Question

To a rough approximation, the Earth can be treated as a greybody sphere with uniform temperature, absorptance ? = 0.70 and emissivity ? = 0.61. The low absorptance is largely due to reflection of the incoming sunlight by clouds and ice, while the low emissivity is largely due to greenhouse gases in the atmosphere.

Q1) Over timescales of decades, the total solar irradiance at the Earth varies between about 1360.3 W/m² and 1362.0 W/m² . Using the greybody Earth approximation, how large of a temperature change can be expected due to these natural changes in solar irradiance? [Answer in K as a positive number.]

Q2) Over the last century, the amount of CO2 in the atmosphere has increased dramatically, which has resulted in a change in ? from 0.62 to 0.61. Using the greybody Earth approximation, how large of a temperature change can be expected due to this change in emissivity? [Answer in K as a positive number.]

Q3) The average temperature of the Earth is about 15 ^oC. Assuming a greybody surface, at what wavelength does the Earth emit most intensely? [Answer in µm].

Answer 1

We are considering Earth as Black Body hence we will use Stefan-Boltzmann law, which says that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:

where

and a body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity,:

Let the irradiance recieved by Earth be ranging between

and assuming Kirchoff's law that absorptivity becomes equal to emissivity at equilibrium

consider average of 0.61 and 0.7 i.e 0.65

and ?=0.65

Now,

Therefore,

| | defines the absolute value.

so the change in temperature is

0.1 Kelvin after putting all the values

(b) now due to carbon dioxide the emmisivity has changed to 0.62 at equilibrium due to Kirchoff's law we will get

?=0.66

and the change in temperature following the equations above as,

0.1 K again.

(c) Using Wein's law

where T is temperature of the body(here it is 15^oC i.e. 288.3 K, b is Wein's constant 2.897771955...×10⁻³ m⋅K and lamda is the wavelength at which spectral radiance peaks.

Therefore