specified an intensity of the radiation or radiance I l ð~ r; tÞat the wavelength l to
the direction ~ r at the moment t, namely:
I l ð~ r; tÞ ¼
dE
dSdOdldt
:
(1.1)
In many cases we are interested not in energy emitted by the object but in energy
of the radiation field, which 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 ¼ dS at an arbitrary distance (Fig. 1.1). Let the system be situated in vacuum,
i.e. radiation is not interacting during the way from dS to dS 2 . Let the element dS 2 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 to the solid angle at which the element dS is
seen from dS 2 at the opposite direction (À ~ 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, dO, dl and dt is called an intensity (radiance) I l ð~ r; tÞ incoming from the
direction~ r to the surface element dS perpendicular to~ r at the wavelength l at the time t
i.e. Eq. 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, mm, cm
2 ) but to the differential scale dt, dO, dl, dS, which is reduced
then to the unit scale. Equation 1.1 is reflecting this obstacle.
Let the surface element dS
0 , which radiation incomes to, be not perpendicular to
the direction ~ r but form the angle ϑ with it (Fig. 1.1). Specify the incident angle (the
angle between the inverse direction À ~ r and the normal to the surface) as
# ¼ ffð~ n; À~ rÞ. In that case we have to use the projection of the element dS
0 on a
plane perpendicular to the direction of the radiation propagation in the capacity
of the surface element dS, when defining the intensity as a factor of the
proportionality. This projection is equal to dS ¼ dS
0 cosϑ. Then the expression
dE ¼ I l ð~ r; tÞdtdldOdS
0 cos y could be obtained from Eq. 1.1. It is suitable to
attribute the sign to energy defined above. Actually, if we fix one concrete side of
the surface dS
0 and assume the normal just to this side as a normal ~ n then the angle
ϑ varies from 0 to p, and the cosine from +1 to À1. 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
0 . Now specify
the irradiance (the radiation flux of energy) F l (t) (often it is called the net
spectral energy flux) as a factor of the proportionality of radiation energy dE
0
1.1 Characteristics of the Radiation Field in the Atmosphere
3
the direction ~ r at the moment t, namely:
I l ð~ r; tÞ ¼
dE
dSdOdldt
:
(1.1)
In many cases we are interested not in energy emitted by the object but in energy
of the radiation field, which 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 ¼ dS at an arbitrary distance (Fig. 1.1). Let the system be situated in vacuum,
i.e. radiation is not interacting during the way from dS to dS 2 . Let the element dS 2 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 to the solid angle at which the element dS is
seen from dS 2 at the opposite direction (À ~ 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, dO, dl and dt is called an intensity (radiance) I l ð~ r; tÞ incoming from the
direction~ r to the surface element dS perpendicular to~ r at the wavelength l at the time t
i.e. Eq. 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, mm, cm
2 ) but to the differential scale dt, dO, dl, dS, which is reduced
then to the unit scale. Equation 1.1 is reflecting this obstacle.
Let the surface element dS
0 , which radiation incomes to, be not perpendicular to
the direction ~ r but form the angle ϑ with it (Fig. 1.1). Specify the incident angle (the
angle between the inverse direction À ~ r and the normal to the surface) as
# ¼ ffð~ n; À~ rÞ. In that case we have to use the projection of the element dS
0 on a
plane perpendicular to the direction of the radiation propagation in the capacity
of the surface element dS, when defining the intensity as a factor of the
proportionality. This projection is equal to dS ¼ dS
0 cosϑ. Then the expression
dE ¼ I l ð~ r; tÞdtdldOdS
0 cos y could be obtained from Eq. 1.1. It is suitable to
attribute the sign to energy defined above. Actually, if we fix one concrete side of
the surface dS
0 and assume the normal just to this side as a normal ~ n then the angle
ϑ varies from 0 to p, and the cosine from +1 to À1. 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
0 . Now specify
the irradiance (the radiation flux of energy) F l (t) (often it is called the net
spectral energy flux) as a factor of the proportionality of radiation energy dE
0
1.1 Characteristics of the Radiation Field in the Atmosphere
3