Microwave Sounding of the Giant Planets

ICARUS

119, 67–89 (1996) 0003

ARTICLE NO.

Microwave Sounding of the Giant Planets R. M. KILLEN Department of Space Science and Instrumentation, Southwest Research Institute, San Antonio, Texas 78228-0510 E-mail: [email protected] AND

F. M. FLASAR Laboratory for Extraterrestrial Physics, Goddard Space Flight Center, Greenbelt, Maryland 20771 Received October 17, 1994; revised August 22, 1995

progressively less of the opacity as the wavelength increases to 1 cm and beyond. The interpretation of microwave observations has generally relied on the assumption that the volatiles NH3 , H2O , and H2S dominate the opacity at centimeter wavelengths (cf., e.g., Berge and Gulkis 1976, de Pater 1990, de Pater et al. 1991). At the temperatures believed to characterize the deep tropospheres of the giant planets, these constituents are all condensible, and their uncertain spatial distribution complicates any interpretation of the observations. Even their abundances below all the clouds are uncertain. It is important to remember that evidence for these species is still largely indirect: NH3 has been spectroscopically observed only on Jupiter and Saturn (Prinn et al. 1984), and H2O has been observed only on Jupiter (Bjoraker et al. 1986). Neither has been identified on Uranus or Neptune, presumably because their cold temperatures restrict the abundances of these gases to small amounts at the atmospheric levels accessible to infrared observation. Sulfur has not been observed in the undisturbed atmospheres of the giant planets. For example, Joiner et al. (1992) placed an upper limit to the jovian stratospheric H2S mixing ratio of approximately 1 3 1026. A large mass of sulfur-bearing material was seen in Jupiter’s stratosphere and upper atmosphere after the P/Shoemaker–Levy 9 impact; however, its origin is controversial (Noll et al. 1995, Weissman 1994). There is, nonetheless, evidence that the H2 –He atmospheres of the giant planets have abundances of heavier elements that are in excess of solar (cf., e.g., Gautier and Owen 1983). This is particularly true of Uranus and Neptune. Spectroscopic observations in the visible and nearinfrared have been used to constrain the abundance of atmospheric methane. The most recent studies (Baines and Bergstralh 1986, Smith and Baines 1990) rely on comparison of a feature at 681.89 nm, identified as an individual CH4 line, and the H2 4–0 S(0) and S(1) quadrupole lines

We systematically explore the sensitivity of the microwave spectra of the giant planets to the vertical profiles of temperature and condensible absorbers. We find that the spectrum at centimeter wavelengths is largely insensitive to vertical profile of temperature, but generally is sensitive to the relative humidities (RH) of the condensible absorbers. Under a wide variety of plausible conditions, gaseous ammonia dominates the microwave opacity, and the spectra can be used to retrieve the vertical profile of its relative humidity. We examine the ammonia relative humidity profiles implied by the spectra of Uranus and Saturn. For moderate enhancements of H2S above its solar value (,10), the microwave spectrum of Uranus depends primarily on the NH3 relative humidity at wavelengths longward of 1 cm, out to at least 5 cm, depending on whether there are thick water clouds. The implied vertical variation in the relative humidity of NH3 is small in the temperature range 140 to 250 K, confined to the range 0.05 to 0.35%. The reported meridional variation in brightness temperatures at 2 and 6 cm can be interpreted as a factor of 10 variation in the relative humidity, with the largest relative humidities occurring near 208S, where temperatures are coldest. Part of the 6-cm brightness temperature variation may be attributable to meridional variations in the opacity of thick water clouds. On Saturn, the vertical variation of relative humidity is greater. At temperatures less than 152 K, ammonia is saturated, but at deeper levels, it follows a curve such at d ln RH/d ln P 5 24.5, which corresponds nearly to a constant mole fraction, down to temperatures .230 K.  1996 Academic Press, Inc.

I. INTRODUCTION

In the absence of in situ measurements, microwave observations offer a unique probe of the deep atmospheres of the giant planets, well below the levels accessible to visible and infrared wavelengths. However, they are an ambiguous probe, because molecular hydrogen provides 67

0019-1035/96 $12.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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near 684 nm to derive [CH4 ]/[H2 ]. These arrive at CH4 mole fractions in the ranges 0.020–0.046 and 0.023–0.037 on Uranus and Neptune, respectively. In addition to assuming that the CH4 line has been correctly identified, these analyses require that the scattering by particulates at these wavelengths has been properly accounted for. Relating the abundances of other volatiles to CH4 is not straightforward. Interior models, which attempt to fit the radius and lowest two gravitational moments, given a planet’s mass and rotation rate, also provide evidence of heavy element enhancement. They require that much of the interior be composed of ‘‘icy’’ material (CH4 , NH3 , H2O, . . .), with the predominantly hydrogen envelope constituting a relatively thin shell (Hubbard and Marley 1989, Hubbard et al. 1991). Although it is plausible to infer that the hydrogen envelope is therefore enriched in these volatiles, the modeling itself is limited in what can be deduced, as only the lowest gravitational moments are utilized. Hubbard and Marley (1989) have recently concluded that models with ‘‘standard’’ CH4 and NH3 abundances of 30 times solar and 1 times solar, respectively, can have up to 100 times solar abundances of H2S and H2O and still fit the observed gravitational harmonics. Analyses of microwave observations have traditionally assumed an adiabatic extrapolation downward from that region of the atmosphere accessible to temperature determination from other observations, such as infrared spectroscopy and radio occultation. With the temperature thereby constrained, the absorbers can be inferred, subject to the constraints of saturation and chemical equilibrium. This problem still remains unconstrained to some degree, because, at many wavelengths, no single constituent dominates the opacity, and their relative abundances are not known a priori. Studies of this type have concluded that both Uranus and Neptune are severely depleted in NH3 through much of their observable atmospheres, relative to the solar abundance (Gulkis et al. 1978, de Pater and Massie 1985). The NH3 depletion is not thought to be characteristic of the entire H2 –He envelope, but may result from reaction of NH3 in thermochemical equilibrium with H2S, to form an NH4SH cloud, and also from dissolution of NH3 in a water cloud at deeper levels (de Pater et al. 1989, Romani et al. 1989). If the NH4SH cloud contributes significantly to this massive depletion, there must be almost as much H2S as NH3 below that cloud, implying a S to N ratio that is at least 4.6 times solar. Alternatively, nitrogen could have been incorporated into Uranus and Neptune in the form of N2 rather than NH3 (Atreya et al. 1995). In that case NH3 would be depleted throughout the envelope, and a smaller water cloud could be a sink for residual NH3. An alternative suggestion is that the microwave opacity in the upper troposphere is due to H2S rather than NH3 (de Boer and Steffes 1994). These studies generally neglect the uncertainties in the

temperature profile itself at the levels accessible to microwave sounding. The use of an adiabatic extrapolation is not unreasonable, given that heat must be transported convectively from the interior to the 300–400-mbar level, where direct radiation to space becomes effective; even Uranus, with a marginal internal heat flux (Pearl et al. 1990), probably has an adiabatic interior that is a remnant of its earlier history, when it did transport a significant flux of heat from the interior. The problem rather is that the deep adiabat is not well known, because it depends on the temperature at the point of departure for the extrapolations. The most recent analyses of Uranus and Neptune use adiabatic extrapolations of the temperature profiles retrieved from the Voyager radio occultation data which penetrated, respectively, to 2.3 and 6.3 bars (Lindal et al. 1987, Lindal 1992). The retrieval of temperature from these data requires a knowledge of the atmospheric composition. In the present context, the largest uncertainty is in the knowledge of the vertical profile of CH4 . Lindal et al. (1987, Fig. 9) illustrate the effect on the retrieved temperatures of different assumed deep abundances of CH4 on Uranus. Their temperature retrievals correspond to mole fractions ranging from 0 to 0.04. (McMillan and Flasar (1989) have considered mole fractions of 0.10 and beyond.) The retrieved temperature profiles are nearly identical above the 1-bar level, because the local CH4 abundance is constrained by saturation to very small values. At deeper levels, however, the retrieved profiles diverge, becoming warmer with higher assumed CH4 mole fraction. The Lindal et al. retrievals have a spread of 16 K at the 2.3-bar temperature. Although those warmer have lapse rates in temperature that exceed the dry adiabat, they are nevertheless statically stable, because the increasing mole fraction of methane with depth results in a molecular weight stratification that is stabilizing. All these retrievals are consistent with existing infrared data, including the Voyager infrared spectroscopy experiment (IRIS), which samples the atmosphere at pressures #1 bar. Lindal et al. did adopt a nominal profile that corresponds to a deep CH4 mole fraction of 0.02, and this has been cited as a determination of the deep methane abundance (cf., e.g., Gautier and Owen 1989), but in reality the radio occultation measurements by themselves can provide only very broad limits on this abundance. The present work represents a departure from previous analyses of microwave data, in that constraining the volatile abundances of the atmosphere is not its principal objective. Instead, it attempts to elucidate the information content that is retrievable from the microwave spectrum and present models in that context. If the observed spectrum of an atmosphere with temperature profile T(P), where T is temperature and P is the barometric pressure, derives from thermal emission in which hydrogen and N condensible constituents with mole fractions qi (P) (!1) contribute significantly to the opacity, then in general the profiles

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MICROWAVE SOUNDING OF THE GIANT PLANETS

of temperature and all N condensibles affect the spectral radiance. (Here and throughout the paper, we assume that the ratio of helium to hydrogen is known. The effects of helium on the hydrogen opacity and temperature retrieval are well known (see Conrath et al. 1991 and references therein) and not the subject of this work. In order to retrieve information on T(P) and qi (P) from the spectral radiance, one must somehow specify all but one unknown. At wavelengths #1 mm, hydrogen provides the dominant opacity, N 5 0, and a temperature profile can be unambiguously retrieved from the spectrum—with the usual caveats attendant on the inversion problem; see Section VI. At longer wavelengths, however, several species, whose abundances are not well known, can contribute to the spectral opacity. Saturation of these volatile species impose additional constraints on the N 1 2 dimensional hyperspace (P, T, qi ), because it relates the partial pressure of the ith species pi (P) to T(P). But this is generally not sufficient to permit the determination of qi (P) or T(P) from the spectrum alone; an ambiguity remains. However, we will demonstrate that there is a range of centimeter wavelengths for which the spectrum can uniquely determine the profile of the relative humidity of a condensible, provided that it dominates the opacity. We will mostly focus on Uranus, because previous work has demonstrated that Uranus, with Neptune, exhibits most of the ambiguities inherent in interpreting the microwave spectrum. Most of our conclusions relating to sensitivities of the brightness temperature spectra apply directly to Neptune, since its atmospheric structure is similar to Uranus’ in many respects. Because our conclusions are fairly general, they are applicable by extension to the other giant planets, Jupiter and Saturn. We briefly consider Saturn. The plan of the paper is as follows: Section II summarizes the microwave observations of Uranus’ atmosphere. Section III discusses the microwave opacities and radiative modeling used in the present analysis. Section IV covers the physical–chemical treatment of atmospheric condensibles. Section V treats the sensitivity of the microwave spectrum to temperature, NH3 , H2S, and H2O and water clouds. We also determine the meridional variation of the relative humidity of NH3 for Uranus implied by the data of Hofstadter (1992), and we derive the global average relative humidity of NH3 for Saturn. Section VI contains a discussion and summary of our results on the information content of microwave spectra, and on the retrieval T(P) and composition. II. OBSERVATIONS

Disk-averaged brightness temperatures of Uranus’ microwave spectrum have been published from the early 1960s to the present. Most of the data used in this study are summarized in the review by Gulkis and dePater

(1984). This data set spans the years 1965–1982.45 and is culled from 19 sources. Additional data from the VLA were taken from Jaffe et al. (1984), from dePater and Gulkis (1988), and from dePater et al. (1991). A general post-1973 warming trend has been discussed by Gulkis and dePater (1984) and others. The warming trend was reported by Jaffe et al. (1984) to have been reversed a year before the north pole reached its closest sunward alignment. The warming trend was attributed by Hofstadter and Muhleman (1989) to a meridional variation in the NH3 molar mixing ratio such that the abundance poleward of 458 is depleted by a factor of 3 relative to midlatitudes. We have concentrated our attention for this study on the 1980s decade disk-averaged observations. The primary motivation for this study is to discuss the information content of microwave data rather than to develop new atmospheric models. In order to avoid confusion with possible temporal variations we have not plotted the 1960s decade observations. Only the post-1970 data are plotted with the synthetic spectra. Disk-resolved microwave data of Uranus at 2 and 6 cm from Hofstadter (1992) are plotted in Fig. 17 along with the corresponding relative humidity of ammonia. III. RADIATIVE TRANSFER The microwave brightness temperature of the outer planets is due to the thermal emission of the atmosphere, and especially in the case of the jovian system, to synchrotron emission. Uranus’ microwave spectrum is unaffected by synchrotron emission. Uranus’ brightness temperature in the microwave region is determined by the mixing ratios of H2 and three condensible gases: H2S, NH3 , and H2O (cf. dePater et al. 1991). Although our opacity sources are the same as those in dePater et al. (1991), Hofstadter (1992), and Grossman (1990), our formulation of the line profiles differs, and our formulation of the liquid water opacity in the water cloud is an empirical fit to data. For liquid water, we consider the extinction without scattering, which is quite reasonable for centimeter wavelengths. The H2 opacity is due to a dipole induced by H2 –H2, H2 –He, and H2 –CH4 collisions (Bachet et al. 1983, Bachet 1986, Dore et al. 1983). Methane opacity is negligible at microwave frequencies, but methane contributes to the collision induced dipole of H2. We assume ‘‘frozen equilibrium’’ H2 with a mixing ratio of 0.85 H2 /(He 1 H2 ) (Conrath et al. 1987, Conrath and Gierasch 1984). The microwave flux is given by a straightforward integral of the thermal emittance without scattering (1) E E eB (T(t)) exp S2 teD d SteD de f F 5 2f E B (T(t))E (t ) dt 5 22f E B (T ) dE (t ),

f Fn 5 2f

1

y

0

0

n

n

y

n

0

n

y

n

2

n

n

0

n

3

n

(2)

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where f Fn is the flux at frequency, n, Bn (T ) is the Planck function at temperature, T, at the level defined by optical depth, t, P is the pressure, E2 is the second exponential integral, and E3 is the third exponential integral (cf. Goody and Yung, 1989). This result for the disk-averaged flux assumes a spherical planet. The contribution function, defined as Cf (T) 5 2Bn (T(t))

dE3 (t) d ln P

somewhere between 50 and 100 bars in Uranus’ atmosphere, depending on the relative abundance of the absorbers, ammonia and water, and T(P). The pressure broadened line profile given by Spilker (1990) for the ammonia inversion line is in the same form as that derived by Ben Reuven based on a quantum-mechanical treatment of the problem. The absorptivity is a sum over the contributions of the individual lines described by the rotational quantum numbers, J and K:

(3) a(n0 ) 5 C

O O A(J, K, m) F (J, K, m, n ), 0

J

(see, e.g., Hanel et al. 1992), gives a measure of the relative contribution of each level in the atmosphere to the outgoing flux at a given wavelength (i.e., the intensity is the integral of the contribution function over pressure). The contribution functions illustrated in this paper are shown normalized to unity at the peak of the function. A. Ammonia The ammonia opacity in the microwave region is due to the inversion of the nitrogen atom through the plane defined by the hydrogen atoms, and to the wings of the rotation lines in the submillimeter region. The inversion lines of the ammonia band centered at 1.3 cm are pressure broadened so that this band significantly influences the opacity in the entire microwave region of interest. Since ammonia may be severely depleted in Uranus’ atmosphere above roughly 30–50 bars (cf., dePater 1990), the broadening at very high pressures becomes critical to the model. We use three different formulations for the lineshape which are most accurate in different wavelength and pressure regions. These are the VanVleck–Weisskopf profile (VanVleck and Weisskopf 1945), the Ben Reuven profile as modified by Berge and Gulkis (Berge and Gulkis 1976), and a second form of the Ben Reuven profile as modified by Spilker (1990). The VanVleck–Weisskopf profile is most accurate at low pressures (,1 bar) because it was derived with the impact approximation, under the assumption that distant encounters make an unimportant perturbation to the energy state of the molecule. For high pressures, distant encounters may appreciably affect the energy state of the atom, thereby affecting the line profile. The Spilker line profile is an empirical fit to data in the frequency range 9–18 GHz (3.3–1.6 cm), spanning the pressure range 1–8 bars and the temperature range 213– 315 K. These pressures and temperatures encompass those which would be probed by microwaves in the atmospheres of Saturn and Jupiter. However, Uranus’ atmosphere is far colder, and ammonia may be severely depleted in the upper atmosphere. The temperature at 10 bars on Uranus is 30 K colder than the coldest data point in Spilker’s study. It is worth noting that the 20 cm wavelength will probe

(4)

K

where a (n0 ) is the absorption coefficient at frequency, n0 , A(J, K, m) gives the line intensity, F (J, K, m, n0 ) gives the lineshape factor, m is a vector describing the physical conditions at which the line is evaluated, and C is an empirical correction factor used by both Berge and Gulkis and by Spilker to fit their data. We use the formulation given by Spilker (1990), with the correction coefficient, C, given by C 5 0.12228 1 0.0039588T, T , 180 K T2 T 2 , 180 , T , 300 K, (5) C 5 20.3364 1 110.4 70600 C 5 1.1, T . 300 K. The absorption coefficient thus approaches that given by the unmodified Ben Reuven profile in the deep atmosphere. Because the data used in Spilker’s study were all at wavelengths greater than 1 cm and pressures greater than 1 bar, we used the Ben Reuven profile as modified by Berge and Gulkis (1976) for wavelengths less than 1 cm. The Ben Reuven lineshape is compared to the measurements shortward of 1 cm at 1 atm of pure NH3 in dePater and Massie (1985). For longer wavelengths, we used the VanVleck–Weisskopf profile (VVW) at pressures less than 1 bar and a modified Spilker profile for pressures greater than 1 bar. The line profiles for the Ben Reuven and Spilker formulations are compared at pressures of 1 and 10 bars in Fig. 1. In Fig. 2 we compare the spectra computed for a solar composition atmosphere using the VVW, Ben Reuven, and modified Spilker line profiles, respectively. For models with greater depletion of ammonia in the upper atmosphere there will be less difference between spectra computed with the Ben Reuven and Spilker line profiles because the two profiles differ the most at low temperature and pressure. In comparison of models it is important to note that both Grossman (1990) and Hofstadter (1992) use the VVW profile in the millimeter region. We used the Ben Reuven profile in the millimeter region because it fits the laboratory data better than the VVW profile (Joiner et al. 1989, Joiner and Steffes 1991).

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MICROWAVE SOUNDING OF THE GIANT PLANETS

magnitude less than that for ammonia (Fig. 3). However, the solar abundance of oxygen is an order of magnitude greater than that for nitrogen, and models with enhanced water are preferred on theoretical grounds (Pollack and Bodenheimer 1991, Hubbard and Marley 1989). For a water/ammonia ratio of 30/1, the water opacity at 20 cm will be the same order of magnitude as that of ammonia. We have used the line profile derived by Waters (Ulaby et al. 1981) a (H2O) 5

F

2n 2r (300/T )3/2c1 300 2644/T e 0.434 T 1 (494.4 2 n 2)2 1 4n 2c 21

(6)

G

1 1.2 3 1026 cm21, where c1 5 2.85

S DS D F P 1013

300 T

0.626

1 1 0.018

G

rT . P

(7)

Here c1 and n are in gigahertz, T is in Kelvins, r is the water vapor density in grams per cubic centimeters, and P is in millibar. The first term in parentheses results from the 22.235 GHz line and the second term represents contributions from higher frequency water vapor absorption lines. The absorption profile given by Waters is very close to that given by Goodman (1969), and there is a negligible FIG. 1. Comparison of the Gulkis–Ben Reuven (solid), Spilker (dotdashed), and VanVleck–Weisskopf (dashed) profiles for NH3 at 1 bar (T 5 80 K), (Bottom) and 10 bars (T 5 270 K) (Top). We have assumed a mole fraction [NH3 ] 5 1.5 3 1026, in both panels. We assumed that [He] 5 0.15 and [H2] 5 0.85.

It is interesting to note that the NH3 inversion transition is inhibited at high pressures by an interaction of NH3 with a colliding molecule which has a dipole moment (Anderson 1949, Margenau 1949, Birnbaum and Maryott 1953, Townes and Schawlow 1975). If the mixing ratios of H2O, CH4 , and H2S are highly enhanced above solar mixing ratios then, due to this effect, the ammonia opacity may become less important than the water opacity below the water condensation level. Bear in mind that there may be an additional uncertainty in the opacity at high pressures beyond that which is due to the uncertainty in the broadening parameters. B. Water Vapor At a wavelength of 20 cm, the absorption coefficient for water at 10 bars total pressure is almost two orders of

FIG. 2. A comparison of the spectra of Uranus computed with the VanVleck–Weisskopf (dashed), Ben Reuven (solid), and Spilker (dotdashed) line profiles, respectively, for solar composition atmosphere. The differences between the Ben Reuven and Spilker profiles is minimal at high pressure because a similar form of the equations is used in both cases.

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ically meaningful mixing ratio for Uranus: [H2 ] 5 0.833, [He] 5 0.147, and [CH4 ] 5 0.02. The others do not correspond to physically meaningful or realistic mixing ratios. The Ben Reuven lineshape was assumed for NH3 . C. Liquid Water The microwave absorption of water clouds and fog has been studied since radar came into use during World War II. The formula which we use for absorption by liquid water is a fit to data given in Van de Hulst (1981) for absorption by clouds and fog at 208 C (293 K). The attenuation per centimeter is given by

a 5 0.050

FIG. 3. The absorption coefficients of hydrogen sulfide, water, and ammonia are compared at 1 and 10 bars total pressure at a mixing ratio of 1 3 1025 for each constituent except H2 for which [H2 ] 5 0.833, [He] 5 0.147, and [CH4 ] 5 0.02 were assumed to correspond with Uranian composition. The individual rotation lines of H2S are evident, but the individual lines of ammonia and water are completely blended. The relative contributions of these constituents to the opacity are model dependent.

difference between the computed spectra using the two formulas. The expression for the linewidth assumes broadening by a terrestrial mix of gases, mostly N2 and O2 . Broadening by a mixture of 85% H2 1 15% He would be about 80% that due to 79% N2 1 21% O2 . Uncertainty in water vapor opacity has minimal impact on the accuracy of models considered here since H2S and NH3 provide the dominant opacity. Water is relatively more important in the far centimeter region. The absorption coefficients of water, ammonia and hydrogen sulfide, and hydrogen are compared in Fig. 3. Each of these constituents except H2 was set to a mixing ratio of 1 3 1025 for this computation so that the absorption coefficients could be compared without abundance differences. The H2 absorption coefficients correspond to a phys-

SD 3 l

2

r /0.434 cm21,

(8)

where l is wavelength in centimeters and r is the density of the water cloud in grams per cubic centimeter. Our formulation, based on total water content rather than number density and drop size, is applicable because the expected drop radius is much less than l/2f, so it falls within the Rayleigh absorption region. It compares within 4% with the formula given by Staelin (1966) which was used by Briggs and Sackett (1989) in their study of Saturn’s atmosphere. It also compares closely to the data given by Bean and Dalton (1966) for microwave attenuation by very small water drops. Although Carlson et al. (1988) computed a typical cloud particle radius of 200 em at the cloud base, they state that, due to precipitation, the actual cloud particle radius is probably about 10 em. In either case the expected drop size is well within the range of applicability of the formula. There may be a larger error in the absorption coefficient due to neglect of the temperature dependence. D. Hydrogen Sulfide Hydrogen sulfide, if present in the giant planets’ atmospheres, may be important both as a reagent for the formation of ammonium hydrosulfide (NH4SH) and the consequent depletion of ammonia (Wildt 1937), and as an opacity source (dePater et al. 1991). We take the opacity as a summation over 55 lines in the AFGL catalog (Rothman et al. 1987) with wavenumbers between 0.06 and 5.95 cm. We used a VVW line profile with a line width, Dn. The absorption coefficient at frequency, n, is given by

a 5 a0

S DH n n0

2

J

Dn /f Dn /f 1 (n 2 n0 )2 1 Dn 2 (n 1 n0 )2 1 Dn 2

(9)

MICROWAVE SOUNDING OF THE GIANT PLANETS

with a line width, n, given by Dn 5 0.08

S D 273 T

2/3

[0.81P(H2 ) 1 0.35P(He)].

(10)

The linewidth is the inverse of t, where t is the mean time between collisions. The partial pressures are in atmospheres, T is in Kelvins. The linewidth and line center frequency, n0 , are in wavenumbers (cm21). The linewidth in the AFGL catalog is 0.15 cm21, based on N2 broadening. We used half that value based on measurements of broadening of water lines by collisions with H2 . This is consistent with the half width used by dePater et al. (1991). We have neglected any possible absorption by ammonium-hydrosulfide clouds. The imaginary part of the index of refraction of NH4SH is unknown, but few crystalline structures have high opacity in the microwave. IV. PHYSICAL–CHEMICAL MODELS

A. Mixing Ratios We assume constant mixing ratios of condensibles in the deep atmosphere below the water solution cloud. Physical or chemical effects which may alter the bulk composition in the deep interior will not be considered here. Above the water cloud base, various condensation processes of interest occur. Ammonia and hydrogen sulfide dissolve within the liquid water cloud. At higher altitudes they combine to form an NH4SH cloud. At still higher altitudes they can condense into NH3 and H2S crystals. The partial pressure of ammonia above a water solution is calculated by interpolating values of coefficients given by Romani (1986). For a given concentration of condensate in the cloud, Romani fit an equation of the form ln Pk 5 A 1 B/T 1 C ln T

(11)

to the data given by Wilson (1925) for vapor pressures of ammonia and water over aqueous ammonia solutions. For a given input concentration, we interpolate between coefficients in the table given by Romani, Appendix B (1986). Since the partial pressures of ammonia, water, and hydrogen sulfide are interrelated, a Newton–Raphson iterative scheme was used to ensure consistency among the partial pressures, the composition of the liquid solution, and mass conservation. The same type of iterative procedure was followed to obtain the partial pressure of H2S in the solution cloud. The equilibrium vapor pressure of H2S over aqueous solution was taken from Briggs and Sackett (1989, Eq. 16). We chose an initial concentration of condensate in liquid water and computed the partial pressure over the liquid solution. The concentration of the solution was then derived from the mole fraction of condensate which would

73

have been removed from the parcel to establish the equilibrium vapor pressure. Our chemical equilibrium model is the same as that followed by dePater et al. (1991). We assume NH4SH forms when ln(PNH3 PH2S ) . ln(Kp ) 5 34.151 2 10834/T,

(12)

where Kp is the equilibrium constant, the partial pressures are in bars, and T is in Kelvins (cf. e.g., Weidenschilling and Lewis 1973, Grossman 1990). NH3 , H2S, and H2O are also constrained by saturation over their own pure condensed phases. The saturation vapor pressure of the ith species is given by ln Pi 5 A1 /T 1 A2 1 A3 ln T 1 A4 T 1 A5 T 2.

(13)

The coefficients for water and ammonia are the same as those given in Briggs and Sackett (1989). Vapor pressures for H2S were taken from the low temperature data of Kraus et al. (1993) and are of the same form as (13). In our modeling, we have often constrained the vapor pressures of NH3 and H2S to not exceed some specified fraction of saturation, or equivalently, some specified relative humidity. This is perhaps a more realistic treatment of the vapor pressures of condensibles. On Earth, the average water vapor content is seldom at saturation, even within cloudy regions (cf., e.g., Ludlam 1980). B. The T(P) Profiles In order to investigate the sensitivity of the microwave spectrum to temperature profiles, T(P), we have used adiabatic extensions of retrieved T(P) from Voyager radio occultation data (Lindal et al. 1987). The radio occultation measurements provide a vertical profile of radio refractivity. To retrieve a temperature profile from this, the atmospheric composition must be specified, because the refractivity properties of a gas depend on the composition. For Uranus, we assumed that the ratio [He]/[H2 1 He] 5 0.15 (Conrath et al. 1987), where the brackets refer to mole fractions, but we treated the abundance of CH4 below its cloud base as a variable parameter. Figure 4 depicts temperature profiles with the assumption that the mixing ratio of CH4 in the deep atmosphere is, respectively, 2 and 10%, and that the gaseous CH4 abundance follows the saturation curve above the cloud base (McMillan and Flasar 1989). We will discuss the contribution functions later. Note that the 10% methane profile is warmer at a given pressure than the 2% profile. This is entirely attributable to the refractivity properties of the atmosphere assumed in the temperature retrieval. Other physical effects, such as the latent heat of condensation, are not relevant to the temperature retrieval. Above the cloud base, gaseous CH4

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FIG. 4. Contribution functions at 2 cm for the 0.05% relative humidity model for the two temperature profiles indicated. The horizontal dotted lines mark the centroids of the contribution functions.

is constrained by the saturation curve and decreases rapidly with altitude; its scale height is small compared with the scale height of total pressure. Because the refractivity per molecule for CH4 is much larger than that for a H2 –He mixture and the abundance of CH4 becomes relatively large (a few percent by number), the refractivity properties vary rapidly with altitude. It is this local variation that controls the vertical gradients in the retrieved temperatures within the saturation region. The larger the abundance of CH4 reached along the saturation curve, the higher the gradients and retrieved temperatures. This effect has been discussed in some detail by Flasar (1983) in retrieving temperatures from radio occultation soundings of Titan’s atmosphere, and Lindal et al. have retrieved several temperature profiles corresponding to different assumptions about the vertical distribution of CH4 on Uranus (see their Fig. 9). Note that in Fig. 4, wet adiabats have not been imposed above the CH4 cloud base. The two temperature profiles in Fig. 4 form the basis of our sensitivity studies. Below the altitudes accessible to radio occultation sounding (P . 2.3 bar), these temperatures have been extrapolated along dry adiabats to 1 kbar. In computing these adiabats, we have assumed that the specific heat of molecular hydrogen corresponds to the ‘‘frozen equilibrium,’’ or ‘‘intermediate’’ state, in which the local para fraction is in thermal equilibrium, but the exchange of energy between the para and ortho states is neglected (Trafton 1967, Conrath and Gierasch 1984.) We have not imposed wet adiabats at the altitudes of condensation for two reasons. First, cloudy regions on Earth are in fact heterogeneous, and the temperature lapse rate typically lies between those of a wet and dry adiabat (cf., Ludlam 1980). Second, the difference between brightness temperatures computed using a dry or wet adiabat is

negligible. Even in the water cloud, where the contribution to the change in internal heat of the latent heat of water may be significant, the computed brightness temperatures at wavelengths less than 20 cm turn out to be insensitive to variations in T(P). One final point requires clarification. A self-consistent treatment would require that, in computing synthetic spectra from the two temperature profiles, we use the appropriate CH4 mole fraction in each case. Instead, we elected to always specify the deep CH4 mole fraction at 0.02, up to the level of saturation, above which methane followed its saturation curve. There were several reasons for this. The most important was that we wished to study the sensitivity of the spectra to the temperature profile, T(P). For reasons that will become apparent later, it was important not to confuse this effect with changes in the absorption cross section. The formulation of the absorption cross sections discussed in Section III only includes the effect of broadening with the absorber, or with H2 and He. The broadening effects of CH4 are not particularly well known. Insofar as the cross sections themselves have some uncertainty (cf., e.g., Spilker 1990), we have kept the H2 –He– CH4 mix the same in all our models—aside from constraints imposed by CH4 saturation, which has little effect on our conclusions on the spectra at microwave wavelengths. C. Water Cloud Density If DqH2O denotes the mole fraction of condensed water at some level in the atmosphere, then the mass density in the water cloud at this level is just

rcloud 5 eH2O DqH2O P/(RT ),

(14)

where R is the universal gas constant and eH2O is the molecular weight of water in atomic mass units. We use three estimates for DqH2O : (1) a deep uplift model, which assumes that DqH2O 5 qdeep 2 DqH2O (i), where qdeep and DqH2O (i) denote, respectively the mole fractions of water vapor below the water cloud and at level i; (2) a shallow uplift model, which assumes that DqH2O 5 qH2O (i) 2 qH2O (i 2 1); (3) the adiabatic uplift cloud as used by Briggs and Sackett (1989), Hofstadter (1992), and others:

rcloud 5

eH2O DqH2O P eRT D ln P

.

(15)

In this model the vapor pressures are in equilibrium and do not convect. In this case the density is that of the shallow cloud divided by (e D ln P), where D ln P is the step size. (For models 2 and 3 the cloud densities are dependent on the integration step size. In all of the models shown in this paper, D ln P 5 0.02).

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When applied to Earth’s atmosphere (U.S. Standard Atmosphere 1976) with this mesh size, the shallow model yields cloud densities that are within the measured range as given in Ludlam (1980). For the deep uplift model, we specify the vertical extent of the cloud in pressure scale heights, typically one scale height. The purpose of these different models is to explore the effects of thin and thick clouds in a situation for which the microphysical processes are poorly known. Clouds on the Earth show variations with respect to the nonconvecting adiabatic model of this order of magnitude. Our deep uplift model overestimates somewhat the cloud densities produced by a true adiabatic uplift, because we evaluate qdeep and qH2O (i), at the ambient temperatures, and hence along a dry adiabat within the water cloud (cf., Section IV.B), instead of a wet adiabat. V. RESULTS

A. Ammonia The microwave spectrum at centimeter wavelengths depends on the vertical profile of temperature and the vertical distribution of condensible gases, which provide much of the opacity. In order to investigate the sensitivity of the spectrum to these, we first consider the microwave spectrum with only H2 and NH3 as opacity sources. Figure 5 depicts spectra for four model atmospheres. In these models two T(P) profiles are used, the 2 and the 10% methane temperature profiles of Fig. 4. For each temperature profile, ammonia is constrained below the tropopause to follow a constant relative humidity of 0.05 and 100% down to the level at which the implied mole fraction becomes

FIG. 5. The spectra for models having uniform ammonia relative humidities of 0.05 and 100% are compared for the two temperature profiles in Fig. 4. The spectra corresponding to the colder profile are solid and those corresponding to the warmer one are dashed. The upper branch of each temperature set corresponds to the lower relative humidity. There is no H2S or H2O in these models.

FIG. 6. Contribution functions at selected wavelengths, renormalized to maximum values of unity, and the vertical profile of the mole fraction of NH3 for the 0.05% relative humidity and colder temperature model of the previous figure. The dashed line gives the NH3 mole fraction vertical profile.

equal to 100 times the solar value: 1.7 3 1022; at deeper levels the mole fraction is fixed at this value. In all cases this level lies well below those contributing to the fluxes at wavelengths #20 cm. Figure 6 illustrates the contribution functions and vertical profiles of gaseous NH3 for the one of the model atmospheres having 0.05% relative humidity. At short wavelengths, &1 mm, the brightness temperatures in Fig. 5 are insensitive to the NH3 abundance and depend primarily on T(P). This is because the opacity at these wavelengths is dominated by molecular hydrogen, which can be verified by noting that the 1-mm contribution function in Fig. 6 lies well above the NH3 condensation level. At longer wavelengths, *3 mm, the brightness temperature is sensitive to both the temperature and the NH3 abundance. However, beginning near 1 cm, ammonia is the dominant opacity source, and the brightness temperature is quite insensitive to T(P) and depends only on the NH3 relative humidity. This insensitivity to temperature is nearly complete out to 5 cm, but it persists in large degree out to 20 cm, the longest wavelength for which the spectra are displayed. To elucidate the behavior of the spectrum at centimeter wavelengths, we compare in Fig. 4 the contribution functions at a wavelength of 2 cm for two of the model atmospheres with the same relative humidity of NH3 , 0.05%, but with different T(P). If the 2-cm contribution function in the two models coincided in barometric altitude, the 2cm brightness temperatures would be lower for the colder atmosphere. However, due to condensation, the colder atmosphere has less NH3 above any specified pressure level, and so its contribution function is centered at higher pressure, where the temperature is higher. Thus, there is

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a compensation that acts to reduce the effects of different temperature profiles. Figure 4 indicates, in fact, that this compensation is complete: the centroids of the contribution functions for the two atmospheres are located at pressure levels having identical temperatures. This phenomenon has long been recognized in infrared sounding of the Earth’s atmosphere, in which the 6.3-em water vapor band has been used to retrieve vertical profiles of relative humidity (c.f., e.g., Mo¨ller 1961, Conrath 1969). Appendix A treats the general problem in some detail: If (i) the opacity at a given wavelength is predominantly from a condensible absorber in the troposphere, and (ii) the absorption cross section is independent of barometric pressure, then the flux (or radiance) at this wavelength is independent of the vertical profiles of temperature and absorber amounts, provided that these profiles are restricted to the class such that (a) the temperature lapse rates are a common function of temperature, and (b) the relative humidities are a common function of temperature. The spectra in Fig. 5 at centimeter wavelengths are a special case of this. The vertical profiles of relative humidity are uniform. The contribution functions at these wavelengths are situated at barometric pressures .2 bar, where the warmer and colder temperature adiabats (Fig. 4) have similar lapse rates at the same temperatures. Figure 7 illustrates two model atmospheres that offer a striking illustration of this phenomena in a situation where the relative humidity varies markedly with altitude. As before, the microwave opacity was restricted to H2 and NH3 , and the two temperature profiles of Fig. 4 were used. In these cases the mole fraction of NH3 was specified to remain constant with altitude until saturated, then follow the (100%) saturation curve at higher altitudes to the tropopause. The mole fractions are such that the temperature dependence of relative humidity is identical for the two atmospheres. As a consequence of this construction, the calculated microwave spectra are nearly coincident from 2 to 6 cm. Figure 8 depicts a series of model atmospheres with uniform relative humidities and illustrates the effect of decreasing the relative humidities. As the relative humidity decreases, there is less congruence of spectra in the centimeter region for atmospheres with a common relative humidity but different temperature adiabats. However, all the spectra shown still exhibit a marked insensitivity to the T(P) profile at centimeter wavelengths. For example, at the p400 K level of the atmosphere the difference between the warm and cold profiles in Fig. 4 is nearly 100 K, but even for the lowest relative humidity case indicated in Fig. 8—0.001%—the difference in brightness temperature is

only 30 K. The increasing divergence at long wavelengths with lower relative humidity is not a consequence of the increasing importance of H2 opacity; NH3 remains the dominant opacity at wavelengths .2 cm for all the cases illustrated. Rather, it results from the dependence of the absorption cross section on the barometric pressure P, violating item (ii), above. With lower relative humidities, the contribution functions move to greater depths and higher pressure, where the absorption cross section decreases as P21. In contrast to the situation that was depicted in Fig. 4 at 2 cm, the colder atmosphere exhibits the higher brightness temperature, because the contribution function at long wavelengths penetrates more deeply than it would if the absorption cross section were independent of P. The pressure dependence of the cross section also accounts for the divergences observed at long wavelengths in Figs. 5 and 8 (see also Section V.E). The Appendix treats this in further detail (cf. Fig. A1). Figures 5 and 8 illustrate the dependence of the spectral flux (brightness temperature) on relative humidity becomes weaker as the relative humidity increases. Factors of 2 change in relative humidity have less and less effect. Equation (A7) of the Appendix indicates that the effect on the centimeter spectrum of increasing the temperature lapse rate is equivalent to that of decreasing the relative humidity. Figure 9 illustrates the effect of varying the lapse rates. Model atmospheric temperatures are given in the upper panel. They are identical to the cooler profile in Fig. 4 down to the 2.3-bar level. At deeper levels they have been extrapolated along dry adiabats with values of Cp ranging from 0.8 to 1.6 times the frozen equilibrium value, C pfe , discussed in Section IV. The NH3 relative humidity has been set to 0.05% in all the models. The lower panel of Fig. 9 depicts the resulting spectra. At wavelengths less than 6 cm the brightness temperatures show a direct relationship to the temperature: the colder profiles have colder brightness temperatures. At longer wavelengths there is an inverse relationship between brightness temperature and temperature/ pressure profiles: the colder profiles have warmer brightness temperatures at a given wavelength. This can be explained by the pressure dependence of the absorption coefficient of ammonia. For pressures greater than 20 bars, the absorption cross section decreases (Fig. A1). A given optical depth would correspond to a larger integrated column for a colder atmosphere. This in turn would correspond to a higher temperature. Between 2 and 10 cm they are remarkably insensitive to the lapse rates. The sensitivity of the spectrum to T(P) in the millimeter region results from the fact that the H2 contributes to the opacity. The upper panel of Fig. 9 also depicts the adiabat for the specific heat corresponding to fully equilibrated ortho–para H2 . It is sufficiently close to the frozen equilibrium value below the 2.3-bar level

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FIG. 7. Two model atmospheres of Uranus with the temperature and NH3 profiles indicated. H2S and H2O are absent. The NH3 relative humidities in the two atmospheres have the same dependence on temperature, and they have been chosen so that the NH3 mole fraction is constant with altitude over much of the range probed by centimeter wavelengths. Also depicted are the contribution functions at 2.5 cm and the microwave flux spectra. In all panels the dashed curves correspond to the warmer atmosphere.

that the spectra do not offer a viable means of distinguishing between the two in this altitude range. B. Hydrogen Sulfide Hydrogen sulfide can also be important as an opacity source if its abundance is large enough (dePater et al. 1991). The opacity of H2S is greatest at wavelengths less than 1 cm; at centimeter wavelengths the H2S cross section is much weaker than that of NH3 (cf. Fig. 3). Figure 10 depicts the microwave spectrum for two model atmospheres, having the temperature profiles of Fig. 4 (and Figs. 5–8), in which H2 and H2S provide the absorption; NH3 and H2O are absent. H2S relative humidities for the warmer atmosphere are kept constant at 100% down to the level at which the H2S mole fraction reaches the rather large value 1.7 3 1022, or 500 times solar. At deeper levels it is constant. The relative humidity of the colder atmosphere was constrained to be the same function of temperature as that for the warmer atmosphere. [Note that the mole fraction of H2S in these models is constant over much of the range of altitudes probed by centimeter wavelengths. Unlike the

case with NH3 in Fig. 5, it was not practical to constrain H2S to constant relative humidity curves at these altitudes. The H2S absorption cross section is so weak (Fig. 3) that the contribution functions over much of the centimeterwavelength region are situated rather deeply in the atmosphere, so that the implied mole fractions of H2S at 100% relative humidity would be close to unity.] The computed spectra in Fig. 10 indicate that the condensate opacity is compensating for the different T(P) profiles over much of the microwave spectrum. The spectra are not as insensitive to T(P) as when NH3 provided the dominant opacity (e.g., in Fig. 5 or 7), but the differences in brightness temperatures at millimeter and centimeter wavelengths are still small. For example, at 1 cm, where the brightness temperature is p160 K, the difference between the two spectra is 9 K; the corresponding difference in atmospheric temperature at the 160-K level is 36 K. At 20 cm, the difference in brightness temperature (p300K) is 30 K, whereas the corresponding difference in atmospheric temperature is p70 K. That the two spectra do not coincide stems from different effects at long and short wavelengths. At the shorter wave-

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tion of NH3 and H2S will be controlled by the formation of an NH4SH cloud via the reaction H2S 1 NH3 } NH4SH—first proposed by Wildt (1937) as a depletion mechanism for NH3 in the upper atmosphere of Jupiter— and also by dissolution in any liquid water cloud that may form (cf. Section V.C, below). When the relative humidity does vary vertically, the foregoing analysis suggests that the emitted flux (or radiance) at centimeter wavelengths depends principally on the relative humidities of the condensible absorbers, averaged over the vertical extent of the contribution function at each wavelength. In Fig. 11 we compare spectra computed with constant relative humidity with one computed assuming that the

FIG. 8. Microwave flux spectra for model atmospheres with the various NH3 relative humidities indicated and the two temperature profiles in Fig. 4. Curves are dashed for the warmer atmosphere. The relative humidities are constant down to a level at which the implied NH3 mole fraction is 1.7 3 1022, 100 times solar. H2S and H2O are absent.

lengths the H2 contribution to the microwave opacity is not negligible. In the first panel of Fig. 10 we have plotted the ratio of the H2S optical depth, measured vertically from the top of the atmosphere, to the total optical depth. This is plotted for 1 cm, but that at 2.5 mm is virtually identical, and that at 10 cm is quite similar. This is not surprising, because, from Fig. 3 the H2S and H2 absorption coefficients for fixed abundances—equivalently the absorption cross sections—exhibit similar rates of decrease with wavelength beyond 2.5 mm. In Fig. 10 the 2.5-mm and 1-cm contribution functions span an altitude range over which H2 contributes significantly to the optical depth at 2.5 mm and not negligibly at 1 cm. At longer wavelengths H2S dominates the optical depth, but the absorption cross section for H2S depends on the pressure, varying nearly linearly with P over the pressure range of interest (cf. the discussion in the Appendix and Fig. A1). This is unlike the behavior of the NH3 , which is independent of pressure over much of the centimeter wavelength range. Hence, when H2S dominates, the colder atmosphere has the lower brightness temperature. From the foregoing discussion and that in the Appendix, it follows that the above conclusions can be extended to mixtures of NH3 and H2S. At centimeter wavelengths, the radiances depend only weakly on the temperature profiles and primarily on the relative humidities of the condensibles. We do not anticipate that the relative humidities of NH3 and H2S necessarily remain constant with altitude in Uranus’ troposphere. To some extent, the spatial distribu-

FIG. 9. (Top) Temperature profiles corresponding to different adiabats for P . 2.3 bar. The adiabats are labeled by the ratio of the assumed Cp to that for the atmosphere with hydrogen in the so-called frozen equilibrium state. The temperature profile with the adiabat labeled 1 is identical to the colder temperature profile in Fig. 4. The dotted curve corresponds to an adiabat for which Cp assumes that hydrogen is fully equilibrated. (Bottom) Spectra corresponding to the temperature profiles depicted as solid curves in the (Top).

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FIG. 10. Two model atmospheres of Uranus with the H2S profiles indicated and the temperature profiles as in Fig. 4. In all panels, the curves corresponding to the warmer atmosphere are dashed. NH3 and H2O are absent. The H2S relative humidities in the two models have the same dependence on temperature. Also depicted are the microwave flux spectra, the contribution functions at 2.5 mm, 1 cm, and 10 cm, and the ratio of the H2S vertical optical depth–measured from top of the atmosphere–to the total optical depth at 1 cm. The corresponding curves for the ratios of optical depths at 2.5 mm and 10 cm are nearly the same.

FIG. 11. Solid curves: Spectra computed analogously to those in Fig. 8 with the colder (2% methane) T(P) profile of Fig. 4 for the four values of ammonia relative humidity indicated. Dashed curve: Spectrum computed using the same temperature profile, but with an equilibrium chemistry model with 4.6 solar H2S, and otherwise solar composition.

vertical profile of ammonia is controlled by equilibrium chemistry with H2S, as well as condensation above the lifting condensation level. The solid curves in Fig. 11 are spectra for four values of NH3 relative humidity (100, 0.35, 0.05, and 0.01%). The dashed curve in Fig. 11 is a spectrum computed with equilibrium chemistry for solar ammonia and 4.6 solar H2S. The opacity from H2S in this model is small at all wavelengths relative to that from H2 or NH3 . The main effect of H2S is to control the vertical distribution of NH3 over the barometric pressure range 30 to 10 bars and above on Uranus for the nominal temperature profile. The chemical equilibrium curve and the 0.35% constant relative humidity curve give almost exactly the same spectrum, although the relative humidity profiles for these two cases, shown in Fig. 12, differ in detail. There is a mild suggestion that the spectrum at 2 cm is better fit with a higher relative humidity (0.35%) than at 6 cm (0.05%), suggesting that the relative humidity decreases with depth. However, the spread in the data and the formal errors preclude any definitive conclusion. For a model with a shallow cloud, and assuming a VVW profile, then H2S does not dominate the opacity unless it is enhanced by a factor

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deep models, respectively. The densities are plotted in the upper panel along with the NH3 mixing ratio profile. The calculated brightness temperatures, bottom panel, illustrate what portion of the difference in temperature could be accounted for by variations in the underlying water cloud opacity. The cloud opacity is small for the shallow cloud model and increases as the cloud becomes thicker. Note that terrestrial clouds can assume liquid water density profiles that resemble any of these models (Ludlam 1980). The bottom panel of Fig. 14 compares the variation in the spectrum caused by varying the water cloud depth with variations caused by varying the H2S mole fraction. The

FIG. 12. For two of the model atmospheres for which spectra are illustrated in Fig. 11, the profile of relative humidity of NH3 and contribution functions at 2 cm are illustrated. These models are (a) constant RH(NH3 ) 5 0.35% and (b) 4.6 solar H2S, solar NH3 , and solar H2O with equilibrium chemistry.

of 30 or more above its solar value. However, if H2S has a Ben Reuven lineshape in the centimeter region, as derived by de Boer and Steffes (1994), then the opacity assumed here may be 40% low, and H2S would dominate if it is enhanced by more than a factor of 18 over solar. C. Liquid Water Liquid water may influence the microwave spectrum in two ways: first, through the effect of the solution cloud on the vertical profiles of the major opacity sources, NH3 and H2S; second, if the cloud top is at sufficiently low pressure, the opacity of the water droplets themselves may become important. Depending on the depth of the cloud base and its vertical extent, an atmosphere with a thicker cloud may be less opaque in the far centimeter region, beyond 5 cm, than one with a thinner cloud as a result of dissolution of the major opacity sources in the cloud. The water cloud will only contribute opacity if the cloud is high enough in the atmosphere to compete with the NH3 opacity. Paradoxically, for large water abundances, the water cloud forms deeper in the atmosphere, and unless strong vertical uplift extends over several scale heights, such a cloud would not contribute to the opacity in the microwave region. A thick water cloud can appreciably alter the vertical distribution of condensibles, as discussed below. We illustrate the effect of the opacity of a water cloud on the microwave spectrum in Fig. 13. Here we have used a simplified model containing only NH3 and H2O and for simplicity we have assumed that NH3 does not dissolve in the water cloud. [NH3 ] follows a constant 0.05% relative humidity profile, and [H2O] 5 1 3 1023 to the cloud base. The cloud models are our shallow, normal adiabatic, and

FIG. 13. Synthetic spectra (Bottom) were calculated for an atmosphere with NH3 following a constant 0.05% relative humidity profile with three cloud models shown in (Top): the shallow (short-dash), adiabatic (dot-dash), and deep (long-dash), respectively. The cloud densities and NH3 mixing ratio profile are shown in the upper panel. There is no dissolution of NH3 in the water cloud in this example to show the effect of cloud opacity only. The solid curve represents a model with no cloud.

MICROWAVE SOUNDING OF THE GIANT PLANETS

FIG. 14. (Bottom) Synthetic spectra for four atmospheric models, using the colder T(P) in Fig. 4. Solar abundance models computed with (a) the deep uplift cloud with vertical extent 1.5 scale heights (solid curve) and (b) the deep uplift cloud with vertical extent 1.0 scale heights (long dash), and (c) the shallow uplift cloud (dot-dashed curve); model (d) is computed assuming the shallow cloud with solar abundance except that the deep H2S abundance is enhanced by a factor of 4.6 with respect to its solar value (short dashed). (Top) Contribution functions at 6 cm (right–hand side) NH3 mole fractions, and cloud densities for models (a) solid, (b) long dash, and (d) short dash. Clouds (a) and (b) coincide except for the vertical extent. The lower portion of the NH3 mixing ratio coincides for models (a) and (b).

models shown have solar composition with a deep cloud having a vertical extent of (a) 1.5 scale heights (solid curve) and (b) 1 scale height (long dashed curve), and (c) a shallow cloud (dot-dashed curve). The figure also depicts (d) a model with H2S enhanced to 4.6 its solar value with a shallow cloud (dashed curve). (The spectra for the model with 4.6 solar H2S with a deep cloud is virtually identical to that of the solar abundance model with a deep cloud

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and therefore is not shown). The upper panel depicts NH3 mole fractions, cloud densities, and the 6-cm contribution functions for models (a), (b), and (d). The 6-cm contribution functions probe cloud opacity for the deep cloud models (a,b) and NH3 opacity for the shallow cloud models (c,d). The brightness temperatures at 6 cm are closer than the corresponding cloud top temperatures because for model (b) enough NH3 vapor remains above the cloud to provide the additional opacity. Although, for clarity, the 6-cm contribution function for (c) is not shown, the shallow cloud is relatively ineffective in dissolving NH3 in a solar, H2S model, so that the contribution function lies at lower pressures, p4 bars, well above the cloud top. This is reflected in the much colder brightness temperatures of model (c) at centimeter wavelengths. The close proximity of the 6-cm contribution functions of models (a), (b), and (d) illustrates the possible ambiguities involved in attempting to use the spectra as diagnostic of NH3 relative humidity and cloud opacity. Model (b) is colder in the millimeter region than models (a) and (d) because enough ammonia remains between the water cloud top and the NH3 condensation levels ([NH3 ] 5 6 3 1027) to provide the extra opacity here. On the other hand, this model is warmer than the latter two in the centimeter region for different reasons. In model (a), the opacity in the centimeter region is mostly provided by liquid water, whereas in model (d) it is provided by ammonia vapor. Note that if [H2S] were enhanced to the extent that [H2S] . [NH3 ] below the NH4SH cloud, H2S rather than NH3 would remain above the NH4SH cloud, and the opacity above the water cloud would be due to H2S instead of NH3 . Whether the H2S opacity is significant depends not only on the [H2S]/[NH3 ] ratio below the water cloud base, but also on the water cloud model, the H2O abundance, and the H2S line strength and broadening. Recent measurements (de Boer and Steffes 1994) indicate that the H2S centimeter opacity may be larger than that assumed herein. De Boer and Steffes (1994) showed that a Ben Reuven formulation fits the H2S data, while we assumed a VVW profile. It is clear from the millimeter data that a small opacity source must remain in the region 5–30 bar. If this opacity is provided by NH3 then [NH3 ] > 5–6 3 1027 between 30 bars and the NH3 condensation level. De Pater et al. (1991) showed that this could be accomplished with H2S. A water cloud is an alternative. Since ammonia dissolves more readily in liquid water in the presence of H2S, a thinner cloud than that in model (b) would suffice for models with H2S enhanced up to 4 solar. D. Meridional Structure of Uranus Disk resolved brightness temperatures of Uranus at 2 and 6 cm (plotted in Fig. 15) were obtained by Hofstadter and Muhleman (1989) and Hofstadter (1992). They were

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[I(e 5 1) 2 I(e)]/I(e 5 1), illustrating that the limb function can vary considerably. In order to reconstruct the uncorrected brightness temperature data, we used the limb darkening curve from Hofstadter’s standard model. We used the resulting brightness temperatures to estimate the relative humidity as a function of latitude, assuming vertical profiles of constant relative humidity. The exact relative humidities computed for selected data points are given under each datum. Although there are differences in detail from the relative humidities implied from using Hofstadter’s corrected brightness temperatures in Fig. 15, the general trend of the relative humidity with latitude is unchanged. These deduced variations in ammonia relative humidity assume that there is no water or H2S opacity at these wavelengths. Comparison of the brightness temperatures at 2258 and 2658 latitude with spectra calculated for a 10solar H2S model using shallow and deep cloud models,

FIG. 15. The disk resolved brightness temperatures, corrected to normal viewing, from Hofstadter (1992) at 2 cm (Bottom) and 6 cm (Top) are illustrated along with curves showing the brightness temperature for models with constant ammonia relative humidity and without H2S or water. The derived relative humidities at various latitudes are shown under the corresponding datum.

published corrected to normal viewing using a limb function for a standard atmosphere model (basically our cold temperature adiabat in Fig. 4) with [NH3 ] 5 8 3 1027, and [H2O] 5 1.0 3 1023, with no H2S. The opacities in his radiative transfer model differ slightly from those we used, as discussed in Section III, and his cloud model is our adiabatic uplift model. The data show a minimum brightness temperature near 2258 latitude and a maximum near 2658 latitude. The horizontal lines in Fig. 15 indicate brightness temperatures for various constant NH3 relative humidity profiles. The figure suggests that the brightness temperatures could be explained by Uranus’ atmosphere having a higher relative humidity near 2208 than at higher latitudes. Note that the 2- and 6-cm limb functions depend on the vertical relative humidity profiles. In Fig. 16 we plot

FIG. 16. Limb-darkening profiles at 2 and 6 cm for Saturn and Uranus are shown for models with constant vertical mixing ratios of NH3 up to saturation (solid), and for various assumed constant relative humidity models: dashed 5 0.01%, dotted 5 0.10%, dot-dash 5 1.0%, long dashdot 5 100%.

MICROWAVE SOUNDING OF THE GIANT PLANETS

FIG. 17. The brightness temperatures of Uranus at 2208 and 2608 latitude are shown with models having solar abundance ammonia with 10 solar H2S and with a shallow water cloud (solid) and a deep water cloud (dashed).

respectively, (Fig. 17) shows that up to half of the latitudinal variation of brightness temperature could be caused by variations in an underlying water cloud. In this model [10-solar H2S, 1-solar NH3, 1-solar H2O], all of the NH3 is depleted by the H2S in the NH4SH cloud. In this case variations in brightness temperature in the region 2–6 cm are exclusively due to cloud opacity, not vapor opacity. This is in contrast with the models illustrated in Fig. 13 in which there is neither H2S nor dissolution of NH3 in the water cloud. In that case, variations in brightness temperature are due to cloud opacity in the presence of the same overlying NH3 abundance. That case is artificial because the overlying NH3 abundance is assumed to be unaffected by the cloud, and the cloud opacity is masked to a large degree by the vapor opacity.

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For Saturn, the 20-cm wavelength probes the region 3–8 bar where the absorption cross section of NH3 increases with increasing pressure. The warmer atmosphere will have an equivalent absorber column at a lower pressure, but the optical depth will be less there because of the pressure dependence of the absorption coefficient. Thus, the warmer atmosphere will have a slightly warmer spectrum even if the relative humidity as a function of depth is the same. Recall that the opposite effect was found for Uranus’ atmosphere, in which the 20-cm wavelength probes the region near 50–70 bars. In this pressure regime (50–70 bars), the NH3 absorption cross section decreases with increasing pressure, and the atmosphere with the warmer temperature profile would have the cooler spectrum, given that the RH profiles were the same. In Fig. 18 the implied RH 5 100% for brightness temperature, TB , 152 K, but decreases rapidly with higher TB . To study this further, we have constructed models in which the RH 5 100% at altitudes for which T , 152 K and d ln RH/d ln P 5 constant at deeper levels with higher temperatures. We then examined several models with different values of the constant until we achieved a reasonably good fit with d ln RH/d ln P 5 24.5. The resulting spectra are shown as the dot-dashed and long dashed curves in Fig. 18. Note that the brightness temperatures are much closer than the difference in kinetic temperature at a given pressure. The bottom panel of Fig. 19 depicts relative humidities and mole fractions of NH3 for the atmospheres with the two temperature profiles. The dot-dash curves in

E. Saturn Figure 18 shows the brightness temperatures for Saturn for two constant relative humidity models with two different temperature profiles, given in Fig. 19. The colder profile is the Saturn temperature profile retrieved by Lindal et al. (1985), extrapolated to altitudes below the 1.3-bar level along a dry adiabat. The warmer profile represents an offset from the Lindal et al. profile at 1.3 bar and extrapolation along a warmer adiabat. The warmer profile yields a spectrum very close to the colder profile when the same RH(T ) is used. Because d ln RH/d ln P 5 G (T )d ln RH/d ln T, where G 5 R/Cp (T ), it follows that this quantity should be identical in the two models. The difference in brightness temperature, which increases with increasing wavelength, is attributable to the pressure dependence of the absorption coefficients, as discussed in the Appendix.

FIG. 18. Brightness temperatures for Saturn’s microwave spectrum for two models having constant relative humidities of NH3 at 1 and 100%, respectively, for the two temperature profiles shown in the next figure: those for the colder profile are solid, the warmer profile dashed. The dot-dashed and long-dashed curves correspond to a model having 100% relative humidity for T , 152 K, and d ln(RH)/d ln(P) 5 24.5 for T . 152 K down to the level at which this would imply that the mixing ratio decreases with increasing pressure for the cold and warm temperature profiles, respectively. The data (starred) are from Grossman (1990).

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for wavelengths less than 15 cm, because the contribution functions at these wavelengths are above the region of constant mixing ratio in the modified model. VI. DISCUSSION AND CONCLUSIONS

We have shown in the preceding section that, under a wide variety of conditions, the microwave spectrum at centimeter wavelengths is not very sensitive to the T(P) profile, but rather to the vertical profile of relative humidities of the condensibles. Figure 20 illustrates this geometrically. The upper panel illustrates the situation at wavelengths shorter than 1 mm. In this region molecular hydrogen dominates the infrared opacity, and a profile of

FIG. 19. (Top) shows the two temperature profiles assumed. The contribution function is shown for 20 cm. For the model depicted with long dash and short-dash lines in Fig. 18, (Bottom) depicts the mixing ratio of ammonia, and the corresponding relative humidity for the cold (solid) and warm (dashed) temperature profiles, respectively. The dotdash curve is the solution assuming d ln(RH)/d ln(P) 5 24.5 for all T . 152 K; the solid and dashed curves assume that q is constant below 3 bars.

the bottom panel of Fig. 19 show the mole fractions and relative humidity for the constant d ln RH/d ln P model with the colder temperature profile. At levels deeper than 3 bars, the assumed simple parametric variation of RH implies that the mole fraction of NH3 decreases with depth, which is probably unphysical if the source of ammonia is the deep atmosphere and interior. The solid and dashed curves (lower panel) represent a modification of the model in which the mole fraction remains constant with depth below the level of maximum mole fractions in the constant d ln RH/d ln P model. The difference between the spectrum in this model and those in which [NH3] is constant below the 3-bar level (solid and dashed curves) is negligible

FIG. 20 (Top) Geometrical representation of retrieved atmospheric parameters at wavelengths l , 1 mm, where hydrogen opacity dominates. (Bottom) Idealized case at centimeter wavelengths, when NH3 dominates the opacity.

MICROWAVE SOUNDING OF THE GIANT PLANETS

temperature vs pressure can be retrieved from the spectrum. (As always, we assume the ratio of helium to hydrogen is known.) In the (N 1 2)-dimensional hyperspace comprising temperature, pressure, and the mole fractions of the N radiatively active condensibles, the solution is an (N 1 1)-dimensional surface, whose properties do not depend on the mole fractions of the condensibles, qi . The intersections of this surface with the (N 1 1)-dimensional planes qi-constant yield identical T(P) curves. It is important to emphasize that in dealing with an actual spectrum, the retrieved T(P) relation is not unique. First, the radiative-transfer process acts as a low-pass filter on the retrievable vertical temperature profile, no matter how finely the spectrum is sampled in wavelength. In practice, spectra are often only at a finite number of wavelengths, often not optimal, thereby limiting the vertical resolution of the retrieved profile even more. Second, real data have measurement errors, which introduce limits to the accuracy to the retrieved temperature at any specified pressure level. In general there is a tradeoff between temperature accuracy and vertical resolution, and any retrieval algorithm necessarily entails a compromise between these factors (Conrath 1972, Hanel et al. 1992, Chapter 7). Hence, the actual hypersurface in Fig. 20 is not uniquely determined in a retrieval, although its shape at lower spatial frequencies is constrained by the spectrum. The lower panel depicts an idealized situation at centimeter wavelengths when a single minor gaseous condensible, e.g., ammonia, dominates the microwave opacity. The results of the previous section suggest that in this situation the profile of relative humidity with temperature can be retrieved, subject to the caveats just given. The solution is again an (N 1 1)-dimensional surface, which now depends on qNH3 , but not on the remaining condensibles. The intersections of this surface with the (N 1 1)-dimensional planes perpendicular to the T-axis define a series of hyperbolas that correspond to constant relative humidity on each of these planes qNH3 P 5 f (T )P, where f is the relative humidity. The solid curves in the lower panel represent a series of temperature profiles within the solution surface that have similar lapse rates but are displaced from one another in the T–P plane. All are possible solutions, and from the microwave data alone, it is not possible to distinguish among them. For this to be an accurate representation of the solution space, the microwave absorption cross section must be independent of the barometric pressure, P, and the lapse rate in temperature must be a single-valued function of temperature. Neither condition is strictly met in the outer planets. The absorption cross section, a, generally does vary with P, although there are important exceptions, such as the insensitivity of the ammonia cross section to P near 2 cm (Fig. A1). The temperature lapse rates in the interiors of the giant planets are not a priori known. Nonetheless, the pressure variations in a are sufficiently

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weak, and the lapse rates in temperature are probably sufficiently constrained from theoretical considerations that the effects of these on the brightness temperature remain secondary. The effect of the relative humidity remains dominant, and the dependence of the spectrum T vs P is still relatively weak. The solution space is therefore more nearly an (N 1 2)-dimensional curved ‘‘pancake’’ rather than an (N 1 1)-hypersurface in the bottom panel. When the microwave absorption is dominated by two gaseous absorbers—as it might be, for example, when H2S is enhanced well above its solar abundance and removes nearly all of the gaseous NH3, forming an NH4SH cloud— the retrieval process becomes ambiguous, since now the spectrum depends on the relative humidities of two absorbers. Some of this ambiguity may be removed by a consideration of a sufficiently large spectral interval, given the different spectral behavior of the NH3 and H2S absorption cross sections, coupled with a consideration of physical processes determining the relative humidity profiles. We have not pursued this question in the present work, but we do note that even in this case, the centimeter spectrum is still primarily sensitive to the relative humidities of the condensibles contributing to the opacity, and it is only weakly dependent on the T(P) profile. At wavelengths where cloud opacities become important, interpretation becomes more difficult. Section V.D illustrates how variations in the opacity of thick water clouds can mimic variations in the relative humidity of NH3 in the resulting spectrum. One could possibly extend the formalism of the Appendix if one were to uniquely relate cloud opacity to, for example, supersaturated relative humidity of water, but this involves the formulation of a physical process that is not at all well understood on the giant planets. Fortunately, there are a wide range of situations in which it is plausible that NH3 dominates at least part of the centimeter spectrum. Water clouds become important contributors at the longer wavelengths. For our thickest clouds, this occurs for l . 4 cm on Uranus (Fig. 13); for the remaining cloud models l . 10 cm. If NH3 does dominate the spectrum, the variation of relative humidity with depth is relatively slow. Given the errors in the available spectral fluxes, the disk-averaged spectrum can be reproduced with an atmospheric model for which the vertical profile of relativity remain constant at 0.05% (Figs. 5, 8). A somewhat better fit is achieved if the relative humidity decreases with depth, being approximately 0.35% when the atmospheric temperature T , 180 K, and 0.05% when T . 200 K. The meridional variation of the 2- and 6-cm brightness temperatures suggest that the relative humidity is less at high latitudes than that at latitudes near 2208, although a variation in the height of thick clouds with latitude could simulate at least part of this behavior at 6 cm. On Saturn, the disk-averaged spectrum is more consistent with a rapid

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variation of NH3 relative humidity with depth. An iterative procedure in which the d ln RH/d ln P was assumed to be constant at levels where T . 152 K gave a best fit to the spectrum when the implied mole fraction of NH3 was nearly constant with depth at these temperatures (Figs. 18 and 19), in agreement with the shape of the NH3 profile used by Grossman (1990). Part of the apparent differences in the vertical profiles of NH3 on Uranus and Saturn may reflect the colder temperatures in the former, but it may also be a consequence of an enhancement in sulfur and water, with the implied enhancement in the chemistry controlling the vapor pressure of NH3 , which several authors have proposed for Uranus’ bulk interior composition. The total problem is very complex, and it is important to appreciate what type of information is retrievable from the microwave spectrum before embarking on an ambitious program of modeling the physical processes of the deep atmosphere and interior. Further progress requires improved microwave spectra, particularly of Uranus and Neptune, not only with respect to measurement accuracy but also wavelength coverage. APPENDIX A

Optical Depth of a Condensible Absorber In this Appendix we first demonstrate that the optical depth measured from the top of the atmosphere to an altitude within the troposphere is simply a function of temperature, provided that: 1. The optical depth only depends on the overlying amount of absorber. 2. The absorber is a condensible, whose column abundance is determined by the saturation constraint. 3. The relative humidity profile is constant. 4. The lapse rate in temperature is constant. These conditions were specified by Conrath (1969) in the context of absorption by infrared water vapor bands in the Earth’s atmosphere, although no explicit proof was given. It follows from Eq. (1) that a family of model atmospheres having the same relative humidity and lapse rates but different temperature profiles, T(P), will have identical spectral fluxes and radiances. After demonstrating these results, we then specify under what conditions (1), (3), and (4) can be relaxed and the conclusions concerning the spectral radiances and fluxes still remain valid. The optical depth at wavenumber n, tn , is

tn 5

E

P

0

a (P9, T )ni H

dP9 , P9

(A1)

where ni is the number density of the condensible gas, and H is the pressure scale height, H5

RT , eg

(A2)

R being the universal gas constant, e the molecular weight of the atmosphere, and g the local gravitational acceleration; a is the absorption cross section, which we initially assume to be constant. From the perfect gas law,

ni 5

Pi , RT

(A3)

where Pi is the partial pressure of the condensible. Hence,

tn 5

1 eg

E

P

0

a Pi

dP . P

(A4)

Defining a temperature lapse rate by ­T , ­ ln p

(A5)

Pi , P si (T)

(A6)

G5 and a relative humidity by f5

where P si (T ) is the saturation vapor pressure, we rewrite (A4) as

tn 5

1 eg

E

T

T0

a fP si (T9)

dT9 , G

(A7)

where T0 represents the temperature at zero barometric pressure. In physical cases of interest, the partial pressure of the condensible will be small through most of the atmosphere above the tropopause or some level in the troposphere, below which it follows a saturation law. From condition (3) this saturation law is assumed to be one of constant relative humidity with altitude. Because the saturation vapor pressure increases rapidly with temperature in this region as exp(2e L(T )/RT 2), L(T ) being the molar latent heat of vaporization, the dominant contribution to the integral in (A6) comes from a region in which Pi is uniquely determined by temperature via the Clausius-Clapeyron relation. Assuming, from condition (4), that the lapse rate G is also constant, tn is only a function of temperature. As already noted, it follows from Eq. (1) that a a family of atmospheres with different temperature profiles T(P), but with identical lapse rates G and relative humidities f, will have identical radiances and fluxes at n. Conditions (3) and (4) can be immediately relaxed to encompass spatially varying relative humidities and G. If we restrict ourselves to a class of profiles for which the relative humidities and lapse rates each have a common functional dependence on temperature, i.e., f 5 f (T ) and G 5 G (T ), then from (A7), all have the same optical depth at n, hence the same spectral radiance and flux, provided that the condensible dominates the opacity. Thus, for example, in Fig. 7, the spectra at centimeter wavelengths are insensitive to T(P) and depend only on the prescribed constant relative humidities, because at the levels probed by the radiances at centimeter wavelengths, the adiabatic extrapolations we have used for our warm and cold temperature profiles have similar lapse rates at the same temperatures. Note further that, from (A7), the effect of varying 1/G (T ) is equivalent to varying f (T ). Condition (1) can also be relaxed somewhat. The conclusions of the theorem remain valid if a, the absorption cross section—i.e., the absorption coefficient per molecule of absorber—depends on T but not on P. For NH3 absorption, when the mole fraction of NH3 is small, the absorption cross section is a function of T, PNH3 , and P (cf. Section III). In the present context, T uniquely determines PNH3 , either because the absorber is constrained to follow a profile of constant relative humidity or because we have restricted our consideration to a family of relative humidity profiles having the same functional dependence on T. We therefore have examined the behavior of the NH3 absorption cross section with P when T and PNH3 remain constant. We first verified that when P and T are

MICROWAVE SOUNDING OF THE GIANT PLANETS held constant and qNH3 is sufficiently small (#1022), the NH3 absorption cross section is independent of qNH3 , and hence PNH3 (because PNH3 5 qNH3 P). From this it follows that the variation of a with P when T and PNH3 are held constant is equivalent to its variation with P when T and qNH3 are held constant—this can be directly verified by application of the chain rule for partial derivatives. The top panel of Fig. A1 depicts the latter variation at two wavelengths, 2 and 10 cm. Included are the locations (in pressure) of the contribution functions corresponding to the temperature profiles in Fig. 4 and the constant NH3 relative humidities of 100 and 0.05% (c.f. Fig. 5), and 0.01% (Fig. 8). At 2 cm the variation of the absorption cross section with pressure is small. From the preceding discussion, we anticipate that the 2-cm fluxes will exhibit no variation with T(P) at a fixed relative humidity, and Fig. 8 indicates that this is indeed the case. The cross section at 10 cm, on the other hand, exhibits a maximum near 20 bar, increasing linearly with P when P , 10 bar and decreasing as P21 when P . 60 bar. This behavior can be easily understood in terms of the Lorentz profile for pressure broadening. Each line in the inversion spectrum contributes a term to the absorption cross section proportional to Dn [(n 2 n0 )2 1 (Dn)2], where n is the wavenumber of observation, n0 is the kth inversion line center

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wavenumber, and Dn is the linewidth from pressure broadening, YP. Because 10 cm is well away from the inversion line centers two distinct regimes emerge. At sufficiently low pressure, un 2 n0 u @ Dn, and the lineshape factor scales as P. At sufficiently high pressures, where un 2 n0 u ! Dn, the lineshape factor scales as P21. The contribution functions corresponding to the 100% relative humidity case for the two T(P) profiles span a region where a increases with pressure. Instead of exhibiting the same brightness temperature, as would be the case if a were independent of P, the warmer atmosphere exhibits the higher brightness temperature (Fig. 5), because the 10-cm contribution function corresponding to the cooler atmosphere does not penetrate as deeply as it otherwise would have. In the lower relative humidity atmospheres, 0.05 and 0.01%, the contribution functions are situated in a pressure range where a decreases with P, and it is now the colder atmospheres that have the higher brightness temperatures (Figs. 5 and 8). In the 0.05% relative humidity case, the contribution functions span a pressure range where the decrease in a with P is small, and the differences between the spectra are also small. The lower panel in Fig. A1 presents the corresponding cross sections for H2S. Here the important line centers are at millimeter wavelengths, un 2 n0 u @ Dn over much of the pressure range at both wavelengths, and a increases linearly with P. In this case the colder atmosphere always has the lower brightness temperature (c.f., e.g., Fig. 10). The analysis presented is primarily useful for insight, rather than actual application, and direct calculation of the spectrum is generally necessary to reach conclusions concerning its sensitivity to T(P). Section V explores this in some detail. Equations (A1), (A4), and (A6) were derived under the assumption that only one condensible absorber is dominant. However, they can easily be extended to the situation in which several condensible absorbers contribute to the opacity, by summing over the i variables.

ACKNOWLEDGMENTS Initial support for this research came from the Universities Space Research Association Visiting Scientists Research Program at the Goddard Space Flight Center (R.M.K.) and the NASA Uranus Data Analysis Program (F.M.F.). F.M.F. also acknowledges support for this work from the NASA Planetary Atmospheres Research Program. We thank B. J. Conrath, P. N. Romani, and T. R. Spilker for several stimulating discussions. P.N.R. also provided valuable comments on setting up the chemical equilibrium calculations, and a code to compute vapor pressures of NH3 and H2S over water. We thank two anonymous referees for valuable comments on the manuscript.

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