The average distance of the planet Mercury from the sun is 0.39 times the average distance of the earth from the sun. How long is a year on Mercury in units of earth years?

Question

The average distance of the planet Mercury from the sun is 0.39 times the average distance of the earth from the sun. How long is a year on Mercury in units of earth years?

Answer 1

Consider an arbitrary planet rotating around the Sun in a circular orbit.

Let

m be the mass of the planet,

M be the mass of the sun,

r be the radius of the planet's orbit,

T be the period, i.e., length of a year on the planet,

v be the planet's speed,

G be the gravitational constant.

I assume you are familiar with the following identities:

By definition of speed

v = 2πr/T (1)

The gravitational attraction between the Sun the planet is

F = GMm/r² (2)

The centripetal acceleration of the planet is

a = v²/r (3)

Since the gravitational force F is what is responsible for the centripetal acceleration

we can apply Newton's Second Law

F = ma (4)

Substituting (2) and (3) into (4)

GMm/r² = mv²/r (5)

Substituting (1) into (5)

GMm/r² = m4π²r²/T²r (6)

Expressing T² from (6)

T² = 4π²r³/GM (7)

Now let planet 1 be Mercury with radius r₁ and period T₁,

and let planet 2 be the Earth with radius r₂ and period T₂.

And assume that the orbits of the two planets are circular, which is approximately true.

We are given r₁/r₂ = 0.39 and are supposed to calculate T₁/T₂.

Write (7) for both planets

T₁² = 4π²r₁³/GM

T₂² = 4π²r₂³/GM

Divide the two equations

T₁²/T₂² = r₁³/r₂³

Substituting actual numbers

T₁/T₂ = √(0.39³) = 0.24