Mars orbits the Sun at a mean distance of 228 million km, in a period of 687 day

Question

Mars orbits the Sun at a mean distance of 228 million km, in a period of 687 days. The Earth orbits at a mean distance of 149.6 million km, in a period of 365.26 days.
(a) Suppose Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. Exactly 365.26 days later, when the Earth has completed one orbit, what is the angle between the Earth-Sun line and the Mars-Sun line?


(b) The initial situation in part (a) is a closest approach of Mars to Earth. What is the time, in days, between two closest approaches? Assume constant speed and circular orbits for both Mars and Earth.


(c) Another way of expressing the answer to part (b) is in terms of the angle between the lines drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle?

Explanation / Answer

(a) angular velocity: w = (2*pi)/P, where P is the period. So mars' angular velocity is w=(2*pi)/228000000 = 2.8*10^-8 radians/sec angle = w*time = (2.8*10^-8 radians/sec)(365.26 days) there's not much time left on this problem so I'll leave the unit conversions up to you! the final answer should be in radians. (b) the angle traveled by earth until the next closest approach will be the angle from (a) added to 2pi radians. Use the relation, angle = w*t, rearranged to t = (angle)/w, with this angle and the EARTH's angular velocity, to find how long between two closest approaches. (c) plug the time you got in (b) into angle = w*t using earth's w

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