Neutron star or black hole?

Adapted from IOAA 2015 (Magelang), theory problem L2. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Two stars A and B, of masses $m_{\text{A}}$ and $m_{\text{B}}$, are a distance $d$ apart and move on circular orbits round their centre of mass, in the $xy$ plane with the centre of mass at the origin.

(a) Find the speed and the angular speed of star A. [30%]

(b) An observer far away in the $yz$ plane looks at the stars along a direction at an angle $\theta$ to the $z$ axis. The velocity of A along the line of sight is $K\cos (\omega t+\epsilon )$, with $K$ and $\epsilon$ positive. Express $K^3/(\omega G)$ in terms of $m_{\text{A}}$, $m_{\text{B}}$ and $\theta$. [30%]

(c) Star A has a mass of 30 solar masses. Star B gives out X-rays and is a neutron star if its mass is less than 2 solar masses, or a black hole if it is more. The observer measures $K^3/(\omega G)=M_{\odot }/250$. The angle $\theta$ is not known. For which values of $\theta$ is B a black hole? [40%]

Answer(a) $v_{\text{A}}=m_{\text{B}}\sqrt{G/((m_{\text{A}}+m_{\text{B}})d)}$ and $\omega =\sqrt{G(m_{\text{A}}+m_{\text{B}})/d^3}$. (b) $K^3/(\omega G)=m_{\text{B}}^3{\sin }^3\theta /(m_{\text{A}}+m_{\text{B}}{)}^2$. (c) B is a black hole if $\sin \theta$ is less than 0.8, that is for $\theta$ less than 53.1° or more than 126.9°.