How will you calculate the mass of a planet?

On the basis of observations it is said that the planet as well as its satellite rotate around a common centre of mass. Let O be the centre. So, both the planet and its satellite form a binary system.

Consider the masses of the satellite and the planet. Suppose M₁ as the mass of the planet and M₂ the mass of the satellite. As we know that the planets are of circular shape. So they must have some radii. So take a₁ and a₂ as the radii of the planet as well as satellite respectively. If M₁ > M₂ then the centre of mass will be shifted towards the mass M₁
The figure is shown below:

According to the definition of centre of mass the equations that will be formed are given below:

M₁ a₁ = M₂ a₂
M₂ / M₁ = a₁ / a₂
M₂ / M₁ +1 = a₁ / a₂ + 1
M₁ + M₂ / M₁ = a₁ + a₂ / a₂ = a /a₂

We can write a₁ + a₂ = a, because the sum of distance of a₁ and a₂ is equal to a as shown in figure.
So substituting the values:
a₂ = aM₁ / M₁ + M₂ —— 4.1

Similar is in case of a1:
a₁= aM₁ / M₁ + M₂ —–4.2

Suppose v₁ and vs₂ are the revolution speeds and T be the Time period of revolution around the centre of mass.
Then the equations formed will be:
v₁ = 2 a₁ / T
Similarly the equation of speed v2 will be:
v₂ = 2 a₂ / T
The force responsible for the circular rotation of satellite is called Centripetal force. In this case, centripetal force is generated due to the gravitational force of attraction of the planet.

GM₁M₂ / a² = M₂v₂² / a₂

The mass of the satellite will be always less than that of planet. So M₁ will be extremely large. In front of planet satellite mass is almost negligible.
Then
M₁ >> M₂
M₁ + M ₂:– > M₁
Therefore the resultant equation will be:

So, the method explained above can be easily used to find the mass of sun, by taking into account a binary system composed of sun and the earth.