Category Archives: Astronomy

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Libration Force

The Libration Force

Kepler coined the term “libration” for the oscillation of a planet’s distance from the Sun, approaching and receding.

He analyzed the libration for the eccentric-equant model, and found it unexpectedly complicated. Stephenson (p.78):

Many absurdities were involved in supposing that a planet could move, … non-uniformly, about the vacant center of the eccentric, with no guide except the apparent magnitude of the solar disk. Such complicated hypotheses, although designed to yield a perfectly simple eccentric circular path, were not physically credible…

Notice the remarkable thing that Kepler was doing here. He was analyzing motion on an eccentric circle, a model that had been in general use for nearly two millenia, apparently the simplest possible model with any empirical accuracy. He took apart this beautifully simple model and showed that as a physical process (and in the absence of solid spheres) it was really quite complicated, so complicated as to raise doubt about whether it could be real. He had performed so radical a reassessment by interpreting astronomy, for the first time, as a physical science.

Eventually Kepler achieved the elliptical orbit. Seeking a physical explanation, he hit on a magnetic force to produce the libration:

What if all the bodies of the planets are enormous round magnets? Of the earth (one of the planets, for Copernicus) there is no doubt. William Gilbert has proved it.

But to describe this power more plainly, the planet’s globe has two poles, of which one seeks out the sun, and the other flees the sun. So let us imagine an axis of this sort, using a magnetic strip, and let its point seek the sun. But despite its sun-seeking magnetic nature, let it remain ever parallel to itself in the translational motion of the globe…

—Astronomia nova, Chapter 57.

The figure at the top of this post (taken from the Epitome of Copernican Astronomy) shows how it works. (The figure in the Astronomia nova has extra clutter.) Kepler explains:

[When] the strip is at A and E, there is no reason why the planet should approach or recede, since it holds its ends at equal distance from the sun, and would undoubtedly turn its point towards the sun if it were allowed to do so by the force that holds its axis straight and parallel. When the planet moves [counterclockwise] away from A, the point approaches the sun perceptibly, and the tail end recedes. Therefore, the globe begins perceptibly to navigate towards the sun. After E, the tail end perceptibly approaches and the head end recedes from the sun. Therefore, by a natural aversion, the whole globe perceptibly flees the sun…

—Astronomia nova, Chapter 57. [I have changed the letters from C and F to A and E to match the diagram from the Epitome.]

Implicit: the magnetic force weakens with distance, so when the head
is closer to the Sun than the tail, the net force is attractive. And vice versa.

Kepler argued that this scheme gave the force a sinusoidal dependence on the longitude, and showed that this agreed with the libration for an elliptical orbit. Some aspects of this demonstration needed special pleading. Stephenson details the strong and the weak points of the reasoning (pp.110–117).

But: “The theory had one glaring flaw, however. The magnetic axis of the planet had to maintain a constant direction, perpendicular to the apsidal line.” (Stephenson, p.117.) The Earth’s rotational axis doesn’t come close to meeting this requirement. So why should we believe it holds for Mars? Kepler acknowledged the problem:

I will be satisfied if this magnetic example demonstrates the general possibility of the proposed mechanism. Concerning the details, however, I have my doubts. For when the earth is in question, it is certain that its axis, whose constant and parallel direction brings about the year’s seasons at the cardinal points, is not well suited to bringing about this reciprocation… And if this axis is unsuitable, it seems there is none suitable in the earth’s entire body, since there is no part of it that rests in one position while the whole body of the globe revolves in a ceaseless daily whirl about that axis.

As one possible out, Kepler appealed to a planetary mind.

Besides the radial libration, planets have a libration in latitude. This enmeshed the theory in further difficulties. Ever inventive, Kepler devised ad hockery around all these rough spots. But we have a contrast: we can trace a direct path from the whirlpool force to the area law. This cannot be said for Kepler’s libration theory. Kepler’s whirlpool speculations came years before the area law. The libration force came after the elliptical orbit.

There is a reason for this. You can justify the whirlpool force (more or less) using the conservation of angular momentum. Kepler’s libration force has no counterpart in Newtonian physics.

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Nature of the Whirlpool Force

Kepler was unsure of the nature of the whirlpool force. Sometimes he compares it to light, sometimes to magnetism. Three successive marginal notes in Chapter 33 lay out the analogy with light: “The kinship of the solar motive power with light”; “Whether light is the vehicle of the motive power”; “The motive power is an immaterial species of the of the solar body”.

The Latin term species calls for a short discussion. Donahue (in his glossary) says:

This word, related to the verb “specio” (see, observe) has an extraordinarily wide range of meaning. Its root meaning is is “something presented to view”, but it can also mean “appearance”, “surface”, “form”, “semblance”, “mental image”, “sort”, “nature”, or “archetype” … I have therefore thrown up my hands, admitted defeat, and declined to translate it at all.

while Stephenson (p.68,footnote) writes:

“Image” is our rendering of Kepler’s species, which has for the most part been left untranslated in other accounts. As Kepler used it the word seems to mean the appearance or visible manifestation of the sun…

Gilbert’s De magnete furnished Kepler with another analogy, and a clue. Chapter 34 is titled “The body of the sun is magnetic, and rotates in its space.” A few quotes from it:

The magnet … has filaments (so to speak) or straight fibers (seat of the motor power) extended throughout its length. …it is credible that [the sun] … has circular fibers all set up in the same direction, which are indicated by the zodiac circle.

… It is therefore plausible, since the earth moves the moon through its species and is a magnetic body, while the sun moves the planets similarly through an emitted species, that the sun is likewise a magnetic body.

Kepler postulates filaments or fibers in the Sun as the source of the whirlpool force. These encircle the Sun along the circles of latitude. The Sun rotates, and the image of the moving fibers acts upon the planet to move it in the same direction.

Inverse Square vs. Inverse Linear

Taking a modern perspective, we have an inverse square law whenever we have a conservation law plus spherical symmetry. But with cylindrical symmetry (like the whirlpool force), we can have an inverse linear dependence.

A nice modern analogy: dipole radiation. Feynman discusses this in his Lectures, Chapter I-28:

The gradually discovered properties of electricity and magnetism … showed that these forces … fell off inversely as the square of the distance… As a consequence, for sufficiently great distances there is very little influence of one system of charges on another.

… Maxwell [to obtain a consistent system] … had to add another term to his equations. With this new term there came an amazing prediction, which was that a part of the electric and magnetic fields would fall off much more slowly with the distance than the inverse square, namely, inversely as the first power of the distance!

It seems a miracle that someone talking in Europe can, with mere electrical influences, be heard thousands of miles away in Los Angeles. How is it possible? It is because the fields do not vary as the inverse square, but only inversely as the first power of the distance.

He goes on to treat the dipole radiator. That is, an antenna. The key point: We have charges moving up and down the antenna. What matters is how that motion looks to a distant observer:

… all we have to do is project the motion on a plane at unit distance. Therefore we find the following rule: Imagine that we look at the moving charge … like a painter trying to paint a scene on a screen at a unit distance … We want to see what his picture would look like. So we see a dot, representing the charge, moving about in the picture. The acceleration of that dot is proportional to the electric field.

At root we have a simple matter of geometry. Substituting an eye for the painter, and the eye’s retina for the screen, we have the diagram below. We have a vertical antenna of height H at distance r from the eye’s lens. The image of the antenna on the retina has height I. The antenna has unit distance 1 from the lens. Thus:

From similar triangles, I/1=H/r. In other words, the length of the image is inversely proportional to the first power of the distance.

Note also that the symmetry is cylindrical and not spherical. If the line of sight is not perpendicular to the antenna, the image will be smaller, vanishing completely when the line of sight passes through the antenna.

Returning from Feynman’s Lectures to Kepler’s Astronomia nova, we can resolve the inverse-square problem. Instead of an antenna, we have the Sun’s circular fibers. Instead of the retina, we have the image (species) of those fibers moving the planet. The whirlpool force results from the sum total of those fiber images. Each image’s contribution diminishes inversely with distance, so the sum does too. For all the difference in the physics, the basic geometry remains the same for Feynman’s dipole and Kepler’s whirlpool.

Stephenson (p.75) puts it this way:

The composite motion was directed along a circle parallel to the sun’s equator (the resultant of the images of all the filaments) and it was therefore weakened—at any latitude—only as this circle expanded, in the simple proportion of distance.

This one sentence states the matter more clearly than Kepler does in Chapter 36. Nonetheless, the ingredients are all there, just scattered through the chapter.

One more thing: is the the whirlpool force is confined to the ecliptic? Some authors say that Kepler claimed this. Stephenson shows, with quotes, that this is not true. But something like it holds effectively. A planet above the solar pole would see the fibers moving in all directions, and the net effect would be complete cancellation. At intermediate latitudes, you get partial cancellation. Only at the ecliptic does the planet get the full effect. Figure 14 (p.75) of Stephenson illustrates this:

A final footnote. In 1645 Ismael Boulliau published his Astronomia philolaica. In it he noted, as Kepler had, that the whirlpool force should exhibit an inverse square dependence on distance. But he ignored Kepler’s solution to this problem. On the strength of this, the noted historian Thony Christie credits him with being The man who inverted and squared gravity. I am far less inclined to award him this accolade.

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The Whirlpool Force: Early Thoughts

In the Astronomia nova, Kepler introduced the whirlpool force this way:

… since there are (of course) no solid orbs, as Brahe has demonstrated from the paths of comets, the body of the sun is the source of the power that drives all the planets around. Moreover, I have specified the manner [in which this occurs] as follows: that the sun, although it stays in one place, rotates as if on a lathe, and out of itself sends into the space of the world an immaterial species of its body, analogous to the immaterial species of its light. This species itself, as a consequence of the rotation of the solar body, also rotates like a very rapid whirlpool throughout the whole breadth of the world, and carries the bodies of the planets along with itself in a gyre, its grasp stronger or weaker according to the greater density or rarity it acquires through the law governing its diffusion.

Voelkel calls this the motive force hypothesis. As we’ve seen, Kepler devised it quite early, inspired by the steady decrease in planetary speeds with increasing orbital radii.

In the Mysterium cosmographicum, in a chapter titled “Why a planet moves uniformly about the center of the equant”, he adduced a new argument: the changing speed in a single orbit. The speed at perihelion is faster than at aphelion; in fact, the speed ratio is the inverse of the distance ratio:

v_peri/v_ap = r_ap/r_peri

We saw in post 4 that all three laws (the equant, the inverse distance, and the area law) give this relation.

So we have two similar phenomena. Jupiter (for example) moves slower than Mars, and Mars at aphelion moves slower than Mars at perihelion. The distance from the Sun seemed to be the common factor.

Kepler delighted at finding a physical cause to replace the equant. He was out of step with most astronomers of the time. True, they despised the equant. But Copernicus had replaced its non-uniform motion with the uniform motion of a small epicycle (often called an epicyclet). This they admired, while rejecting heliocentrism.

Kepler’s old teacher Maestlin discovered a geometrical demonstration for the near-equivalence of the Copernican epicyclet with Ptolemy’s equant; he communicated this to Kepler in a letter in 1595. (See Voelkel (p.19) or Evans (p.1013) for the proof.) The next year, in the Mysterium cosmographicum, Kepler wrote:

The path of the planet is eccentric, and it is slower when it is further out, and swifter when it is further in. For it was to explain this that Copernicus postulated epicycles, Ptolemy equants… Therefore at the middle part of the eccentric path … the planet will be slower, because it moves further away from the Sun and is moved by a weaker power; and in the remaining part it will be faster, because it is closer to the Sun and subject to a stronger power…

Nowadays we know that these two phenomena stem from different physics. Kepler’s second law reflects the conservation of angular momentum; it would hold with any central force. Kepler’s third law comes from the inverse square law for gravity plus the formula for centripetal force: F ∝ 1/r² and F ∝ v²/r. For orbits with small eccentricities, we have approximately

From the start, Kepler favored an inverse distance law for the whirlpool force. In a letter to Maestlin in 1595, he suggested a way to derive the orbital radii from the much more accurately known periods. (He needed the radii to test his polyhedral hypothesis.) He noted that two factors contributed to the longer periods of the more distant planets. First, they have to traverse a longer orbit. Second, they do so at a slower speed. Now, the motive force originates in the Sun and spreads out evenly over the orbits, so it should diminish in inverse proportion. Here is the passage, quoted in Voelkel (p.39). (Kepler uses ‘motion’ to mean motive force (proportional to speed), ‘orbs’ to mean orbits, and ‘circles’ to mean circumferences.)

There is, as I said, a moving spirit [motrix anima] in the Sun. If equal motion and the same strength came from the Sun into all orbs, one would still circulate more slowly than another on account of the inequality of the orbs. The periodic times would be as the circles. For quantity measures motion. However, circles [go] as the radius, namely as the distance. Thus from the certainly-known mean motions we could easily construct also the mean distances. But another cause enters which makes the more remote slower. Let us take the experience [experimentum] of light. For as both light and motion are connected in their origin so also [are they connected] in their actions, and perhaps light itself is the vehicle of motion. Therefore, in a small orb and also in a small circle near the Sun, there is as much light as there is in a large and more remote sphere. Therefore the light is thinner in the large, and denser and stronger in the narrow. And this strength is in inverse proportion to the circles, or the distances.

Note the last sentence. In the Mysterium cosmographicum Kepler repeated the argument:

Let us suppose, then, as is highly probable, that motion is dispensed by the Sun in the same proportion as light. Now the ratio in which light spreading out from a center is weakened is stated by the opticians. For the amount of light in a small circle is the same as the amount of light or of the solar rays in the great one. Hence, as it is more concentrated in the small circle, and more thinly spread in the great one, the measure of this thinning out must be sought in the actual ratio of the circles, both for light and for the moving power. Therefore in proportion as Venus is wider than Mercury, so Mercury’s motion is stronger, or swifter, or brisker, or more vigorous than that of Venus, or whatever word is chosen to express the fact. But in proportion as one orbit is wider than another, it also requires more time to go round it, although the force of the motion is equal in both cases. Hence it follows that one excess in the distance of a planet from the Sun acts twice over in increasing the period; and conversely, the increase in the period is double the difference in the distances.

Perhaps you already see two problems with this. First, the analogy with light indicates an inverse square dependence, not inverse linear. Second, neither of these are the right law. Let T be the period and r the orbital radius. An inverse linear dependence for the whirlpool force dictates that T is proportional to r²; an inverse square, to r³. But T is proportional to r^3/2, as Kepler would eventually discover.

Kepler convinced himself that the inverse distance law fit the available data. Note the end of the passage above: “the increase in the period is double the difference in the distances”. For example (with made-up numbers): Say Planets 1 and 2 have T₁=100, T₂=400, and r₁=1000. T₂−T₁=300, or 3 times T₁. The increase in the radii, i.e., r₂−r₁, should then be 1.5 times r₁, or 1500. So r₂ should be 2500. Compare this with the correct result of assuming T proportional to r²: r₂=2000. (Kepler carries out a similar computation with Mercury and Venus.)

Kepler applied this procedure to adjacent pairs of planets, using the Copernican periods to calculate the ratios of the distances. The results agreed (sort of¹) with the Copernican ratios for the distances. Still, the procedure makes no sense. In the second edition of the Mysterium cosmographicum (twenty five years later) Kepler added this footnote:

Here the mistake begins… Now what I ought to have inferred … is that the ratio of the periods is the square of the ratio of the distances, not because I hold it to be true, for it is only the 3/2th power, as we shall hear, but because it was the legitimate conclusion from this line of argument. You see how at this point the arithmetic mean was taken, by halving the difference, when the geometrical mean should have been taken.

Next: inverse square or inverse linear? Voelkel remarks

whereas light propagates spherically, Kepler confined his attention to the plane of the orbit… Only much later did he reconsider the spherical propagation of the motive virtue and address the problem of whether the strength ought, as light, to decrease as the square of the distance. [p.40]

And in a footnote Voelkel adds, “In his first thoughts about the propagation of motor virtue, he appears to have thought only about the plane of the orbit.”

Kepler’s Pars Optica (1604) clearly stated the inverse square law for light. By the time of the Astronomia nova, Kepler realized he had a problem. In his letter to Maestlin, he had said “perhaps light itself is the vehicle of motion”. In Chapter 33 of the Astronomia nova, he asserts

…although this light of the sun cannot be the moving power itself, I leave it to others to see whether light may perhaps be so constituted as to be, as it were, a kind of instrument or vehicle, of which the moving power makes use.

This seems gainsaid by the following: first, light is hindered by the opaque, and therefore if the moving power had light as a vehicle, darkness would result in the movable bodies being at rest; again, light spreads spherically in straight lines, while the moving power, though spreading in straight lines, does so circularly; that is, it is exerted in but one region of the world, from east to west, and not the opposite, not at the poles, and so on. But we shall be able to reply plausibly to these objections in the chapters immediately following.

In Chapter 36 he amplifies the second objection before resolving it:

This objection wearied me for a long time without offering any prospect of a solution.

It was demonstrated in Chapter 32 that the intension and remission of a planet’s motion [i.e., the time taken to traverse a given length] is in simple proportion to the distances. It appears, however, that the power emanating from the Sun should be intensified and remitted in the duplicate or triplicate ratio of the distances or lines of efflux. [I.e., as the square or cube of the distances.]

As this post is long enough, I’ll resume the story next time.

[1] But as Stephenson notes, the Copernican distances were not that accurate; Kepler’s third law would not have fit very well either.

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Kepler’s Physics

Let me repeat the motto, already implicit in the Mysterium cosmographicum, but come into full bloom in the Astronomia nova:

Forces emanating from the Sun guide or drive the planets in their orbits.

A word about the word ‘force’. F=ma this is not. Kepler’s forces are Aristotelian, satifying F∝v (force is proportional to speed). Not only that: we must be careful (as Stephenson writes (p.187)) not to “impute to Kepler a conceptual exactness which his physics—in contrast with his astronomy—did not possess …The word force, vis, did not yet have the clarity of an accepted technical term.”¹

I wrote “forces” above. Kepler ends up with three forces in the Astronomia nova: (a) Gravity, which plays no role in determining the orbits. (b) A whirlpool-like force that sweeps the planets around the Sun. Kepler calls this the Sun’s species; I will call it the whirlpool force. (c) A force responsible for the elliptical shape of the orbit. Kepler calls this the magnetic virtue; I will call it the libration force, for reasons that will become apparent².

Souls and Minds

Ptolemy in the Planetary Hypotheses endowed the planets with souls, the source of their motion. This seems odd to us. But remember that for the ancient Greeks, the planets were celestial beings, perhaps even gods. On Earth, what moves under its own power? Animals and people, of course. Add to this the authority of Aristotle. Here’s how Kepler recaps Aristotle’s Metaphysics (Book 12, part 8) in the Astronomia nova, Ch.2:

…having thus accumulated 49 orbs in all (or 53 or 55, following Callipus), he attributed to each its own mover … In this way, he introduced us to separate minds which, it turned out, were gods, as the perpetual administrators of the heavens’ motions. They also bestowed a moving soul [anima motrix], more closely attached to the orbs and giving them form, so that the mind would only have to give assistance … They therefore transferred this potentiality for creating motion to a soul …

Now this coupling of mind and soul is indeed quite in agreement with the detailed considerations of the astronomers, even though the philosophers’ mode of argument is chiefly metaphysical. For it is the same in humans: the moving faculty is one thing; that which makes use of the moving faculty according to the indications of the senses—the Will—is another … Similarly, if we should propose these Aristotelian orbs as objects of contemplation, two things will present themselves to us: (1) the motive force, from whose activity and constant strength the time of revolution arises; (2) the direction in which it acts. The former is more correctly ascribed to the animate faculty, and the latter to its intelligent or remembering nature.

More succinctly, Stephenson writes:

In these early chapters of the Astronomia nova, souls furnished the motion and minds the guidance.

—Stephenson (p.30)

From the first, Kepler preferred a different explanation for the planets’ motion. The decisive clue: the farther a planet is from the Sun, the slower it moves. In Kepler’s earliest extant astronomical writing (a disputation he wrote while still at Tübingen, six years before the Mysterium cosmographicum), he proposed that the motive cause resided in the Sun, “whence as if from the center and the certain heart of the world it would extend itself through its effect most equally to all surrounding orbs” (quoted in Voelkel, p.29). Five years later, in a letter to his old teacher Maestlin, Kepler repeated this idea:

…there is in the Sun a moving spirit [anima movens] and an infinite motion … that strength of motion (as light in optics), which is further from the source is that much weaker.

—quoted in Voelkel (p.36)

The next year, in the Mysterium cosmographicum, he wrote

… one of two conclusions must be reached: either the moving souls are weaker the further they are from the Sun; or, there is a single moving soul in the center of all the spheres, that is, in the Sun, and it impels each body more strongly in proportion to how near it is. In the more distant ones on account of their remoteness and the weakening of its power, it becomes faint, so to speak.

—quoted in Voelkel (p.54)

He immediately opts for the second conclusion. Notice that Kepler has not dispensed with souls, just relocated and united them all in the Sun. This line of thought led to the whirlpool force, culminating in the Second Law—as we will see.

Twenty-five years later Kepler published a second edition of the Mysterium cosmographicum with new notes. Here is the note for the phrase “there is a single moving soul”:

If for the word “soul” you substitute the word “force”, you have the very same principle on which the Celestial Physics is established in the Commentaries on Mars [i.e., Astronomia nova], and elaborated in Book IV of the Epitome of Astronomy. For once I believed that the cause which moves the planets was precisely a soul, as I was of course imbued with the doctrines of J.C. Scaliger on moving intelligences. But when I pondered that this moving cause grows weaker with distance, and that the Sun’s light also grows thinner with distance from the Sun, from that I concluded, that this force is something corporeal, that is, an emanation which a body emits, but an immaterial one.

The planets do not revolve in circular orbits concentric with the Sun. Here Kepler thought that planetary minds might come into play. Even with an eccentric circular orbit, the direction is sometimes partly towards the Sun, sometime away from it. As Stephenson puts it,

[W]hen Kepler spoke of a mind in the heavens he was considering the problem of controlling motion; more specifically, the problem of obtaining sufficient information to constrain motion into the regular path that was observed.

—Stephenson (p.30)

What computations would the mind have to make, what data would it need, and how would it obtain it? Even for a circular orbit, this perplexed him. The notion of the libration force, along with the possibility of an oval orbit, gradually took shape in Kepler’s thought. Although references to souls and minds occur throughout the Astronomia nova, physics began to displace planetary “psychology”. Stephenson again:

Kepler thus proposed two quite different theories in Chapter 57 to account for the planet’s [oval orbit]. One relied upon natural forces, magnetic or quasi-magnetic, while the other supposed some kind of primitive planetary mind.

—Stephenson (p.120)

By this time, in fact, Kepler was decidedly less enthusiastic than he had once been about planetary minds. Years of physical speculation had brought an increasingly confident belief that natural means alone could explain the patterns of motion for which he had, hesitantly, once invoked mental control. … The change in Kepler’s attitude was a gradual one, occurring over a period of years and never really completed. Celestial minds remained a part of Kepler’s universe; only from his astronomy can one see them discreetly withdrawing.

—Stephenson (p.132)

Souls, too, did not entirely disappear:

Kepler believed that a motive soul was necessary to account for the persistence of the solar rotation… He offered several reasons why it should be thought plausible that there was a soul in the body of the sun … Generally he thought that a soul was necessary to sustain the rotation, but that no mind or intelligence was needed, because the motion was constant in direction and speed.

—Stephenson (p.141)

[1] Kepler also employed the Latin virtus and species, sometimes translated ‘power’ and ‘image’, not with a clear distinction from vis. I will mostly steer clear of these subtleties. (For his translation of the Astronomia nova, Donahue (pp.107–108) wrote about species: “The translator, in the end, has thrown up his hands, admitted defeat, and has declined to translate it at all.”)

[2] In a previous version of the post, I called this the quasi-magnetic force, but I now feel that libration force is better. Stephenson calls it the libratory force.

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The Astronomia nova

The full title of the Astronomia nova is The New Astronomy, based upon causes, or celestial physics, treated by means of commentaries on the motions of the star Mars, from the observations of Tycho Brahe. This hits all the high spots: the treasure trove of Tycho’s observations, Kepler’s new physics, and the “battles with Mars”.

Historians have discovered that the Astronomia nova, although ostensibly a blow-by-blow account of Kepler’s process of discovery, in fact was (as Stephenson puts it) “one sustained argument”, written and rewritten carefully. So I will not initially deal with the topics in the same order.

Tycho’s Observations

The observational accuracy available to Ptolemy and Copernicus is generally taken to be about 10′, i.e., 10 arcminutes (Babb; Thoren, p.11), although errors of 40′ and more in his star catalog are frequent (Pedersen, p.252). Tycho’s observations rarely erred by more than 4′, and often were accurate to 1′ (Thoren, pp.11–12).

But just as crucial was Tycho’s systematic approach, following each planet throughout its orbit:

Previous astronomers had been clever, observing chiefly at critical moments, such as oppositions to the sun, when their observations gave clear answers to the questions they wanted to ask. Tycho’s less directed program proved its worth, naturally enough, when Kepler’s research reached the point where he had to ask questions that had not been thought of before.

—Stephenson (p.51)

We saw in post 4 that with a suitable choice of parameters, Ptolemy’s equant speed law and Kepler’s area law give the same speeds at perihelion and aphelion, and give nearly the same times for the planet’s arrivals at the quadrants. We will see that the data for the octants played a critical role in Kepler’s discoveries.

Actual Orbits

Thanks to the Planetary Hypotheses, we know that Ptolemy regarded his planetary models as physically real. By the time of Copernicus, however, most astronomers subscribed to the viewpoint expressed in Osiander’s (in)famous preface to De revolutionibus:

…it is the duty of an astronomer to compose the history of the celestial motions through careful and expert study. Then he must conceive and devise the causes of these motions or hypotheses about them. Since he cannot in any way attain to the true causes, he will adopt whatever suppositions enable the motions to be computed correctly from the principles of geometry for the future as well as for the past. The present author has performed both these duties excellently. For these hypotheses need not be true nor even probable. On the contrary, if they provide a calculus consistent with the observations, that alone is enough.

Cardinal Bellarmine enunciated this viewpoint in a letter to Paolo Foscarini, warning him not to advocate the Copernican hypothesis as reality:

To say that on the supposition that the Earth moves and the Sun stands still all the appearances are saved better than on the assumption of eccentrics and epicycles, is to say very well—there is no danger in that, and it is sufficient for the mathematician: but to wish to affirm that in reality the Sun stands still in the center of the world, and that the Earth is located in the third heaven and revolves with great velocity about the Sun, is a thing in which there is much danger.

Historians later called this attitude instrumentalism, in contrast with realism.¹

Kepler, like Copernicus (and Ptolemy and Peurbach) was a realist. The Astronomia nova includes, for the first time, a diagram of the actual orbit of Mars according to Ptolemy:

Actual Orbit of Mars in the Ptolemaic System

Donahue comments:

The figure on the facing page [see above] is a momentous diagram. Nothing like it had ever before been published. Astronomers had become so accustomed to thinking of celestial motions as compounded circular motions that it had apparently not occurred to anyone to consider the actual path traversed by a planet. Once the reality of the celestial spheres came into question, however, the actual path traversed came to be a matter of great interest.

— Donahue (p.35)

Expanding on the last sentence: the Planetary Hypotheses provides a physical mechanism for Ptolemy’s geometrical models, as we’ve seen. For a number of reasons (among them the discovery that comets would barrel right through the spheres) this mechanism seemed more and more dubious. The instrumentalists didn’t lose any sleep over this.

But for the realist Kepler, this “pretzel orbit” (as he termed it) constituted prima facie evidence againt the Ptolemaic and Tychonic systems².

The Astronomia nova puts it this way:

But, with arguments of the greatest certainty, Tycho Brahe has demolished the solidity of the orbs, which hitherto was able to serve these moving souls [motrices animae, the beings that kept the orbs rotating], blind as they were, as walking sticks for finding their appointed road; and hence the planets complete their courses in the pure aether, just like birds in the air.

The realist approach has a couple of consequences. First, the longitude and latitude models must mesh properly to form a single consistent three-dimensional orbital geometry.

Second, if crystal spheres don’t guide the planets, what does?

[1] Centuries earlier, the Arabic polymath Averroös (Ibn Rushd, 1126–1198) complained, “The astronomy of our time offers no truth, but only agrees with the calculations and not with what exists.” [Wikipedia, longer quote in “The Search for a Plenum Universe” in Gingerich], p.140]

[2] The Tychonic system kept the Earth motionless but had the planets revolve about the Sun. It also suffered from ‘pretzelosis’.

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Kepler

Kepler wrote five major astronomical works. Chronologically:

the Mysterium cosmographicum (1596)

the Astronomia nova (1609)

the Epitome astronomiae Copernicanae (1618–1622)

the Harmonice mundi (1619)

and the Tabulae Rudolphinae (1627).

The Mysterium cosmographicum (Cosmographical Mystery) expresses Kepler’s youthful enthusiasm and sounds the leading notes to themes that would persist throughout his career. Kepler’s elliptical orbits and the area speed law make their debut in the Astronomia nova (New Astronomy). (Although at this point Kepler regarded the area law as just an approximation to the inverse speed law.) The Epitome astronomiae Copernicanae (Epitome of Copernican Astronomy) completes and refines his theory. The Harmonice mundi (Harmonies of the World) contains the statement of Kepler’s 3rd law, its main scientific claim to fame. The Tabulae Rudolphinae (Rudolphine Tables) ultimately led to the widespread acceptance of Keplerian astronomy.

He also wrote several lesser astronomical works, and treatises on optics, on computing volumes, on the philosophy of science, a pamplet on snowflakes… The critical edition of his collected works runs to 22 volumes. I will focus just on the Mysterium cosmographicum and the Astronomia nova.

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The Planetary Hypotheses

In the Planetary Hypotheses, Ptolemy lays out his cosmology: that is, the structure and arrangement of the universe. This work answers the question, did Ptolemy believe in the physical truth of the Almagest’s celestial geometry?—with an unambiguous Yes. Contrary to an opinion often expressed by earlier historians, he did not regard it just as a calculational scheme for predicting planetary positions.

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Cycle Counts

You may have heard that Ptolemaic systems grew to have 80 spheres or cycles, while the Copernican system had only 34. This is a myth.

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Latitudes

First thing to note about Ptolemy’s latitude theory: its decoupling from the longitude theory. For longitudes, one projects the orbits into the ecliptic plane. The actual speeds will differ from the projected speeds. However, the effect is small because the inclinations are fairly small (see the table below), and Ptolemy’s longitude computations ignore it. The latitude algorithms use the longitude as an input.

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Mercury

Mercury refused to cooperate with Ptolemy’s basic paradigm. You might guess that the fault lies with Mercury’s larger eccentricity, but studies show that bad data bears most of the blame. Mercury hugs the Sun, only appearing near the horizon close to sunrise or sunset, hardly ideal observation conditions.

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