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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Astronomia nova | 9/11 | https://en.wikipedia.org/wiki/Astronomia_nova | reference | science, encyclopedia | 2026-05-05T16:09:34.057453+00:00 | kb-cron |
=== Part 4 === In part 4, Kepler develops an accurate theory to account for the motion of Mars based on the observations and the physical hypotheses that were laid out in the previous section. In chapters 41-44, Kepler proves that the orbit of Mars is not a circle. The procedure once again uses the fact that the Earth, sun and Mars form a triangle. The Earth-sun distance can now be calculated accurately from the theory developed in the previous section, and the heliocentric longitude of Mars is determined from the vicarious hypothesis. For any given observation of Mars, the position of Mars can now be plotted accurately using the following procedure: plot the position of Earth using the theory developed in the previous section. Then, draw a line extending from the sun in the direction given by the vicarious hypothesis. Draw a line from the Earth in the direction corresponding to the heliocentric longitude that Mars is observed. The intersection between these two lines is the position of Mars. Finally, a correction is made for the fact that Mars is not in the plane of the ecliptic when observed, by using the latitude determined from the sun. By plotting several points of Mars on its orbit, Kepler shows the path of Mars is smaller at the sides than the best fit circle. thus, the path is an oval. In chapters 45-50, Kepler attempts to find the physical cause of deviation of the planet from a perfect circular path. He considers the following model: the magnetic rays from the rotating sun move the planet in a circular path. But the planet's own internal magnetic force causes it to move on a circle of its own, creating an epicycle. The motion of the planet on this epicycle is uniform, while the motion of the planet around the sun is non-uniform, its speed being given by the law of area. This motion should create an oval path. Constructing this oval is extremely difficult however, so Kepler settles on another idea: compute the distances of the planet from its epicycle and use the vicarious hypothesis to determine the direction of the planet from the sun. The oval path that is constructed by this method is slightly wider at the perihelion than at aphelion, so this orbit is properly an egg shape. In order to make use of his law of areas, Kepler needs to determine the area of this egg shape, which is not a trivial problem. Kepler approximates the oval as an ellipse, noting that the area should not differ significantly from the oval. When Kepler compares this model to the observations, however, he finds an error of 8 minutes of arc in predicted longitudes. This is the same error which was found in the bisected eccentricity model. However, where the bisected eccentricity predicted the planet ahead of its true position, the oval would predict it behind, so the errors were in the opposite direction. After rejecting various possible sources of error in his calculations, Kepler comes to the conclusion that the real path of the planet must lay halfway between bisected eccentricity model and the oval path. This also brings into question physical principles on which this hypothesis is based. In chapters 51-55, Kepler takes several pairs of observations of Mars that are symmetric along the line of apsides. These observations confirm that the distances to Mars are the same on either side and thus confirms that the line of apsides drawn through the sun is correct, which confirms his physical hypothesis. By taking several of these observations, spaced 687 days apart, Kepler is able to adjust the parameters of Mars orbit until the distances match. Doing this allows him to find more accurate distances for Mars. But the observations also force him to question the accuracy of the vicarious hypothesis outside of opposition observations. So, Kepler takes observations of Mars close to opposition, where the vicarious hypothesis could be trusted. After adjusting the parameters of the orbit until the distances line up, he finds that the distances at the sides are exactly halfway between what is predicted by the oval and the bisected eccentricity model. In chapters 56-60, Kepler tells the story of how he finally arrived at the correct path for the orbit of Mars. He had noticed that the maximum deviation of the true anomaly and the eccentric anomaly was
5.3
∘
{\displaystyle 5.3^{\circ }}
; he refers to this as the optical equation. The secant of this is
1.00429
{\displaystyle 1.00429}
, which represented an accurate fit to the deviation of Mars' path from a circle, which he had earlier determined from the observations to be about
0.43
%
{\displaystyle 0.43\%}
. He considers the possibility that the distances might be given by the secant of optical equation at other points in its orbit. When computing the numbers, he realized that he had seen them before in an earlier calculation which involved projecting the orbit of Mars on the diameter of an epicycle. Thus, Kepler declares that the Mars moves as if it is oscillating on the diameter of an epicycle. He examines a possible physical mechanism that could cause such a thing, and he finds that the same mechanism he outlined in Chapter 39 works: the planets' magnetic force pushes or pulls depending on the orientation of its poles. This oscillating motion is shown to be proportional to
cos
(
E
)
{\textstyle \cos(E)}
, so that the radial distance from the sun is given by
r
=
1
−
e
cos
(
E
)
{\textstyle r=1-e\cos(E)}
, where
E
{\textstyle E}
is the eccentric anomaly, and
e
{\textstyle e}
is the eccentricity. What Kepler had just described here is essentially the formula for an ellipse in polar coordinates. However, when he attempted the construction, he made an error, resulting in a completely different orbit which did not match the observations. After returning to his method from earlier, he once again stumbled on the ellipse, only then did he realize his error. He writes: