with no, they could not have possibly deduced the mass ratios from Kepler's law in the early 1600's.

But I am not a historian of science, and as this pretty field of scholarship keeps bringing surprises and new insights, I would say that the informed opinion of a historian would be mandatory at this point.

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Yes, once you have Newton, the origin of Kepler's laws and the (postulated) role of mass are clear. My question is whether, prior to Newton's laws, anyone (a) did this calculation to (b) arrive at the conclusion (or at least the suspicion) that the computed ratio says something about a relation between an attribute of Jupiter and the Sun. – raxacoricofallapatorius Sep 6 '12 at 23:42
We would need some tough historical literature to answer this question at the level of your expectations. Pierre Duhem may have wrote about this in his 10-volume history or somewhere else (Le Système du monde. Histoire des doctrines cosmologiques, de Platon à Copernic, published 1913-1958). Besides Arthur Koestler's "Lunatics" my readings in history are very meager. – Lupercus Sep 6 '12 at 23:53
+1 Thanks for this concise and well written answer. Mathemetics is the key :) – Stefan Bischof Dec 28 '13 at 4:33