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 these problems, we need to use some basic geometry and knowledge about orbits.
a) To find the angle between the Earth-Sun line and the Mars-Sun line, we can consider the triangle formed by the Sun, Earth, and Mars. Since Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun, this triangle is a straight line. The angle between the Earth-Sun line and the Mars-Sun line can be found by subtracting the angle between the Earth-Sun line and the Earth-Mars line from π radians (180 degrees).
To find this angle, we need to calculate the angle between the Earth-Sun line and the Earth-Mars line. We can use the concept of angular velocity:
Angular velocity (ω) = 2π / T,
where T is the period of the orbit. For Earth, T = 365.26 days, and for Mars, T = 687 days.
The angle θ can be calculated using the formula:
θ = ω * t,
where t is the time. In this case, since we are considering 365.26 days later, the time (t) will be 365.26 days.
Using these formulas, we can find the angle between the Earth-Sun line and the Mars-Sun line:
ω_earth = 2π / 365.26 = 0.017202 radians/day,
ω_mars = 2π / 687 = 0.009157 radians/day.
θ_earth = ω_earth * 365.26 = 6.2832 radians,
θ_mars = ω_mars * 365.26 = 3.1673 radians.
Therefore, the angle between the Earth-Sun line and the Mars-Sun line (α) is:
α = π - (θ_earth - θ_mars) =π - (6.2832 - 3.1673) ≈ 0.2745 radians.
So, the angle between the Earth-Sun line and the Mars-Sun line is approximately 0.2745 radians.
b) To find the time between two closest approaches, we need to consider that the angles between the Earth-Sun line and the Mars-Sun line will be equal at those points. Let's call this time period T_ca.
So, we need to calculate the time taken by both Earth and Mars to cover an angle of π radians. We can use the formula:
T_ca = T_mars / π,
where T_mars is the period of Mars' orbit.
T_mars = 687 days,
T_ca = 687 / π ≈ 218.89 days.
Therefore, the time between two closest approaches is approximately 218.89 days.
c) To express the answer in terms of the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations, we need to find this angle. Let's call it θ_ca.
Using the formula for angular velocity:
ω_earth_ca = 2π / T_ca,
θ_earth_ca = ω_earth_ca * T_ca = 2π.
Therefore, the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations (θ_ca) is equal to 2π radians.