Chapter 19: Problem 38

As a red giant, the Sun's luminosity will be about 2000 times greater than it is now, so the amount of solar energy falling on the Earth will increase to 2000 times its present-day value. Hence, to maintain thermal equilibrium, each square meter of the Earth's surface will have to radiate 2000 times as much energy into space as it does now. Use the Stefan-Boltzmann law to determine what the Earth's surface temperature will be under these conditions. (Hint: The present-day Earth has an average surface temperature of \(14^{\circ} \mathrm{C}\).)

Short Answer

Expert verified

Using the Stefan-Boltzmann law and considering the Sun's increased luminosity, the future average surface temperature of the Earth will be approximately 72.85 degrees Celsius.

Step by step solution

Convert temperature to Kelvin

The Stefan-Boltzmann law requires temperature to be in Kelvin. So, convert Earth's current temperature (14°C) to Kelvin by adding 273.15. The result, 287.15 K, is the current temperature of the earth in Kelvin.

Apply the Stefan-Boltzmann Law

Since the energy radiated by a body is proportional to the fourth power of its temperature according to the Stefan-Boltzmann law, the future temperature (T2) can be related to the present temperature (T1) using the formula \(T2 = (2000)^{1/4} * T1\). Substituting T1 with 287.15 we obtain the future temperature of Earth in kelvin.

Convert Back To Celsius

For a more understandable result, we convert the final temperature from Kelvin to Celsius by subtracting 273.15 from the solution obtained in step 2.

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Short Answer

Step by step solution

Convert temperature to Kelvin

Apply the Stefan-Boltzmann Law

Convert Back To Celsius

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