Almost everything we know about the masses of stars comes from binaries, pairs of stars in orbit around each other. In a visual binary a telescope separates the two stars, and over years or decades you can watch one trace an ellipse around the other. In Binary Stars and Stellar Masses you saw the idea; here you will do the full calculation, using Kepler's third law from Newton.
Why it works
Newton's version of Kepler's third law for two bodies is $$P^2 = \frac{4\pi^2a^3}{G(M_1 + M_2)},$$ where $a$ is the semi-major axis of the relative orbit (one star around the other). For Earth and the Sun the same law gives $(1\text{ yr})^2 = 4\pi^2(1\text{ AU})^3/(GM_\odot)$. Divide one equation by the other, and $4\pi^2$ and $G$ cancel: with $a$ in AU, $P$ in years and masses in solar masses, $M_1 + M_2 = a^3/P^2$. The catch is that a telescope measures the orbit as an angle $\alpha$ in arcseconds. The parsec is defined so that 1 AU seen from 1 pc spans exactly 1″, so the small-angle rule becomes $a\,(\text{AU}) = \alpha\,('') \times d\,(\text{pc}) = \alpha/p$.
Key idea
Convert the angle first, then weigh: $$a = \frac{\alpha}{p}\ \text{AU}, \qquad M_1 + M_2 = \frac{a^3}{P^2}\ M_\odot.$$ Each star circles the center of mass, so $M_1a_1 = M_2a_2$: the heavier star moves on the smaller ellipse.
Worked examples
Worked example: Weighing Sirius
Sirius A and its white dwarf companion Sirius B have a relative orbit with angular semi-major axis $7.50''$ and period $50.1$ years. The parallax of Sirius is $0.379''$. Find the total mass.
- Semi-major axis: $a = 7.50/0.379 = 19.8$ AU.
- Kepler: $M_1 + M_2 = a^3/P^2 = 19.8^3/50.1^2$.
- $= 7740/2510 \approx 3.09$.
Answer: About $3.1\,M_\odot$ for the pair.
Worked example: Splitting the mass
Measured against the background stars, Sirius B's orbit around the center of mass is about twice as large as Sirius A's. Find each mass.
- $M_A a_A = M_B a_B$, so $M_A/M_B = a_B/a_A = 2.0$.
- $M_A + M_B = 3.09$ and $M_A = 2.0\,M_B$, so $3.0\,M_B = 3.09$.
- $M_B = 1.03\,M_\odot$ and $M_A = 2.06\,M_\odot$.
Answer: Sirius A has about $2.1\,M_\odot$; Sirius B, only Earth-sized, has about $1.0\,M_\odot$.
Worked example: A pair like our neighbors
The two bright stars of Alpha Centauri orbit each other with $a = 23.4$ AU and $P = 79.9$ yr. What is their total mass?
- $a^3 = 23.4^3 = 12{,}800$ and $P^2 = 79.9^2 = 6380$.
- $M_1 + M_2 = 12{,}800/6380 \approx 2.0$.
Answer: About $2.0\,M_\odot$, two stars much like the Sun.
Common mistake
Putting the angle straight into Kepler's law. $\alpha = 7.50''$ is not 7.50 AU; Kepler's law needs a length. Using arcseconds as if they were AU here would give $7.5^3/50.1^2 = 0.17\,M_\odot$ instead of $3.1\,M_\odot$. Convert with $a = \alpha/p$ (or $\alpha \times d$ in parsecs) first. Also use the relative orbit, $a = a_1 + a_2$, not one star's orbit around the center of mass.
This is how the mass–luminosity relation was first measured: binaries give masses, and the same stars' distances and brightness give luminosities.