Question

Halleys comet has period of 75.3 years. Using Kepler’s third law, find it’s semimajor axis expressed in astronomical units?

Answer 1

Answer: 17.83 AU

Explanation:

According to Kepler’s Third Law of Planetary motion “The square of the orbital period of a planet is proportional to the cube of the semi-major axis (size) of its orbit”.  

$T^{2}\propto a^{3}$  (1)

Talking in general, this law states a relation between the orbital period $T$ of a body (moon, planet, satellite, comet) orbiting a greater body in space with the size $a$ of its orbit.

However, if $T$ is measured in years, and $a$ is measured in astronomical units (equivalent to the distance between the Sun and the Earth: $1AU=1.5(10)^{8}km$), equation (1) becomes:

$T^{2}=a^{3}$  (2)

This means that now both sides of the equation are equal.

Knowing $T=75.3years$ and isolating $a$ from (2):

$a=\sqrt[3]{T^{2}}=T^{2/3}$  (3)

$a=(75.3years)^{2/3}$  (4)

Finally:

$a=17.83AU$  (5)