18.2 Measuring Stellar Masses

Binary Star Systems and Stellar Mass Measurement

Types of Binary Star Systems

You can't just put a star on a scale, so how do astronomers figure out how massive stars are? The answer is binary star systems: pairs of stars orbiting each other under mutual gravity. By studying their orbits, astronomers can calculate stellar masses directly. About half of all star systems in our galaxy are binaries or multiples, so there's no shortage of targets.

There are three main types, classified by how we detect them:

• Visual binaries are pairs where both stars can be seen separately through a telescope. Albireo in Cygnus is a classic example. Because you can watch them orbit over years or decades, you can measure the orbital size and period directly, then plug those into Newton's version of Kepler's third law to get the combined mass.
• Spectroscopic binaries are too close together to resolve visually, even with powerful telescopes. Mizar A in Ursa Major is one example. We detect them through periodic Doppler shifts in their spectral lines: as each star moves toward and away from us during its orbit, its light shifts slightly bluer or redder. Measuring these radial velocities lets astronomers work out orbital properties and estimate masses. The spectra also reveal each star's temperature and composition.
• Eclipsing binaries have orbits tilted so that, from Earth's perspective, one star periodically passes in front of the other. Algol in Perseus is the best-known example. The resulting dips in brightness produce a light curve that reveals the stars' relative sizes and orbital geometry. When combined with spectroscopic data, eclipsing binaries let astronomers determine individual masses and radii with high precision.

Mass Calculation in Binary Stars

The fundamental tool here is Newton's version of Kepler's third law:

$\frac{a^3}{P^2} = \frac{G(M_1 + M_2)}{4\pi^2}$

• $a$ = semi-major axis of the orbit
• $P$ = orbital period
• $G$ = gravitational constant
• $M_1$ and $M_2$ = masses of the two stars

When $a$ is measured in AU, $P$ in years, and masses in solar masses, the equation simplifies nicely to:

$a^3 = (M_1 + M_2) \times P^2$

Here's how astronomers use this in practice for a visual binary:

1. Observe the system over time to determine the orbital period $P$ (this can take years or even decades for wide pairs).
2. Measure the angular separation between the two stars on the sky.
3. Determine the distance to the system, often using parallax. Combining angular separation with distance gives the physical size of the orbit, $a$.
4. Plug $a$ and $P$ into the equation and solve for the combined mass $(M_1 + M_2)$.

To find individual masses, you need additional information. If you can track how much each star moves relative to the center of mass (from visual or spectroscopic data), the star that moves more is the less massive one. The ratio of their distances from the center of mass is inversely proportional to the ratio of their masses.

Mass-Luminosity Relationship

Binary star observations have revealed one of the most important patterns in stellar astronomy: more massive main-sequence stars are dramatically more luminous. This is the mass-luminosity relationship:

$L \approx M^{3.5}$

where $L$ is luminosity in solar luminosities ($L_{\odot}$) and $M$ is mass in solar masses ($M_{\odot}$).

This is a steep relationship. A star with 10 times the Sun's mass doesn't shine 10 times brighter; it shines roughly $10^{3.5} \approx 3{,}162$ times brighter. Compare Rigel (about 18 solar masses, luminosity ~120,000 $L_{\odot}$) to Proxima Centauri (about 0.12 solar masses, luminosity ~0.0017 $L_{\odot}$).

Why does this happen? More massive stars have stronger gravitational compression in their cores, which drives core temperatures higher and nuclear fusion rates way up. The result is far more energy output.

This relationship also works in reverse: if you measure a main-sequence star's luminosity (from its apparent brightness and distance), you can estimate its mass without needing a binary companion.

Stellar Evolution and Mass

Mass is the single most important property determining a star's fate. It controls:

• Surface temperature: More massive stars are hotter, which is why the most massive stars are spectral types O and B, while the least massive are type M.
• Size: More massive main-sequence stars are generally larger in radius.
• Lifetime: This is the counterintuitive part. Even though massive stars have more fuel, they burn through it so much faster that they live shorter lives. A star with 25 solar masses might last only a few million years, while a 0.5 solar mass star can shine for over 100 billion years.
• Evolutionary path: High-mass stars end their lives as supernovae, leaving behind neutron stars or black holes. Low-mass stars like the Sun shed their outer layers gently and become white dwarfs.

Stars also lose mass over their lifetimes through stellar winds, and this mass loss can alter their evolutionary path, especially for the most massive stars.