Black-Body Radiation

Planck's law

Spectral radiance (power per unit area per solid angle per wavelength):

B_λ(T) = (2hc²/λ⁵) · 1/(exp(hc/(λkT)) - 1)

Or equivalently per unit frequency:

B_f(T) = (2hf³/c²) · 1/(exp(hf/(kT)) - 1)

Plotted as function of λ at fixed T, this gives a peak at λ_max with characteristic shape.

Important derived laws

Law	Formula	Description
Stefan-Boltzmann	P/A = σ·T⁴	Total emission, all wavelengths
Wien's displacement	λ_max · T = 2.898 × 10⁻³ m·K	Peak wavelength shifts with T
Rayleigh-Jeans (low f limit)	B ≈ 2cf²kT/c²	Long wavelengths; classical
Wien (high f limit)	B ≈ (2hf³/c²)·exp(-hf/kT)	Short wavelengths

Real-world blackbody temperatures

Object	T (K)	λ_max	Notes
CMB	2.725	1.063 mm	Microwave
Liquid helium	4	720 µm	Far IR
Liquid nitrogen	77	38 µm	Mid-far IR
Earth (avg)	288	10 µm	Mid IR (thermal camera)
Human body	310	9.4 µm	Mid IR
Hot stove (red)	~1000	2.9 µm	Near IR + visible red
Light bulb filament	~2700	1073 nm	Mostly near IR + some visible (inefficient)
Sun's surface	5778	502 nm	Peak in visible (green/yellow)
O-class star (hottest main sequence)	30,000+	97 nm	UV
Sirius B (white dwarf)	25,200	115 nm	UV

JavaScript — blackbody calculations

const SIGMA = 5.67e-8;
const h = 6.626e-34;
const c = 3e8;
const k = 1.381e-23;
const WIEN_B = 2.898e-3;  // m·K

// Stefan-Boltzmann power
function blackbodyPower(T_K, area = 1, emissivity = 1) {
  return emissivity * SIGMA * area * Math.pow(T_K, 4);
}

console.log(`Sun surface (5778 K): ${(blackbodyPower(5778) / 1e6).toFixed(1)} MW/m²`);
console.log(`Human body (310 K): ${blackbodyPower(310).toFixed(0)} W/m²`);
console.log(`Earth avg (288 K): ${blackbodyPower(288).toFixed(0)} W/m²`);

// Wien's peak wavelength
function wienPeak(T_K) {
  return WIEN_B / T_K;  // meters
}

console.log(`Sun peak λ: ${(wienPeak(5778) * 1e9).toFixed(0)} nm`);  // 502
console.log(`Human peak λ: ${(wienPeak(310) * 1e6).toFixed(2)} µm`); // 9.35

// Planck spectral radiance (per λ)
function planckSpectralRadianceLambda(T_K, lambda_m) {
  const exponent = h * c / (lambda_m * k * T_K);
  return (2 * h * c * c / Math.pow(lambda_m, 5)) / (Math.exp(exponent) - 1);
}

// Sun output at peak wavelength
console.log(`Sun at 502 nm: ${planckSpectralRadianceLambda(5778, 502e-9).toExponential(2)} W/(m²·sr·m)`);

// Find effective temperature from peak wavelength (T from λ)
function tempFromPeak(lambda_max_m) {
  return WIEN_B / lambda_max_m;
}

console.log(`Star with peak 350 nm: T = ${tempFromPeak(350e-9).toFixed(0)} K`);

// Earth's effective temperature (rough)
function planetEquilibriumTemp(albedo, distance_AU) {
  // Solar flux at distance d: 1361/d² W/m²
  const solar_flux = 1361 / (distance_AU * distance_AU);
  // Absorbed: solar_flux × (1-albedo) × π·R² (area)
  // Emitted: σ·T⁴ × 4π·R² (full sphere)
  // Equilibrium: solar_flux·(1-albedo)/4 = σ·T⁴
  return Math.pow(solar_flux * (1 - albedo) / (4 * SIGMA), 0.25);
}

console.log(`Earth effective T (no greenhouse): ${planetEquilibriumTemp(0.3, 1).toFixed(0)} K`);
// 254 K (-19°C); actual surface 288 K due to greenhouse warming

// CMB temperature evolution with redshift
function cmbTemp(z) {
  return 2.725 * (1 + z);
}

console.log(`CMB at z=1: ${cmbTemp(1).toFixed(2)} K`);  // 5.45
console.log(`CMB at recombination (z~1100): ${cmbTemp(1100).toFixed(0)} K`);  // ~3000

// Compute total emission
function totalEmissionRate(T_K, surfaceArea_m2 = 1) {
  return SIGMA * Math.pow(T_K, 4) * surfaceArea_m2;
}

// Sun's total luminosity (radius 6.96e8 m, T 5778 K)
const sunLuminosity = totalEmissionRate(5778, 4 * Math.PI * Math.pow(6.96e8, 2));
console.log(`Sun's luminosity: ${(sunLuminosity / 1e26).toFixed(2)} × 10²⁶ W`);
// ~3.83 × 10²⁶ W (matches measured value)

Where blackbody radiation matters

• Stellar physics. Star surface temperatures inferred from spectra (color-color analysis).
• Cosmic microwave background. Best blackbody spectrum measured; key probe of early universe.
• Earth's energy balance. Greenhouse effect, climate models.
• Thermal imaging. Cameras detect IR radiation from objects.
• Lighting. Incandescent bulbs are blackbody-like; LEDs and fluorescents are not.
• Material design. Solar absorbers, radiative cooling materials use spectral tuning.
• Quantum physics origin. Planck's law required quantization → birth of QM.

Common mistakes

• Treating real objects as ideal blackbodies. Real objects have emissivity < 1; modify Stefan-Boltzmann formula. Color, surface texture matter.
• Using Wien's law in wavelength but forgetting frequency form differs. λ_max · T constant in wavelength view; f_max · T constant for frequency view (different constants!).
• Forgetting T⁴ scaling. Doubling T → 16× radiated power. Massive sensitivity.
• Using Celsius for Stefan-Boltzmann. Always Kelvin. Power scales with T⁴.
• Confusing absorptivity with emissivity. At thermal equilibrium, ε = α (Kirchhoff's law). Same wavelength, same magnitude.
• Missing greenhouse effect. Earth's surface is hotter than blackbody equilibrium because atmosphere traps IR. Greenhouse gases absorb in IR but transmit visible (sunlight in, IR less out).