The temperature of a planet irradiated by solar radiation can be estimated by balancing the amount of **radiation absorbed** (**Ra**) against the amount of **outgoing radiation** (**Ro**). The **Ra** will be the product of:

(i) Solar irradiance (**I**)

(ii) *Area of planet*. Area of planet relevant for such calculations is the area of the planet as seen by incoming radiation which is given by r^{2} where r = radius of the planet.

(iii) *Absorbed fraction of radiation*. The fraction of radiation that is absorbed is given by (1 – A where A = albedo of planet. This albedo represents the fraction of radiation that is reflected back from the planet.

Figure. 1. Energy balance of the Earth. (Components values in kcal/cm^{2}/year).

Thus the energy absorbed by the planet will be:

**Ra = I ^{} (1 – A)**

## Intensity of outgoing radiation of a body is given by Stefan-Boltzmann Law i.e.

## Io = T^{4}

where ****= Stefan-Boltzmann Constant = 5.6 x 10^{-8} Wm^{-2} K^{-4}. The total energy radiated by the planet will be the product of the intensity of outgoing radiation (**Io**) and the area of the whole planetary surface giving out radiation (4 r ^{2}). Thus, the outgoing radiation (**Ro**) from the planet is given by:

**Ro = 4r ^{2} T^{4}**

**Effective planetary temperature**

Since **Ra** = **Ro** i.e. system is assumed to be in steady-state where radiation absorbed and outgoing radiation are equal, an expression for the **effective planetary temperature** (**Te**) can be obtained from the above equation and it may be given by:

## Te = [I – (1 – A)/4 ]^{0.25}

In this expression of effective planetary temperature, effect of atmosphere has not been taken into account. For Earth, solar irradiance (**I**) at the top of atmosphere is about 1.4 x 10^{3}/m^{2}/s and albedo of Earth as a whole is about 0.33. From these values, the calculated equilibrium temperature of Earth comes to be 254 K. However, the actual observed average ground level temperature of Earth is about 288 K. This higher effective temperature of Earth from the calculated value is due to the greenhouse-effect of atmosphere.

The black-body spectrum of Earth at 288 K shows that radiation from Earth is of much longer wavelength and is at much lower intensity than radiation from Sun. The absorption spectrum of Earth’s atmosphere overlaps fairly well with the solar emission spectrum. Except for a very narrow window in the absorption bands, much of the long-wave radiations from Earth correspond with the region of absorption in the atmosphere. This means that much of the incoming radiation reaches the Earth’s surface while the outgoing thermal radiation is largely absorbed by the atmosphere rather than being lost to space. Thus, the effect of atmosphere is to trap the outgoing thermal radiation. This effect is termed **green-house effect**.. The thermal radiation i.e. the heat trapped by the atmosphere due to green-house effect is responsible for the effective temperature of Earth being higher than the temperature calculated without taking into account the effect of atmosphere. In general, absorption of re-emitted long-wave radiation and vertical mixing processes determine the temperature profile of the lower part of atmosphere (troposphere) which in turn determine the Earth’s temperature.

### Optical depth of atmosphere and Earth’s surface temperature

The atmosphere is not transparent to the outgoing long-wave radiation and much of this radiation is absorbed in the lower part of the atmosphere, which is warmer than the upper parts. Simple *radiative equilibrium models* have been developed for Earth and to account for this effect, these models divide the atmosphere into layers that are just thick enough to absorb the outgoing radiation. These atmospheric layers are said to be optically thick and the atmosphere is discussed in terms of its **optical depth** based on the number of these atmospheric layers of different optical thickness. Earth’s atmosphere is sometimes said to have two layers while that of planet Venus has almost 70 layers which are largely due to enormous amount of CO_{2} in the atmosphere of Venus. The radiation equilibrium model indicates that the effective planetary temperature (**Te**) is thus related to **ground-level planetary temperature** (**Tg**) by the equation:

**Tg ^{4 }= (1 – )Te^{4}** (where

****= optical depth of atmosphere)

The optical depth of atmosphere increases with increase in atmospheric concentrations of carbon dioxide and water vapor because both these are principal atmospheric absorbers of outgoing long-wave radiation. With increasing concentrations of CO_{2} in lower layers of atmosphere, other such gases that are responsible for radiating heat to outer space are pushed to slightly higher and colder levels of atmosphere. The radiating gases will radiate heat less efficiently because they are colder at higher altitudes. Thus, the atmosphere becomes less efficient radiator of heat and this results in rise of atmospheric temperature. This rise in atmospheric temperature, in turn, leads to more evaporation and increase in atmospheric water vapor, which is a greenhouse gas and further increases the absorption of outgoing long-wave thermal radiation. This positive feedback results in further increase in atmospheric temperature. The model also suggests that increase in atmospheric CO_{2} is associated with decrease in temperatures of upper (stratospheric).

### Vertical heat transport and Earth’s surface temperature

Simple models of radiation balance of atmosphere do not take into account various other processes that transport heat vertically in the atmosphere and, therefore, overestimate the surface temperature of Earth. Convection is major process of vertical heat transport and is very important in lowering the surface temperature. Convection occurs because warm air is lighter than cool air and so rises upwards carrying heat from Earth’s surface to the upper atmosphere. As warm air rises up, it expands due to fall in pressure and work done in expansion causes it to cool adiabatically. Thus,

**C _{v} T = – P V** (where Cv = molar heat capacity at constant volume)

Ideal gas equation PV = RT takes the differential form P dV + V dP = R dt which may be rearranged in incremental form as:

###### – P V + R = V P

This equation may be combined with equation C_{v} T = – P V using the fact that C_{p} – C_{v} = R, where C_{p} = molar heat capacity at constant pressure = 29.05 J/mol/K. This results in following equation:

**C _{p} T = (C_{v} + R) T = – P V + R T = V P = (RT/P) P** ………….(a)

It can be shown that P/P = – M_{m}g z /RT where M_{m} = mean molecular weight of air = 0.028966 kg/mol; g = acceleration due to gravity = 9.8065 m/s/s; z = altitude. This gives:

**RT/P = – Mmg z/ P**……………………………………………….(b)

Substitution of the above equation (b) in equation (a) gives:

**C _{p} T = – M_{m}gz**

or,** T/ z = – (M _{m}g/Cp)**

For Earth’s atmosphere, the **lapse rate** ( **T/ z**) works out to be -9.8 k/km for dry air. However, the air is usually wet and as it rises up, it releases latent heat so the measured lapse rate is -6.5 K/km.

If atmospheric temperature falls much less slowly with height than the lapse rate (or even rises with height) then inversion conditions exist and air is very stable with respect to vertical convective mixing. Conversely, if temperature falls very rapidly with height, at a rate greater than lapse rate, then the atmosphere is unstable and convective mixing will be active.

**Short-wave radiation and temperature**

The discussion till now has assumed total transparency of atmosphere to incoming solar radiation. Though it is true for visible range of radiation, it is not true for ultra-violet region of the solar spectrum. Though the amount of such short-wave radiation is very small, it has important consequences for the temperature of Earth-atmosphere system.

Various ultra-violet wavelengths are absorbed in the atmosphere at different heights. At just over 40 km, absorption of ultra-violet radiation by ozone results in considerable warming of stratosphere and in this zone, temperature rises with altitude. Average temperature of stratosphere is 250 K. Considering it to be a black-body radiator, maximum power radiation would be expected at 11.5 µm. This value is very close to absorption band of carbon dioxide which means that this gas also plays important role in stratospheric temperature. Increase in concentration of carbon dioxide in stratosphere might allow more effective radiation from stratosphere and, therefore, its cooling. This effect is quite opposite to that noted for troposphere.

Further, at the altitude of thermosphere, atmosphere is very thin. In this zone, molecules are exposed to unattenuated solar radiation of extremely short wavelength i.e. of high energy. This radiation arises from the outer region of Sun. At wavelengths below 50 nm, effective emission temperature exceeds 10,000 K. High-energy solar protons of such wavelengths are absorbed by gas molecules giving them high transitional energies i.e. high temperatures. The energies may be large enough to dissociate oxygen and nitrogen. Temperatures in thermosphere undergo wide variations depending upon the state of Sun. During solar disturbances, output of high-energy protons is very much enhanced that results in very high atmospheric temperatures. Temperature in this zone may further be increased by another mechanism. The temperature is normally defined in terms of transitional energy but absorption and emission of radiation occur through vibrational and rotational changes. In upper atmosphere, the frequency of molecular collisions is relatively low and so exchange of translational, vibrational and rotational energies is infrequent. Hence the cooling of thermosphere by re-radiation is very inefficient. The temperature of thermosphere increases with height so it is also stable against convection. Heat can be lost only by very inefficient diffusion processes and as a result, thermospheric temperatures are extremely high.

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