Blackbody Radiation and Specific Intensity

Every quantitative statement about a star begins with the radiation it emits. Before any of it can be interpreted, the radiation field itself needs a precise description: how much energy crosses a given area, in a given direction, at a given frequency, per unit time. That quantity is the specific intensity. Its angular averages generate the energy density, the flux that carries a star's luminosity, and the pressure that radiation exerts on matter. In thermodynamic equilibrium the specific intensity takes one universal form, the Planck function, and the classical radiation laws follow from it as limits, integrals, and derivatives.

Specific intensity

Consider energy flowing through a small surface element of area $d A$ with unit normal $\overset{n}{^}$ . Let $d E_{ν}$ be the energy in the frequency interval $[ν, ν + d ν]$ that crosses $d A$ in time $d t$ , travelling within a cone of solid angle $d Ω$ about a direction making angle $θ$ with $\overset{n}{^}$ . The specific intensity $I_{ν}$ is defined by

d E_{ν} = I_{ν} cos θ d A d Ω d ν d t .

The factor $cos θ$ projects $d A$ onto the plane perpendicular to the propagation direction: a beam sees only the foreshortened area $d A cos θ$ . The intensity therefore has SI units of $W m^{- 2} Hz^{- 1} sr^{- 1}$ . It is a function of position, direction, frequency, and time; the whole apparatus of radiative transfer tracks how $I_{ν} (r, \overset{n}{^}, ν, t)$ evolves.

The pencil of radiation: energy in frequency band d-nu crossing the area dA into the solid angle d-Omega at angle theta to the normal defines the specific intensity through the cosine-projected area.

A parallel definition uses wavelength, $I_{λ}$ , with $I_{λ} d λ = I_{ν} d ν$ over the corresponding intervals. Since $ν = c / λ$ gives $∣ d ν ∣ = (c / λ^{2}) d λ$ , the two forms relate by $I_{λ} = (c / λ^{2}) I_{ν}$ . The frequency form is more convenient for thermal physics; the wavelength form matches how spectra are recorded.

Invariance along a ray

In empty space, with no matter to emit or absorb, the specific intensity is constant along a ray. Take two surfaces $d A_{1}$ and $d A_{2}$ separated by distance $r$ along the line joining them, oriented normal to the ray. The energy leaving $d A_{1}$ toward $d A_{2}$ equals the energy arriving at $d A_{2}$ from $d A_{1}$ , since none is lost in between. The solid angle subtended by $d A_{2}$ at $d A_{1}$ is $d Ω_{1} = d A_{2} / r^{2}$ , and that subtended by $d A_{1}$ at $d A_{2}$ is $d Ω_{2} = d A_{1} / r^{2}$ . Writing the shared energy two ways,

d E_{ν} = I_{ν, 1} d A_{1} d Ω_{1} d ν d t = I_{ν, 2} d A_{2} d Ω_{2} d ν d t,

and substituting the solid angles, both sides carry the factor $d A_{1} d A_{2} / r^{2}$ . The distance cancels and

I_{ν, 1} = I_{ν, 2} .

Specific intensity does not fall off with distance. What decreases as $1/ r^{2}$ is the flux, because the source subtends a shrinking solid angle. A resolved surface, such as the solar disk, has the same surface brightness whether viewed from Mercury or from Earth; only the number of resolution elements it fills changes. The result is the radiative form of Liouville's theorem: the quantity $I_{ν} / ν^{3}$ is proportional to the photon phase-space density and is conserved in free propagation.¹

Moments of the radiation field

Most applications do not need the full angular dependence of $I_{ν}$ ; they need angular averages weighted by successive powers of $cos θ$ . These moments are the mean intensity, the flux, and the radiation pressure.

The mean intensity $J_{ν}$ is the simple average of $I_{ν}$ over all solid angles,

J_{ν} = \frac{1}{4 π} \oint I_{ν} d Ω = \frac{1}{4 π} 0 \int 2 π 0 \int π I_{ν} sin θ d θ d ϕ .

It fixes the radiation energy density $u_{ν}$ , since a photon of energy $h ν$ travelling at speed $c$ contributes energy density $I_{ν} / c$ per unit solid angle,

u_{ν} = \frac{1}{c} \oint I_{ν} d Ω = \frac{4 π}{c} J_{ν} .

The radiative flux $F_{ν}$ is the net energy per unit area per unit frequency flowing across a surface, weighting each direction by $cos θ$ ,

F_{ν} = \oint I_{ν} cos θ d Ω.

For an isotropic field $I_{ν}$ is independent of direction and the flux vanishes: the $cos θ$ weighting sends equal amounts through the surface in both directions. A net flux requires anisotropy, which is why the deep interior of a star, where the field is very nearly isotropic, transports energy only through a tiny residual asymmetry.

The radiation pressure $P_{ν}$ is the momentum flux. A photon carries momentum $h ν / c$ ; the component along $\overset{n}{^}$ is $(h ν / c) cos θ$ , and the rate at which it crosses the surface again carries a factor $cos θ$ , so

P_{ν} = \frac{1}{c} \oint I_{ν} cos^{2} θ d Ω.

The three moments differ only in the power of $cos θ$ in the weight: zeroth for $J_{ν}$ , first for $F_{ν}$ , second for $P_{ν}$ .

The zeroth, first, and second angular moments of the intensity weight each direction by 1, cos-theta, and cos-squared-theta, yielding the energy density, the flux, and the radiation pressure.

For an axially symmetric field, common in a plane-parallel atmosphere where $I_{ν}$ depends on direction only through $μ = cos θ$ , the azimuthal integral gives $2 π$ and the moments reduce to one-dimensional integrals over $μ \in [- 1, 1]$ :

J_{ν} = \frac{1}{2} - 1 \int 1 I_{ν} d μ, H_{ν} = \frac{1}{2} - 1 \int 1 I_{ν} μ d μ, K_{ν} = \frac{1}{2} - 1 \int 1 I_{ν} μ^{2} d μ .

Here $H_{ν} = F_{ν} /4 π$ is the Eddington flux and $K_{ν} = c P_{ν} /4 π$ is the second moment; the notation $(J, H, K)$ is standard in atmosphere theory.

The Planck function

A cavity in thermodynamic equilibrium at temperature $T$ fills with radiation whose intensity is isotropic, unpolarized, and independent of the cavity walls. That universal intensity is the Planck function $B_{ν} (T)$ . Counting photon states in a box and populating them with the Bose-Einstein occupation number $(e^{h ν / k T} - 1)^{- 1}$ gives

B_{ν} (T) = \frac{2 h ν ^{3}}{c ^{2}} \frac{1}{e ^{h ν / k T} - 1},

with $h$ Planck's constant and $k$ Boltzmann's constant.² The prefactor $2 h ν^{3} / c^{2}$ counts the two polarization states and the density of modes; the occupation factor supplies the mean number of photons per mode. In wavelength form, using $B_{λ} = (c / λ^{2}) B_{ν}$ with $ν = c / λ$ ,

B_{λ} (T) = \frac{2 h c ^{2}}{λ ^{5}} \frac{1}{e ^{h c / λk T} - 1} .

A blackbody is any body that absorbs all radiation incident on it; in equilibrium Kirchhoff's law forces it to emit with intensity $B_{ν} (T)$ . Stars are not perfect blackbodies, but their continuum radiation is close enough that $B_{ν} (T)$ is the reference against which departures are measured.

Planck curves at three temperatures. Higher temperature raises the curve everywhere and shifts the peak to shorter wavelengths; the dashed locus traces the Wien displacement peak.

The Rayleigh-Jeans and Wien limits

Two limits bracket the Planck curve. When photon energies are small compared with the thermal energy, $h ν ≪ k T$ , the exponential expands as $e^{h ν / k T} - 1 \approx h ν / k T$ , and

B_{ν} (T) \approx \frac{2 ν ^{2}}{c ^{2}} k T (h ν ≪ k T) .

This is the Rayleigh-Jeans law, the classical result with no $h$ . It rises as $ν^{2}$ without bound; integrated over all frequencies it diverges, the ultraviolet catastrophe that the quantum occupation factor removes. Radio observations of thermal sources sit deep in this regime, which is why radio astronomers report brightness in temperature units.

When photon energies greatly exceed the thermal energy, $h ν ≫ k T$ , the $- 1$ in the denominator is negligible and

B_{ν} (T) \approx \frac{2 h ν ^{3}}{c ^{2}} e^{- h ν / k T} (h ν ≫ k T) .

This is the Wien law. The exponential cutoff makes the high-frequency tail fall steeply, so the short-wavelength side of a stellar spectrum is exquisitely temperature-sensitive: a small change in $T$ moves the tail by orders of magnitude.

Log-log Planck spectrum bracketed by its two limits. The Rayleigh-Jeans law follows the nu-squared rise at low frequency and the Wien law the exponential fall at high frequency; the true curve turns over between.

Integrated laws

Integrating and differentiating the Planck function reproduces the two laws known before Planck derived his formula.

Stefan-Boltzmann law

The total intensity radiated by a blackbody is the integral of $B_{ν}$ over all frequencies. Substituting $x = h ν / k T$ ,

B (T) = 0 \int \infty B_{ν} (T) d ν = \frac{2 h}{c ^{2}} (\frac{k T}{h})^{4} 0 \int \infty \frac{x ^{3}}{e ^{x} - 1} d x .

The dimensionless integral is a standard result, $\int_{0}^{\infty} x^{3} / (e^{x} - 1) d x = π^{4} /15$ . Hence $B (T) = (2 π^{4} k^{4} /15 c^{2} h^{3}) T^{4}$ . The energy emerging from a blackbody surface into a hemisphere carries an extra factor $π$ from the angular integral of $cos θ$ , giving the emergent flux

F = π B (T) = σ T^{4}, σ = \frac{2 π ^{5} k ^{4}}{15 c ^{2} h ^{3}} = 5.670 \times 1 0^{- 8} W m^{- 2} K^{- 4} .

This is the Stefan-Boltzmann law.³ It fixes a star's luminosity from its radius and effective temperature, $L = 4 π R^{2} σ T_{e}^{4}$ , and it defines the effective temperature $T_{e}$ as the blackbody temperature that reproduces the star's surface flux even when the true spectrum departs from Planck.

Wien displacement law

The wavelength at which $B_{λ}$ peaks follows from $d B_{λ} / d λ = 0$ . Writing $x = h c / λk T$ , the condition becomes the transcendental equation

x = 5 (1 - e^{- x}),

whose nonzero root is $x = 4.965$ . Therefore $h c / λ_{max} k T = 4.965$ , or

λ_{max} T = \frac{h c}{4.965 k} = 2.898 \times 1 0^{- 3} m K .

This is the Wien displacement law: the peak wavelength scales inversely with temperature. For the Sun, $λ_{max} = 2.898 \times 1 0^{- 3} /5780 = 501 nm$ , in the green, consistent with the peak of the solar spectrum. A subtlety worth stating: the peak of $B_{ν}$ in frequency satisfies a different equation, $x = 3 (1 - e^{- x})$ with root $x = 2.821$ , so $ν_{max}$ and $c / λ_{max}$ do not coincide. The location of the peak depends on whether the spectrum is plotted per unit wavelength or per unit frequency, because the two differ by the Jacobian $c / λ^{2}$ .

Brightness temperature

Any measured intensity can be assigned a brightness temperature $T_{b}$ , the temperature at which a blackbody would produce that intensity at the observed frequency,

I_{ν} = B_{ν} (T_{b}) .

In the Rayleigh-Jeans regime the definition inverts cleanly. Setting $I_{ν} = (2 ν^{2} / c^{2}) k T_{b}$ ,

T_{b} = \frac{c ^{2}}{2 ν ^{2} k} I_{ν} = \frac{λ ^{2}}{2 k} I_{ν} .

For a true blackbody $T_{b} = T$ at every frequency. For a real source $T_{b}$ varies with frequency, and its departure from a constant value is a compact diagnostic of how far the emission is from thermal. Radio astronomers use $T_{b}$ as the natural intensity unit precisely because thermal sources at long wavelengths sit in the Rayleigh-Jeans limit, where $T_{b}$ is a direct linear measure of $I_{ν}$ .

A resolved source has the same specific intensity, hence the same brightness temperature, at any distance; only the solid angle it subtends, and therefore the received flux, shrinks as one over distance squared.

The specific intensity, its three moments, and the Planck function together form the vocabulary for everything that follows. The next lesson lets matter emit and absorb along the ray, turning the constancy of $I_{ν}$ in free space into the transfer equation that governs how a stellar spectrum is formed.⁴

Carroll & Ostlie, §9.1 — the description of the radiation field: specific intensity, its invariance along a ray, and the mean intensity, flux, and radiation pressure as angular moments. ↩
Carroll & Ostlie, §3.4 — the Planck function, its frequency and wavelength forms, and the Rayleigh-Jeans and Wien limits. ↩
CODATA/NIST recommended values of the Stefan-Boltzmann constant, Planck constant, and Boltzmann constant — https://physics.nist.gov/cuu/Constants. ↩
Maoz, Ch. 2 — basic radiative quantities and the blackbody spectrum as the reference for stellar continua. ↩