How often are mars and earth diametrically apart

[epicbattle]

Homework Statement

Mars orbits the sun at a mean orbital radius of 228 Gm ( 1 G = 10^9 m) and has an orbital radius of 687 days. Earth orbits the sun at a mean orbital radius of 149.6 Gm.
The earth-sun line sweeps out an angle of 360 during one Earth year. Approximately what line is swept out by the Mars-Sun line during one Earth-year?
How frequently are Mars and the Sun in opposition (on diametrically opposite sides of the earth?)

Homework Equations

365 days = 360 degrees for earth
687 days = 360 degrees for Mars

The Attempt at a Solution

(360 degrees/ 687 days)x 365 day = 191 degrees for Mars sweeping.
That's cool. I get that.

It's the diametrically opposed part that I am having issues with comprehending. Yahoo answers gave a solution:

"Mars and the Sun are in opposition every T days where T is given by 1/T = 1/E - 1/M where E =365 days is the Earth year and M=687 days is the Mars year. This works out to T = 779 days, slightly more than 2 years. "

This works. But this formula seemed out of the blue to me. Where did they come up with this? Am I missing something really simple? (Probably...)

No need to solve the problem, it's on where they got that formula. Thank you very much.

 

[tiny-tim]

hi epicbattle!

1/T = 1/E - 1/M where E =365 days is the Earth year and M=687 days is the Mars year.

hint: angular speed = 2π/D radians per day …

and how do you find the relative angular speed of two bodies? ​

 

[epicbattle]

Ah hah, thank you.

[2pi/Days(Earth) - 2pi/Days(Mars)]*180/2pi degrees per day. The 2pi cancels leaving the formula. Thank you.

So .2311 degrees per day is relative angular velocity

So how often are they diametrically apart 180/.2311 which solves for 779 days. Thank you very much. I won't forget this.