Modern Physics

Equation

T = 2,897,000 / λmax

7.1

More evidence about atoms and light came from blackbodies, any object that absorbs all the radiation that falls on it. A blackbody must also emit the energy it absorbs or else it would not be in equilibrium and would absorb an almost infinite amount of energy! Blackbodies can’t control the radiation that falls on to them, but they re-radiate or emit radiation at a wavelength that is dependent on their temperature and is completely independent of the wavelengths of radiation absorbed. This physical fact becomes a powerful tool for astronomers wishing to understand the distant universe. If you can’t take a thermometer to the planet or star, you can still derive its temperature by looking at the blackbody radiation it emits. This light spectrum has a characteristic shape and is called a blackbody curve or Planck curve. This figure above shows the Planck curve for four stars with surface temperatures between 3,000 and 6,000 K. The Planck curve for each star peaks at a different wavelength. The hotter the star, the shorter the wavelength of its peak emission. It turns out that a star’s surface temperature can be derived from a very simple formula called Wien’s Displacement Law. As you probably can image, a German physicist named Wilhelm Wien discovered the relationship. An astronomer measures the wavelength of maximum energy emission (λmax) and uses this equation:

Temperature = constant / wavelength of maximum emission
T = 2,897,000 / λ
max

Where T is the Kelvin surface temperature of the star, λmax is the wavelength of maximum intensity in nanometers (one billionth of a meter), and 2,897,000 nm·K is the Wien’s Displacement Constant.

The Sun has an emission spectrum that peaks at around 500 nm. What is the surface temperature of the Sun?

K

λmax = b/T; so T = b / λ(max) = 2,897,000 nm·K / 500 nm = 5,795 K

1 nm = 1 x 10-9 m

The figure below illustrates this very convenient property of radiation.

Above: Image from: http://docs.kde.org/stable/en/kdeedu/kstars/ai-colorandtemp.html

The white curves in the top graphs show the brightness of the stars against a visible light spectrum. The bottom images show colors matching each brightness distribution curve.

Another example:
Betelgeuse is a red super giant star in the constellation of Orion and the eighth brightest star in the sky. It’s blackbody radiation spectrum peaks at around 970 nm. What is its surface temperature? Is it hotter or cooler than our sun?

T = 2,897,000 / λmax

Applying it to Betelgeuse:

T = 2,897,000/970 = 2,987 K

Our Sun has a surface temperature of 5,795 K, almost twice as hot as the red star Betelgeuse.

Rigel is a blue super giant star also in Orion. It’s spectrum peaks at about 145 nm. What is Rigel’s surface temperature? How does this temperatures compare with our sun?

K