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)

The earth is an average. 150 million km . from the sun calculate its average speed in orbit.

a) To find the angle between the Earth-Sun line and the Mars-Sun line, we need to first determine the position of Mars after 365.26 days.

Given that Mars completes one orbit in 687 days, the fraction of Mars' orbit completed after 365.26 days can be calculated as:
fraction = 365.26 / 687

Now, we can find the angle between the Earth-Sun line and the Mars-Sun line by multiplying the fraction of Mars' orbit completed by 2π. This is because Mars completes a full orbit of 2π radians in its orbital period.

angle = fraction * 2π

Therefore, the angle between the Earth-Sun line and the Mars-Sun line after 365.26 days is angle.

b) The time between two closest approaches can be found by determining the time it takes for the angle between the Earth-Sun line and the Mars-Sun line to be the same on two consecutive closest approaches.

Since Mars completes one orbit in 687 days and Earth completes one orbit in 365.26 days, we can find the time between two closest approaches by finding the least common multiple (LCM) of 365.26 and 687.

LCM(365.26, 687) = time between two closest approaches

Hence, the time between two closest approaches is the LCM(365.26, 687).

c) The angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations can be found using the concept of supplementary angles.

Since the angles between the Earth-Sun line and the Mars-Sun line on two consecutive closest approaches are equal (as stated in the question), the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations would be the supplementary angle of this angle.

Supplementary angles add up to a straight angle of π radians (180 degrees). Therefore, the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations is π - angle.

Hence, the required angle is π - angle.