Black-body spectra and Wien’s law

A black body is an ideal object that absorbs all radiation incident on it and emits radiation. The spectrum of the radiation it emits depends on its temperature.

At a given temperature, a black body emits radiation across a continuous range of wavelengths. On a graph of spectral intensity against wavelength, the intensity rises to a peak and then falls. The wavelength at the peak, $$\lambda_{\max}$$, is the wavelength at which the wavelength spectrum has its greatest intensity.

When the temperature increases, the emitted intensity increases and the peak moves to a shorter wavelength. A hotter black body therefore has a shorter peak wavelength than a cooler one.

Wien’s displacement law

Wien’s displacement law relates the peak wavelength of a black-body spectrum to its absolute temperature:

Here, $$\lambda_{\max}$$ is in metres and $$T$$ is in kelvin. Their product is constant, so increasing the temperature decreases the peak wavelength. For example, doubling the absolute temperature halves the peak wavelength.

Estimating a star’s surface temperature

A star can be treated approximately as a black body. Measuring the peak wavelength of its spectrum therefore allows its surface temperature to be estimated using Wien’s law.

For a star with a peak wavelength of $$500,\mathrm{nm}$$, or $$5.0\times10^{-7},\mathrm{m}$$:

The estimated surface temperature is $$5800,\mathrm{K}$$. A star with a shorter peak wavelength would have a higher estimated surface temperature; one with a longer peak wavelength would have a lower estimated surface temperature.

Stefan’s law for stellar power

A hot surface emits energy as radiation. Its radiated power is the total energy it emits each second. Stefan’s law gives this power for a black body, an idealised emitter of radiation, from its surface area and absolute temperature:

Here, $$P$$ is power in watts, $$A$$ is emitting surface area in square metres, and $$T$$ is absolute temperature in kelvin. The Stefan–Boltzmann constant is $$\sigma = 5.67 \times 10^{-8}\ \mathrm{W,m^{-2},K^{-4}}$$.

Power is directly proportional to area: at the same temperature, doubling the emitting area doubles the power. Power is also proportional to the fourth power of temperature: for the same area, doubling the absolute temperature increases the power by a factor of $$2^4 = 16$$.

Stefan’s law for a spherical star

A star can be approximated as a spherical black body. For a star of radius $$R$$, the emitting surface area is $$A = 4\pi R^2$$. Substituting this into Stefan’s law gives its total radiated power:

The star’s power is therefore proportional to $$R^2T^4$$. At the same surface temperature, a star with twice the radius has four times the emitting area and radiates four times the power. At the same radius, a star with twice the absolute surface temperature radiates 16 times the power.

Comparing stellar power outputs

For two stars treated as spherical black bodies, the constants cancel when their powers are divided:

The subscripts identify the two stars. Use their radii in the same units and their surface temperatures in kelvin. The ratio accounts for both the difference in emitting area and the difference in temperature.

Inverse square law for radiation

Radiation from a source spreads out as it travels. If the source emits equally in all directions, its power is distributed over an expanding spherical surface centred on the source. The same power is spread over a larger area at greater distances, so the intensity decreases.

Intensity is the power received per unit area when the receiving surface is normal to the radiation. At a distance $$r$$ from the source, the spherical surface has area $$4\pi r^2$$. For a source emitting total power $$P$$ uniformly in all directions, the intensity is therefore:

Here $$I$$ is intensity in watts per square metre ($$\mathrm{W,m^{-2}}$$), $$P$$ is emitted power in watts ($$\mathrm{W}$$), and $$r$$ is distance from the source in metres ($$\mathrm{m}$$).

How intensity changes with distance

For a fixed emitted power, intensity is inversely proportional to the square of distance: $$I \propto 1/r^2$$. Doubling the distance makes the spherical area four times as large, so the intensity falls to one quarter. Tripling the distance makes the area nine times as large, so the intensity falls to one ninth.

Conditions for the inverse square law

The equation assumes isotropic emission: the source emits power uniformly in all directions. It also assumes negligible absorption or scattering between the source and receiver, so the power spreading through successive spherical surfaces remains approximately the same.

An extended source, such as a star, can be treated approximately as a point source when the distance from it is sufficiently large compared with its size.