(a) Integrate Eq. ( 27 ) over all wavelengths to obtain an...

(a) Integrate Eq. ( 27 ) over all wavelengths to obtain an expression for the total luminosity of a blackbody model star. Hint: $$\int_{0}^{\infty} \frac{u^{3} d u}{e^{u}-1}=\frac{\pi^{4}}{15}$$ $$=\frac{8 \pi^{2} R^{2} h c^{2} / \lambda^{5}}{e^{h c / \lambda k T}-1} d \lambda$$ (b) Compare your result with the Stefan-Boltzmann equation $(17),$ and show that the StefanBoltzmann constant $\sigma$ is given by $$\sigma=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}}$$ $$L=4 \pi R^{2} \sigma T_{e}^{4}$$ (c) Calculate the value of $\sigma$ from this expression.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann law establishes that the total energy radiated per unit surface area of a blackbody across all wavelengths is proportional to the fourth power of its absolute temperature. This law is crucial in linking the integrated emission from Planck's law to observable quantities like a star's luminosity.

Stefan-Boltzmann Constant

The Stefan-Boltzmann constant is the proportionality constant in the Stefan-Boltzmann law that relates the radiated energy flux to the fourth power of temperature. It is derived from fundamental physical constants by integrating Planck's law over all wavelengths, illustrating the deep connection between quantum theory and macroscopic thermodynamic radiation.

Integration over Wavelengths

Integrating the spectral distribution given by Planck's law over all wavelengths yields the total luminosity of a blackbody. This process involves variable substitution and the use of standard integrals, such as the provided integral, to sum the contributions of all wavelength components into one comprehensive measure of total energy output.

Blackbody Radiation

Blackbody radiation refers to the thermal electromagnetic radiation emitted by a body in thermodynamic equilibrium. The spectrum and intensity of this radiation depend solely on the body's temperature, making it a fundamental concept in understanding energy emission in stars and other celestial objects.

Planck's Law

Planck's law provides the description of the spectral energy distribution of radiation emitted by a blackbody as a function of wavelength (or frequency) and temperature. It introduces the idea that energy is quantized, which was a pivotal concept in the development of quantum mechanics.

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