a. Find the solar power received by the bright side of the planet. · OlympiadBench #1458

Physics · thermodynamics

Problem

a. Find the solar power PP received by the bright side of the planet. (Hint: the Stefan-Boltzmann law states that the power emitted by a blackbody with area AA is σAT4\sigma A T^{4}.) Given constants: g=9.8 N/kgG=6.67×10−11 N⋅m2/kg2k=1/4πϵ0=8.99×109 N⋅m2/C2km=μ0/4π=10−7 T⋅m/Ac=3.00×108 m/skB=1.38×10−23 J/KNA=6.02×1023( mol)−1R=NAkB=8.31 J/(mol⋅K)σ=5.67×10−8 J/(s⋅m2⋅K4)e=1.602×10−19C1eV=1.602×10−19 Jh=6.63×10−34 J⋅s=4.14×10−15eV⋅sme=9.109×10−31 kg=0.511MeV/c2(1+x)n≈1+nx for ∣x∣≪1sin⁡θ≈θ−θ3/6 for ∣θ∣≪1cos⁡θ≈1−θ2/2 for ∣θ∣≪1\begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\ c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} & k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K} \\ N_{\mathrm{A}}=6.02 \times 10^{23}(\mathrm{~mol})^{-1} & R=N_{\mathrm{A}} k_{\mathrm{B}}=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) \\ \sigma=5.67 \times 10^{-8} \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right) & e=1.602 \times 10^{-19} \mathrm{C} \\ 1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J} & h=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.14 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s} \\ m_{e}=9.109 \times 10^{-31} \mathrm{~kg}=0.511 \mathrm{MeV} / \mathrm{c}^{2} & (1+x)^{n} \approx 1+n x \text { for }|x| \ll 1 \\ \sin \theta \approx \theta-\theta^{3} / 6 \text { for }|\theta| \ll 1 & \cos \theta \approx 1-\theta^{2} / 2 \text { for }|\theta| \ll 1 \end{array} Extra Supplementary Reading Materials: Green Revolution 1{ }^{1} In this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius RR, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature T1T_{1}, while the dark side has a constant uniform temperature T2T_{2}. The orbit radius of the planet is R0R_{0}, the Sun has temperature TsT_{s}, and the radius of the Sun is RsR_{s}. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies.
Intuitive explanation

The Sun radiates power uniformly in all directions as a blackbody; only a tiny fraction of that power intercepts your planet, which presents a circular cross-section to the incoming radiation. Use the Stefan-Boltzmann law to find the Sun's total power output, then calculate what fraction actually hits the planet's disk based on the geometry of the orbit.

Intuitive explanation

The Sun radiates power uniformly in all directions as a blackbody; only a tiny fraction of that power intercepts your planet, which presents a circular cross-section to the incoming radiation. Use the Stefan-Boltzmann law to find the Sun's total power output, then calculate what fraction actually hits the planet's disk based on the geometry of the orbit.

Related theory

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