Question

At our distance from the Sun, the intensity of solar radiation is 1370 W/m^2. As discussed in class and in the textbook, the temperature of the Earth is affected by the greenhouse effect of the atmosphere. This phenomenon describes the effect of absorption of infrared radiation by the surface so as to make the surface temperature of the Earth higher than if it were airless. The average global surface temperature of the Earth is ~288 K. Calculate the surface temperature of the Earth if it did not have an atmosphere. You may assume an emissivity of 1. The radius of the Earth is 6370 km and its distance from the Sun is 1.496 × 10^11 m.

Answer 1

The total amount of solar radiation energy striking Earth per second is equal to the solar radiation times the Earth’s cross sectional area.

Some of the radiation is reflected by clouds and  does not contribute energy to raise Earth’s temperature. The fraction which is reflected back is called Albedo.

For earth, albedo = 0 .3

so,

we can write

F * pi * r² * (1 - 0.3)

where F is solar radiation, r is earth's radius

Also,  Earth and its atmosphere radiate in all directions and the radiation can be described by the Stefan–Boltzmann Law.

assuming the emissivity as 1 , we have

T⁴

where is boltzmann constant and is equal to 5.67 x 10^–8 W m^–2 K^–4

we must account for surface area of earth (assuming it as a sphere)

so, we have

F * pi * r² * (1 - 0.3) = 4*pi*r² * T⁴

r will cancel on both sides,

F * (1 - 0.3) = 4 * T⁴

T⁴ = 0.7 * F / 4

T⁴ = 0.7 * 1370 / 4 * 5.67 x 10^–8

T = 255 K

In ^oC

T = 255 - 273

T = -18 ^oC

Life would not be possible