Some exoplanets possess atmospheres that absorb a fraction of the infrared radiation…
PHYSICS · Multi-part · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter
Problem
Some exoplanets possess atmospheres that absorb a fraction g of the infrared radiation emitted by the planetary surface and re-radiate that absorbed energy isotropically, sending half back toward the surface and half out to space. The atmosphere is nearly transparent to incoming stellar radiation. The planet reflects a fraction A of the incident stellar radiation and absorbs the remainder.
Consider a planet receiving an incident stellar flux S (power per unit area measured at the top of the atmosphere). The planet’s surface and atmosphere each reach their own steady temperatures. Assume the surface and the atmosphere each behave like gray bodies with the same infrared emissivity, which you may call ε.
Respond to the tasks below.
(1) Part 1: Draw and label an energy-flow diagram that identifies all significant energy exchanges among the star, the planetary surface, the atmosphere, and space. Include absorbed, emitted, and transmitted energy flows, and indicate the fractional coefficients where appropriate.
(2) Part 2: Using your diagram, derive an algebraic expression for the steady-state surface temperature Ts of the planet in terms of S, A, g, ε, and fundamental constants.
(3) Part 3: Evaluate your expression for the special case ε=1, A=0.30, g=0.40, and S=1350W m−2. Present your numerical answer for Ts in kelvins.
(4) Part 4: Explain qualitatively how increasing the parameter g affects the planet’s surface temperature and give a physical interpretation of this result.