How do you calculate the distances of the perihelion and aphelion if you know that the period is 76 years and the semi major axis is 17.94 AU?

Question

Owen Atkinson · Accepted Answer

You can't as you are missing either the semi-minor axis or eccentricity.

Kepler's 3rd law relates the period to the semi major axis as #T^2 = a^3#. Now given the figures #75^2# does equal #17.94^3#. In fact we only require one of these values as the other can be calculated from it. The semi-major and semi-minor axes can be used to calculate the orbit's eccentricity, which is necessary to calculate perihelion and aphelion.

Claire Anctil · Answer

undefined The formula for calculating the perihelion distance, which is the closest point to the Sun, is perihelion distance = semi-major axis × (1 - Eccentricity). Similarly, the formula for calculating the aphelion distance, which is the farthest point from the Sun, is aphelion distance = semi-major axis × (1 + Eccentricity). Using the period (P) and semi-major axis (a), you can use Kepler's third law to calculate the eccentricity (e): $P^2 = a^3$. Finally, enter the eccentricity into the formulas to obtain the values.