Solution

Statement

$2.2.7.$ During observations from Earth, it is possible to determine only the radial velocity of partner stars that are part of a binary star (i.e., the projection of the velocity onto the Earth-star line). Upon measurements, the values of the radial velocity $v_1$ and $u_1$ of the partner stars of the binary star were obtained. During repeated measurements taken a year later, the values of this velocity turned out to be $v_2$ and $u_2$. Find the ratio of the masses of the partner stars comprising this binary star. Why is it necessary to change the calculations if the repeated measurement is taken after a month or half a year?

Solution

From the law of conservation of momentum $\vec{p}_1 = \vec{p}_2$:
$$m_1 v_1 + m_2 u_1 = m_1 v_2 + m_2 u_2$$
$$m_1/m_2 = (u_2 - u_1)/(v_1 - v_2)$$

If measuring not after a year, it is necessary to introduce a correction for the change in the position and velocity of the Earth, since the radial velocities are relative to the velocity of the Earth.