Question #92044
Assuming earth re emits all the radiations it receives from the sun.Calculate the temperature of the earth(given stefan’s constant and E)
1
Expert's answer
2019-07-29T08:18:00-0400

An easy calculation is to start with the solar constant, the power (energy per unit time) produced by solar radiation at a distance of one astronomical unit. This is 1.361 kilowatts per square meter. The surface area of the Earth is 4πR24\pi R^2, where RR is the radius of the Earth, while the cross section of the Earth to solar radiation is πR2\pi R^2. Thus the Earth as a whole receives 1/4 of that solar constant.

Therefore the magnitude of the incident radiation flux is equal to:


J0=πR2EJ_0=\pi R^2 E


where EE is Solar constant.

Due to the fact that the Earth reflects part of the radiation, taking into account the average Earth’s albedo over the entire spectrum, the energy flux absorbed by the planet will be equal to:

J1=πR2E(1a)J_1=\pi R^2 E (1-a)


where aa is geometric albedo of the Earth.

In equilibrium, the flux of absorbed energy is equal to the radiated flux (expressed from the Stefan-Boltzmann law), therefore we obtain the equality

πR2E(1a)=4πR2σT4\pi R^2 E (1-a)=4\pi R^2 \sigma T^4

where σ\sigma is Stefan–Boltzmann constant and TT is effective temperature.So, we have

T=(1a)E4σ4T=\sqrt[4]{\frac{(1-a)E}{4\sigma}}

Assuming Earth re emits all the radiations it receives from the sun we have a=1a=1. So, T=0T=0


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