The mean distance of Mars from the Sun is 142 million miles. If the perihelion of Mars is 128.5 million miles, what is the aphelion? Find an equation for the orbit of Mars about the Sun.
The correct answer and explanation is :
Mars’s orbit around the Sun is elliptical, characterized by two key distances: perihelion (closest approach) and aphelion (farthest point). The mean distance from Mars to the Sun is approximately 142 million miles, which represents the semi-major axis of its elliptical orbit. Given that the perihelion distance is 128.5 million miles, we can calculate the aphelion distance using the relationship:
Aphelion Distance (r_max) = 2 × Semi-Major Axis (a) – Perihelion Distance (r_min)
Applying the provided values:
r_max = 2 × 142 million miles – 128.5 million miles = 155.5 million miles
This calculation indicates that Mars’s aphelion distance is 155.5 million miles.
To derive an equation for Mars’s orbit, we utilize the standard form of the equation for an elliptical orbit in polar coordinates:
r(θ) = (p) / (1 + ε × cos(θ))
Where:
- r(θ) is the distance from the Sun at a given true anomaly (θ).
- p is the semi-latus rectum, related to the semi-major axis (a) and eccentricity (ε) by:
p = a × (1 – ε²)
- ε is the orbital eccentricity, defined as:
ε = (r_max – r_min) / (r_max + r_min)
Calculating the eccentricity:
ε = (155.5 million miles – 128.5 million miles) / (155.5 million miles + 128.5 million miles) ≈ 0.0934
With ε determined, we can compute p:
p = 142 million miles × (1 – 0.0934²) ≈ 141.3 million miles
Thus, the equation governing Mars’s orbit is:
r(θ) = 141.3 million miles / (1 + 0.0934 × cos(θ))
This equation describes the varying distance between Mars and the Sun at different positions along its orbital path.
Understanding these orbital parameters is crucial for predicting Mars’s position in its orbit at any given time, which has significant implications for mission planning and astronomical observations.