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

Question

Lydia Allen · Accepted Answer

There is not enough information to calculate the apses distances.

The period #P# and semi-major axis #a# are not sufficient to calculate the apses distances as the eccentricity of the orbit can't be determined. If the eccentricity #e# is known then the perihelion distance is #a(1-e)# and the aphelion distance is #a(1+e)#. In fact the period and semi-major axis can be calculated from each other using Kepler's third law. For our solar system if #P# is in years and #a# is in AU then #P^2=a^3#.

Hailey Alexander · Answer

undefined To calculate the distances of the perihelion and aphelion for an orbit with a known period and semi-major axis:

1. Use Kepler's third law to find the orbit's major axis length $ a $.
$$ T^2 = \frac{{4\pi^2}}{{GM}} a^3 $$
Where:
$ T $ = period in seconds
$ G $ = gravitational constant (approximately $ 6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} $)
$ M $ = mass of the central body (usually the Sun)

2. Calculate the eccentricity $ e $ of the orbit using the formula:
$$ e = \sqrt{1 - \frac{{b^2}}{{a^2}}} $$
Where:
$ b $ = semi-minor axis length

3. Determine the distance of the perihelion ($ r_p $) and aphelion ($ r_a $) using the formulas:
$$ r_p = a(1 - e) $$
$$ r_a = a(1 + e) $$

Substitute the values of $ a $ and $ e $ calculated from steps 1 and 2 into the respective formulas to find $ r_p $ and $ r_a $.