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Here Comes the (True) Sun
I mentioned earlier “Kepler’s zeroth law”, a corollary to the motto:
Refer everything to the true sun, and not the mean sun. Nor should the earth receive any special treatment.
Kepler plucked several fruits from this tree. Here are two fruits of the zeroth law.
Apsides
Wrong Apsides
The solid circle is the true orbit for Mars. The dashed circle has the wrong apsidal line, passing through the mean sun instead of the true sun. X and Y are the positions of Mars in the two models at the same time. Note that the longitudes of X and Y differ little, whether viewed from the true sun or the mean sun. But the distances differ significantly.
Gingerich summarizes the situation nicely:
[T]he eccentricities assigned by Ptolemy to each of the planets are essentially vector sums of the earth’s and the planet’s eccentricities, with the result that in each case the line of apsides is slightly wrong…Copernicus did not bother to sort out the individual eccentricities, so each planetary eccentricity implicitly combines the earth’s. In addition, his apsidal lines are drawn through the mean sun… [Gingerich, Chap.19 “Kepler’s Place in Astronomy”]
So far as heliocentric longitudes are concerned, the wrong apsidal line won’t pose too serious a problem. See the figure above, which shows that solar distances are not so forgiving.
To understand this better, we need to look at acronychal risings. These are observations taken when the planet (here Mars) is in opposition to the sun. (See the figure below. ‘Acronychal’ means ‘night rising’.)
Key point: for an acronychal observation, the heliocentric and geocentric longitudes are the same. The astronomer can directly measure only geocentric longitudes. Except at opposition and conjunction, deducing the heliocentric longitude requires an assumption about the earth’s position. At conjunction, the brightness of the sun makes the planet invisible.
Kepler analysed the situation in Chapters 5 and 6. (These make slow reading; Stephenson (pp.31–39) gives a detailed discussion.)
This was a “must pass” test for the switch to the true sun. Tycho’s chief assistant Longomontanus (aka Christian Severinus) had worked out a model of Mars’s orbit that accounted for the acronychal data with an accuracy of 2 minutes, for a period of over 40 years. As Kepler explains in Chapter 7:
A hypothesis was invented which, it was proclaimed, represented all these oppositions within a distance of two minutes in longitude… It was only in the latitude at achronychal positions and also the parallax of the annual orb [i.e., observations out of opposition] that Christian got stuck… At the beginning there was great controversy between us as to whether it were possible to set up another sort of hypothesis which would express to a hair’s breadth so many positions of the planet, and whether it were possible for the former hypothesis to be false despite its having accomplished this so far over the entire circuit of the zodiac.
Chapter 6 showed the near agreement of the true sun and mean sun models for acronychal longitudes. But observations “from the side” (i.e., out of opposition) could amount to as much as 1° 20′, fourteen times the error at opposition and easily observable.
Latitudes
In post 13, I noted (following Swerdlow) that poor data bore most of the blame for Ptolemy’s complicated latitude theory. But having the orbital planes pass through the mean sun is just asking for trouble. The figure below illustrates the problem.
Latitudes and the Mean Sun
The diagram shows a cross-section of the three-dimensional geometry. Oaphelion and Operihelion are Earth’s aphelion and perihelion, so the line through them is the ecliptic, and the mean sun is midway between Oaphelion and Operihelion. The solid line at an angle is the orbital plane of Mars, with the inclination greatly exaggerated.
Shifting Mars’s orbital plane to pass through the mean sun, while keeping the inclination the same, gives the dashed line. Mars shifts from M to M; exactly where M lands depends on the longitude theory. Viewed from Earth’s aphelion its predicted latitude would be too large, from Earth’s perihelion too small, and from other points in the earth’s orbit also incorrect (except for two points). To fix this problem, one might naturally change the inclination, resulting in the dotted line. This dotted line represents a plane passing through the true position of Mars (M) and the mean sun. However, this fixes things only at one position M; at other points in Mars’s orbit, like M′, you’d need a different inclination. We see the cause of the rocking orbital plane.
This scheme still has a lot of leeway, since it takes three (non-collinear) points to determine a plane; an infinite number of planes pass through M and the mean sun. But it is impossible to avoid a varying inclination, simply because the mean sun does not lie in the actual orbital plane of Mars. Any plane passing through M and the mean sun will intersect Mars’s orbital plane in a line, which means it can correct matters for at most one other point of Mars’s orbit.
Copernicus’s latitude theory did not follow this scheme, but was essentially a transcription of the Ptolemy theory into heliocentric terms. It suffered from an additional drawback: the inclination depended on Earth’s position. This violates Kepler’s zeroth law.
Kepler showed if the orbital plane of Mars passes through the true sun, then you can use a fixed inclination. (The line of nodes does rotate very slowly. Kepler knew this.) He demonstrated this in three ways. I’ll outline only the first two.
Both involve finding the geocentric latitude for observations when this will equal the heliocentric latitude. When Mars is “at a limit”—that is, as far above or below the ecliptic as it gets—this will give the inclination.
Kepler noted that the highest accuracy is not required for these methods. Say M is Mars, and M′ is M projected onto the ecliptic. The distance MM′ doesn’t change that much for M near a limit.
Determining the Inclination, First Method
For the first method, suppose that the earth and the sun are equidistant from Mars, hence likewise from M′. Then MM′ appears the same whether viewed from the sun or the earth. So the geocentric and heliocentric latitudes are equal. (See Kepler’s diagram, reproduced above. Here A=Earth, B=Sun, E=Mars (my M), and C=projection of E to the ecliptic (my M′).)
How do we know when Sun-Mars-Earth form an isosceles triangle? Given the ratio of the Sun-Earth and Sun-Mars sides, (i.e., the orbital radii), basic trigonometry tells us the Sun-Earth-Mars angle. It turns out that you can use any observation where that angle is between (roughly) 60° and 72°.
Determining the Inclination, Second Method
For the second method, Kepler noted that when two planes cut one another, any two lines drawn in the respective planes to a point on the line of intersection, and perpendicular to that line, always include the same angle. (See the diagram above, also Kepler’s.) So if B is the earth, F is Mars, and D is its projection to the ecliptic, then ∠ DBF is the inclination. Therefore we need an observation when the earth lies in the line of nodes, Mars is at a limit, and the sun and Mars appear at right angles from the earth. For this method, we make no assumptions about the orbital radii. However, it requires very special observations.
Kepler consistently found an inclination close to 1.8° with all three methods and many observations. He concluded that not only is the inclination fixed, but it’s rather small. Tycho’s assistants had made a mess of things, finding not only that the orbit of Mars wobbled, but was also “fractured”, with a maximum latitude of 4.6° at one limit and 6.4° at the other.
I noted at the beginning of post 13 that Ptolemy “decoupled” longitudinal computations from latitudinal ones. Now, the speed of planet differs from the speed of its projection to the ecliptic. Moreover, the ratio of these two speeds is not constant. Ptolemy could safely ignore this issue because the effect is small; that’s true because because the inclinations are small. But with Tycho’s much greater accuracy, the matter could no longer be swept under the rug. (It’s known as the “reduction to the ecliptic”.) So in cleaning up the latitude theory, Kepler laid a crucial foundation stone for everything else.
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Diagonal Argument 