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which is similar to the nadir radiative transfer equation, Equation (6.38). In Figure 6.10 the nadir transmission and transmission weighting function is compared with  Weighting function Transmission, and 2E3(t)

Figure 6.10. Calculated weighting functions for Jupiter at 600 cm-1 for nadir-viewing (solid line) and for disk-averaged (dotted line) conditions.

2^3 (r) and 2 J£3 (r) / Jz. As can be seen the curves are very similar with the only real difference being that the disk-averaged weighting function peaks at slightly higher altitudes, roughly at the altitude of the transmission weighting function calculated for a zenith angle of ^50°, but is slightly broader.

6.4.3 Limb viewing

We saw in Section 6.4.1 that for nadir, or near-nadir, viewing geometries the weighting functions are rather broad. This low vertical resolution can be a problem when the spectra are used to assess the vertical profiles of gases whose abundances vary rapidly with altitude. One way to increase the vertical resolution is to view the limb of the planet. The radiative transfer equation is very similar to Equation (6.4Q), but instead of integrating over height we instead integrate directly along the path /

where

Since density varies exponentially with height in an atmosphere, the density of the slant path is highest near the tangent altitude, and thus at wavelengths where the gas absorption is low the transmission weighting function is an exponential shape function with a sharp base at the tangent altitude zq. As gas absorption becomes stronger, more and more emission comes from molecules between the observer and the tangent point and the weighting function becomes broader. Example limb-weighting functions are shown for Jupiter at 6QQ cm-1 in Figure 6.11. At a tangent pressure of ~1Qmbar, the weighting function can be seen to be very narrow with a sharp lower boundary. However, as the tangent pressure increases, and thus the absorption increases, the weighting function becomes broader, as can be seen in the second weighting function of Figure 6.11, where the pressure at the tangent altitude is ~Q.38 bar.

In addition to increased vertical resolution, limb sounding also provides greater sensitivity to the detection of trace atmospheric constituents by providing much longer pathlengths. For an atmosphere whose number density varies with height as n = nQ exp(-(z - zq)/H) above some reference altitude zq, where H is the scale height, it may easily be shown that the column amount (i.e., molecules per square meter) of molecules above this altitude for nadir viewing is = nQH. However, if limb viewing the atmosphere with a tangent altitude at zq, the total amount of molecules in the path may be shown to be well approximated by = nQ(2^RH)1/2, where R is the planetary radius at the tangent altitude zq. Since the radius of all planets is very large compared with H, the limb path contains much greater amounts, and thus absorption features of trace constituents are much easier to observe in limb paths than nadir ones. For the giant planets, the increase in path amount ^¿/^ is of the order of 1QQ. Figure 6.11. Calculated limb weighting functions for Jupiter at 600 cm 1 and tangent height pressures of 9.7mbar (solid lines) and 387mbar (dotted line).

The disadvantage of limb viewing it that it places considerably tighter constraints on telescope field-of-view and pointing accuracy and is hence much more difficult to achieve in practice. However, the possible improvement in vertical resolution and sensitivity to trace species makes it a very attractive technique and it is widely used in terrestrial remote sensing. The field of view of the Voyager IRIS spectrometers was too broad to allow limb sounding of the giant planets, although it was achieved for Titan. The Cassini CIRS instrument, however, was specifically designed with limb viewing in mind and has been able to make limb-sounding observations of both Titan and Saturn since it arrived in the Saturnian system in 2004.

The balance between radiative heating and cooling affects the temperature profile in upper tropospheres and stratospheres of the giant planets and thus affects the circulation of air at these altitudes. At an altitude z, or optical depth r, in the atmosphere, the upward spectral flux of radiation is from Equation (6.45) equal to

E3(ro-r

and the downward spectral flux is f0.5

If the fluxes are then integrated over all wavelengths, the difference between the two may be used to calculate the heating rate as

where »is the mass density; and 'p is the specific heat capacity at constant pressure. In the stratospheres of the giant planets it is of particular interest to know how long it takes for temperature perturbations introduced by effects such as dynamics to relax back to zero (i.e., for the temperature profile to return to its equilibrium radiative balance state). Near the emitting level, a simple estimate of the radiative time constant is (Allison et al., 1991)

where Te is the effective emitting temperature. This expression is basically the heat stored per unit area of the atmosphere above the level at pressure p divided by the outgoing thermal radiation flux. Once an estimate for the radiative time constant has been derived, the local heating rates Q implied by local departures of temperature from equilibrium can be estimated from (e.g., Conrath et al., 1990)

where Teqm is the local radiative equilibrium temperature.

6.4.5 Local thermodynamic equilibrium

For radiative transfer calculations, we normally assume that the atmosphere is in local thermodynamic equilibrium (LTE). This means that we can use the Planck function (z), as we have done in our previous equations. However, at very low pressures the time between collisions becomes equal to, or greater than, the time of interaction of the molecules with photons, and thus the population density of states deviates from a simple Boltzmann distribution (Goody and Yung, 1989). Under these conditions of non-LTE, the calculation of the sour'e fun'tion, used instead of the Planck function in the previous radiative transfer equations, becomes very complicated and is a whole research area in its own right (e.g., Lopez-Puertas and Taylor, 2001).

6.4.6 Transmission calculations

Given the absorption coefficients of the various gases and aerosols in a planetary atmosphere, it can be seen that the emerging radiance for a real atmosphere may be calculated. Since the line strengths and linewidths are functions of temperature and pressure, and since atmospheres are extremely inhomogeneous in both respects, the monochromatic transmission at wavenumber v of a path through an atmosphere between two levels z1 and z2 at zenith angle d is given by where n(z) is the number density of the atmosphere at altitude z (molecules per cubic meter); p(z) and T(z) are the atmospheric pressure and temperatures; qj(z) is the mole fraction of gas j at altitude z; and the summation over i is over all the absorption lines of gas j that contribute to the optical depth at this wavenumber. There are a number of databases available that list the molecular absorption lines of important gases, including HITRAN (Rothman et al., 2005), which is extensively used for terrestrial atmospheric calculations, and GEISA (Jacquinet-Husson et al., 2005), which also includes lines of exotic gases only found in the atmospheres of the giant planets. For each line of each gas, the absorption coefficient kij must be calculated from the line strengths and line shape, both of which are functions of temperature and pressure. Since thousands of absorption lines may contribute to the absorption at a particular wavelength, it can easily be seen that such line-by-line calculations are computationally expensive and thus slow, although they are clearly the most accurate method available.

In most cases, spectral calculations are made to compare with real measured spectra which have limited spectral resolution. Hence, the transmissions used in the radiative transfer equation must be integrated over the instrument function. While it is still most accurate to smooth line-by-line calculated spectra to the required spectral resolution, there are alternative methods of simulating finite-resolution spectra that are much faster and only slightly less accurate. Since the line parameters used to generate line-by-line spectra are themselves accurate to only 10% in some cases, the use of these lower accuracy models is very common and perfectly defendable. There are two main approaches, band models and correlated-k models, which will now be discussed.

Band model approximation

A number of possible band models exist, including the Goody-Lorentz, Godson-Lorentz, Malkmus-Lorentz, and Goody-Voigt band models. A full discussion of these models is beyond the scope of this book, but they are discussed in detail in a number of more general radiative transfer books such as Goody and Yung (1989). The basic idea of such models is that if we have an atmosphere of fixed temperature, pressure, and composition, and if we then measure (or calculate using a line-by-line model) the mean transmission over the wavenumber range v to v + Av of a number of

paths of different lengths through this atmosphere, then the mean transmission as a function of the path amount m (molecules per square meter), due to individual gases, may be well approximated by a smoothly varying analytical function of just a few parameters. For example, the band transmission as a function of the path amount for the Goody-Voigt approximation due to a certain gas of mole fraction q is given by

dx where V(x, y) is the Voigt function and where aL a L p vTO (