Stellar Spectra and the Hertzsprung-Russell Diagram

The spectrum of a star shows its temperature, composition and motion, and the Hertzsprung-Russell (HR) diagram plots the luminosity (or absolute magnitude) of stars against their surface temperature (or spectral class), revealing the main sequence, giants, supergiants and white dwarfs; the luminosity of a star is given by L = 4πR²σT⁴. In brief: spectral classes O, B, A, F, G, K, M run from hot (above 30,000 K, blue) to cool (below 3,700 K, red); the Sun is G2 (5,772 K). About 90 percent of stars lie on the main sequence, where hydrogen is fused in the core, and on it the luminosity rises steeply with mass (L ∝ M^3.5 approximately). The HR diagram is the key to understanding stellar evolution.

The light of a star carries information on its surface temperature, size, composition and motion. This guide explains how spectra reveal temperature and composition, how spectral classes are defined, how the HR diagram is organized, compares the stellar groups in a table, works through numerical examples of radius, luminosity and lifetime, and corrects common mistakes.

What do stellar spectra show?

The light of a star is almost a continuous black-body spectrum, from the hot dense photosphere, with a peak wavelength given by Wien’s law (λ_max = 2.898 × 10⁻³ m·K ÷ T). Cooler gas in the star’s atmosphere absorbs light at specific wavelengths, producing dark absorption lines (Fraunhofer lines). Each element has its own pattern of lines, which gives the chemical composition of the star (about 74 percent hydrogen and 24 percent helium by mass for the Sun, and about 2 percent heavier elements). The strengths of the lines depend on temperature, because the energy levels that are populated depend on it: hydrogen Balmer lines are strongest at about 10,000 K (class A), while at higher temperatures hydrogen is ionized, and at lower temperatures, few atoms are in the n = 2 state; ionized helium appears in very hot stars, and molecular bands (titanium oxide) in cool red stars. The Doppler shift of lines gives the radial velocity (the motion toward or away from us, the rotation of the star, and the orbit of binary stars and exoplanets), and the width of lines gives information about rotation, pressure and magnetic fields.

What are the spectral classes?

Stars are classified by surface temperature in the Harvard sequence: O, B, A, F, G, K, M (a mnemonic: “Oh Be A Fine Girl/Guy, Kiss Me”), each divided into subclasses 0 to 9, with the Sun as G2. O stars: above 30,000 K, blue, rare, very luminous, helium lines; B: 10,000 to 30,000 K, blue-white (Rigel); A: 7,500 to 10,000 K, white (Sirius, Vega); F: 6,000 to 7,500 K, yellow-white; G: 5,200 to 6,000 K, yellow (the Sun); K: 3,700 to 5,200 K, orange (Arcturus); M: below 3,700 K, red (Betelgeuse, Proxima Centauri), the most numerous. A luminosity class (I to V) is added: V is a main-sequence star (dwarf), III giants, I supergiants.

What is the HR diagram?

Ejnar Hertzsprung and Henry Norris Russell plotted stars by luminosity (vertical axis, in solar luminosities or absolute magnitude) against temperature (horizontal axis, decreasing to the right, from O to M). The stars are not spread randomly, but fall into groups:

Main sequence: a diagonal band from hot, luminous stars (top left) to cool, dim stars (bottom right). Stars spend about 90 percent of their lives here, fusing hydrogen to helium in the core.
Red giants (upper right): cool but very luminous, so they must have large radii (10 to 100 R_☉); they are evolved stars with hydrogen shell burning or helium burning. Supergiants (top): extremely luminous and large, up to 1,000 R_☉ (Betelgeuse).
White dwarfs (lower left): hot but dim, so very small (Earth-sized): the cores of dead low-mass stars.

Lines of constant radius run diagonally on the diagram, since L = 4πR²σT⁴: at the same temperature, a larger star is more luminous.

Schematic HR diagram (approximate positions) with temperature decreasing to the right. The main sequence runs from hot, luminous stars at upper left to cool, dim stars at lower right. Giants and supergiants are cool but luminous, so they must be large. White dwarfs are hot but faint, so they are tiny. The Sun is at about 5,772 K and 1 L☉.

How do mass and luminosity relate on the main sequence?

For main-sequence stars, the luminosity is approximately L ∝ M^3.5 (the exponent is between 3 and 4 depending on the mass range). A star of 2 M_☉ is 11 times as luminous as the Sun. The nuclear fuel available is proportional to the mass, so the lifetime on the main sequence is t ∝ M ÷ L ∝ M^−2.5: t ≈ 10¹⁰ years × (M/M_☉)^−2.5. Massive stars live fast and die young; red dwarfs of 0.1 M_☉ live for trillions of years, longer than the present age of the universe (13.8 billion years).

Stellar groups compared

Typical properties of stars in the HR diagram
Group	Temperature (K)	Luminosity (L_☉)	Radius (R_☉)
Main sequence O	30,000 to 50,000	10⁵ to 10⁶	6 to 15
Sun (G2 V)	5,772	1	1
Red dwarf (M V)	2,500 to 3,700	0.001 to 0.1	0.1 to 0.5
Red giant	3,500 to 5,000	100 to 1,000	10 to 100
White dwarf	8,000 to 40,000	10⁻⁴ to 0.1	about 0.01

Worked example: luminosity, radius and lifetime

Luminosity from radius and temperature. L ÷ L_☉ = (R ÷ R_☉)² (T ÷ 5,772 K)⁴. A star of radius 2.0 R_☉ and temperature 5,000 K: L = 4 × (0.866)⁴ = 4 × 0.563 = 2.25 L_☉.
White dwarf. Sirius B has T = 25,000 K and R = 0.0084 R_☉: L = (0.0084)² × (25,000 ÷ 5,772)⁴ = 7.06 × 10⁻⁵ × 352 = 0.025 L_☉. It is hot, but its tiny surface area makes it faint.
Radius from luminosity and temperature. A red giant has L = 400 L_☉ and T = 4,000 K. R ÷ R_☉ = √(L ÷ L_☉) × (5,772 ÷ T)² = 20 × 2.082 = 41.6 R_☉, about 0.19 AU, roughly half of the distance of Mercury from the Sun (0.39 AU).
Wien’s law. A star at 10,000 K peaks at 2.898 × 10⁻³ ÷ 10,000 = 290 nm in the ultraviolet; at 3,500 K it peaks at 828 nm, in the infrared, and appears red.
Mass-luminosity relation. A main-sequence star of 2.0 M_☉: L ≈ 2^3.5 = 11.3 L_☉. Its main-sequence lifetime: 10¹⁰ ÷ 2^2.5 = 1.8 × 10⁹ years. A star of 10 M_☉: L ≈ 3,160 L_☉, lifetime about 3 × 10⁷ years.
Doppler shift of a spectral line. The hydrogen-alpha line (656.28 nm) in a star is observed at 656.50 nm. Δλ ÷ λ = 0.22 ÷ 656.28 = 3.4 × 10⁻⁴, so the radial velocity is v = 3.4 × 10⁻⁴ × 3.00 × 10⁵ km/s = 100 km/s, receding.

How was the diagram developed?

Joseph von Fraunhofer mapped the dark lines in the solar spectrum in 1814. Gustav Kirchhoff and Robert Bunsen identified elements in the Sun in 1859 by their spectral lines. Annie Jump Cannon classified over 200,000 stellar spectra at Harvard Observatory (1910s to 1920s) and arranged the sequence OBAFGKM; Cecilia Payne showed in her 1925 thesis that stars are mostly hydrogen and helium. Hertzsprung (1911) and Russell (1913) plotted the diagram. Arthur Eddington established the mass-luminosity relation in the 1920s.

Which mistakes are common?

Mistake 1: Reading the temperature axis from the wrong end

On the HR diagram, temperature increases to the left.

Mistake 2: Thinking that a hot star must be bright

White dwarfs are hot but dim because they are tiny. Luminosity depends on R²T⁴.

Mistake 3: Believing that the main sequence is a path in the life of a star

It is the location where stars spend most of their lives. Stars move off it when they run out of hydrogen in the core.

Mistake 4: Saying that massive stars live longer because they have more fuel

They burn their fuel much faster. The lifetime decreases with mass.