# Calculating distance of Mars from the Sun based on two elongations

• danunicamp
In summary, the triangle E1, S, M and E2, S, M have an angle of 43 degrees between them. The triangle E2, S, M will reach the angle E1, S, M in 42.41 days. The angle between E1 and E2 is 57.41 degrees.
danunicamp
Homework Statement
Hello, This is problem 13b, from chapter 1 of Zeilik, Gregory, Introductory Astronomy and Astrophysics.
Mars has a synodic period of 779.9 days and a sidereal period of 686.98 days.
On February 11, 1990, Mars had an elongation of 43° West.
The elongation of Mars 687 days later, on December 30, 1991, was 15° West.
What is the distance of Mars from the Sun in astronomical units?
Relevant Equations
Law sins, law cosines
I called the point E1 the point where Earth was at Feb,11 1990 and E2 at 30, Dec 1991, S for Sun and M for Mars and r for the Mars-Sun distance.
Since we got a whole sidereal period between both alongation, I assumed Mars was on the same point in space (wrt Sun).

I think I got the triangles wrong, since I already redid the calculations twice.
I defined the triangles E1, S, M and E2, S, M.
E1, S, M with a side r opposite to the angle of 43 deg, a side 1 opposite to the angle SME1.
E2, S, M with a side r opposite to the angle of 15 deg, a side 1 opposite to the angle SME2.
The sides SE1 and E2M crossed at a point P.

I calculated that in (730-687) 43 days, or 42.41 degrees, the point E2 would reach E1. And with that got the angle of 42.41 in E2SP.
From that i got the angle of 122.59 at SPE2 and the same angle at E1PM.
From that I got the angle of 57.41 at SPM.
The angle E1MP was 14.41 (180 - 43 - 122.69)
From the law of sins (1/sin 122.59 = SP/sin 15) I calculated the side SP to be 0,3 and PE2 to be 0.7 (1 - 0.3).
From the law of sins I calculated the side PM to be 1.98 (0.7/sin 14.41 = PM/sin 43).
With the sides 0.3, 1.98 and r and the angle 57.41 i used the law of cosines:
r^2 = 0.3^2 + 1.98^2 - 2 (0.3)(1.98)(cos57.41)
and got a r = 1.83, which is wrong, since the Sun-Mars distance is 1.52 AU.

I know it is hard to visualize without the picture. Is there any way to add it here?
Any help would be very welcome. Thank you in advance

There are so many steps to the calculation I would worry about rounding errors.
Try writing those steps in a spreadsheet so that much more precision is carried through.

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