---
title: Blackbody Radiation and Specific Intensity
draft: false
module: Radiation and Matter
moduleNumber: 3
lessonNumber: 1
order: 301
summary: >
  Specific intensity is the fundamental measure of a radiation field: energy per
  unit area, time, frequency, and solid angle. It is conserved along a ray in
  empty space, and its angular moments give the mean intensity, flux, and
  radiation pressure. In thermal equilibrium the intensity equals the Planck
  function, whose limits and integrals reproduce the Rayleigh-Jeans law, the Wien
  law, Stefan-Boltzmann, and Wien's displacement law.
topics: [Radiation and Matter]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 3 — The Continuous Spectrum of Light; §3.4 The Planck Function; Ch. 9 — Stellar Atmospheres; §9.1 The Description of the Radiation Field"
  - book: Maoz
    ref: "Ch. 2 — Basic Physics of Stars"
---

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 $\hat n$. Let $\d E_\nu$ be the energy in the frequency interval
$[\nu, \nu + \d\nu]$ that crosses $\d A$ in time $\d t$, travelling within a cone
of solid angle $\d\Omega$ about a direction making angle $\theta$ with $\hat n$.
The **specific intensity** $I_\nu$ is defined by

$$
\d E_\nu = I_\nu \, \cos\theta \, \d A \, \d\Omega \, \d\nu \, \d t .
$$

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

$$
% caption: The pencil of radiation: energy in frequency band d-nu crossing the
% area dA into the solid angle d-Omega at angle theta to the normal defines the
% specific intensity through the cosine-projected area.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% surface element
\draw[thick] (-1.5,0) -- (1.5,0) -- (2.1,0.7) -- (-0.9,0.7) -- cycle;
\node[black!70, anchor=north] at (0.3,-0.05) {area dA};
% normal
\draw[->, black, thick] (0.3,0.35) -- (0.3,2.6);
\node[black!70, anchor=east] at (0.3,2.5) {normal};
% ray direction into cone
\draw[->, acc, very thick] (0.3,0.35) -- (2.9,2.35);
\node[acc, anchor=west] at (2.9,2.35) {ray};
% cone edges
\draw[acc, thin] (0.3,0.35) -- (2.4,2.5);
\draw[acc, thin] (0.3,0.35) -- (3.2,2.05);
\draw[acc, thin] (2.4,2.5) .. controls (2.9,2.4) .. (3.2,2.05);
\node[acc, anchor=south west] at (2.9,2.5) {solid angle};
% angle arc
\draw[black] (0.3,1.2) arc (90:37.5:0.85);
\node[black] at (0.85,1.55) {angle};
\end{tikzpicture}
$$

> **Definition (Specific intensity).** $I_\nu$ is the energy per unit time, per
> unit frequency, per unit solid angle, crossing unit area oriented
> perpendicular to the beam. It is the most detailed macroscopic description of a
> radiation field short of the photon distribution function itself.

A parallel definition uses wavelength, $I_\lambda$, with
$I_\lambda \, \d\lambda = I_\nu \, \d\nu$ over the corresponding intervals. Since
$\nu = c/\lambda$ gives $\lvert \d\nu \rvert = (c/\lambda^2)\,\d\lambda$, the two
forms relate by $I_\lambda = (c/\lambda^2) I_\nu$. 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 $\d\Omega_1 = \d A_2 / r^2$, and that subtended by $\d A_1$ at
$\d A_2$ is $\d\Omega_2 = \d A_1 / r^2$. Writing the shared energy two ways,

$$
\d E_\nu = I_{\nu,1}\, \d A_1 \, \d\Omega_1 \, \d\nu \, \d t
        = I_{\nu,2}\, \d A_2 \, \d\Omega_2 \, \d\nu \, \d t ,
$$

and substituting the solid angles, both sides carry the factor
$\d A_1 \, \d A_2 / r^2$. The distance cancels and

$$
I_{\nu,1} = I_{\nu,2} .
$$

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
$I_\nu / \nu^3$ is proportional to the photon phase-space density and is
conserved in free propagation.[^bob-field]

## Moments of the radiation field

Most applications do not need the full angular dependence of $I_\nu$; they need
angular averages weighted by successive powers of $\cos\theta$. These **moments**
are the mean intensity, the flux, and the radiation pressure.

The **mean intensity** $J_\nu$ is the simple average of $I_\nu$ over all solid
angles,

$$
J_\nu = \frac{1}{4\pi} \oint I_\nu \, \d\Omega
      = \frac{1}{4\pi} \int_0^{2\pi}\!\!\int_0^\pi I_\nu \sin\theta \, \d\theta \, \d\phi .
$$

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

$$
u_\nu = \frac{1}{c} \oint I_\nu \, \d\Omega = \frac{4\pi}{c} J_\nu .
$$

The **radiative flux** $F_\nu$ is the net energy per unit area per unit frequency
flowing across a surface, weighting each direction by $\cos\theta$,

$$
F_\nu = \oint I_\nu \cos\theta \, \d\Omega .
$$

For an isotropic field $I_\nu$ is independent of direction and the flux vanishes:
the $\cos\theta$ 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** $P_\nu$ is the momentum flux. A photon carries
momentum $h\nu/c$; the component along $\hat n$ is $(h\nu/c)\cos\theta$, and the
rate at which it crosses the surface again carries a factor $\cos\theta$, so

$$
P_\nu = \frac{1}{c} \oint I_\nu \cos^2\theta \, \d\Omega .
$$

The three moments differ only in the power of $\cos\theta$ in the weight: zeroth
for $J_\nu$, first for $F_\nu$, second for $P_\nu$.

$$
% caption: The zeroth, first, and second angular moments of the intensity weight
% each direction by 1, cos-theta, and cos-squared-theta, yielding the energy
% density, the flux, and the radiation pressure.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% three panels
\foreach \x/\w/\lab/\mom in {0/1/{weight 1}/{energy density}, 4/2/{weight cos}/{f\/lux}, 8/3/{weight cos sq}/{pressure}} {
  \begin{scope}[shift={(\x,0)}]
  \draw[black] (0.9,0) arc (0:180:0.9);
  \draw[black] (-0.9,0) -- (0.9,0);
  \draw[->, thick] (0,0) -- (0,1.15);
  \draw[->, thin] (0,0) -- (0.78,0.78);
  \draw[->, thin] (0,0) -- (-0.78,0.78);
  \node[black!70, anchor=north] at (0,-0.15) {\lab};
  \node[anchor=south] at (0,1.55) {\mom};
  \end{scope}
}
\draw[->, black, thick] (1.6,0.45) -- (2.5,0.45);
\draw[->, black, thick] (5.6,0.45) -- (6.5,0.45);
\end{tikzpicture}
$$

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

$$
J_\nu = \tfrac{1}{2}\!\int_{-1}^{1} I_\nu \, \d\mu, \quad
H_\nu = \tfrac{1}{2}\!\int_{-1}^{1} I_\nu \, \mu \, \d\mu, \quad
K_\nu = \tfrac{1}{2}\!\int_{-1}^{1} I_\nu \, \mu^2 \, \d\mu .
$$

Here $H_\nu = F_\nu / 4\pi$ is the **Eddington flux** and $K_\nu = c P_\nu / 4\pi$
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** $B_\nu(T)$. Counting photon
states in a box and populating them with the Bose-Einstein occupation number
$(e^{h\nu/kT} - 1)^{-1}$ gives

$$
B_\nu(T) = \frac{2h\nu^3}{c^2} \, \frac{1}{e^{h\nu/kT} - 1} ,
$$

with $h$ Planck's constant and $k$ Boltzmann's constant.[^bob-planck] The
prefactor $2h\nu^3/c^2$ 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 $B_\lambda = (c/\lambda^2) B_\nu$ with $\nu = c/\lambda$,

$$
B_\lambda(T) = \frac{2hc^2}{\lambda^5} \, \frac{1}{e^{hc/\lambda kT} - 1} .
$$

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

$$
% caption: Planck curves at three temperatures. Higher temperature raises the
% curve everywhere and shifts the peak to shorter wavelengths; the dashed locus
% traces the Wien displacement peak.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {wavelength};
\draw[->, black] (0,0) -- (0,5.0) node[above, black!70] {spectral radiance};
% hottest: peak at short wavelength, tall
\draw[very thick] (0.3,0.2) .. controls (0.8,4.6) and (1.3,4.7) .. (2.0,3.6)
  .. controls (3.2,2.0) and (5.0,0.9) .. (8.2,0.35);
\node[anchor=south west] at (1.2,4.6) {hot};
% medium
\draw[thick, densely dashed] (0.4,0.1) .. controls (1.6,2.9) and (2.2,3.0) .. (3.0,2.35)
  .. controls (4.4,1.5) and (5.8,0.8) .. (8.2,0.3);
\node[anchor=south] at (2.6,2.95) {warm};
% coolest
\draw[black, thick] (0.6,0.05) .. controls (2.6,1.5) and (3.4,1.55) .. (4.4,1.25)
  .. controls (5.8,0.9) and (6.8,0.55) .. (8.2,0.25);
\node[black, anchor=south] at (4.2,1.5) {cool};
% Wien locus
\draw[black, dotted, thick] (1.0,4.15) -- (3.0,2.75) -- (4.4,1.4);
\node[black, anchor=west] at (3.1,3.2) {peak locus};
\end{tikzpicture}
$$

### The Rayleigh-Jeans and Wien limits

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

$$
B_\nu(T) \approx \frac{2\nu^2}{c^2} \, kT \qquad (h\nu \ll kT).
$$

This is the **Rayleigh-Jeans law**, the classical result with no $h$. It rises as
$\nu^2$ 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, $h\nu \gg kT$, the $-1$ in
the denominator is negligible and

$$
B_\nu(T) \approx \frac{2h\nu^3}{c^2} \, e^{-h\nu/kT} \qquad (h\nu \gg kT).
$$

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.

$$
% caption: Log-log Planck spectrum bracketed by its two limits. The
% Rayleigh-Jeans law follows the nu-squared rise at low frequency and the Wien
% law the exponential fall at high frequency; the true curve turns over between.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {log frequency};
\draw[->, black] (0,0) -- (0,5.0) node[above, black!70] {log radiance};
% Planck curve
\draw[acc, very thick] (0.4,0.6) .. controls (2.4,3.4) and (3.6,4.3) .. (4.8,4.3)
  .. controls (5.9,4.3) and (6.4,2.0) .. (7.0,0.3);
% RJ asymptote (straight rising line)
\draw[black, densely dashed] (0.4,0.6) -- (5.6,4.9);
\node[black, anchor=south east, rotate=38] at (3.2,2.9) {Rayleigh-Jeans};
% Wien asymptote (steep falling)
\draw[black, densely dotted, thick] (5.5,4.9) -- (7.3,0.1);
\node[black, anchor=west] at (6.5,3.2) {Wien};
\node[acc, anchor=south] at (4.8,4.4) {Planck};
\end{tikzpicture}
$$

## 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 $B_\nu$ over all
frequencies. Substituting $x = h\nu/kT$,

$$
B(T) = \int_0^\infty B_\nu(T) \, \d\nu
     = \frac{2h}{c^2}\left(\frac{kT}{h}\right)^4 \int_0^\infty \frac{x^3}{e^x - 1}\, \d x .
$$

The dimensionless integral is a standard result,
$\int_0^\infty x^3/(e^x - 1)\,\d x = \pi^4/15$. Hence
$B(T) = (2\pi^4 k^4 / 15 c^2 h^3)\, T^4$. The energy emerging from a blackbody
surface into a hemisphere carries an extra factor $\pi$ from the angular integral
of $\cos\theta$, giving the emergent flux

$$
F = \pi B(T) = \sigma T^4, \qquad
\sigma = \frac{2\pi^5 k^4}{15 \, c^2 h^3} = 5.670 \times 10^{-8}\ \text{W}\,\text{m}^{-2}\,\text{K}^{-4} .
$$

This is the **Stefan-Boltzmann law**.[^nist] It fixes a star's luminosity from its
radius and effective temperature, $L = 4\pi R^2 \sigma T_e^4$, 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.

> **Worked example.** The Sun radiates $L_\odot = 3.83 \times 10^{26}\ \text{W}$
> from a radius $R_\odot = 6.96 \times 10^8\ \text{m}$. Solving
> $L_\odot = 4\pi R_\odot^2 \sigma T_e^4$ for the effective temperature,
>
> $$
> T_e = \left( \frac{L_\odot}{4\pi R_\odot^2 \sigma} \right)^{1/4}
>     = \left( \frac{3.83 \times 10^{26}}{4\pi (6.96 \times 10^8)^2 (5.67 \times 10^{-8})} \right)^{1/4}
>     = 5.78 \times 10^3\ \text{K} .
> $$
>
> The value $T_e = 5780\ \text{K}$ is not the temperature of any single layer; it
> is the single blackbody temperature that carries the observed surface flux.

### Wien displacement law

The wavelength at which $B_\lambda$ peaks follows from $\d B_\lambda / \d\lambda = 0$.
Writing $x = hc/\lambda kT$, the condition becomes the transcendental equation

$$
x = 5\left(1 - e^{-x}\right),
$$

whose nonzero root is $x = 4.965$. Therefore
$hc/\lambda_{\max} kT = 4.965$, or

$$
\lambda_{\max} \, T = \frac{hc}{4.965\, k} = 2.898 \times 10^{-3}\ \text{m}\,\text{K} .
$$

This is the **Wien displacement law**: the peak wavelength scales inversely with
temperature. For the Sun, $\lambda_{\max} = 2.898 \times 10^{-3} / 5780 = 501\ \text{nm}$,
in the green, consistent with the peak of the solar spectrum. A subtlety worth
stating: the peak of $B_\nu$ in frequency satisfies a different equation,
$x = 3(1 - e^{-x})$ with root $x = 2.821$, so $\nu_{\max}$ and $c/\lambda_{\max}$
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 $c/\lambda^2$.

## 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,

$$
I_\nu = B_\nu(T_b) .
$$

In the Rayleigh-Jeans regime the definition inverts cleanly. Setting
$I_\nu = (2\nu^2/c^2) kT_b$,

$$
T_b = \frac{c^2}{2\nu^2 k} \, I_\nu = \frac{\lambda^2}{2k} \, I_\nu .
$$

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 $I_\nu$.

$$
% caption: A resolved source has the same specific intensity, hence the same
% brightness temperature, at any distance; only the solid angle it subtends, and
% therefore the received flux, shrinks as one over distance squared.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% source
\draw[fill=acc!12, draw=acc, thick] (0,1.4) circle (0.85);
\node[acc, anchor=south] at (0,2.35) {source};
% near observer
\fill[black!70] (4.0,1.4) circle (2pt);
\node[black!70, anchor=south] at (4.0,1.7) {near};
\draw[black, thin] (0,2.2) -- (4.0,1.4);
\draw[black, thin] (0,0.6) -- (4.0,1.4);
% far observer
\fill[black!70] (7.8,1.4) circle (2pt);
\node[black!70, anchor=south] at (7.8,1.7) {far};
\draw[black, thin] (0,2.15) -- (7.8,1.4);
\draw[black, thin] (0,0.65) -- (7.8,1.4);
\node[black, anchor=north, text width=3.6cm, align=center] at (4.5,0.4)
  {same intensity, smaller solid angle, f\/lux falls as inverse square};
\end{tikzpicture}
$$

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 $I_\nu$ in free space into the
transfer equation that governs how a stellar spectrum is formed.[^maoz-rad]

[^bob-field]: 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.
[^bob-planck]: Carroll & Ostlie, §3.4 — the Planck function, its frequency and wavelength forms, and the Rayleigh-Jeans and Wien limits.
[^nist]: CODATA/NIST recommended values of the Stefan-Boltzmann constant, Planck constant, and Boltzmann constant — https://physics.nist.gov/cuu/Constants.
[^maoz-rad]: Maoz, Ch. 2 — basic radiative quantities and the blackbody spectrum as the reference for stellar continua.