Question

A resonance orbit is one in which planets have a fixed ratio of orbital periods. For example, for 2 planets with an orbital resonance of 3:1, the outer planet's orbital period would be 3 times that of the inner one. That is, the inner planet can complete 3 exact orbits whilst the outer one completes only one. This is a special type of orbital "locking". The two planets orbiting the nearby star are observed to be in a 2:1 resonance (i.e., the period of one is twice that of the other). The inner planet has an orbital period of 76 days. If the star's mass is the mass of the Sun, calculate the semi-major axis of the outer planet's orbit. Answer in AU.

Answer 1

The relation between the semi-major axis and the period of revolution is given by

The period of the inner planet is

And the period of the outer planet is

The ration between the periods is given as

Squaring both sides

The mass of the sun is

And the period of the inner planet is

Putting the values in the equation

So, the semi-major axis of the outer planet's orbit is 5.31 x 10¹³ m.