Stars — Parallax, Luminosity & the H-R Diagram

This lecture covers how we measure stellar properties: distance, brightness, temperature, size, and mass.

Distance is measured for nearby stars by stellar parallax — the small angular shift in a star's apparent position as Earth orbits the Sun. The parsec is defined so a star with a 1-arcsecond parallax sits at 1 pc (= 3.09 × 10¹³ km = 3.26 light-years). The formula is d (pc) = 1 / parallax (arcsec). Proxima Centauri, the closest star, has parallax 0.77″ → 1.3 pc.

Luminosity is a star's total energy output per second — an intrinsic property. Apparent brightness is how bright it looks from Earth and falls off as 1/d². Two stars that appear equally bright can be at very different distances. Apparent brightness is reported on the inverted, logarithmic magnitude scale (smaller = brighter; a 5-magnitude difference = 100× in brightness).

Temperature is determined from color (Wien's law on the blackbody peak) or from spectral lines. The seven classical spectral classes — hottest to coolest — are O, B, A, F, G, K, M (mnemonic: "Oh, Be A Fine Girl/Guy, Kiss Me!"). Modern additions L, T, Y cover brown dwarfs.

Size is calculated from luminosity and temperature using the Stefan-Boltzmann law. Stars range from dwarfs (≤ R☉) through giants (10–100 R☉) to supergiants (> 100 R☉).

The Hertzsprung-Russell (H-R) diagram plots luminosity vs. temperature. About 90 % of stars lie on the Main Sequence, a diagonal band from hot/luminous (upper left) to cool/dim (lower right) where they fuse hydrogen. A star's position on the Main Sequence is set primarily by its mass.

Masses come from analysis of binary star orbits — visual, spectroscopic, or eclipsing — using Newton's modified version of Kepler's third law.