About half of the stars like the Sun have at least one companion star. Pairs bound by gravity, binary stars, are a gift: by weighing with orbits they give the masses of stars, the one property we cannot read from light alone. And when those masses are put on the H–R diagram, they explain the main sequence.
Why it works
The two stars orbit their common center of mass, the balance point between them. Like children on a seesaw, the heavier star sits closer: $M_1 r_1 = M_2 r_2$. Both go around in the same time, so each one's speed is proportional to its distance from the center: $\frac{M_1}{M_2} = \frac{r_2}{r_1} = \frac{v_2}{v_1}$. Newton's form of Kepler's third law gives their total mass from the size $a$ (in AU) and period $P$ (in years) of their orbit: $M_1 + M_2 = \frac{a^3}{P^2}$. With the sum and the ratio, each mass follows. The speeds come from Doppler shifts; the sizes from watching the stars move across the sky, or from eclipses when the orbit is edge-on.
Key idea
For a binary: $M_1 + M_2 = \dfrac{a^3}{P^2}$ (solar masses, AU, years) and $\dfrac{M_1}{M_2} = \dfrac{r_2}{r_1} = \dfrac{v_2}{v_1}$. Binaries show that the main sequence is a sequence of mass: luminosity rises roughly as $M^{3.5}$.
Worked examples
Worked example: Sirius A and B
Sirius B orbits Sirius A with $a = 19.8$ AU and $P = 50.1$ years. Find their total mass. Sirius A stays about half as far from the center of mass as Sirius B; find each mass.
- Total: $\frac{19.8^3}{50.1^2} = \frac{7762}{2510} = 3.09$ solar masses.
- Seesaw: $r_B = 2r_A$, so $M_A = 2M_B$.
- $M_A + M_B = 3M_B = 3.09$, so $M_B = 1.03$ and $M_A = 2.06$ solar masses.
Answer: About $3.1\,M_\odot$ in all: $2.06$ and $1.03$.
Worked example: From Doppler speeds
In a binary, star A moves at $20$ km/s and star B at $60$ km/s around the center of mass. How do their masses compare?
- $\frac{M_A}{M_B} = \frac{v_B}{v_A} = \frac{60}{20} = 3$.
- Star A, the slower one, is three times as massive.
Answer: Star A has three times the mass of star B.
Common mistake
It is tempting to read the main sequence as a life story, with stars sliding down it as they age. Binary masses show something else: along the main sequence, mass changes. A $10\,M_\odot$ star is thousands of times more luminous than the Sun, and a $0.2\,M_\odot$ red dwarf less than a hundredth as luminous. A star stays near one spot on the main sequence for most of its life. Check: in binaries, the brighter main-sequence star is always the more massive one.