Black body radiation (Topic 9C)
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A black body radiator is a theoretical object that absorbs and emits radiation at all wavelengths. While true black bodies are ideal and do not exist in reality, stars provide the closest real-world approximation.
The spectrum of radiation emitted by a black body is determined solely by its temperature.
The intensity–wavelength graph for black bodies shows the relationship between the temperature and the peak wavelength of emitted radiation for different objects. As the temperature in kelvin rises, the peak wavelength reduces, and the intensity increases.
Wien’s displacement law relates the peak wavelength of radiation emitted by an object to its surface temperature. It states that the wavelength at which the radiation curve peaks is inversely proportional to the object’s temperature:
Where:
- is the peak wavelength (),
- is the surface temperature (), and
- is Wien’s constant (; metres kelvin).
Based on Wein’s displacement law:
- Hotter objects emit radiation with shorter peak wavelengths, meaning they appear white or blue.
- Cooler objects have longer peak wavelengths, giving them a red or yellow appearance.
- Hotter objects also emit greater intensity at each wavelength compared to cooler ones.
Recall how wavelength varies along the electromagnetic spectrum, so reducing wavelength means the radiation moves from the radio end of the spectrum towards the gamma end of the spectrum. Within the visible light region, a lower wavelength means light moves from red to violet.
Do not forget to convert all temperatures given to kelvin instead of Celsius. This is done by adding 273 to the temperature in Celsius.
Question walkthrough
Wien's Law, Temperature, and Star Colour
Uses Wien's displacement law to find a star's surface temperature in Celsius from its peak emission wavelength, then links that wavelength to the star's apparent colour.
Intensity is the power per unit area carried by a wave and is proportional to the square of the amplitude. This means that if the amplitude of a wave doubles, its intensity increases by a factor of four.
Intensity represents the amount of energy transmitted by the wave per second over a given area. In a progressive wave, intensity decreases as the wave spreads:
Where:
- is intensity (in watts per square metre,
- is the power carried by the wave (in watts, W), and
- is area over which the wave is spread (in square metres, ).
Luminosity is the total amount of energy that a star (or any radiating object) emits per second in the form of electromagnetic radiation.
The luminosity of an object is determined by two main factors:
- Its surface temperature
- Its surface area
The relationship between these factors is described by the Stefan-Boltzmann law (or Stefan’s law). This states that the total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body:
Where:
- is the luminosity of the star ,
- is the radius of the star ,
- is the Stefan-Boltzmann constant , and
- is the surface temperature of the star .
From the Stefan-Boltzmann law:
We can see that the luminosity of a star is proportional to:
- Its radius:
- Its surface area:
- Its absolute surface temperature:
Remember that the surface area of a star (or any spherical object) can be calculated using the following formula:
Question walkthrough
Star Luminosity from Radius and Temperature
Uses the Stefan-Boltzmann law to calculate a star's luminosity from its radius and surface temperature, rounding the answer to an appropriate number of significant figures.