How can the orbital semi-major axis, eccentricity, and period be calculated if perihelion and aphelion are known?

To calculate the orbital semi-major axis, eccentricity, and period of an orbit when you know the perihelion and aphelion distances, you can follow these steps:

Semi-Major Axis Calculation

The semi-major axis (a) is the average of the perihelion (q) and aphelion (Q) distances. You can calculate it using the formula:

a = (q + Q) / 2

Eccentricity Determination

Eccentricity (e) measures how much the orbit deviates from being circular. It can be found using the formula:

e = (Q - q) / (Q + q)

Orbital Period Calculation

The orbital period (T) can be calculated using Kepler's Third Law, which relates the period to the semi-major axis. The formula is:

T = 2π * √(a³ / μ)

Here, μ is the standard gravitational parameter, which is the product of the gravitational constant (G) and the mass of the central body (M).

Example Calculation

• If perihelion (q) = 1 AU and aphelion (Q) = 3 AU:
• Calculate semi-major axis: a = (1 + 3) / 2 = 2 AU
• Calculate eccentricity: e = (3 - 1) / (3 + 1) = 0.5
• Assuming μ = 1 AU³/year², calculate period: T = 2π * √(2³ / 1) = 2π * √(8) ≈ 17.8 years

By following these steps, you can effectively determine the key orbital parameters from the perihelion and aphelion distances.