In the center of mass coordinate, $m_1 r_1 = m_2 r_2$, which is straightforward. Yet in this detailed deviation of radial velocity page 27, it says that the $r_1$, which is the magnitude of the vector pointing from the CM to the star, is simply the semi‐major axis of the star’s orbit around the mutual CM, $a_1$. With the same reasoning, the $r_2$, which is the magnitude of the vector pointing from the CM to the planet, is the semi‐major axis of the planet’s orbit around the mutual CM, $a_2$. Therefore $m_1 a_1 = m_2 a_2$.
However, I do not see why $r_1 (r_2)$ can be identical to $a_1(a_2)$ since the $r's$ are both changing (in a elliptical orbit for example) while the $a's$ are fixed. Or the other way to ask this question is I do not see why $r_1/a_1 = r_2/a_2$?