Stellar Spectra and Spectral Classification

Broadband colors place a star roughly on a temperature scale, but the spectrum itself — the flux resolved wavelength by wavelength — carries far more. It divides into a smooth continuum and a forest of narrow absorption lines, and the pattern of those lines classifies the star. The classification turns out to be a temperature sequence, and the reason lies in two statistical relations: the Boltzmann factor, which sets how atoms distribute over excited states, and the Saha equation, which sets how they distribute over ionization stages. Together they explain why a given line strengthens, peaks, and fades as temperature rises, and why the historical letter classes reorder into O B A F G K M.

Kirchhoff's laws and the origin of lines

Three empirical rules, Kirchhoff's laws, describe when a spectrum shows a continuum, bright lines, or dark lines:

• Continuous spectrum: a hot, dense body (a solid, liquid, or optically thick gas) emits a continuum that approaches the blackbody form.
• Emission-line spectrum: a hot, diffuse gas emits bright lines at wavelengths set by its atomic transitions.
• Absorption-line spectrum: a continuous source seen through a cooler diffuse gas shows dark lines at those same wavelengths.

A star produces an absorption spectrum because the deep, dense layers radiate a near-continuum, and the cooler, more transparent gas above removes photons at its line wavelengths along the line of sight. The lines are dark relative to the continuum, not truly black; the physics of exactly how deep a line goes belongs to radiative transfer. Here the question is which lines appear and how strong they are.¹

The Boltzmann factor: populating excited states

An atom absorbs a line photon only from the lower level of the corresponding transition. The strength of an absorption line therefore depends on how many atoms sit in that lower level, which is a question of statistical mechanics. In thermal equilibrium at temperature $T$, the ratio of the number of atoms in state $b$ (energy $E_b$) to the number in state $a$ (energy $E_a$) is the Boltzmann factor, weighted by the statistical degeneracies $g_a$ and $g_b$ of the levels:

where $k_B=1.381\times 10^{-23}\text{J}{\text{K}}^{-1}$ is the Boltzmann constant. The degeneracy of hydrogen level $n$ is $g_n=2n^2$. As $T$ rises, the exponential climbs toward unity and higher levels fill; at low $T$ nearly all atoms sit in the ground state.

For hydrogen the Balmer absorption lines (the visible series H$\alpha$, H$\beta$, and beyond) arise from the first excited state, $n=2$, at $E_2-E_1=10.2\text{eV}$ above the ground state. The fraction of neutral hydrogen in $n=2$ relative to $n=1$ is

At the solar temperature, $k_{B}T\approx 0.5\text{eV}$, the exponent is about $-20$ and only one hydrogen atom in $10^8$ sits in $n=2$. The Balmer lines are weak in the Sun not because hydrogen is scarce but because almost none of it is in the level that produces them. Raising the temperature pushes more atoms into $n=2$ and strengthens the lines — but only up to a point, because a second process removes neutral hydrogen entirely.²

The Saha equation: ionization balance

At high enough temperature, collisions and radiation strip the electron off the atom, and an ionized hydrogen atom has no bound levels to absorb Balmer photons. The balance between an ion in stage $i$ and the next stage $i+1$ (with a free electron released) is governed by the Saha equation. For the ratio of number densities,

where ${\chi }_i$ is the ionization energy from stage $i$, $n_e$ is the free-electron number density, $Z_i$ and $Z_{i+1}$ are the partition functions of the two stages, $m_e$ is the electron mass, and $h$ is Planck's constant. The factor of 2 counts the two spin states of the freed electron. The partition function is the degeneracy-weighted Boltzmann sum over all bound levels of a stage,

and reduces to the ground-state degeneracy when only the lowest level is populated.

Two features of the Saha equation matter for line strengths. First, the exponential $e^{-\chi /k_{B}T}$ makes ionization rise steeply once $k_{B}T$ approaches the ionization energy $\chi$ (13.6 eV for hydrogen). Second, the $1/n_e$ prefactor means ionization also depends on density: a diffuse atmosphere, with few electrons to recombine with, is more ionized at fixed temperature than a dense one. This density sensitivity is the physical basis of the luminosity classes introduced below, since a giant's extended atmosphere has a far lower gas pressure than a dwarf's.³

The Balmer maximum: Boltzmann times Saha

The strength of the hydrogen Balmer lines is proportional to the number of neutral hydrogen atoms in the $n=2$ level. That number is the product of two competing fractions: the fraction of hydrogen that is still neutral (from Saha) and the fraction of that neutral hydrogen sitting in $n=2$ (from Boltzmann). Write the neutral fraction as $N_{\text{I}}/N_{\text{total}}$ and combine:

At low temperature the Boltzmann factor is tiny — almost no atoms are excited to $n=2$ — so the lines are weak. At high temperature the Saha factor is tiny — almost all hydrogen is ionized — so the lines are again weak. The product peaks at an intermediate temperature near $T\approx 9500\text{K}$, where enough hydrogen is both neutral and excited. That temperature marks an A0 star, which is why the Balmer lines reach their maximum strength in A stars and weaken toward both hotter B/O stars and cooler F/G/K stars.

The same reasoning, applied to other atoms and ions, explains the whole classification. Each species has its own ionization and excitation energies, so each line peaks at its own temperature. The lines present in a spectrum therefore act as a thermometer far more precise than a broadband color.

The OBAFGKM sequence

The letters of the classical Harvard classification were originally assigned by Balmer-line strength alone, in alphabetical order A, B, C, and so on. Once the Boltzmann-Saha analysis showed the true ordering variable was temperature, the classes were reordered and pruned into the temperature sequence

from hottest (O, above $30,000\text{K}$) to coolest (M, below $3900\text{K}$). Each class is subdivided by a digit 0-9, so the Sun is a G2 star. The diagnostic lines shift systematically along the sequence:

• O: ionized helium (He II) lines, only excitable above $\sim 25,000\text{K}$; hydrogen weak because it is largely ionized.
• B: neutral helium (He I) at maximum; hydrogen strengthening.
• A: hydrogen Balmer lines at maximum near $9500\text{K}$; He I gone.
• F, G: Balmer weakening; ionized metals (Ca II H and K) strengthening.
• K: neutral metals dominant; Ca II very strong; molecular bands appearing.
• M: molecular bands (titanium oxide, TiO) dominate; too cool for most atomic lines.

The temperature calibration of the sequence is the relation later lessons invoke: a spectral type maps onto an effective temperature to within a few percent, and the digit subdivides that mapping. Because the lines respond to temperature and not to composition (to first order, all stars are mostly hydrogen and helium), two stars of very different metal content but the same temperature look nearly alike in the sequence.

Luminosity classes and the MK system

Temperature alone does not fix a star's spectrum. Two stars at the same temperature but different sizes — a main-sequence dwarf and a bloated giant — have very different atmospheric densities, hence different electron pressures, and the Saha $1/n_e$ term makes them differ in ionization. A giant's low-density atmosphere is more ionized, and its lines are narrower because collisional (pressure) broadening is weaker. The MK (Morgan-Keenan) system adds a Roman-numeral luminosity class encoding this second dimension:

• I — supergiants (subdivided Ia, Ib)
• II — bright giants
• III — giants
• IV — subgiants
• V — main-sequence dwarfs (the Sun is G2V)
• VI / sd — subdwarfs
• wd / D — white dwarfs

A full MK type combines the two axes: a temperature letter with digit and a luminosity class, such as G2V for the Sun, M2Ia for a red supergiant like Betelgeuse, or B0V for a hot main-sequence star. The two-dimensional grid separates the giant branch from the main sequence on a spectrum alone, without needing a distance — which is what makes spectroscopic parallax possible: read the luminosity class to get the absolute magnitude, compare to the apparent magnitude, and solve the distance modulus for distance.

The classification of this lesson is the bridge from raw spectra to physical parameters: temperature from the spectral type, surface gravity (and thus luminosity) from the class. Those parameters feed the interior physics of later modules and, through spectroscopic parallax and main-sequence fitting, supply distances for the cosmic distance ladder. Reading the spectrum in detail, rather than counting broadband colors, requires collecting the light in the first place, which is the subject of the next lesson on telescopes and detectors.