# Astronomy - synodic/sidereal periods

1. Aug 30, 2012

### accountkiller

1. The problem statement, all variables and given/known data
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.)

2. Relevant equations
$\frac{1}{Synodic_Mars} = \frac{1}{Sidereal_Earth} - \frac{1}{Sidereal_Mars}$
Law of sines
Law of cosines

3. 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!

2. Aug 30, 2012

### voko

Between the two observations, both planets traverse some parts of their trajectories. You know the length of Earth's arc in terms of AU. The one of Mars can also be expressed in terms of the "Martian AU".

The two observations also form some triangles that could be used to figure out the length of Mars' arc in terms of AU.

3. Sep 1, 2012

### accountkiller

I guess this problem is hard to get help with over the internet with no drawings but I just wanted to add that I finally got it - it was just geometry. I just made sure I drew my diagram correctly then used law of sines to get all the angles and sides I needed then used the law of cosines once at the end to find the final distance.

Thanks!