The first term on the right side of Eq. (6.22) provides information about the addition of energy by the atmosphere while the second term gives insights into how the atmosphere absorbs energy. The extinction coefficient (av) is related to the optical depth (sv) of the atmospheric gases:
The ratio of extinction and absorption coefficients is called the source function (Sv):
Applying these substitutions to Eq. (6.22) gives the equation of radiative transfer:
If Sv does not depend on the optical depth of the atmosphere, we can integrate Eq. (6.25) to get
An optically thick atmosphere has sv ^ 1 and Eq. (6.26) reduces to Iv = Sv, indicating that the intensity of the emitted light entirely results from energy emitted from within the atmosphere. An optically thin atmosphere has sv ^ 1, reducing Eq. (6.26) to Iv = Iv(0). In this case, the radiation is minimally affected by its passage through the atmosphere and its intensity is defined by the incident radiation. The amount of atmosphere through which the radiation passes varies with zenith angle (angular distance of object from overhead). Thus, the change in intensity can be related to both the optical depth (sv) and the zenith angle (z) through Beer's law:
The temperature gradient can never exceed the dry adiabatic lapse rate under equilibrium conditions, so we can determine whether radiation or convection dominates within a particular region of an atmosphere by comparing these two values. Superadiabatic conditions occur when the observed thermal gradient exceeds the dry adiabatic lapse rate:
Convection will drive the temperature gradient into an adiabatic lapse rate under superadiabatic conditions, so if Eq. (6.28) is satisfied we can say that convection is the primary energy transport mechanism within that part of the atmosphere.
Present-day martian atmosphere Radiation will dominate when
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