Using Spectra to Measure Stellar Radius, Composition, and Motion

Core Concepts

Composition: reading the elements

Every absorption line is produced by a specific atom or ion, identified by its wavelength pattern, so a spectrum is in principle a chemical inventory. Two cautions apply. First, as in the previous topic, line strength reflects temperature as much as abundance — a cool star shows strong metal lines because its atoms are in the right states, not because it is metal-rich. Second, only the outer layers are sampled; the interior is invisible. The message is nonetheless robust: stars are overwhelmingly hydrogen and helium — commonly taught reference values put hydrogen near 73% of the mass, helium near 25%, with all heavier elements (the "metals") making up a couple of percent. The Sun shows lines of dozens of elements but is still dominated by H and He.

Motion: the Doppler effect as a speedometer

When a star moves along our line of sight, its lines shift: receding → longer wavelengths (redshift); approaching → shorter wavelengths (blueshift). The fractional shift equals the radial velocity (motion along our line of sight) via the commonly taught relation:

v = c × Δλλ

where v is the radial velocity, c the speed of light, λ the rest wavelength, and Δλ the measured shift. A star moving perpendicular to our line of sight shows no shift — the effect senses only the toward/away component; sideways drift (proper motion) is measured as a change of position on the sky. Since laboratory wavelengths of common lines are well known, measuring a shift is straightforward: redder means receding, bluer means approaching.

Rotation: the broadening of lines

The same Doppler logic reveals rotation. One limb of a spinning star approaches us while the opposite limb recedes, so the star's light is a blend of blueshifted and redshifted contributions: each spectral line is broadened, and faster spin means wider lines. Line width therefore gives the rotation speed projected along our line of sight (we see the full speed only roughly equator-on). Our Sun is a slow rotator, but many young massive stars spin so fast their lines are visibly broad. Thermal motion (thermal broadening) and pressure in dense atmospheres (pressure broadening) also widen lines, so width is a composite effect — rotation usually dominates.

Radius: putting temperature and luminosity together

The most widely applicable method uses the Stefan–Boltzmann law: the energy a hot surface radiates per second grows with the fourth power of temperature and with surface area. For a spherical star:

L = 4 πR² σT⁴

where L is luminosity (total power), R radius, T surface temperature, and σ the Stefan–Boltzmann constant. Given luminosity and temperature, we solve for radius: a star luminous at a given temperature must have a large surface — it must be big. This is how astronomers separate giants from dwarfs: a red giant and a red dwarf can have nearly the same temperature (hence similar spectral class), but the giant is hundreds of times more luminous, so its radius must be hundreds of times larger. Temperature comes from the spectral class; luminosity from apparent brightness plus distance (Chapter 19).

How It Works / Step-by-Step Process

1. Record the spectrum and identify the wavelengths of known lines (Balmer hydrogen, Ca II, etc.).
2. Compare observed wavelengths to laboratory values: a systematic shift toward red or blue gives radial velocity via v = c·Δλ/λ.
3. Measure the width of the lines: broad lines indicate fast rotation (plus thermal and pressure contributions).
4. Use the spectral class to assign temperature; combine with luminosity (apparent brightness + distance) in L = 4πR²σT⁴ and solve for radius.
5. Cross-check: a large derived radius for a cool star is the signature of a giant; a small radius for a hot star points to a white dwarf.

Common Confusions