Answer to Question #92044 in Astronomy | Astrophysics for Parul Kaushik

Question

Assuming earth re emits all the radiations it receives from the sun.Calculate the temperature of the earth(given stefan’s constant and E)

Answer 1

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 $4\pi R^2$, where $R$ is the radius of the Earth, while the cross section of the Earth to solar radiation is $\pi R^2$. Thus the Earth as a whole receives 1/4 of that solar constant.

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

$$J_0=\pi R^2 E$$

where $E$ 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:

$$J_1=\pi R^2 E (1-a)$$

where $a$ 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

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

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

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

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