Galileo’s Law of Planetary Motion

Just as all the parts of the earth mutually cooperate to form its whole, from which it follows that they have equal tendencies to come together in order to unite in the best possible way and adapt themselves by taking a spherical shape, why may we not believe that the sun, moon, and other world bodies are also round in shape merely by a concordant instinct and natural tendency of all their component parts? If at any time one of these parts were forcibly separated from the whole, is it not reasonable to believe that it would return spontaneously and by natural tendency? And in this manner we should conclude that straight motion is equally suitable to all world bodies.  [Galileo, 1629]

Galileo’s two main published works were “Dialogue Concerning the Two Chief World Systems” in 1629 and “Discourses and Demonstrations Concerning Two New Sciences” in 1638.  The first of these was fully ten years after Kepler published his third law of planetary motion, and twenty years after the publication of Kepler’s first and second laws, yet Galileo seemed oblivious to those developments – despite the fact that he was very familiar with Kepler’s works and had high regard for him (referring to him as “a person of independent genius”).  Einstein described Galileo’s failure to take account of Kepler’s laws as “a grotesque illustration of the fact that creative individuals are often not receptive”.  Of course, Einstein himself exhibited this same quality, so he knew it when he saw it.  (Following an embarrassing attempt to design a “better” aerodynamic wing during the first world war, disregarding all the literature on the subject, Einstein explained sheepishly that it was “an example of what can happen to a man who thinks a lot but reads little”.)

As Einstein noted, almost all of Galileo’s arguments in favor of the Copernican system were qualitative, as opposed to Kepler’s more quantitative work.  However, Galileo did present one quantitative proposition about planetary motion in the Dialogues, and he repeated it ten years later in the Discourses.  This “law of planetary motion” is seldom discussed, perhaps due in part to the obscurity of Galileo’s description of it, as well as to the erroneousness of the “law” itself.  Nevertheless, it’s interesting to consider Galileo’s “law” to see if it sheds any light on his thinking.  If nothing else, his devotion to this “law” - which he may well have first conceived prior to the appearance of Kepler’s laws - might explain why he was “not receptive” to the latter.  In addition, there are some aspects of Galileo’s reasoning, vis a vis Kepler, that could actually be seen (with some imagination) as precursors of general relativity and its geometric view of inertial motion.

The inspiration for Galileo’s law - like the inspiration for his use of the dialogue as a literary device for presenting his arguments - seems to have come from Plato.  Early in the First Day of the Dialogues, Galileo attributes to Plato the idea that

…these world bodies, after their creation and the establishment of the whole, were for a certain time set in straight motion by their Maker. Then later, reaching certain definite places, they were set in rotation one by one, passing from straight to circular motion, and have ever since been preserved and maintained in this. A sublime concept, and worthy indeed of Plato…

Unfortunately no precise reference is given, but it is generally surmised that Galileo was alluding to Plato’s description of God’s creation of the universe in Timaeus, which reads in part

The sun and moon and five other stars, which are called the planets, were created by him… and when he had made their several bodies, he placed them in the orbits… Now, when all the stars … had attained a motion suitable to them … and learnt their appointed task, moving in the motion of the diverse, which is diagonal and passes through and is governed by the motion of the same, they revolved, some in a larger and some in a lesser orbit, those which had the lesser orbit revolving faster, and those which had the larger more slowly. [Trans by Benjamin Jowett]

Another translation of the same passage reads as follows

…. he brought into being the Sun, the Moon and five other stars … These are called "wanderers," … When the god had finished making a body for each of them, he placed them into the orbits traced by the period of the Different … Now when each of the bodies … had come into the movement prepared for carrying it, and when … these bodies had … learned their assigned tasks, they began to revolve along the movement of the Different, which is oblique and which goes through the movement of the Same, by which it is also dominated.  Some bodies would move in a larger circle, others in a smaller one, the latter moving more quickly and the former more slowly.  [Trans. by Donald J. Zeyl].

It’s not easy to see (from either translation) how Galileo inferred that Plato thought the planets were first moved in straight lines to their appropriate positions and then began to revolve in their circular orbits.  In fact, it isn’t clear that Plato thought all these orbits were about the same center, since he clearly states that the Moon orbits the Earth, whereas it had been suggested even in Plato’s time that Venus and Mercury moved in circles around the Sun, which of course was believed to move in a circle around the Earth.  Nevertheless, Galileo somehow managed to discern from this an idea of planetary motion relevant to the Copernican model.  The essence of the idea is summarized in Plato’s assertion that the planets in smaller orbits move more rapidly than the planets in larger orbits.  By the time Galileo’s Dialogues were published, Kepler had already published his Third Law, which gives the precise quantitative relationship rv² = constant for the speed v of a planet orbiting at the radius r, but Galileo never mentions this.  Instead, he outlines his own theory for how the velocity of each planet is determined:

Let us suppose God to have created the planet…upon which he had determined to confer such-and-such a velocity, to be kept perpetually uniform thereafter.  We may say with Plato that at the beginning he gave it a straight and accelerated motion; and later, when it had arrived at that degree of velocity, converted its straight motion into circular motion whose speed thereafter was naturally uniform… And here I wish to add one particular observation … which is quite remarkable. Let us suppose that … the divine Architect … thought of creating in the universe those globes which we behold continually revolving, and of establishing a center of their rotations in which the sun was located immovably. Next, suppose all the said globes to have been created in the same place, and there assigned tendencies of motion, descending toward the center until they had acquired those degrees of velocity which originally seemed good to the Divine mind. These velocities being acquired, we lastly suppose that the globes were set in rotation, each retaining in its orbit its predetermined velocity. Now, at what altitude and distance from the sun would have been the place where the said globes were first created, and could they all have been created in the same place?

Galileo goes on to review the sizes and periods of the planetary orbits, and concludes that the speeds of the planets are all consistent with the premise that they all “descended from the same height” in a linear motion (beginning at rest) to the places of their orbits, at which point the speed they each achieved during this linear descent was converted into circular motion.  Thus the inner planets move at a greater speed, because they descended a further distance toward the Sun.  Galileo claims that “the size of the orbits and the velocities of the motions agree so closely with those given by the computations that the matter is truly wonderful”.

Unfortunately he gives no details of the calculation, but this idea was definitely more than just a passing fancy, because he repeated it ten years later in the Discourses.  In both works the inspiration for the idea is said to have been Plato, but the idea itself is attributed not to any of the interlocutors, but to “our Author” and the “Lincean Academician”, both of which refer to Galileo himself.  Presumably Galileo was particularly proud of this idea, and took care to secure his claim to it.  Could this have been an idea from the beginning of his scientific career, pre-dating the publication of Kepler’s laws?  (The apparent lack of receptiveness is reminiscent of Poincare’s attitude toward Einstein’s version of relativity after 1905, since Poincare already has his own version.)

The discussion of this topic in the Discourses is almost identical to the account given in the Dialogues, but it does contain a few additional details, presented in dialogue form as follows:

Sagredo: Allow me, please, to interrupt in order that I may point out the beautiful agreement between this thought of the Author [regarding natural acceleration of bodies] and the views of Plato concerning the origin of the various uniform speeds with which the heavenly bodies revolve… Plato thought that God, after having created the heavenly bodies… made them start from rest and move over definite distances under a natural and rectilinear acceleration such as governs the motion of terrestrial bodies. He added that once these bodies had gained their proper and permanent speed, their rectilinear motion was converted into a circular one… This conception is truly worthy of Plato; and it is to be all the more highly prized since its underlying principles remained hidden until discovered by our Author, who removed from them the mask and poetical dress and set forth the idea in correct historical perspective.   In view of the fact that astronomical science furnishes us such complete information concerning the size of the planetary orbits, the distances of these bodies from their centers of revolution, and their velocities, I cannot help thinking that our Author (to whom this idea of Plato was not unknown) had some curiosity to discover whether or not a definite "sublimity" might be assigned to each planet, such that, if it were to start from rest at this particular height and to fall with naturally accelerated motion along a straight line, and were later to change the speed thus acquired into uniform motion, the size of its orbit and its period of revolution would be those actually observed.

Salviati: I think I remember his having told me that he once made the computation and found a satisfactory correspondence with observation. But he did not wish to speak of it, lest in view of the odium which his many new discoveries had already brought upon him, this might be adding fuel to the fire. But if anyone desires such information he can obtain it for himself from the theory set forth in the present treatment.

So here Galileo is claiming to have suppressed one of his discoveries, and a “truly wonderful” one at that, to avoid public disapproval.  This seems a bit odd, considering all the other equally scandalous material he managed to publish in support of the heliocentric view.  Also, he didn’t actually refrain from publishing it… because we’ve seen that the idea appeared prominently in both of his major works.  The only thing that has been suppressed is the actual calculation, which surely was not the most scandalous element.  It seems that Galileo could easily have included the “truly wonderful” calculation in one or both of those works if he had been so inclined.  If the idea itself was ever actually suppressed, it must have prior to the writing of the Dialogues, which was during the years from 1616 to 1629.  If so, then it pre-dated the publication of Kepler’s third law.

In any case, what does Galileo’s Law (if we may call it that) actually tell us? It’s tempting to apply the inverse-square force law to try to reproduce Galileo’s wonderful computation, but this law was presumably unknown to Galileo, as well as to Kepler, although the latter seems to have had some hint of it).  Notice that Galileo refers to “a natural and rectilinear acceleration such as governs the motion of terrestrial bodies”, i.e., a motion of constant acceleration.  On this basis, if an object begins at rest and falls under constant acceleration g for a period of time t, then its final speed if v = gt and it has traversed a distance of s = v²/(2g).  A planet in a circular orbit of radius r moves at the squared speed v² = GM/r, so it must (according to the Plato/Galileo theory) have fallen a distance of s = GM/(2gr), which implies that it must have started at the height r + GM/(2gr).  Given the orbital radii r_A and r_B for two planets, we can say they fell from the same height with constant acceleration g only if we set g = GM/(2r_Ar_B).  Obviously this doesn’t give a consistent acceleration (under Galileo’s assumption that the planets all fell from the same height) for more than two planets, so this doesn’t look promising at all.

Is it possible that Galileo actually did take account of something like an inverse-square acceleration?  This question was posed by Bentley in a letter to Isaac Newton.  Bentley wondered if Galileo’s claim about the orbital speeds of the planets could be confirmed on the basis of Newton’s theory of gravity.  In reply, Newton pointed out that Galileo’s claim is not correct.  He wrote:

As for the passage from Plato, there is no common place from whence all the planets being let fall and descending with uniform and equal gravities (as Galileo supposes) would at their arrival to their several Orbs acquire their several velocities with which they now revolve in them. If we suppose the gravity of all the Planets towards the Sun to be of such a quantity as it really is and that the motions of the Planets are turned upwards, every Planet will ascend to twice its height from the Sun. Saturn will ascend till he be twice as high from the Sun as he is at present and no higher. Jupiter will ascend as high again as at present, that is, at little above the orb of Saturn. Mercury will ascend to twice his present height, that is to the orb of Venus and so of the rest. And then by falling down again from the places to which they ascended they will arrive again at their several orbs with the same velocities they had at first and with which they now revolve.

Newton’s point follows directly from the fact that the potential energy of a planet of mass m (in the Sun’s gravitational field) between two radial distances is GMm(1/r₂ – 1/r₁) and the kinetic energy of the planet is mv²/2, so if a planet begins at rest at a distance r1 from the Sun and falls to a distance r₂, its speed will be given by v² = 2GM(1/r₂ – 1/r₁).  Kepler’s third law for a circular orbit of radius r₂ is r₂v² = GM, so if it weren’t for the factor of “2” we could say that the planets all have the speeds they would acquire by falling to their present orbital radii from infinity.  However, given the factor of 2, the best we can do is to substitute for v² from Kepler’s law into the equation of motion to give GM/r₂ = 2GM(1/r₂ – 1/r₁), from which we get r₁ = 2r₂, just as Newton told Bentley.

Still, Galileo’s assertion is tantalizingly close to being consistent with an inverse-square acceleration and Kepler’s third law.  If, instead of saying all the planets fell from the same height, Galileo had said they all fell from twice their orbital heights, it would have clearly indicated knowledge of the inverse-square law of gravity.  As it is, we seem forced to conclude that the “truly wonderful” agreement between the planetary orbits and his computations was based on some kind of mistake.  It’s easy to imagine that a factor of 2 could have been lost when considering the ratios of the squares of the velocities.  Essentially Galileo’s Law of planetary motion (under this interpretation) was rv² = 2GM, whereas the correct statement of Kepler’s third law is rv² = GM.  Needless to say, neither the mass of the Sun nor the value of the gravitational constant was known, so we might forgive Galileo for thinking the factor of 2 was not sufficient grounds for denying him priority in the discovery of this remarkable pattern in the planetary orbits. It also seems plausible that the inconvenient factor of 2 was the reason he never published his wonderful computation, preferring instead to just indicate vaguely the outline of his idea.

Of course, this all assumes he applied a linear acceleration inversely proportional to the square of the distance, which he never states.  There are, however, at least two reasons for thinking he might have applied such a rule.  First, it is the only rule that gives anything close to “wonderful agreement” with the actual planetary motions (and his enduring enthusiasm for the idea clearly suggests that he found some correct pattern). Second, it is consistent with his reasoning about how it is most natural for a body beginning at rest to acquire all the intermediate speeds as it accelerates to a given speed.  Applying this same principle to the acceleration would lead to the idea of a continuously increasing acceleration. It’s also worth noting that, at the very least, Galileo had correctly intuited the existence of a link between the linear acceleration of gravity and the speeds of objects in circular orbits.  Ironically, the actual parametric path of linear free-fall under an inverse-square acceleration is a cycloid, which is a mathematical curve that Galileo studied, and which he suggested would be the optimum shape for the arch of a bridge.

Another point on which Galileo has often been faulted is his belief that the bound orbital paths of the planets represented natural inertial motion.  It is said that this belief showed he did not fully grasp the principle of inertia, which asserts that inertial motion must be rectilinear.  This criticism was certainly valid from the Newtonian perspective, according to which gravity is a force that impels objects away from their inertial paths.  However, ironically, the theory of general relativity vindicates Galileo’s view (if not his teleological reasoning), because according to Einstein’s theory the planets actually do follow inertial paths.  This is another example - like Newton’s “photons” - of how discredited ideas and points of view from the past are sometimes rehabilitated on a more sophisticated level.

Perhaps the best indication of Galileo’s level of confidence in the quantitative aspect of his idea is given in the Dialogues after Salviati has stated that the actual planetary orbits are in wonderful agreement with Galileo’s computations. The discussion continues

Sagredo: I have heard this idea with extreme delight, and if I did not believe that making these calculations accurately would be along and painful task, and perhaps one too difficult for me to understand, I should ask to see them.

Salviati: The procedure is indeed long and difficult, and besides I am not sure I could reconstruct it offhand. Therefore we shall keep it for another time.

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