I know that the mass of a binary star system is given by Kepler's Law: $$\mathrm{m_1 + m_2 = \frac{4 \pi^2 r^3}{GT^2}}$$
Further we know that: $$\frac{r_2}{r_1} = \frac{v_2}{v_1} = \frac{m_1}{m_2}$$
Therefore if we are able to determine the period and velocity of the stars, we can then determine their mass. The period of the stars can be easily determined by the period of splitting of the spectroscopic binary's spectral lines. Also, it is possible to determine the velocity of the stars by the extent of red-shift/blue-shift of the spectral lines.
However, what if the binary stars weren't orbiting in a plane parallel to the observer, but rather on an angle? Can the velocity of the binary stars still be determined, and hence can its mass still be determined?
If this is not possible, is there any other means in which their mass can be determined?
Best Answer
In general, yes you need to know the orbital inclination angle $i$ in order to fully solve the orbit. The radial velocity amplitude $K$ is just modified to $K \sin i$ (where $i=0$ is a face-on orbit). Combining this with the orbital period and Keplerian orbits gives you the "mass function" $$ \frac{M_1^3 \sin^3 i}{\left(M_1 + M_2\right)^2} = \frac{K_{2}^3 \sin^3 i\ P_{orb}}{2\pi G},$$ where the right hand side can be measured from radial velocity data in a spectroscopic binary. If you have a velocity amplitude for both stars, then there is a similar expression with the labels reversed. Without $i$ this can then only tell you the mass ratio $M_1/M_2$.
There are several ways to break this degeneracy depending on what kind of binary system it is.
In a visual binary system where you can observe the orbits, then the orbital path of both objects can be observed and the inclination of the orbit is directly measured. However, radial velocity amplitudes are not usually measurable (too small) and one relies on the absolute size of the orbit, which in turn requires a distance (parallax) estimate.
In an eclipsing binary, then the shape and depth of the eclipses can be uniquely solved to give the inclination and hence the masses of the individual stars.
In non-eclipsing close binary systems, or when one component is not seen, then ellipsoidal modulation of the seen component depends on the mass ratio and the inclination. Together with the radial velocity curve, this can then give unique masses for the components.
In general it is not possible to get any more than a mass ratio for the components of a double lined spectroscopic binary system (SB2), or the "mass function" (see above) of a single lined spectroscopic binary system (SB1).
To make further progress in these general cases you need an estimate of the primary mass. This can be done with reference to stellar evolutionary models. In principle, for an SB2, the mass ratio and the combined appearance of an object in the Hertzsprung-Russell diagram contain enough information to determine the masses of the individual components and the age of the system. In practice this is hard and there are degeneracies. A better way is to fit a combination of spectral type templates to the measured spectrum and hence estimate the spectral types and hence masses.
In an SB1 you really are stuck. The spectral type and position in the HR diagram give you $M_1$, but you will only have a lower limit to the unseen secondary mass. This is why it is difficult to estimate the masses of black holes in binaries - you need to know the inclination.