4.2 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Astronomia nova | 5/11 | https://en.wikipedia.org/wiki/Astronomia_nova | reference | science, encyclopedia | 2026-05-05T16:09:34.057453+00:00 | kb-cron |
In chapters 18-21, Kepler compares the theory to observations. First, he compares the longitude of the remaining 8 oppositions and finds that they all fit the predicted position of Mars to within Tycho's observational accuracy of two minutes of arc. This means that the vicarious hypothesis can be taken as an accurate theory for the true anomaly. Despite this remarkable accuracy, however, Kepler shows that the theory is false. He remarks:Who would have thought it possible? This hypothesis, so closely in agreement with the observations, is nevertheless false. Using the latitudes of the opposition and the latitude triangle from figure 1, Kepler is able to find the ratio of the Earth and Mars distances from the sun. The Earth-Sun distances
G
H
{\textstyle GH}
are taken from an existing theory given by Tycho Brahe in his Progymnasmata, even though these values are not precisely correct, and the goal of the next part will be to determine a more accurate theory for the Earth's motion. The angles
∠
I
H
E
{\textstyle \angle IHE}
are determined by the latitude of the observations, and the angle
I
G
H
{\textstyle IGH}
is determined from the orbital inclination and the angle between Mars and the node. From this, the remaining sides can be determined, and the distances. By computing such distances, he obtained a lower and upper estimate for the eccentricity of Mars:
0.08
{\displaystyle 0.08}
–
0.09943
{\displaystyle 0.09943}
. The eccentricity found in the vicarious hypothesis is outside this range. Kepler then examines another method for determining distances to Mars, by using observations of Mars when it is not at opposition and determining the longitude of Mars. The angle between the sun and Mars as viewed from Earth can be determined from observations. Tycho made many observations of Mars when it is not at opposition and determined the difference in ecliptic longitude between the sun and Mars in the sky. The angle between Mars and the Earth as viewed from the sun can be determined by calculating the heliocentric longitude of Mars from the vicarious hypothesis, and that of the Earth from Tycho's theory and taking the difference. And the distance from the Earth to the sun is given from Tycho's theory. Thus, the Earth, sun and Mars form a triangle, where two angles are known, and one side is given, the remaining sides and angles can be computed. In particular, we can determine the Earth-Mars distance. Computing these distances, Kepler once again finds an eccentricity closer to
0.09
{\displaystyle 0.09}
, half the value of the total eccentricity (sum of that of the equant plus that of the circle).
Kepler repeats the calculation where he substituted the mean sun in place of the true sun, to show that exactly the same thing arises in such case. So, the hypothesis of the true sun cannot be at fault. As a final recourse, Kepler considers what would happen if we substituted bisected eccentricity into the vicarious hypothesis (i.e. let the eccentricity of the circle be half the total eccentricity), which is
0.09282
{\textstyle 0.09282}
. When he compares this model to the observations of oppositions, he finds the error now increases to 8 minutes of arc, which is greater than Tycho's observational error. He writes:Now, because they could not be disregarded, these eight minutes alone will lead us along a path to the reform of the whole of Astronomy, and they are the matter for a great part of this work.The inconsistency in determining the eccentricity means that at least one of the assumptions that went into constructing the vicarious hypothesis must be false: either the orbit is not circular, or there is no equant point a fixed distance away from the center of the circle.