Astrophysics Part II: Stars as Black Bodies and Wien’s Law

In the earlier post, I described the solar system, and introduced concepts such as luminosity and brightness. These two qualities, denoted as L and b, are also used in other equations, which describe the characteristics of stars. But first the concept of black body radiators must be introduced.

A black body is a hypothetical object which absorbs all incident electromagnetic radiation

The emission spectra of black bodies are related to the temperature of the body. As the temperature of the body is increased:

• the intensity of each wavelength increases
• the total energy emitted is higher
• the increase in intensity of shorter wavelengths is greater than the increase in intensity of longer wavelengths

A black body is hypothetically black at room temperature, however, at higher temperatures, the object would begin to emit thermal radiation and glow. Mathematically, this can be expressed as:

E∝T⁴

This equation shows the proportionality and is referred to as Stefan’s Law. A commonly referred to example of a black body is that of a box which has had its insides painted completely black. The box is completely sealed, but for a small hole made in one of the sides. When placed in a bright room, light would be able to enter the box. However when inside, the black walls would absorb the majority of the light when the light ray bounced off the walls. Only a small portion of the light would be reflected each instance, so there would only be a small chance of any of the light ever being reflected back out of the small hole. A diagram aids the explanation:

When written completely, the equation is:

L=4πR²σT⁴

Here, R refers to the radius of the concerned star, sigma σ refers to the Stefan-Boltzmann constant, which has the value 5.67×10^-8 W m^-2 K^-4. The units cancel when multiplied together, so the luminosity has the units of watts, denoted as W. Using the equation, we can determine the luminosity of a star.

Again using the concept of a black body, another equation called ‘Wien’s law‘ shows a relationship between the wavelength which is emitted with maximum intensity from the star, and the temperature of the star. This equation is used for all black bodies, and as stars are considered to be black bodies, the Wien’s Displacement Law can be used for stars as well.

Here, the intensity peaks when the wavelength denoted by the symbol lambda is at just over 2μm. This value of wavelength is used as the λ_max. Whenever we see such a graph, by observing the wavelength with peak intensity, this value is the λ_max. The relationship between λ_max and temperature T is inversely proportional.

T∝k/λ_max

here ‘k’ refers to a constant.

The actual value of this constant is 2.9×10^-3 m K, thus the equation can be written as an expression of T in terms of λ_max. The value for λ_max must be expressed in m. 

Using this equation, by observing a star’s spectrum, the temperature of it can be predicted.