Radiation From the Stars

This is an Applied Project from James Stewart's Calculus 6E - Instructor's Edition. It might take a long time, so don't get into it if you're in a hurry. Have fun.

Any object emits radiation when heated. A blackbody is a system that absorbs all the radiation that falls on it. For instance, a matte black surface or a large cavity with a small hole in its wall (like a blastfurnace) is a blackbody and emits blackbody radiation. Even the radiation from the sun is close to being a blackbody radiation.

Proposed in the late 19th century, the Rayleigh-Jeans Law expresses the energy density of blackbody radiation of wavelength λ\lambda as

f(λ)=8πkTλ4f(\lambda)=\frac{8\pi kT}{\lambda^4}

where λ\lambda is measured in meters, TT is te temperature in kelvins KK, and kk is Boltzmann's constant. The Rayleigh-Jeans Law agrees with experimental measures for long wavelengths but disagrees drastically for short wavelengths. [The law predicts that f(λ)f(\lambda)\rightarrow\infty as λ0%+\lambda \rightarrow 0\%+ but experiments have shown that fλ0f\lambda \rightarrow 0.] This fact is known as the ultraviolet catastrophe.

In 1900 Max Planck found a better model (known now as Planck's Law) for blackbody radiation:

f(λ)=8πhcλ5ehc(λKT)1f(\lambda)=\frac{8\pi hc\lambda^{-5}}{e^{\frac{hc}{(\lambda KT)}}-1}

where λ\lambda is measured in meters, TT is the temperature (in kelvins), and

h=h= Planck's constant ==6.6262×1034Js6.6262\times 10^{-34}J\cdot s

c=c= speed of light =2.997925×108m/s=2.997925\times 10^8m/s

k=k= Boltzmann's constant =1.3807×1023J/K=1.3807\times 10^{-23}J/K.

I. Use L'Hôpital's Rule to show that

limλ0+f(λ)=0\lim _{ \lambda\rightarrow0^+ }{ f(\lambda) }=0

for Planck's Law. So this law models blackbody radiation better than the Rayleigh-Jeans Law for short wavelengths.

II. Use a Taylor polynomial to show that, for large wavelengths, Planck's Law gives approximately the same values as the Rayleigh-Jeans Law.

III. Graph ff as given by both laws on the same screen and comment on the similarities and differences. Use T=5700KT=5700K (the temperature of the sun (on surface)). (You may want to change from meters to the more convenient unit of micrometers: 1μ=106m1\mu=10^{-6}m).

IV. Use your graph in Problem 3 to estimate the value of λ\lambda for which f(λ)f(\lambda) is a maximum under Plack's Law.

V. Investigate how the graph of ff changes as TT varies. (Use Planck's Law.) In particular, graph ff for the stars Betelgeuse (T=3400KT=3400K), Procyon (T=6400KT=6400K), and Sirius (T=9200KT=9200K) as well as the sun. How does the toal radiation emitted (the area under the curve) var with TT? Use the graph to comment on why Sirius is known as a blue star and Betelgeuse as a red star.

Problem credit: James Stewart: _Calculus: _6th Edition, Instructor's Edition.

Note by John M.
6 years, 3 months ago

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