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Chapter 9: Problem 17

Treating Figure \(9.9\) somewhat literally, in which one out of every four infrared photons escapes without being absorbed by a greenhouse gas molecule, what would the effective upward radiation be if \(400 \mathrm{~W} / \mathrm{m}^{2}\) left the ground, and re-radiation of the absorbed fraction was split equally between upward radiation and radiation returning to the ground? \(^{83}\)

Short Answer

Expert verified
Answer: To find the effective upward radiation, we follow these steps: Step 1: Calculate the absorbed radiation: Absorbed radiation = 0.75 * 400 W/m² = 300 W/m². Step 2: Calculate the re-radiated upward portion: Re-radiated upward portion = 0.5 * 300 W/m² = 150 W/m². Step 3: Calculate the effective upward radiation: Effective upward radiation = 150 W/m² (re-radiated upward portion) + 100 W/m² (escaped radiation) = 250 W/m². So, the effective upward radiation is 250 W/m².

Step by step solution

01

Calculate the absorbed radiation

Given that one out of every four infrared photons escapes without being absorbed, so the absorption rate is 75%, meaning that 75% of the radiation is absorbed by the greenhouse gases. To find the absorbed radiation, we can find 75% of 400 W/m²: Absorbed radiation = (75% of 400 W/m²) = 0.75 * 400 W/m².
02

Calculate the re-radiated upward portion

Now that we have the absorbed radiation, we can find the re-radiated portion that goes upward. The absorbed radiation is split equally between upward radiation and radiation returning to the ground, so half of the absorbed radiation is radiated upward: Re-radiated upward portion = (50% of absorbed radiation) = 0.5 * absorbed radiation.
03

Calculate the effective upward radiation

To find the effective upward radiation, we can add the re-radiated upward portion (from Step 2) to the escaped radiation (25% of 400 W/m²). So: Effective upward radiation = Re-radiated upward portion + Escaped radiation. Now substituting the values from Steps 1 and 2 to find the effective upward radiation.

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