Weighing Stars

Section 1 of 4

The Hidden Variable

By the end of this reading, you will be able to:

Explain
why mass is the most fundamental stellar property and why it cannot be measured from a single, isolated star’s light alone
Describe
visual, spectroscopic, and eclipsing binary systems and explain what each reveals
Apply
Newton’s version of Kepler’s third law to determine the total mass of a binary
Use
the center-of-mass condition and radial-velocity amplitudes to determine individual masses
Interpret
the mass-luminosity relation ( $L \propto M^{3.5}$ ) and explain why it makes mass the master variable for main-sequence stars
Connect
mass measurement to the full Module 2 inference chain

Mass is the most important thing about a star — and the one thing you can’t see. Every other property — luminosity, temperature, radius, lifetime, and how a star dies — follows from mass. But mass leaves no direct imprint on a star’s light. To weigh a star, you need to catch it in a gravitational dance with a partner. Binary stars are nature’s gift to astronomers: two bodies orbiting under gravity, revealing their masses through the physics you already know.

The Big Idea

Mass determines a star’s fate — but you can’t weigh a star from its light alone.

In Lectures 1–3, you learned to measure a star’s distance, luminosity, temperature, radius, and composition from photons. But none of these tells you the most consequential property: mass. A star’s mass determines how brightly it shines, how hot its surface is, how long it lives, and how it dies. Yet mass is invisible — no spectral line, no color, no brightness measurement reveals it directly.

What you’ll gain: after this reading, you’ll know how astronomers exploit gravitational physics — binary star orbits — to measure the one property that controls all the others. And you’ll see that the resulting mass-luminosity relation is the most important empirical relation in stellar astrophysics.

Part 1: The Hidden Variable

Why Mass Matters Most

Consider two stars on the main sequence — the band where most stars spend most of their lives. One has mass $M = 0.5\,M_\odot$ ; the other $M = 10\,M_\odot$ . How do they compare?

Main sequence

The band on the H–R diagram where stars spend most of their lives in stable hydrogen-core fusion. A main-sequence star’s position is set almost entirely by its mass.

Property	$0.5\,M_\odot$ star	$10\,M_\odot$ star	Ratio
Luminosity	$\sim 0.09\,L_\odot$	$\sim 3 \times 10^3\,L_\odot$	$\sim 3 \times 10^4$
Surface temperature	$\sim 3{,}800\,\text{K}$	$\sim 2.5 \times 10^4\,\text{K}$	$\sim 7$
Radius	$\sim 0.5\,R_\odot$	$\sim 6\,R_\odot$	$\sim 12$
Main-sequence lifetime	$\sim 60\,\text{Gyr}$	$\sim 20\,\text{Myr}$	$\sim 3 \times 10^3$
Spectral type	M	B	—
Death	White dwarf	Core-collapse supernova	—

A factor of 20 in mass increases luminosity by $3 \times 10^4$ , cuts lifetime by $3 \times 10^3$ , and leads to completely different endpoints. Mass is the master variable. Change the mass, and everything else follows.

Why Mass Is Hidden

So why can’t we just “read” mass from a star’s spectrum, the way we read temperature from spectral type or composition from line wavelengths?

The problem is fundamental: mass affects a star’s light only indirectly, through its influence on internal structure. A spectrum encodes surface properties — temperature, composition, surface gravity. Mass determines those, but the mapping from mass to surface isn’t unique without a physical model connecting them. A red giant and a red dwarf can have similar surface temperatures but wildly different masses.

Put differently: luminosity, temperature, and composition are observables — encoded directly in the photons. Mass is a derived quantity — you need a physical model (or a dynamical measurement) to get it. The solution is gravity: if a star has a gravitational companion, orbital dynamics gives the mass directly, through physics you already know from Module 1.

Quick check

Recall from Module 1 (Lecture 3): Newton showed that Kepler’s third law contains the mass of the central body. For a planet orbiting the Sun we used $P^2 = 4\pi^2 r^3 / GM$ . What changes when both objects have comparable mass — like two stars orbiting each other?