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) To find the angle between the Earth-Sun line and the Mars-Sun line after 365.26 days, we need to calculate the difference in angles for each planet's orbit.
The angle (θ) traveled by a planet in its orbit can be calculated using the formula:
θ = (2π * t) / T
Where:
θ is the angle traveled,
t is the time elapsed,
T is the period of the orbit.
For Earth:
θ_earth = (2π * 365.26) / 365.26 = 2π
For Mars:
θ_mars = (2π * 365.26) / 687 = ~3.339
Now, we need to find the difference in angles between the Earth and Mars:
angle_difference = θ_mars - θ_earth
angle_difference = ~3.339 - 2π
Since we want the angle to be in the range (0, 2π), we need to adjust the angle_difference:
angle_difference = angle_difference % (2π)
angle_difference = (~3.339 - 2π) % (2π)
Let's calculate the value of angle_difference:
angle_difference = (~3.339 - 6.283) % 6.283
angle_difference = ~(-2.944) % 6.283
angle_difference = 3.339
Therefore, the angle between the Earth-Sun line and the Mars-Sun line after 365.26 days is approximately 3.339 radians.
b) The time between two closest approaches occurs when the angles traveled by both planets are equal.
Let's assume that the time between two closest approaches is T_closest.
For Earth:
θ_earth_closest = (2π * T_closest) / 365.26
For Mars:
θ_mars_closest = (2π * T_closest) / 687
Since we want these angles to be equal:
θ_earth_closest = θ_mars_closest
Substituting the equations:
(2π * T_closest) / 365.26 = (2π * T_closest) / 687
Now, we solve for T_closest:
365.26 * (2π * T_closest) = 687 * (2π * T_closest)
365.26 = 687
T_closest = (365.26 * 687) / 365.26
T_closest = 687
Therefore, the time between two closest approaches is approximately 687 days.
c) The angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations can be found using the angle_difference calculated in part a).
Since the angle_difference is the difference between the angles traveled by Earth and Mars when they are in a straight line, it represents the angle between the line drawn through the Sun, Earth, and Mars.
Therefore, the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations is approximately 3.339 radians.
a) To find the angle between the Earth-Sun line and the Mars-Sun line, we need to consider the positions of Earth, Mars, and the Sun after exactly 365.26 days.
Given that the Earth orbits the Sun in a period of 365.26 days, it means that after this time period, the Earth will complete one orbit and return to its starting position.
Since Earth lies on a straight line between Mars and the Sun initially, after one orbit, the angle between the Earth-Sun line and the Mars-Sun line will be the same as the angle between the Earth-Mars line and the Sun-Mars line.
To calculate this angle, we can use the concept of angular displacement. The angular displacement is given by the formula:
angular displacement = (time * angular velocity) + initial angle
In this case:
- The time is 365.26 days
- The angular velocity is the rate at which Mars moves in its orbit, which can be calculated as 2pi radians divided by the period (687 days)
- The initial angle is 0 radians since the initial position is a straight line
Plugging in these values, we can calculate the angular displacement:
angular displacement = (365.26 * (2pi / 687)) + 0
Simplifying, we find:
angular displacement ≈ 1.0609 radians
Therefore, the angle between the Earth-Sun line and the Mars-Sun line after one orbit is approximately 1.0609 radians.
b) The time between two closest approaches can be found by considering that the angle between the Earth-Sun line and the Mars-Sun line is equal in these situations.
Let's assume that the Earth and Mars start at a closest approach position. The time it takes for Earth to complete one orbit is 365.26 days, as given. For Mars, we need to find the time it takes to complete one orbit.
The period of Mars' orbit is given as 687 days. Since the angle covered in one orbit is 2pi radians, we can set up the following proportion to find the time it takes for Mars to complete one orbit:
365.26 days / 2pi radians = T / (2pi) radians
Solving for T (the time it takes for Mars to complete one orbit), we find:
T ≈ (365.26 * 2pi) / (2pi) ≈ 365.26 days
Therefore, it takes approximately 365.26 days for Mars to complete one orbit.
Since both Earth and Mars take the same amount of time to complete one orbit, and assuming that they start at a closest approach position, the time between two closest approaches would be the time it takes for Earth to complete one orbit, which is 365.26 days.
c) To express the answer to part (b) 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 that angle.
In part (b), we found that the time between two closest approaches is 365.26 days, which corresponds to one orbit for Earth.
Since Earth orbits the Sun at a distance of 149.6 million km, the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations can be found using trigonometry. Specifically, we can use the inverse sine (arcsine) function.
The formula to calculate this angle is:
angle = arcsin(distance / mean distance)
Plugging in the values, we have:
angle = arcsin(149.6 / 228) ≈ 0.6746 radians
Therefore, the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations is approximately 0.6746 radians.