In a binary system, how is the mass ratio $\frac{M_1}{M_2}$ determined based on the distances from the center of mass ($a_1$ and $a_2$)?
Answer
The ratio is equal to the inverse ratio of the separations, $\frac{a_2}{a_1}$
The center of mass dictates that the distance each star is from the center is inversely proportional to its mass, leading to the relationship $M_1 a_1 = M_2 a_2$, which rearranges to the mass ratio $\frac{M_1}{M_2} = \frac{a_2}{a_1}$.
Frequently Asked Questions
What system configuration is used by astronomers for the most direct method of weighing stars?What physical law, in its general form, connects the sum of the masses to the orbital period and semi-major axis in a binary system?When using standard astronomical units ($P$ in years, $a$ in AU), what quantity does the equation $(M_1 + M_2) = a^3 / P^2$ yield directly?In a binary system, how is the mass ratio $\frac{M_1}{M_2}$ determined based on the distances from the center of mass ($a_1$ and $a_2$)?If an orbital plane is tilted by an angle $i$ relative to the line of sight, what value represents the observed, foreshortened semi-major axis?What concept arises because the unknown inclination $i$ causes the mass equation to depend on $a^3$?If an astronomer calculates a minimum mass for an unseen companion that exceeds $3 M_ ext{sun}$, what is strongly suggested about that unseen object?How are orbital motions detected in Spectroscopic Binaries?For Eclipsing Binaries, why is the mass calculation generally more direct and accurate?What two conservation laws underpin the physics validating the Newtonian form of Kepler's Third Law in two-body orbits?What observational method produces a light curve by monitoring the total light from a system?