law

Planck's law

Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium.
6 related articles · 6 formulas · 1 who worked here

Description

Planck's law explains why heated objects glow, from embers to stars. It tells how much light of each 'rainbow' band an ideal black body emits at a given temperature. The hotter it is, the more light there is and the bluer it gets.

History

In the late 19th century, the problem of thermal radiation was investigated by Gustav Kirchhoff, Josef Stefan, Ludwig Boltzmann, Wilhelm Wien. The classical Rayleigh-Jeans theory predicted infinite energy in the ultraviolet. Max Planck, trying to reconcile theory with experiment, reported a formula on October 19, 1900, and in December gave its theoretical justification, introducing the concept of the quantum of action h. Later, the law was experimentally confirmed and became a cornerstone of quantum mechanics. The cosmic microwave background radiation, discovered in 1965, matches the Planck spectrum with high precision.

How it works

It is applied in astrophysics to measure the temperature of stars and interstellar dust, in cosmology — to analyze the spectrum of the cosmic microwave background, in pyrometry, thermal imaging, and calibration of radiation sources. It is applicable to all bodies close to black bodies over a wide temperature range. It is generalized to gray bodies through the emissivity coefficient.

Fun fact

If our eyes could see microwaves, we would notice a faint glow across the entire sky — this is the cosmic microwave background radiation, the cooled light of the Big Bang, obeying Planck's law at a temperature of just 2.7 K.

Concept graph →

Formulas

$$L_{\rm Edd} = \frac{4\pi G M c}{\kappa}$$
Radius where radiation pressure balances gravity defines Eddington limit.
What is in the formula
L_{\rm Edd}variableEddington luminosity — luminosity watt
MvariableMass of the object — mass kilogram
cvariableSpeed of light — speed metre per second
GconstantNewtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
\kappaconstantOpacity of the stellar material — opacity = 0.034 square metre per kilogram
r_{\rmqs}\simeq\sqrt{\frac{l_{\rmedd}}{4πσt_{\rmeff}^4}} · 2606.02539v1
$$f_\lambda \propto \lambda^\beta$$
Spectral flux density follows a power law in wavelength.
What is in the formula
f_λvariablespectral flux density per unit wavelength — spectral flux density watt per metre cubed
λvariablewavelength — wavelength metre
βvariablespectral index — spectral index
∝operatoris proportional to — proportional to
f_{λ,{rest}}=f_{λ,0}(\frac{λ}{1500{\aa}})^{β_{{uv}}} · 2606.03522v1
s_ν\proptoν^α · 2606.22354
\mathcal{p}_\mathcal{r}(k)\proptok^{-5} · 2606.23307
\frac{de_b}{dω}\propto\frac{b_0^2m_0^2}{τ_h^{2/3}}ω^{-8/3} · 2607.16880v1
$$L_{\mathrm{Edd}} = \frac{4\pi G M c}{\kappa}$$
Maximum luminosity where radiation pressure balances gravity.
What is in the formula
L_EddvariableEddington luminosity — luminosity watt
MvariableMass of the object — mass kilogram
κvariableOpacity of the stellar material — opacity square metre per kilogram
GconstantConstant of universal gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
cconstantSpeed of light in vacuum — speed of light = 299792458 metre per second
πconstantMathematical constant pi — pi = 3.14159
m<\frac{\kappa_{es}l_{bol}}{4πgc} · 2606.30711
$$B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/k_B T} - 1}$$
Describes spectral radiance of blackbody radiation.
What is in the formula
B_νvariableSpectral radiance per unit frequency — spectral radiance watt per metre squared per steradian per
νvariableFrequency of radiation — frequency hertz
TvariableAbsolute temperature of blackbody — temperature kelvin
hconstantPlanck constant, relating energy of a photon to its frequency — planck constant = 6.62607015e-34 joule second
cconstantSpeed of light in vacuum, fundamental constant — speed of light = 299792458 metre per second
k_BconstantBoltzmann constant, relating temperature to energy — boltzmann constant = 1.380649e-23 joule per kelvin
νi_ν=\frac{2hν^4}{c^2}\frac{1}{\exp(hν/k_bt)-1} · 2607.14276
$$\rho(r) = \begin{cases} \rho_1 & r < r_0 \\ \rho_1 \left(\frac{r_0}{r}\right)^\alpha & r \geq r_0 \end{cases}$$
Density profile with inner constant and outer power law.
What is in the formula
\rhovariablemass density — mass density kilogram per cubic metre
rvariableradial distance from center — length metre
\rho_1variableconstant inner density — mass density kilogram per cubic metre
r_0variablebreak radius — length metre
\alphavariablepower-law index — dimensionless
ρ(r)=\begin{cases}ρ_0&r<r_0\ρ_0(\frac{r_0}{r})^2&r\geqr_0\end{cases} · 2607.24934v1
$$n(r) \propto r^{-\beta}$$
Density decreases with radius following a power law.
What is in the formula
nvariablenumber density — number density per metre cubed
rvariableradial distance — radius metre
βvariablepower-law index — power law index
∝operatoris proportional to — proportionality
n_h\proptor^{-β},\quadβ<2 · 2606.04711v1

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