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)
Mars
has
a
semi-‐major
axis
of
1.5
AUs
(228
million
km).
What
is
the
period
of
Mars
a) To find the angle between the Earth-Sun line and the Mars-Sun line after 365.26 days, we need to calculate the angle swept by each planet during that time.
The Earth completes one orbit in 365.26 days, so it sweeps an angle of 2π radians.
Mars, on the other hand, completes one orbit in 687 days. To find the angle it sweeps in 365.26 days, we can use the formula:
angle = (time / period) * 2π
Substituting in the values, we have:
angle = (365.26 / 687) * 2π
Now, we can calculate the angle:
angle ≈ 1.8158 radians
So, the angle between the Earth-Sun line and the Mars-Sun line is approximately 1.8158 radians.
b) To find the time between two closest approaches, we need to consider when the angles between the Earth-Sun line and the Mars-Sun line are equal.
Since we know the orbital periods, we can calculate the time it takes for both planets to reach this position again.
For Earth, it takes 365.26 days to complete one orbit.
For Mars, it takes 687 days to complete one orbit.
We want to find the time it takes for both planets to reach the same angle again. Let's call this time t.
Using the formula:
angle = (time / period) * 2π
For Earth, the angle after time t is:
angle = (t / 365.26) * 2π
For Mars, the angle after time t is:
angle = (t / 687) * 2π
Since we want the angles to be equal, we set these two equations equal to each other:
(t / 365.26) * 2π = (t / 687) * 2π
Now, we can solve for t:
(t / 365.26) = (t / 687)
Cross-multiplying:
687t = 365.26t
Simplifying:
321.74t = 0
t = 0
So, the time between two closest approaches is 0 days. This means that the Earth and Mars will never be in the same position again.
c) Since the time between two closest approaches is 0, there is no angle between the line drawn through the Sun, Earth, and Mars. Therefore, the angle is 0 radians.