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12.16 Stefan-Boltzmann Law

The size of Betelgeuse compared to our solar system. Click here for original source URL

The smooth part of the Sun’s spectrum is due to thermal emission from the photosphere, at a temperature of 5700 K. We see stars in the sky that are bluer or redder than the Sun - they have photospheres that are respectively hotter or cooler than the Sun’s. But how large are these stars compared to the Sun? The Stefan-Boltzmann law relates a star’s size to its temperature and luminosity; it applies not just to stars but to any object emitting a thermal spectrum (this includes the glowing metal burners on electric stoves, and filaments in light bulbs). The mathematical form of the law states that the luminosity L is proportional to the star’s surface area and the fourth power of its surface temperature:

L = 4 π R2 σ Τ 4

If the radius R of the star is given in meters and the temperature is in Kelvin, the numerical constant σ = 5.67 x 10-8 and the luminosity will come out in Watts. (Unsurprisingly, σ is called the Stefan-Boltzmann constant.)

Hertzsprung-Russell diagram showing color and size of stars. Click here for original source URL.

A star’s luminosity is related to its surface area (4 π R2) and the amount of energy emitted by each square meter of the surface (σ Τ 4). Consider changing only the temperature or radius of a star to see what effect this has on luminosity. If the temperature of a star doubles, the amount of energy radiated increases by 24, or a factor of 16. Thus while doubling the radius of a star would increase the output by a factor of four, doubling its temperature would increase its output 16 times. Hotter stars not only radiate bluer light than cooler stars (a result that was predicted by Wien's law) but also more light per unit area at every wavelength. Remember that blue light has a higher energy that red light. One way to think of this is that if you heat an iron poker, it goes from glowing red (at cooler temperatures) to yellow to white-hot (at higher temperatures).

Here is the Stefan-Boltzmann equation applied to the Sun. The Sun’s luminosity is 3.8 x 1026 Watts and the surface (or photosphere) temperature is 5700 K. Rearranging the equation above: R = √ (L / 4 π R2 σ Τ 4) = √ (3.8 x 1026 / 4 π x 5.67 x 10-8 x 57004) = 7 x 108 meters. This works for any star. Just plug in the luminosity and the surface temperature and you can calculate the radius. Astronomers do a lot of diagnostics of stellar evolution on a Hertzsprung-Russell diagram, or HR diagram. This plots the luminosity of stars against their temperatures, and then we can use the Stefan-Boltzmann law to calculate a radius for any star on the diagram.

Here is what the Stefan-Boltzmann law implies for stars of luminosity and temperature quite different from the Sun, of any value. Rearranging the normal form of the equation to put radius on the left-hand side gives:

R ∝ √ (L / T2)

In this equation, the symbol ∝ means "proportional to." This allows us to form ratios in comparing the Sun to other stars, in which case the numerical constants in the Stefan-Boltzmann law cancel out. If we let the subscript "*" refer to any star and the subscript ☉ (or a circle with a dot in it) refer to the Sun, we can write:

R*/R = √ (L*/L) / (T*/T)2

There is a whole family of stars, called main sequence stars that are producing energy by the fusion of hydrogen into helium in their centers. The most massive of these stars is about a million times as luminous as the Sun and has a surface temperature of about 40,000 K. Therefore, L*/L ☉ = 106 and T*/T ☉ = 40,000/5700 = 7. Substituting in the last equation, we get R*/R ☉ = √ (106) / 49 = 20. So, there are stars 20 times larger than the Sun that fuse hydrogen into helium. The least massive of these stars is about one thousandth the luminosity of the Sun and has a surface temperature of only 2300 K. We see that L*/L ☉ = 10-3 and T*/T ☉ = 2300/5700 = 0.4. By the same reasoning, we get that R*/R☉ = √ (10-3) / 0.16 = 0.2. That is, there are stars one fifth the size of the Sun that fuse hydrogen into helium.

Calculations of star size also yield remarkable results if we consider stars that have the same spectral type (temperature) as the Sun but very different values of luminosity. Stars with the same spectral type have the same surface temperature. In this case, the Stefan-Boltzmann law simplifies even further to R ∝ √(L). There are stars with the same color as the Sun but 100,000 times the luminosity. These giant stars must be √ (105) = 300 times the size of the Sun. There are stars with the same color as the Sun with 1/10,000 the luminosity. These dwarf stars must be √ (10-4) = 0.01 times the size of the Sun. Although the Sun is a typical star, the range of stellar types is enormous! In every case, the Stefan-Boltzmann law allows us to estimate the size without a direct measurement.