# Astronomy - synodic/sidereal periods

accountkiller

## Homework Statement

Mars has a synodic period of 779.9 days and a sidereal period of 686.98 days. On 2/11/1990 Mars had an elongation of 43-deg W. The elongation of Mars 687 days later on 12/30/1901 was 15-deg W. What is the distance of Mars from Sun in AU units.
(Hint: This is a multi-step problem that requires a carefully drawn diagram. You are not allowed to use Kepler's 3rd Law.)

## Homework Equations

$\frac{1}{Synodic_Mars} = \frac{1}{Sidereal_Earth} - \frac{1}{Sidereal_Mars}$
Law of sines
Law of cosines

## The Attempt at a Solution

My drawing consists of the Sun, Mars, Earth, and the planetary orbits. From a top view, I have the situation starting with Earth at the bottom of its orbit and Mars is more left in its orbit so that the Sun-Mars line and the Mars-Earth line make a 43-degree angle. Then I draw Earth's path as it makes almost 2 revolutions but stops to the left of Earth's original position, and Mars is still in its same place it was before, os now the Sun-Mars line and the Mars-Earth line make a smaller 15-degree angle. So now I have this little arc of space between Earth's two different spots in its orbit, I know that the Sun-Earth line's value is 1 AU, I have these two angles, and I know the synodic and sidereal periods.

I just am not sure how to start calculating anything now. I've been told by a classmate that I should use the law of sines first to get as much information as I can and then use the law of cosines at the end to find the Sun-Mars distance but I am not sure how to begin. Could anyone tell me the direction I need to be going in? Thanks!