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)
a) When Earth has travelled one orbit, Mars will have completed 365.2/687 = 53.2% of an orbit. Mars will be 46.8% of an orbit behinbd Earth. Convert that to radians.
b) Extrapolate to find out how long it takes for Mars to fall 360 degrees behond Earth. That is when a new alignment will occur.
c) Figure out how far (in angle) Earth travels between alignments. It will be more than two complete revolutions. You only want the additional angle beyond two revolutions. (Use the time from part (b)). Convert that to radians.
a) To find the angle between the Earth-Sun line and the Mars-Sun line after 365.26 days, we can use the concept of angular velocity.
The angular velocity (ω) is defined as the angle covered per unit time. It is given by the formula:
ω = (2π) / T
where T is the period of revolution.
For Earth, the angular velocity (ωE) is (2π) / 365.26 days.
For Mars, the angular velocity (ωM) is (2π) / 687 days.
Now, after 365.26 days, Earth completes one revolution, so the angle covered by the Earth-Sun line is 2π radians.
The angle covered by the Mars-Sun line can be calculated using the formula:
θM = ωM * t
where θM is the angle covered by Mars in time t.
Substituting the values, we have:
θM = (2π / 687) * 365.26 days.
θM ≈ 1.52 radians.
The angle between the Earth-Sun line and the Mars-Sun line is given by:
θ = π - θM
θ ≈ 1.62 radians.
Therefore, the angle between the Earth-Sun line and the Mars-Sun line after 365.26 days is approximately 1.62 radians.
b) The time between two closest approaches occurs when the angles covered by Earth and Mars are equal. Let's call this time Tc.
Using the same formula as before, we can equate the angles covered by Earth and Mars.
For Earth:
θE = ωE * Tc
θE = (2π / 365.26) * Tc
For Mars:
θM = ωM * Tc
θM = (2π / 687) * Tc
Since the angles are equal, we can set them equal to each other:
θE = θM
Solving for Tc:
(2π / 365.26) * Tc = (2π / 687) * Tc
The Tc on both sides cancels out, leaving us with:
365.26 = Tc / 687
Simplifying further:
Tc ≈ 687 * 365.26
Therefore, the time between two closest approaches is approximately 251,337.62 days.
c) The angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations can be found by subtracting the angle covered by both Earth and Mars from 2π (a full rotation).
From the previous calculations:
θE = (2π / 365.26) * Tc ≈ 2π - θM
Substituting the values:
(2π / 365.26) * Tc ≈ 2π - (2π / 687) * Tc
Solving for Tc:
(2π / 365.26) * Tc + (2π / 687) * Tc ≈ 2π
Simplifying further:
Tc * [(2π / 365.26) + (2π / 687)] ≈ 2π
Solving for Tc:
Tc ≈ (2π) / [(2π / 365.26) + (2π / 687)]
Therefore, the angle between the line drawn through the Sun, Earth, and Mars in two closest approach situations is approximately (2π) / [(2π / 365.26) + (2π / 687)] radians.