Only need help with part c
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 part (c), we need to find the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations.
Let's start by finding the time it takes for Mars to complete one orbit around the Sun. We are given that Mars has a period of 687 days. Since Mars completes one orbit in this time, we can say that the angle it sweeps in one orbit is 2π radians.
Next, let's find the time it takes for Earth to complete one orbit around the Sun. We are given that Earth has a period of 365.26 days. Similar to Mars, Earth completes one orbit in this time, so the angle it sweeps in one orbit is also 2π radians.
Now, let's consider the closest approach situation. At this point, Earth and Mars lie on a straight line between Mars and the Sun. Since the angles between the Earth-Sun line and the Mars-Sun line are equal, the angle between the line drawn through the Sun, Earth, and Mars is the sum of the angles Earth sweeps in its orbit and Mars sweeps in its orbit during the closest approach period.
To find the time between two closest approaches, we need to find the least common multiple (LCM) of the periods of Earth and Mars. The LCM of 365.26 days and 687 days is the time it takes for both planets to be in a closest approach situation again.
Now that we have the LCM, let's find the angle between the line drawn through the Sun, Earth, and Mars. To do this, we divide the LCM by the period of Mars and multiply by the angle Mars sweeps in one orbit. This will give us the angle Mars has swept during the closest approach period.
Finally, the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations is the sum of the angles Earth and Mars have swept during the closest approach period.
Let's go ahead and calculate the value of the angle between the line drawn through the Sun, Earth, and Mars in part (c) using the given values.