5.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Astronomia nova | 4/11 | https://en.wikipedia.org/wiki/Astronomia_nova | reference | science, encyclopedia | 2026-05-05T16:09:34.057453+00:00 | kb-cron |
In the diagram above, let
A
{\textstyle A}
be the sun,
B
{\textstyle B}
be the center of the circular orbit, and
C
{\textstyle C}
be the equant point. The points
I
{\textstyle I}
and
H
{\textstyle H}
are the perihelion and aphelion respectively. Let
G
{\textstyle G}
be the position of Mars at a particular observation. The angle
∠
G
A
H
{\textstyle \angle GAH}
Kepler refers to as the true anomaly, and the angle
∠
G
C
H
{\textstyle \angle GCH}
the mean anomaly. For any observation, the true anomaly could be deduced if we knew the longitude of aphelion, by finding the difference between this and the longitude of the observation. The mean anomaly could be deduced if we knew the time when Mars it at aphelion, and by using the fact the Mars, viewed from the equant, traverses equal angles in equal times. If the true anomaly and the mean anomaly were known, we could likewise determine the location of the point
G
{\textstyle G}
by finding where the lines drawn from
B
{\textstyle B}
and
A
{\textstyle A}
intersect. For the purposes of calculation, we can take the length of the line
C
A
¯
{\textstyle {\overline {CA}}}
to be
1
{\textstyle 1}
. Kepler's procedure is to take for 4 observations of Mars at opposition. By taking an initial guess for the longitude of aphelion and the time of aphelion, values could be computed for the mean anomaly and true anomaly of each observation, from which the location of Mars at each observation could be determined by the intersection of the lines
A
G
¯
{\textstyle {\overline {AG}}}
and
C
G
¯
{\textstyle {\overline {CG}}}
. If the 4 points do not lie on a circle, then the line of apsides
H
I
{\textstyle HI}
is rotated about the point
A
{\textstyle A}
; this shifts the values for the true anomalies, until all 4 points lie on a circle. Then if the center of the circle
B
{\textstyle B}
is not on the line of apsides, the line
H
I
{\textstyle HI}
is rotated about the point
C
{\textstyle C}
until the point
B
{\textstyle B}
falls on the line of apsides; this shifts the values for the mean anomalies. But doing this also shifts the position of the points so that they no longer fall on a circle. This procedure is repeated again and again until all 4 points fall on a circle and the center of the circle
B
{\textstyle B}
falls on the line of apsides
H
I
{\textstyle HI}
. This iterative process takes a long time to converge. In describing the procedure, Kepler writes:
If thou art bored with this wearisome method of calculation, take pity on me, who had to go through with at least seventy repetitions of it, at a very great loss of time. At the end of the procedure, Kepler calculates the parameters for the model. He determines the longitude of aphelion as
148
∘
9
{\textstyle 148^{\circ }9}
. The eccentricity of the circle is defined to be the distance from the center of the circle to the sun
A
B
¯
{\textstyle {\overline {AB}}}
, divided by the radius of the circle
H
B
¯
{\textstyle {\overline {HB}}}
, the value Kepler determines to be
0.11332
{\textstyle 0.11332}
. The eccentricity of the equant is defined as
C
B
¯
{\textstyle {\overline {CB}}}
divided by the radius of the circle, which he finds to be equal to
0.07232
{\textstyle 0.07232}
. The sum of these values is referred to as the total eccentricity. In chapter 17, Kepler makes a small correction for the fact that the longitude of aphelion and nodes are not constant but shift slowly over time.