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...

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.