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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Astronomia nova | 6/11 | https://en.wikipedia.org/wiki/Astronomia_nova | reference | science, encyclopedia | 2026-05-05T16:09:34.057453+00:00 | kb-cron |
=== Part 3 === In the third part, Kepler aims to determine an accurate theory for the motion of the Earth, which will be the steppingstone for determining a more accurate theory for Mars in the next section. In chapters 22-27, Kepler shows that the Earth does not move uniformly about the center of its orbit. The primary observations for determining the Earth's motion around the sun are direct observations of the sun; its ecliptic longitude as seen from the Earth is the opposite to that seen from the sun. Remarkably, uniform circular motion can match these observations to within an accuracy of one minute of arc, less than the accuracy of Tycho's observation. Tycho himself determined the eccentricity of the Earth's orbit based on his observations and the assumption of uniform circular motion to be
0.03584
{\textstyle 0.03584}
. Using the observations of Mars, Kepler finds several methods to show that the eccentricity of the Earth's orbit calculated, on the basis of uniform circular motion, cannot possibly be correct. One method that Kepler uses is based on the fact that after one complete orbit, Mars returns to exactly the same physical location. Since the orbital period of Mars is 687 days, we find observations of Mars that are spaced 687 days apart. If we assume the existence of a fixed point in space from which the motion of the Earth appears uniform, then the distance from Mars to that point is fixed (since the position of Mars is also fixed) . The angle between Earth and Mars as seen from the equant, can be computed from the Earth's mean anomaly and the fact that the observed angle from the equant changes at a uniform rate. The angle between the Mars and the equant as seen from Earth can be computed from the observed position of Mars, and the fact that this imaginary equant point would appear to move uniformly when viewed from Earth; we can therefore use the Earth's mean anomaly. For this purpose, Kepler makes use of Tycho's tables which are calculated from the sun's mean position. From a given distance, and two angles we can solve the triangle. From the observations Kepler shows that the distance from the Earth to the equant is not fixed. Therefore, if the Earth's orbit is circular, it cannot be centered on the point from which its motion is uniform. The equant is thus distinct from the center of the Earth's orbit. Kepler additionally uses various other constructions to show that the real eccentricity of the Earth's orbit is close to
0.018
{\textstyle 0.018}
, precisely half the value computed from the assumption of uniform circular motion. In chapter 28, Kepler shows a method to test the correctness of the hypothesis for the Earth's orbit. This is essentially the
687
{\textstyle 687}
days method in reverse. Compute the distance and angle to the Earth from our hypothesis. Use the observed angles of Mars, and the computed angles from our theory to make the same triangle, except this time the distance to the Earth is given, and we solve for the distance and the heliocentric longitude of Mars. If our hypothesis is correct, then for each observation, the calculated distance to Mars and heliocentric longitude must be exactly the same. This method also allows us to test the critical assumption that Mars really does return to the exact same position after one revolution in its orbit. In chapters 29-30, Kepler briefly mentions two other ways he had shown that the eccentricity of the Earth should be bisected. First he had measured the angular diameter of the sun in the winter (near perihelion) and summer (near aphelion) and computed the relative distances, which gives an eccentricity consistent with half of Tycho's value. He had also shown in his Mysterium Cosmographicum that the distances of his nested polyhedra hypothesis would match the observations better if he assumed the eccentricity is half what Tycho proposes. He then proceeds to construct the table for computing the Earth's position based on the eccentricity of 0.018. He admits, in constructing this table, the use of a non-circular orbit, but the theory for that is developed later on. In chapters 31-36, Kepler considers the reason for this bisection of the eccentricity. The bisection has been shown accurate for the Earth and Mars from the observations. The bisection is also used for the planets Jupiter and Saturn in all theories since Ptolemy. Likewise, Tycho Brahe had shown that this model works well for the moon too. When constructing a theory for Venus and Mercury, Copernicus had added a small epicycle to the orbit that had a period of revolution equal to the orbit of the Earth. Kepler shows that this epicycle can be removed if we bisect the eccentricities of Venus and Mercury as well. Thus the hypothesis of bisected eccentricity is valid for all planets and for the moon. Since this constitutes a universal law, valid for all planets, Kepler finds it necessary to seek the physical cause of this bisection. Kepler starts by computing the speed of the planet at aphelion and perihelion from this bisected eccentricity model; the result shows that the ratio of the speeds at these points is equal to the inverse ratio of the distances. From this, he introduces the hypothesis that the speed of the planet is inversely proportional to its distance from the sun. Kepler then argues this variation in speed must be the result of a force from the sun. To explain why each planet has a different speed than would otherwise be predicted from extending this inverse distance law to all the planets, Kepler postulates that each planet has its own resistance to the force generated by the sun (a concept similar to inertia). Finally, Kepler establishes magnetism as the likely cause for this force, because it has a similar property of a force weakening with distance. In addition to this, the existence of a magnetic field had recently been discovered for Earth. He therefore suggests the Earth's rotation causes the motion of the moon, and likewise, if the sun too rotates and has a magnetic field, this will be the cause of the planets motion. In chapter 37, Kepler briefly touches on the subject of Lunar theory. The orbit of the moon required two additional inequalities to explain its motion, these are evection and variation. Kepler argues that both these can be explained by the fact that the moon speeds up in its orbit when it forms a straight line with the Earth and sun. Thus, both the forces from the sun and Earth combine together to move the moon when it is aligned with them in this configuration. In chapters 38-39, Kepler gives an explanation for why the orbits of the planets are not concentric with the sun. He considers that each planet has its own magnetic force which pushes and pulls it away from the sun, depending on how its poles are oriented with respect to the sun. The physical line of reasoning given also hints at the possibility that the orbit is not circular.