An individual spectral line is described by a line position (i.e. the wavenumber at the center), a line shape, a line strength (or intensity), and a line width. The line shape is described by a nondimensional function of nondimensional argument, f (x), normalized so that the total area under the curve is unity. The contribution of a single spectral line to the absorption coefficient for substance G can then be written
where vc is the frequency of the center of the line, S is the line intensity and y is the line width. Note that f KGdv = S. As a line is made broader, the area remains fixed, so that the absorption in the wings increases at the expense of decreased absorption near the center.
The pressure and temperature dependence of kg enters almost entirely through the pressure and temperature dependence of S and Y. The line center vc can be regarded as independent of pressure and temperature for the purposes of computation of planetary radiation balance. At very low pressures (below 1000Pa), one may also need to make the line shape dependent on pressure.
Every line has an intrinsic width determined by the charactistic time for spontaneous decay of the higher energy state (analogous to a radioactive half-life). This width is far too narrow to be of interest in planetary climate problems. In addition, the lines of a molecule in motion will experience Doppler broadening, associated with the fact that a molecule moving towards a light source will see the frequency shifted to higher values, and conversely for a molecule moving away. For molecules in thermodynamic equilibrium, the velocities have a Gaussian distribution, and so the line shape becomes f (x) = exp(— x2)/y/n. The width is y = Y(T) = vcJC, where v = y2RT, R being the gas constant for the molecule in question. v is a velocity, which is essentially the typical speed of a molecule at temperature T. For CO2 at 250K, the Doppler line width for a line with center 600cm-1 is only about .0006cm-1.
The type of line broadening of primary interest in planetary climate problems is collisional broadening, alternatively called pressure broadening. Collisional broadening arises because the kinetic energy of a molecule is not quantized, and therefore if a molecule has experienced a collision sufficiently recently, energy can be borrowed from the kinetic energy in order to make up the difference between the photon's energy and the energy needed to jump one full quantum level. The theory of this process is exceedingly complex, and in many regards incomplete. There is a simple semi-classical theory that predicts that collision-broadened lines should have the Lorentz line shape f (x) = 1./(n • (1 + x2)), and this shape seems to be supported by observations, at least within a hundred widths or so of the line center. For the Lorentz shape, absorption decays rather slowly with distance from the center; 10 half-widths y from the center, the Lorentz absorption has decayed to only ^ of its peak value, whereas the Gaussian doppler-broadened line has decayed to less that 10-43 of its peak. There are both theoretical and observational reasons to believe that the very far tails of collision broadened lines die off faster than predicted by the Lorentz shape. A full discussion of this somewhat unsettled topic is beyond the level of sophistication which we aspire to here, but the shape of far-tails has some important consequences for the continuum absorption, which will be taken up briefly in Section 4.4.8.
In the simplest theories leading to the Lorentz line shape, the width of a collision-broadened line is proportional to the mean collision frequency, i.e. the reciprocal of the time between collisions. The Lorentz shape is valid in the limit of infinitesimal duration of collisions; it is the finite time colliding molecules spend in proximity to each other that leads to deviations from the Lorentz shape in the far tails, but there is at present no general theory for the far-tail shape. For many common planetary gases the line width is on the order of a tenth of a cm-1 when the pressure is 1 bar and the temperature is around 300K. For fixed temperature, the collision frequency is directly proportional to pressure, and laboratory experiment shows that the implied proportionality of line width to pressure is essentially exact. Holding pressure fixed, the density goes down in inverse proportion to temperature while the mean molecular velocity goes up like the square root of temperature. This should lead to a collision frequency and line width that scales like 1/%/T. Various effects connected with the way the collision energy affects the partial excitation of the molecule lead to the measured temperature exponent differing somewhat from its ideal value of 1. Putting both effects together, if the width is known at a standard state (po,To), then it can be extrapolated to other states using
where n is a line-dependent exponent derived from quantum mechanical calculations and laboratory measurements. It is tabulated along with standard-state line widths in spectral line databases. One must typically go to very low pressures before Doppler broadening starts to become important. For example, for a collision-broadened line with width .1cm-1 at 1 bar, the width doesn't drop to values comparable to the Doppler width until the pressure falls to 6mb - comparable to the middle stratosphere of Earth or the surface pressure of Mars. Even then, the collision broadening dominates the absorption when one is not too close to the line center, because the Lorentz shape tails fall off so much more gradually than the Gaussian.
Another complication is that the collision-broadened line width depends on the molecules doing the colliding. Broadening by collision between molecules of like type is called self-broadening, while that due to dissimilar molecules is called foreign broadening. The simple Lorentz theory would suggest a proportionality to collision rate, which is a simple function of the ratio of molecular weights. Not only does the actual ratio of self to foreign broadened width deviate from what would be expected by this ratio, but the ratio actually varies considerably from one absorption line to another. For CO2, for example, some self-broadened lines have essentially the same width as for the air-broadend case, whereas others can have widths nearly half again as large. For water vapor, the disparity is even more marked. Evidently, some kinds of collisions are better at partially exciting energy levels than others. There is no good theory at present that enables one to anticipate such effects. Standard spectral databases tabulate the self-broadened and air-broadened widths at standard temperature and pressure, but if one were interested in broadening of water vapor by collisions with CO2 (important for Early Mars) or broadening of by collisions with H2 (on Jupiter or Saturn) , one would have to either find specialized laboratory experiments or extrapolate based on molecular weights and hope for the best.
The line intensities are independent of pressure, but they do increase with temperature. For temperatures of interest in most planetary atmospheres, the temperature dependence of the line intensity is well described by
where n is the line-width exponent defined above and hvg is the energy of the lower energy state in the transition that gives rise to the line. This energy is tabulated in standard spectroscopic databases, and is usually stated as the frequency v^. Determination of the lower state energy is a formidable task, since it means that one must assign an observed spectral line to a specific transition. When such an assignment cannot be made, one cannot determine the temperature dependence of the strength of the corresponding line.
Now let's compute the average transmission function associated with a single collision-broadened spectral line in a band of wavenumbers of width A. We'll assume that the line is narrow compared to A, so that the absorption coefficient can be regarded as essentially zero at the edges of the band. Without loss of generality, we can then situate the line at the center of the band. The mean transmission function is
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