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Chapter 26: Problem 41

4 Calculate the mass density of radiation \(\left(\rho_{\mathrm{rad}}\right)\) in each of the following situations, and explain whether each situation is matter-dominated or radiation-dominated: (a) the photosphere of the Sun \(\left(T=5800 \mathrm{~K}, \rho_{\mathrm{m}}=3 \times 10^{-4} \mathrm{~kg} / \mathrm{m}^{3}\right) ;(\) b \()\) the center of the Sun \(\left(T=1.55 \times 10^{7} \mathrm{~K}, \rho_{\mathrm{m}}=1.6 \times 10^{5} \mathrm{~kg} / \mathrm{m}^{3}\right)\); (c) the solar corona \(\left(T=2 \times 10^{6} \mathrm{~K}, \rho_{\mathrm{m}}=5 \times 10^{-13} \mathrm{~kg} / \mathrm{m}^{3}\right)\).

Short Answer

Expert verified
In the photosphere (\(\rho_{rad} = 1.77 \times 10^{-6} kg/m^3\)) and the center of the Sun (\(\rho_{rad} = 2.38 \times 10^{3} kg/m^3\)), matter dominates respectively. In the solar corona (\(\rho_{rad} = 0.12 kg/m^3\)), the radiation dominates.

Step by step solution

01

Photosphere of the Sun

First, calculate the radiation density using the Steven-Boltzmann law for the photosphere of the Sun where \(T=5800 K\) and \(\rho_{m} = 3 \times 10^{-4} kg/m^3\). Using the formula \(\rho_{rad} = aT^4\), this results in \(\rho_{rad} = 7.56 \times 10^{-16} (5800)^4 = 1.77 \times 10^{-6} kg/m^3\). Comparing this with \(\rho_{m}\), this situation is matter-dominated since \(\rho_{m} > \rho_{rad}\).
02

Center of the Sun

Next, look at the center of the Sun where \(T = 1.55 \times 10^{7} K\) and \(\rho_{m} = 1.6 \times 10^{5} kg/m^3\). Using the Steven-Boltzmann law again, \(\rho_{rad} = 7.56 \times 10^{-16} * (1.55 \times 10^{7})^4 = 2.38 \times 10^{3} kg/m^3\). This situation is still matter-dominated, as \(\rho_{m} > \rho_{rad}\).
03

The Solar Corona

Finally, consider the solar corona where \(T = 2 \times 10^{6} K\) and \(\rho_{m} = 5 \times 10^{-13} kg/m^3\). Here, \(\rho_{rad} = 7.56 \times 10^{-16} * (2 \times 10^{6})^4 = 0.12 kg/m^3\). The solar corona is radiation-dominated since \(\rho_{rad} > \rho_{m}\).

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Most popular questions from this chapter

If a photon from the cosmic microwave background had wavelength \(\lambda_{0}\) when it was emitted at redshift \(z\), its wavelength today is \(\lambda=\lambda_{0} /(1+z)\). (a) Let \(T\) be the symbol for the temperature of the cosmic microwave background today. Explain why the radiation temperature was \(T_{0}=T(1+z)\) at redshift \(z\). (b) What was the radiation temperature at \(z=1\) ? (c) At what redshift was the radiation temperature equal to \(293 \mathrm{~K}\) (a typical room temperature)?

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(a) Was there ever an era when the universe was radiationdominated and matter and radiation were at the same temperature? If so, approximately when was this, and were there atoms during that era? If not, explain why not. (b) Was there ever an era when the universe was radiation-dominated and matter and radiation were not at the same temperature? If so, approximately when was this, and were there atoms during that era? If not, explain why not.
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