ABSORPTION

1

• ABSORPTION
• Beers Law
• Optical thickness
• Examples

BEERS LAW Note Beers law is also attributed
to Lambert and Bouguer, although, unlike Beer,
they did not recognize that it applies only to
monochromatic radiation. Absorption occurs in
the atmosphere for both solar and terrestrial
radiation. Beers law describes the absorption
of a monochromatic bean of radiation passing
through a gas. Other conditions for the validity
of Beers law are that T, p are constant along
the path, and that the radiant energy density is
not too high, otherwise non-linear effects will
occur. Beers law in differential form can be
written in terms of the (spectral) absorptivity,
da?
(30.1)
where ?a? is called the mass absorption
coefficient (m3kg-1 ?m-1 ), that is, it is
the absorption cross-section per unit mass. One
can think of the mass absorption coefficient in
the following manner. Suppose one were to remove
one kilogram of the absorbing gas and replace it
with an ensemble of blackbodies with total
cross-sectional area, normal to the beam, equal
2
to ?a? . Then the absorption would remain the
same as if one had not removed the gas. The
shape of individual spectral lines may be
approximated by the Lorentz line profile
(30.2)
3
is called the line strength.
The volume absorption coefficient is simply the
product of the mass absorption coefficient
and the gas density
(30.3)
This can be interpreted in two ways. The first is
analogous to the interpretation of the
mass absorption coefficient as given above. That
is, it is the absorption cross-section per
unit volume. The second interpretation is that it
is the absorptivity per unit length along the
path of the beam. The optical path, u, is
defined to be the absorber mass per unit area
along the path, that is
(30.4)
The optical path is a density-weighted path
length. For a vertical path, the optical path
for water vapour is simply the precipitable water.
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OPTICAL THICKNESS Optical thickness, ?a?, is the
product of the optical path and the mass
absorption coefficient (alternatively, the
product of the path length and the volume
absorption coefficient). It is dimensionless.
(30.5)
Note optical thickness along a vertical path is
known as the optical depth. Beers law can be
expressed particularly simply in terms of optical
thickness, viz
(30.6)
Eq. 30.6 can be integrated to give the integral
version of Beers law
(30.7)
where L?0 is the incident radiation. The
transmissivity of a gas over path length, l, is
therefore
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(30.8)
Eqs. 30.7 and 30.8 demonstrate clearly that the
effect of absorption over a finite path is
an exponential attentuation of the beam. The
absorptivity over a finite path is
(30.9)
Clearly, the absorptivity approaches unity with
increasing optical thickness.
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• EXAMPLES
• THE VERTICAL PROFILE OF ABSORPTION (Chapman
  Profile see Wallace and Hobbs)
• Where does the maximum radiant heating occur in
  the atmosphere? There is no general answer to
• this question since the absorption profile (and
  hence the heating profile) depends upon the total
• optical thickness (depth) along the path, and
  also upon the profile of absorber concentration.
• In order to simplify the problem and come up with
  an answer to the question, we will consider
• here an isothermal atmosphere, in hydrostatic
  equilibrium, with a constant absorber mixing
• ratio, r?/?a, where ? and ?a are, respectively,
  the absorber density and the air density.
• We already know that pressure and air density
  vary exponentially with height in an isothermal
• atmosphere. If the mixing ratio of the absorber
  is constant, then its density must also vary
• exponentially with height, viz

(30.10)
where HRT/g is the scale height of the
atmosphere. Consider a downward beam passing
through such an atmosphere. Since the heating
rate is determined by dL?/dz, we use Beers law
(30.11)
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where . If we assume
the mass absorption coefficient to be independent
of height, and ? given by Eq. 30.10, then it is
straightforward to integrate and show that
(30.12)
that is, the optical thickness increases in
proportion to the density, as one moves down
through the atmosphere. Solving Eq. 30.12 for
?a? and substituting into Eq. 30.11 leads to
(30.13)
Eq. 30.13 is known as the Chapman profile of
absorption (or equivalently of heating rate). The
physical explanation for the maximum in heating
rate in the middle of the atmosphere is simple.
Eq. 30.11 states that the absorption is
proportional to the product of the
absorber density and the incident radiance. At
high altitudes the radiance is high but the
density low. at low altitudes, the density is
high but the radiance is low because of
absorption higher up. Hence the absorption must
be low at both high and low altitudes in the
atmosphere. The maximum in absorption must
therefore occur at some intermediate altitude. It
is straightforward to show by setting
(30.14)
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that the maximum absorption occurs at an optical
thickness of unity. It turns out that, even when
the assumptions of constant temperature and
constant mass absorption coefficient are relaxed,
the maximum absorption still occurs around unit
optical depth. Unit optical depth is also known
as the penetration depth, that is the depth at
which the radiance is diminished by a factor of
1/e. Absorption of ultraviolet radiation by
ozone can also be used to explain the temperature
maximum at the top of the stratosphere (at an
altitude of about 50 km).
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• ABSORPTION AND EMISSION
• Schwarzschilds equation
• Remote sensing
• Planetary equilibrium temperatures
• Greenhouse effect

SCHWARZSCHILDS EQUATION Let us consider
simultaneous absorption and emission in an
atmospheric layer. We will continue to use Beers
Law to describe the absorption, but we will need
to add a term to describe the contribution of
emission to the change in the radiance passing
through the layer. For a layer of differential
thickness, the emission may be written
(31.1)
From Kirchoffs law, we have d??da?, and from
Beers law da??a??dsd?a?. Hence Eq. 31.1 may be
combined with the differential form of Beers Law
(Eq. 30.1) to give
(31.2)
Eq. 31.2 is known as Schwarzschilds equation. We
will now derive a formal solution to it for a
finite path. For convenience, we will drop the
subscript ?. Nevertheless, we must keep in mind
that the results are valid only for monochromatic
radiation.
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In order to integrate Eq. 31.2, we will first
multiply by
(31.3)
Integrating the second equation in Eq. 31.3
between 0 and l (small L!)
(31.4)
Multiplying the second equation in Eq. 31.4 by
, we have finally
(31.5)
Keeping in mind that the transmissivity can be
related to the optical thickness by Eq. 31.5 may
be written more simply as
(31.6)
Note the second equation in 31.6 follows from
the fact that which
leads to
Eq. 31.6 may be interpreted physically as
follows. The radiance at the end of a finite path
is composed of the sum of two parts. The first is
the initial radiance attenuated over the entire
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path (this part is simply the solution to Beers
law). The second is the sum over the entire
path of the radiance emitted at each point,
attenuated over the distance between that point
and the end of the path. Note A business
analogy may be helpful here. Consider the future
value of an annuity, L(l), that begins with a
lump sum payment, L(0), and is followed by
subsequent monthly payments, LB(s)d?. The annuity
receives no interest. Rather, each payment is
diminished in value with time due to the effects
of inflation (the transmissivities). REMOTE
TEMPERATURE SENSING (see Wallace and Hobbs for
details) We will describe the principle behind
remote sensing, from satellites, of the
atmospheric vertical temperature profile.
Consider the atmosphere to be divided into N
thin, isothermal layers. Then Eq. 31.6 may be
written for these layers, approximately, as