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 rv^{2}
= 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 suchandsuch 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, predating 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 predated 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 inversesquare 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 = v^{2}/(2g). A planet in a
circular orbit of radius r moves at the squared speed v^{2} = 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_{A}r_{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 inversesquare 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/r_{2} – 1/r_{1}) and the kinetic
energy of the planet is mv^{2}/2, so if a planet begins at rest at a
distance r1 from the Sun and falls to a distance r_{2}, its speed
will be given by v^{2} = 2GM(1/r_{2} – 1/r_{1}). Kepler’s
third law for a circular orbit of radius r_{2} is r_{2}v^{2}
= 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 v^{2} from Kepler’s law into the equation of motion
to give GM/r_{2} = 2GM(1/r_{2} – 1/r_{1}), from which
we get r_{1} = 2r_{2}, just as Newton told Bentley.


Still, Galileo’s assertion is tantalizingly close to being
consistent with an inversesquare 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 inversesquare 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 rv^{2} = 2GM, whereas the correct
statement of Kepler’s third law is rv^{2} = 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 freefall under an inversesquare 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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