Stellar Spectra and the Stefan and Wien Laws

A photograph of a star tells you it is bright; a spectrum of a star tells you what it is. Stellar spectra encode the surface temperature (from the peak wavelength and the relative strengths of absorption lines), the chemical composition (from the wavelengths of the absorption lines themselves), the radial velocity (from the Doppler shifts of the lines), and — combined with the luminosity — the radius (from the Stefan-Boltzmann law). This lesson develops the two black-body laws that underpin stellar physics, Wien's displacement law and the Stefan-Boltzmann law, the continuous, emission and absorption spectra produced by different physical situations, and the O B A F G K M spectral classification scheme that organises every star known.

Spec mapping. This lesson covers AQA 7408 section 3.9.2 on stellar spectra and black-body radiation: continuous, emission-line and absorption-line spectra; the OBAFGKM spectral classification by temperature and absorption-line strength; Wien's displacement law λ_max T = 2.898 × 10⁻³ m K; the Stefan-Boltzmann law L = 4πr²σT⁴; energy-level transitions and the formation of absorption lines in stellar atmospheres. (Refer to the official AQA specification document for exact wording.)

Synoptic links. Wien's law and the Stefan-Boltzmann law are applications of black-body radiation introduced in thermal physics (AQA 3.6); the absorption lines link directly to atomic energy levels and photon emission/absorption (AQA 3.2.1.4). The Doppler shifts referenced in the spectra connect forward to the cosmological redshift lesson later in this course.

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Black-body radiation — a quick reminder

A black body is an idealised object that absorbs all incident electromagnetic radiation and re-emits a continuous spectrum determined entirely by its temperature. The spectral radiance B(λ, T) — the power emitted per unit area per unit wavelength per unit solid angle — is given by Planck's law:

B(λ, T) = (2 h c² / λ⁵) × 1 / (e^(hc / λkT) − 1)

We will not need Planck's law in raw form for AQA examination, but two consequences of it are central:

1. Wien's displacement law — relates the peak wavelength to temperature.
2. Stefan-Boltzmann law — relates the total power radiated per unit area to T⁴.

Stars are good approximations to black bodies, especially when we average over their broad spectral features. The "surface" — the photosphere — is the thin layer from which most of the light we see is emitted, and its temperature is what black-body laws return.

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Wien's displacement law

Wien's displacement law. The wavelength λ_max at which a black body's spectral radiance peaks is inversely proportional to its absolute temperature:

λ_max × T = 2.898 × 10⁻³ m K

A hotter body has a shorter peak wavelength (bluer); a cooler body has a longer peak wavelength (redder).

Body	T / K	λ_max	Where it peaks
Cosmic microwave background	2.7	1.07 mm	Microwaves
Liquid water (room temp)	293	9.9 µm	Far-IR
Human skin	310	9.4 µm	Far-IR
Tungsten lamp filament	3000	0.97 µm	Near-IR (just out of visible)
Sun	5800	500 nm	Visible (green-yellow)
Sirius A	9940	292 nm	UV
Naos (O5 supergiant)	42 000	69 nm	Far-UV

Wien's law is a workhorse: given any spectrum, the position of the peak gives the temperature.

Worked example: Rigel's temperature

The supergiant Rigel has a peak emission wavelength of 263 nm. Calculate its surface temperature.

Solution. T = 2.898 × 10⁻³ / λ_max = 2.898 × 10⁻³ / (263 × 10⁻⁹) = 2.898 × 10⁻³ / 2.63 × 10⁻⁷ = 1.10 × 10⁴ K. Rigel is therefore an early-B-type blue supergiant — consistent with its blue-white colour.

Worked example: a cool red dwarf

A red dwarf has peak emission at λ_max = 1.07 µm. Calculate T.

Solution. T = 2.898 × 10⁻³ / 1.07 × 10⁻⁶ = 2710 K. Such a star peaks in the near-infrared and is a late M dwarf. Most of its light energy is in the infrared, which is why visible-only photometry can wildly underestimate the luminosity of cool stars.

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The Stefan-Boltzmann law

Stefan-Boltzmann law. A black body of surface temperature T radiates a total power per unit area equal to σT⁴, where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.

For a spherical star of radius r:

L = 4πr²σT⁴

The fourth-power dependence is striking. Double the temperature and the radiated power increases sixteenfold; ten-fold and it goes up by 10⁴. This is why high-mass O-type stars, with surface temperatures of 40 000 K, can be 10⁶ times more luminous than the Sun despite being only ten times larger in radius.

Worked example: the Sun's luminosity

The Sun has T = 5800 K and r = 6.96 × 10⁸ m. Verify the accepted luminosity L_⊙ ≈ 3.85 × 10²⁶ W.

Solution. L = 4π × (6.96 × 10⁸)² × 5.67 × 10⁻⁸ × (5800)⁴. Compute step by step:

• 4πr² = 4π × 4.84 × 10¹⁷ = 6.09 × 10¹⁸ m²
• T⁴ = (5800)⁴ = 1.13 × 10¹⁵ K⁴
• L = 6.09 × 10¹⁸ × 5.67 × 10⁻⁸ × 1.13 × 10¹⁵ = 3.91 × 10²⁶ W.

Agreement with the accepted 3.85 × 10²⁶ W is to within 2 % — good for an approximate black-body treatment.

Worked example: a red giant

Betelgeuse has a surface temperature of about 3500 K and a luminosity of 1.3 × 10⁵ L_⊙. Calculate its radius and compare with the Sun's.

Solution. L = 4πr²σT⁴, so r = √(L / (4πσT⁴)). Substitute L = 1.3 × 10⁵ × 3.85 × 10²⁶ = 5.0 × 10³¹ W and T⁴ = (3500)⁴ = 1.50 × 10¹⁴ K⁴.

4πσT⁴ = 4π × 5.67 × 10⁻⁸ × 1.50 × 10¹⁴ = 1.07 × 10⁸ W m⁻². Then r² = 5.0 × 10³¹ / 1.07 × 10⁸ = 4.7 × 10²³ m², so r = 6.85 × 10¹¹ m. In solar radii (r_⊙ = 6.96 × 10⁸ m): r / r_⊙ = 984. Betelgeuse's radius is roughly a thousand times the Sun's — if placed at the centre of our solar system it would engulf the orbit of Jupiter.

This calculation also illustrates how astronomers measure stellar radii: combine the luminosity (from distance + apparent magnitude) with the surface temperature (from Wien or spectral type) to back out r via Stefan-Boltzmann.

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Black-body spectra and stellar temperature

Three features are visible in the figure (and matter for exam questions):

1. Wien's law shifts the peak: hotter stars peak at shorter wavelengths (further left).
2. Stefan's law scales the area: the area under each curve is the total power per unit area σT⁴, so the hottest body emits dramatically more power per unit area than the cooler ones.
3. At every wavelength, a hotter body emits more than a cooler body (the curves do not cross).

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The three types of spectrum

Kirchhoff (1859) formulated the three rules that explain the basic types of astronomical spectrum:

1. Continuous spectrum — a hot dense gas or solid emits a continuous spectrum with no breaks. A tungsten lamp filament is a good example. The deep interior of a star also emits a continuous spectrum.

2. Emission-line spectrum — a hot, low-pressure gas emits light only at specific wavelengths corresponding to transitions between bound electronic energy levels. Examples include neon signs, the Lyman, Balmer and Paschen series of hydrogen, and nebular emission lines. The wavelengths obey the Rydberg formula 1/λ = R(1/n_lower² − 1/n_upper²).

3. Absorption-line spectrum — when continuous radiation passes through a cooler, low-pressure gas, photons of just the right energies to excite bound transitions are absorbed. The result is a continuous spectrum with dark lines at the wavelengths of those transitions. This is the dominant pattern in stellar spectra.

How stars produce absorption spectra

The interior of a star is opaque — radiation diffuses outward through an ionised plasma. By the time it reaches the photosphere, the gas has thinned enough for photons to escape directly. The photons that escape with no further absorption form a continuous black-body-like background. The cooler outer atmosphere (the chromosphere and upper photosphere) contains atoms and ions whose bound transitions absorb specific wavelengths, removing them from the spectrum and producing the dark Fraunhofer lines observed by Joseph von Fraunhofer in 1814.

A spectrum of the Sun reveals thousands of Fraunhofer lines. Each one identifies an element — Ca, Fe, Na, Mg, Ti, and (most prominently) the Balmer series of hydrogen at 6563 Å (Hα), 4861 Å (Hβ), 4340 Å (Hγ), and so on.

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The OBAFGKM classification

Stars are classified by surface temperature into seven principal spectral classes, conventionally remembered by the mnemonic "Oh Be A Fine Girl/Guy, Kiss Me".

Class	Colour	T_surface (K)	Dominant absorption lines	Examples
O	Blue	> 30 000	Ionised helium (He II), some N III, weak hydrogen	Naos, Mintaka
B	Blue-white	10 000 – 30 000	Neutral helium (He I), strong hydrogen	Rigel, Spica
A	White	7 500 – 10 000	Strong hydrogen (Balmer series), some metal lines	Vega, Sirius A
F	Yellow-white	6 000 – 7 500	Weakening hydrogen, ionised calcium (Ca II), metal lines	Procyon, Polaris
G	Yellow	5 200 – 6 000	Many ionised and neutral metals (Ca II H & K, Fe I)	Sun, Alpha Cen A
K	Orange	3 700 – 5 200	Neutral metals (Fe I, Na I), TiO bands begin in late K	Arcturus, Aldebaran
M	Red	2 400 – 3 700	TiO molecular bands dominate, weak hydrogen	Betelgeuse, Proxima Cen

Each class is subdivided into ten subclasses 0 through 9, giving e.g. G2V for the Sun (G-type, subclass 2, luminosity class V = main sequence). Beyond M lie the L and T classes for brown dwarfs (substellar objects too small to fuse hydrogen).

Why the line strengths change with temperature

It is not just the colour that changes from class to class — different absorption lines dominate at different temperatures, because different temperatures produce different ionisation states. Hydrogen Balmer lines, for example, are strongest in A-type stars (~10 000 K). At lower temperatures, most hydrogen atoms are in the ground state (n=1) and absorb in the ultraviolet Lyman series, not in the Balmer series (n=2 → higher). At higher temperatures, the hydrogen is fully ionised and absorbs nothing. The line strength therefore peaks at an intermediate temperature — for hydrogen, around 10 000 K — and falls off either side.