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- Beer’s Law
- Optical thickness
- Examples

BEER’S LAW

Note: Beer’s 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. Beer’s law

describes the absorption of a monochromatic bean of radiation passing through a gas.

[Other conditions for the validity of Beer’s 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.]

Beer’s 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

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)

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.

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.

Beer’s 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 Beer’s law:

(30.7)

where L0 is the incident radiation.

The transmissivity of a gas over path length, l, is therefore;

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.

- 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 aare, 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 H=RT/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 Beer’s law:

(30.11)

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)

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).

- Schwarzschild’s equation
- Remote sensing
- Planetary equilibrium temperatures
- Greenhouse effect

SCHWARZSCHILD’S EQUATION

Let us consider simultaneous absorption and emission in an atmospheric layer. We will continue

to use Beer’s 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 Kirchoff’s law, we have d=da, and from Beer’s law da=ads=da. Hence Eq. 31.1

may be combined with the differential form of Beer’s Law (Eq. 30.1) to give:

(31.2)

Eq. 31.2 is known as Schwarzschild’s 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.

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

path (this part is simply the solution to Beer’s 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:

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