Question

Mars orbits the Sun at a mean distance of 228 million \(\mathrm{km},\) in a period of 687 days. The Earth orbits at a mean distance of 149.6 million \(\mathrm{km}\), in a period of 365.26 days. 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? b) The initial situation in part (a) is a closest approach of Mars to Earth. What is the time, in days, between two closest approaches? Assume constant speed and circular orbits for both Mars and Earth.

Short answer

Question: Calculate the angle between the Earth-Sun line and the Mars-Sun line after Earth completes one orbit (365.26 days) and compute the time in days between two closest approaches of Mars to Earth, assuming constant speed and circular orbits for both planets. Answer: After Earth completes one orbit, the angle between the Earth-Sun line and the Mars-Sun line is approximately \(\Delta \theta\) radians. The time between two closest approaches of Mars to Earth is approximately \(t\) days.

Step-by-step solution

(Step 1: Calculate the angular speeds of Earth and Mars)

(To calculate the angular speeds, use the orbital period information to compute how much angle each planet covers in a day. Let \(\omega_E\) and \(\omega_M\) represent the angular speeds of Earth and Mars, respectively. Use the formula: \(\omega = \frac{2\pi}{T}\), where \(T\) is the orbital period in days. Calculate \(\omega_E = \frac{2\pi}{365.26}\) and \(\omega_M = \frac{2\pi}{687}\).)

(Step 2: Find the angle covered by each planet in 365.26 days)

(Now, using the angular speeds of Earth and Mars computed in Step 1, find how much angle each planet covers in 365.26 days. For Earth, the angle is one complete orbit, which is \(360^\circ\) or \(2\pi\) radians. For Mars, use the formula: angle = angular speed × time, and calculate the angle covered by Mars in 365.26 days, which is \(\theta_M = \omega_M \times 365.26\).)

(Step 3: Calculate the angle between Earth-Sun line and Mars-Sun line)

(After 365.26 days, the angle between the Earth-Sun line and the Mars-Sun line is equal to the difference of angles covered by Earth and Mars. Earth has completed one orbit, so it has covered \(2\pi\) radians. Calculate the angle difference: \(\Delta \theta = 2\pi - \theta_M\).)

(Step 4: Calculate the speed of Earth and Mars in their orbits)

(In order to find the time between two closest approaches, we first need to calculate the speeds of Earth and Mars in their orbits. We can do this by using their mean distances from the Sun, \(r_E = 149.6\times10^6\) km and \(r_M = 228\times10^6\) km, and their orbital periods. The speed of a planet in its orbit is given by the formula: \(v = \frac{2\pi r}{T}\). Calculate \(v_E = \frac{2\pi \times 149.6\times10^6}{365.26}\) and \(v_M = \frac{2\pi \times 228\times10^6}{687}\).)

(Step 5: Determine the time between two closest approaches)

(To find the time between two closest approaches, we need to determine how long it takes for Earth to "catch up" to Mars in their orbits. The relative speed between the two planets is \(\Delta v = v_E - v_M\). In other words, this is the speed at which Earth "catches up" to Mars. When Earth and Mars are at a closest approach, the angle difference between them is \(\Delta \theta\). Thus, the distance between two closest approaches, measured along Earth's orbit, is \(\Delta s = r_E \times \Delta \theta\). Finally, use the formula time = distance/speed, and calculate the time between two closest approaches: \(t = \frac{\Delta s}{\Delta v}\).)