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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 \(\lambda\) as

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

where \(\lambda\) is measured in meters, \(T\) is te temperature in kelvins \(K\), and \(k\) 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 \(f(\lambda)\rightarrow\infty\) as \(\lambda \rightarrow 0%+\) but experiments have shown that \(f\lambda \rightarrow 0\).] 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:

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

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

\(h=\) Planck's constant \(=\)\(6.6262\times 10^{-34}J\cdot s\)

\(c=\) speed of light \(=2.997925\times 10^8m/s\)

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

I. Use L'Hôpital's Rule to show that

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

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

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

V. Investigate how the graph of \(f\) changes as \(T\) varies. (Use Planck's Law.) In particular, graph \(f\) for the stars Betelgeuse (\(T=3400K\)), Procyon (\(T=6400K\)), and Sirius (\(T=9200K\)) as well as the sun. How does the toal radiation emitted (the area under the curve) var with \(T\)? 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 Muradeli
2 years, 1 month ago

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