Only need help with part c--answer for part a is 2.94 radians and for b it is 777 days
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. All answers should be in the range (0, 2pi)
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? (in rad)
b) The initial situation in part a) is a closest approach of Mars to the Earth. What is the time between 2 closest approaches? Assume constant orbital speeds and circular orbits for both Mars and Earth. (Hint: when angles are equal) (in days)
c) Another way of expressing the answer to part (b) is in terms of the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle? (in rad)
To solve part c, we need to find the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations.
First, let's consider the situation at the time of closest approach. In this scenario, Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. We already know from part a) that the angle between the Earth-Sun line and the Mars-Sun line is 2.94 radians.
Now let's move forward in time to the next closest approach between Earth and Mars. Both planets will complete one orbit, and Earth will once again be in the same position as it was at the time of the first closest approach.
Since Mars has a larger orbit and takes longer to complete one orbit, it will not be in the same position as it was during the first closest approach. However, the angle between the line drawn through the Sun, Earth, and Mars will be the same.
So, the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations is still 2.94 radians.
Therefore, the answer to part c) is 2.94 radians.