# Two stars orbiting each other are separated by 10^9km and revolve around their common centre of gravity in 10 years. what is the combined mass of the 2 stars in solar mass?

## Expert Answers

Hello!

Kepler's Third law helps us here. It states, in the general form, that for two bodies orbiting each other

`T^2/a^3=(4pi^2)/(G(m_1+m_2)),`

where `T` is the period, `a` is the (mean) distance, `m_1` and `m_2` are the masses and `G` is the gravitational constant.

We'll use this law twice, for the given system of two stars and for the system Earth+the Sun. For the latter we can neglect the mass of Earth. So we have

`(T_s^2/a_s^3)*M=(T_E^2/a_E^3)*m_S,`

where `T_s` is the given stars' period, `a_s` is the distance between stars, `M` is their combined mass which we have to find, `T_E` is the Earth's period (1 year), `a_E=1.5*10^8 km` is the distance between Earth and the Sun, and `m_S` is the mass of the Sun.

Actually we are asked to find `M/m_S` which is equal to

`(T_E/T_s)^2*(a_s/a_E)^3 = ((1.5*10^8)/10^9)^2*10^3=22.5` (times, dimensionless).

So the answer is: the combined mass of the 2 stars is 22.5 solar masses.

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