I'm a teaching assistant for a class on Newtonian mechanics, and was confronted with a way of solving for the larger mass in this two-star system that gets the right symbolic solution, but seems to forget the factor of $\frac{1}{2}$ for kinetic energy.
The problem gives a distance between the two stars r, a velocity v, and states that the larger has such a large mass that the orbit of the smaller star is nearly circular.
Here's their solution:
$$U_g = E_k$$
Gravitational and kinetic energy must be equal if the star is in orbit.
$$U_g = -G \frac{m_1m_2}{r}$$
$$E_k = mv^2$$
This is missing its factor of $\frac{1}{2}$ but we ignore this and set the two energies equal to each other and solve.
$$m_2v^2 = -G \frac{m_1m_2}{r}$$
We will ignore the minus sign on gravitational potential energy, and $m_2$ cancels out.
$$v^2 = G \frac{m_1}{r}$$
We solve for $m_1$ and get:
$$\frac{v^2r}{G} = m_1$$
Which happens to be the same symbolic solution that is in the answer key.
How is this correct when kinetic energy is equal to $\frac{1}{2} mv^2$ ?