Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 3

Hulst 1980; Minin 1988), namely:

dE

= I ( r , t ) (1.1)

λ

Ω λ dSd d dt

In many cases, we are interested not in energy emitted by the object but in
energy of the radiation field that is coming to the object (for example to the
instrument input). Then it would be easy to convert the above specification of
radiance. Consider the emitting object and set the second surface element of
the equal area dS 2 = dS at an arbitrary distance (Fig. 1.1). Let the system to

be situated in a vacuum, i. e. radiation is not interacting during the path from
dS to dS 2 .Lettheelement dS 2 to be perpendicular to the direction r , then the
solid angle at which the element dS 2 is seen from dS at the direction r is equal
tothesolidangleatwhichtheelement dS is seen from dS 2 at the opposite
direction (− r ). The energies incoming to the surface elements dS and dS 2 are
equal too thus; we are getting the consequence from the above definition of the
intensity. The factor of the proportionality of emitted energy dE to the values
dS , d

Ω λ ( r , t )incomingfromthe

direction r to the surface element dS perpendicular to r at the wavelength

, d and dt is called an intensity (radiance) I

λ

λ at

the time t , i. e. (1.1). Point out the important demand of the perpendicularity
of the element dS to the direction r in the definition of both the emitting and
incoming intensity.
The definition of the intensity as a factor of the proportionality tends to
have some formal character. Thus, the “physical” definition is often given:
the intensity (radiance) is energy that incomes per unit time, per unit solid
angle, per unit wavelength, per unit area perpendicular to the direction of
incoming radiation, which has the units of watts per square meter per micron
per steradian. This definition is correct if we specify energy to correspond not
to the real unit scale (sec, sterad,

µ

m, cm 2 ) but to the differential scale dt , d

Ω

,

λ d , dS , which is reduced then to the unit scale. Equation (1.1) is reflecting this

obstacle.
Let the surface element dS , which radiation incomes to, not be perpen-
dicular to the direction r but form the angle ϑ with it (Fig. 1.1). Specify the

incident angle (the angle between the inverse direction − r and the normal to
the surface) as

ϑ = ( n ,− r ). In that case defining the intensity as a factor of
theproportionalitywehavetousetheprojectionoftheelement dS' on a plane
perpendicular to the direction of the radiation propagation in the capacity of
the surface element dS .Thisprojectionisequalto dS = dS cos ϑ . Then the

following could be obtained from (1.1):

= λ Ω dS cos ϑ dE I ( r , t ) dt d d . (1.2)

λ

It is suitable to attribute the sign to energy defined above. Actually, if we fix
one concrete side of the surface dS and assume the normal just to this side
as a normal n then the angle

ϑ π ,andthecosinefrom+1

to −1. Thus, incoming energy is positive and emitted energy is negative. It
has transparent physical sense of the positive source and the negative sink
of energy for the surface dS .Nowspecify theirradiance(theradiationflux

varies from 0 to