How do you calculate the aphelion and perihelion of a comet's orbit if it has an orbital period of 83 years and an eccentricity of .875?

Question

Asher Cohen · Accepted Answer

Using Kepler's 3rd law the perihelion distance is about 2.375AU and the aphelion distance is about 35.625AU.

The comet's orbit can be roughly calculated using Kepler's third law; the gravitational pull of other bodies in the solar system prevents the comet's actual orbit from being a perfect ellipse. Kepler's 3rd law states that the square of the orbital period divided by the cube of the cube of the semi-major axis of the orbit is a constant. For our solar system if the period T is in years and the semi-major a axis is in Astronomical Units, then the constant is one. Then #T^2=a^3#. The Astronomical Unit is based on the distance of the Earth from the Sun. Earths period #T=1# year which means that Earth's semi major axis is #a=1# AU. For the comet with a period of #t=83# years then the semi major axis #a^3=83^2#. Squaring and taking the cube root gives approximately #a=19# AU. The perihelion distance #P=a(1-e)# and the aphelion distance #A=a(1+e)# where #e=0.875# is the eccentricity. This gives a perihelion distance of 2.375AU and an aphelion distance of 35.625AU.

Levi Anderson · Answer

undefined The semi-major axis, $ a $ can be calculated using Kepler's third law as follows: $$ T^2 = a^3 $$
$$ a = \left(\frac{T^2}{P^2}\right)^{\frac{1}{3}} $$

The perihelion distance, $ q $, can be calculated using the following formulas: $$ q = a \times (1 - e) $$
 Finally, by substituting the values into the formulas for $ Q $ and $ q $, we can determine the aphelion and perihelion distances.