Radiative-convective equilibrium models of Uranus and Neptune

ICARUS 65, 383-405 (1986)

Radiative-Convective Equilibrium Models of Uranus and Neptune JOHN F. APPLEBY Earth & Space Sciences Division, Jet Propulsion Laboratory, California Institute o f Technology, 4800 Oak Grove Drive, Pasadena, Califi~rnia 91109 Received July 9, 1985; revised November 25, 1985 A study of radiative-convective equilibrium models for Uranus and Neptune is presented, with particular emphasis on the stratospheric energy balance, including the influence of aerosol heating and convective penetration. A straightforward numerical method is employed (J. F. Appleby and J. S. Hogan (1984). Icarus 59, 336-366) along with standard opacity formulations and the assumption of local thermodynamic equilibrium. A range of models was considered for Uranus, reflecting uncertainties in observational constraints on the middle stratospheric temperatures. The results indicate that a "continuum absorber" could be significant in the stratosphere, despite Uranus' great distance from the Sun. Also, test runs are presented to illustrate the influence of uncertainties in the gas composition and changes in the effective mean insolation. A long-standing theoretical problem for Neptune has been to explain the unexpectedly high stratospheric temperatures without invoking supersaturation of CH 4. The results show that a "'continuum absorber" could contribute significantly to the energy balance within a localized stratospheric region; however, it probably cannot provide enough power to explain the observed infrared spectrum, regardless of its vertical distribution. One alternative is "convective penetration" which could arise if, for example, vertical mixing is so rapid that CH4 condensation cannot occur before the gas is swept upward, above the condensation region. In the example considered here, the CH4 mixing ratio in the middle and upper stratosphere is equal to that below the condensation region in the troposphere. The infrared emission from this model was found to be in generally good agreement with the observations. Such a model could also apply to Uranus, in lieu of aerosol or other "additional" heating mechanisms, to an extent that is commensurate with weaker convective uplifting. ~,: 1986AcademicPress. Inc.

INTRODUCTION

This investigation presents mean atmospheric profiles which focus on the middle and lower stratospheric energy balance of Uranus and Neptune, on the possible role of aerosol heating at these levels, and on uncertainties in the principal observational constraints currently available. The corresponding model infrared spectra are compared with recent measurements for both planets. It has been apparent for many years that the stratospheric thermal structures of the Giant Planets do not represent a concise set of radiative-convective (R-C) equilibrium states. Neptune's 8-p~m emission provides a striking example: it is so high that a conven-

tional model calculation analogous to Uranus underestimates the flux by a factor of several hundred. The structure of the Uranian stratosphere has proved to be equally challenging. Prior to the important 8- to 13-/zm observations in 1977 (Gillett and Rieke; Macy and Sinton), CH4 supersaturation was examined by Wallace (1975) because (barring questions of absolute calibration) the 17- to 33-~m data then available favored a considerably warmer stratosphere than was consistent with a model incorporating saturation equilibrium. By 1980, additional observational constraints coupled with better, more complex models (Wallace; Appleby) indicated that supersaturation was probably not needed. The purpose of this article is to examine global

383 0019-1035/86 $3.00 Copyright © 1986by AcademicPress, Inc. All rightsof reproductionin any form reserved.

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mean models for Uranus and Neptune in the light of recent observational data. These results will aid the development of more sophisticated models (now under study) which will take into account the effects of spatial and temporal stratospheric variabilities. This work is concerned primarily with stratospheric levels at P >~ 0.1 mb. Although the model calculations presented below (all of which assume LTE) actually extend upward to much higher levels than indicated, CH4 vibrational relaxation is likely to be of considerable importance at P <~ 0,1 mb (Appleby, 1980). This region of the Uranian atmosphere has been the focus of much attention in recent years due to an extensive, high-quality, and very complex set of stellar occultation measurements. Ultimately, models for the radiative balance in the middle and upper stratosphere must also take into account the effects of photochemical processes and diffusive separation. In any case, a discussion of baseline N L T E models for Uranus (and for the upper stratospheres of the other Jovian planets) will be presented separately, in a forthcoming article. METHOD

The results presented here were obtained using a model atmosphere computer code for the Giant Planets discussed by Appleby and Hogan (1984). This section outlines the method and supplements the information given by Appleby and Hogan with a discussion of those facets of the modeling that are relevant to Uranus and Neptune. Overall, results based on this method compare favorably with those of Wallace, Cess, and collaborators, when the same model is computed; examples and further discussion are given elsewhere (Appleby, 1980).

Technique and Basic" Assumptions Radiative equilibrium implies that the net flux is independent of height. Setting the

divergence of the net flux equal to zero at each level in the model, temperature is obtained by computing the total energy absorbed per unit volume and then determining (numerically) the volumetric emission that is required for a precise balance, taking full account of radiative exchange between layers. The procedure requires as input a zeroth-order temperature profile and specification of P, T, and the composition (in~ eluding mixing ratios) at the base of the model (which is a level sufficiently deep in the convective zone that the inclusion of deeper levels has no influence on the overlying structure; the boundary conditions at the bottom of the model are set a priori). The full atmospheric structure is calculated level-by-level, starting at the lower boundary, and the entire procedure is repeated until the profile has converged to within a small fraction of a degree at all levels. In this method, the effective temperature of the model is obtained as output. Multiple scattering was neglected here and these models assume local thermodynamic equilibrium throughout. The principal constituents of these models are H2, He, and CH4. The earliest models included NH~, however, due to condensation, this gas has a negligible influence on the Uranus-Neptune profiles presented here. Each temperature iteration includes a recalculation of the number density distributions of all constituents, the partial and total pressure distributions, and the corresponding atmospheric opacities. The partial pressure of CH4 was constrained at each level by its saturated vapor pressure. Two-parameter saturation laws from the International Critical Tables f 1928) were used for CH4.

The models also include C2H6 and C2Hz in the thermal inversion layers (P ~< 100 mb), where these gases presumably originate. Mixing ratios were assigned for each model corresponding to these upper levels. The assumptions of complete mixing and

URANUS-NEPTUNE EQUILIBRIUM MODELS saturation equilibrium lead to C2H6 and C2H2 condensation regions extending far above the 100-mb levels in Uranus and Neptune. Saturation number densities were inserted at and below the condensation levels, as in the calculation of CH4 condensation regions (moving upward) in the troposphere. The saturation laws for C2H6 and C2H2 given by Ziegler (1959) and as collected in the Handbook of Chemistry and Physics (49th ed., 1968) were found to be sufficiently accurate for present purposes. Ziegler's review article presented the lowtemperature data of Tickner and Lossing (1951; who studied CH4, C2H6, C2H2 and a number of other hydrocarbons) for comparison with the Antoine equation. Proceeding downward from the levels at which saturation begins in Uranus and Neptune, the C2H6 and C2H2 concentrations are so severely reduced that these gases are virtually absent at the temperature minima. Superadiabatic lapse rates were suppressed by inserting the convective gradient level-to-level as necessary on each temperature iteration. Assuming CH4 is the only condensate in the tropospheric regions of interest here,

×

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following Trafton's (1967) Eq. (18)./:v is the specific heat at constant pressure for the mixture, R = k / ~ (erg deg -j g l) is the specific gas constant, and ~2 is the CH4 mole fraction. /3 is a dimensionless variable proportional to the heat of transformation: for the CH4 saturation laws specified above, /3yap-sol = l I90.1/T and /3vap-liq = 1024.2/T. The quantity in brackets in Eq. (1) is typically about 0.5 to 0.6 at the condensation levels in Uranus and Neptune; it increases steadily upward until a value of 1.0 is recovered at the top of the condensation regions. Note that the effects of convection

385

and subsidence are not taken into account by Eq. (1), since, in a real atmosphere, the global mean gradient would be somewhere between the wet and dry adiabats. On the other hand, this correction would be minor compared with uncertainties regarding, for example, Te, the gas composition, and aerosol heating. b-p was calculated using Trafton's (1967) Table 1 for H2, together with the values 5.196 x 107 erg deg -I g-~ (constant) for He and 1.871 x 107 erg deg -1 g i (T = 158 K) for CH4 (Handbook of Chemistry and Physics, 1977). This value for CH4 was used at all levels. ~-p is impacted by the considerable uncertainties regarding H2 ortho-para ratios in the Giant Planets. The current dialogue (Wallace, 1980; Massie and Hunten, 1982; Conrath and Gierasch, 1984) indicates a need for better laboratory data, including reaction rates for ortho-para relaxation catalyzed on aerosol surfaces. Complications also arise because the ortho-para ratios in the Giant Planets vary horizontally and vertically and they are sensitive to rates for convective transfer. As discussed by Orton and Appleby (1984), infrared data for Uranus and Neptune are in best agreement with synthetic spectra that assume "intermediate" H2, which corresponds to Traflon's (1967) approach.

Opacities Collision-induced absorption coefficients were tabulated for mixtures of H2 and He on an appropriate grid of frequency versus temperature (Appleby and Hogan, 1984). These values are in excellent agreement with laboratory measurements (and related theoretical studies) by Birnbaum and coworkers. The largest uncertainties are in contributions to the H2 rotational-translational (R-T) opacity arising from H2-He induction. This source of error is potentially important for Uranus and Neptune models due to the enhancement of the R-T band at low temperatures. Fortunately, this is a minor problem in the nominal models

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presented below, since [He]/[H2] = 0.1 is assumed. These models do not include pressureinduced opacities related to CH4. Given the CH4 mole fractions assumed here (<2%), laboratory data (see Orton et al., 1983) indicate that H2-CH4 collisions contribute less than about 10% to the pressure-induced opacity. In any case, uncertainties are still relatively large regarding H2-CH4 coefficients, due in part to extrapolation of the laboratory data (Tl~b --> 195 K) to the low temperatures in Uranus and Neptune. Radiative modeling of the hydrocarbons in Uranus and Neptune is more complex than in Jupiter and Saturn due to nonuniform vertical distributions resulting from condensation. In the Uranus models presented below, the CH4 mixing ratio (f,0 drops by factors of ~ l02 to ~ 103 within the condensation regions. In a conventional model, the value OfJm obtained at the top of the condensation region is applied to the overlying levels (apart from other processes which alter the mixing ratio, such as photochemistry). Temperatures in the inversion region are therefore relatively insensitive to changes in the tropospheric composition deep in the convective zone (below the condensation level). For example, in test runs for Uranus for a given value of Te, a nearly 30-fold increase in J~ in the deep atmosphere elevated temperatures in the stratosphere no more than 4 K. By contrast, a two-fold increase for Jupiter produced changes of up to 12 K (Appleby and Hogan, 1984). Radiative modeling for CH4 is based on the random band model. As discussed by Appleby and Hogan (1984), the expressions used here are comparable overall to those in Wallace et al. (1974), with the addition of mean absorption coefficients measured in the laboratory between 0.44 and 1.05 p~m. The formulation includes a simple empirical treatment for combining Doppler and Lorentz line shapes. Opacities for the m and ~'4 fundamentals, and for the 1.7- and

2.3-/zm band groups, require specification of an amount for the absorber, a temperature for the path, and for the Lorentz shape, a broadening pressure. The approach adopted here was to use for all opacities the true CH4 amount in the path, together with scaled quantities for the temperature and broadening pressure. If the CH4 mixing ratio is constant at all levels above Z, then the column density Nm(Z) = n m ( Z ) H ( Z ) , where " m " designates methane and H ( Z ) is the mean pressure scale height. Column amounts within the condensation region were calculated numerically, with the approximation that between adjacent grid points T ( Z ) ~ ~-~ cn 4 c2Z, where c~ and c~ are constants. This procedure was checked by changing the layer thicknesses to insure the temperature profile was invariant to a finer grid in altitude. The effective broadening pressure Pc for CH4 Lorentz lines was defined as in Cess (1974), P c ( Z ) - P ( Z ) / [ I + H m ( Z ) / H ( Z ) ] , where P ( Z ) is the total pressure and Hm(Z) is the methane scale height at Z. Thus, whenever, ttm = H , this expression reduces to the Curtis-Godson approximation for a vertical pressure-varying path at a fixed temperature: P~(Z) - P ( Z ) / 2 . Within the condensation regions of Uranus and Neptune, Hm < H , so P~ varies smoothly between P/2 and P. For example, in Uranus model c presented in Fig. 2 below, H m / t t reaches a minimum value of 0.15 at P - 638 mb (the CH4 condensation region in c encompasses (I.214 ~ P ~ 1.57 bar). The temperature employed for CH4 opacity calculations was an absorberweighted mean, as defined by T ( Z ) = f), T( Z ' )nm( Z ' )dZ' / Nm( Z ) .

Flux divergences for C2H6 and C~H~ were computed directly from Cess and Chen's (1975) parameterizations, which are applicable to strong, nonoverlapping Lorentz lines and to uniformly mixed absorbers. The Cess-Chen expressions do not take into account condensation effects, nor any

URANUS-NEPTUNE EQUILIBRIUM MODELS other process which alters the mixing ratio. As described above, deep C2H6 and C2H2 condensation regions occur above the temperature minima in the Uranus and Neptune models. However, by routinely checking each model, using the lists of emitted energies produced throughout the atmosphere, it was found that these condensation regions were always far below levels where C2H6 and C2H2 are of potential importance to the total thermal emission, even considering a wide range of mixing ratios. Consequently, the Cess-Chen formulas are applicable in this regard. The upper portions (P ~< 1 mb) of the profiles presented below are somewhat too warm because the Cess--Chen formulas do not include Doppler broadening. Thus, radiative cooling by C2H6 and C2H2 is underestimated. These errors are of little consequence here, however, since this study examines constraints on the atmospheric structure from P --- 0.01 to 1.0 bar imposed by recent observational data acquired at wavelengths longward of 17 /xm. Apparently, C2H6 and C2H2 are removed from this region of the atmosphere by condensation (see below). Note, in any event, that a more comprehensive treatment of opacities for the upper stratospheres of Uranus and Neptune would suffer from lack of constraints on appropriate mixing ratios (among other things). Several important strides in atmospheric modeling are forthcoming in the near future, stimulated in part by the prospect of high-quality Voyager 2 observations at Uranus in 1986. Better opacity formulations (encompassing nonuniform vertical distributions) will be developed in that context. Aerosol Heating

Evidence of aerosols in the stratospheres of Jupiter and Saturn was reviewed by Appleby and Hogan (1984). The role of particles in the stratospheric energy balance of these planets is not clear, however, be-

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cause estimates are still wedded to a broad range of uncertainties. These include the atmospheric gas composition, aerosol vertical and horizontal distributions, time-dependent effects, and aerosol heating and cooling properties. Many of these problems cannot be resolved with current data sets. The situation for Uranus and Neptune is even less clear. Of course, there is longstanding indirect evidence suggesting the presence of aerosols somewhere in the upper Uranian atmosphere. For example, Sinton (1972) found that his observations of limb and polar brightening at 8870 A (which is within a relatively strong CH4 band) were best explained as due in part to a high-altitude haze. Prinn and Lewis (1973) showed that the geometric albedos then available were compatible with a haze model in which the particles are probably solid CH4. A series of papers by Price and Franz (see Price, 1978, and references therein) indicated that an aerosol haze was consistent with much of their limb and polar brightening data. Price, however, stated that a clear atmosphere in which the CH4 mixing ratio increases with depth was a plausible alternative model. Pilcher et al. (1979) found that this assumption provided a good quantitative interpretation of their 7250 ,~ limb brightening observations. They calculated the CH4 distribution assuming saturation equilibrium, as it is calculated here, and they also chose [CH4]/[H2] = 0.02 below the condensation level. Not all strong CH4 bands exhibit limb brightening, however. The 6190-A band intensity is nearly uniform over the disk, while the 8000- and 8500-A bands show " . . . pronounced apparent limb darkening. Either polar haze in the upper atmosphere or the visibility of a deep dense cloud layer could be responsible for the effect" (Price and Franz, 1979). For Neptune, Price and Franz (1980), for example, observed limb brightening within the 7300-,~ CH4 band. They concluded that their results " . . . appear to require the presence "of an optically thin layer of

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brightly scattering aerosol particles high in the Neptune atmosphere." There are several aerosol components of potential importance to stratospheric processes. One is associated with condensates formed in the uppermost layers of the convective zones: NH3 ice for Jupiter and Saturn, and CH4 ice for Uranus and Neptune. Depending on particle sizes and the strength of vertical motions, some of this material could extend far above the radiative-convective boundary. Another component is an optically thin high-altitude haze thought to originate from the degradation of CH4 (by photodissociation or other high-energy mechanisms). Finally, C2H6 and C2H2 ice probably form in the middle stratospheres of Uranus and Neptune, below the levels where these constituents originate as gases. The models presented here include aerosol heating in the region of the atmosphere extending upward from the radiative-convective (R-C) boundary. As described by Appleby and Hogan (1984), each model requires specification of a parameter ""v'" (see their Eq. (11)), which expresses the total (bolometric) aerosol heat deposition throughout the radiative zone as a fraction of the planet's solar constant, along with a normalized height distribution function for this absorbed energy. For example, a uniform density profile implies that the total aerosol heat deposition above pressure level P is y & ~ P / 4 P R - c , where S , is the solar constant, ~ implies a global mean, and PR ~' is the pressure at the R - C boundary. In principle, limiting values for v can be estimated as follows, Y <--

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assuming contributions in the wavelength range )tj ~ )t ~ ?,2. In this expression, Ax.ob~ is the observed Bond albedo versus wavelength, as derived from geometric albedo measurements and knowledge of the plane-

tary phase integral qa. Approximate values for qa are now available for Uranus, based in part on preliminary cruise data from the Voyager 2 imaging experiment. A~,c~,,, is the theoretical Bond albedo for Rayleigh plus Raman scattering in the same column of gas corresponding to Ax,obs, except with the aerosol removed. Ax.d~, can be calculated very accurately, although the depth of the layer must be specified, and, untbrtunately, this is not known a priori. Thus, Ax.d~,, will contribute a model-dependent portion to the integral in (2) at wavelengths where the planet's reflectivity is sensitive to optical properties assumed tow-the underlying cloud deck. Alternatively, synthetic spectra constrained to fit the observed reflectivities will also furnish y (above an arbitrary pressure level), based on the aerosol parameterizations, although uniqueness problems arise, leading to a range of possible values. These studies will nonetheless provide important constraints on y and on the vertical distribution of the absorber. Aerosol heating models for Jupiter and Saturn are consistent with y ~< 0.15, on the basis of comparisons with spatially resolved data (Appleby and Hogan, 1984). Most of this energy is absorbed at P ~> 100 rob. For Uranus and Neptune, observed geometric albedos have historically indicated smaller values: y ~< 0.05 to 0.10 (e.g., Savage and Caldwell, 1974; Macy and Trafton, 1975; Savage et al., 1980). In a study based on IUE data, Caldwell et al. (1981) questioned the need for a "continuum" absorber in the inversion layers of Uranus and Neptune. They argued that systematic errors (for example, in the Sun's color and V magnitude) could explain differences between the observed geometric albedos and a clear-atmosphere (Rayleigh-Raman) continuum. More recently, however, a completely independent analysis (Caldwell et al., 1984), based on new 1UE observations, has confirmed the 1981 geometric albedos. Caldwell et al. (1981) were also concerned

URANUS-NEPTUNE EQUILIBRIUM MODELS that, on one hand, a "continuum" absorber would need to have a comparable effect on UV albedos in Uranus and Neptune, suggesting common properties and distributions, whereas the 8- to 13-~m data imply much higher stratospheric temperatures in Neptune than in Uranus. A set of equilibrium models by Appleby (1980; Fig. 37) suggested that aerosol heating with y = 0.1 was not nearly sufficient to power Neptune's thermal inversion. Macy and Trafton (1975) came to the same conclusion on the basis of semiquantitative arguments. Those studies indicated that the UV albedo problem cannot be linked to heating in Neptune's stratosphere to the exclusion of other mechanisms; i.e., aerosol heating may provide a portion, though not most of the required energy. The situation today is less certain: the infrared data used here, with downward revisions in the flux at 1820 /zm, are better than those used in the earlier studies, and values of y somewhat larger than 0.1 cannot be ruled out. RESULTS

The basic constraints for these models

389

are the bulk composition given by [He]/ [H2], assuming these gases are the principal constituents, and the bolometric flux given by T¢. These important parameters have been discussed recently by Orton and Appleby (1984). They find that the UranusNeptune observations shown in Fig. 1 provide a useful data set for quantitative studies because these measurements, corresponding to relatively narrow bandwidths and all taken since 1978, avoid some of the problems inherent in the older data. For example, Rieke and Low's (1974) 33-/zm observations are significantly lower (by more than 5 K in brightness temperature) compared to those of Loewenstein et al. (1977a,b) for b o t h Uranus and Neptune. Systematic effects clearly offer the most plausible explanation for these discrepancies. Even the restricted data sets shown in Fig. l indicate some inconsistencies. For example, the Uranus results from 300 to 500/zm vary between TI3 -~ 62 K and TB = 74 K (and the corresponding error estimates include a somewhat wider range of temperatures). It is unlikely that these differences arise from secular changes in the

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JOHN F. APPLEBY

planet's flux since these data were all taken within a period of four years (1979 to 1983). Also, Moseley and his collaborators (1985; personal communication) have additional measurements from 28 to 38 /zm, supplementing their earlier results (shown in Fig. 1) from 32 to 52 /~m. These data suggest that the Uranus spectrum from 28 to 32/xm is relatively flat, with ATBr <-%2 K. The statistical errors indicate good internal accur a c y - w i t h i n much larger systematic errors related to the Mars standard (these, in effect, would move the spectrum uniformly up or down). Moseley et al. conclude that their data from 28 to 32/~m are inconsistent with the 18- to 20-/~m measurements in Fig. 1 since both spectral intervals correspond to about the same atmospheric levels (P 100 rob), whereas the brightness temperatures differ by 3 to 4 K. Questions regarding temporal changes and instrumental effects in the two data sets here (18-20/~m vs 2832/zm) may be resolved in the near future: Moseley and his colleagues will undertake new measurements in I985 in both spectral intervals. Bolometric fluxes calculated fi'om the data in Fig. 1 indicate 7"~ = 58.3 +_ 2.0 K (Uranus) and 60.3 +_ 2.0 K (Neptune), with absolute calibration uncertainties estimated at 10% (Hildebrand et al., 1985). Verification of calibration standards is forthcoming from the use of other solar system bodies such as Callisto and G a n y m e d e instead of Mars, which was used for the data in Fig. 1 at ~ < 400 cm I. All of the models presented here are consistent with these values for Te. [He]/[H2] is usually assumed to be about 0.1 for Uranus and Neptune, although the data shown in Fig. I, when compared with synthetic spectra, do not strongly favor a particular value. Unless otherwise indicated, the models presented below are based on 0.1. [CH4]/[H2] = 0.02 was employed in all the nominal models for atmospheric layers beneath the CH4 condensation regions in Uranus and Neptune. This important mix-

ing ratio remains uncertain by a factor of about two. Studies by Baines (1983) and Bergstralh and Baines (1984) indicated 3 to 4% CH4; however, these authors have more recently determined a range of about 2 to 5%, using significantly better observations and including several improvements in their theoretical calculations of synthetic spectra (see Baines and Bergstraih, 1986). The use of [CH4]/[H2] = 0.02 is therefore satisfactory, given the many uncertainties which arise in interpreting the visible data. These problems include multiple scattering effects, the temperature dependence of the

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URANUS-NEPTUNE EQUILIBRIUM MODELS

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CH4 absorption coefficients, the determina- ture considerably. F o r example, more comtion of a continuum I/F level between CH4 plex models for the stratospheric structures bands (for constraints on the aerosol ab- will be o f value in interpreting the radio ocsorber's optical properties), and the recon- cultation profiles. ciliation o f CH4 absorption depths with H2 Uranus spectral features. These models also employ g = 830 cm All of the models in Fig. 2 correspond to sec -2 (Uranus) and 1100 cm sec 2 (Nep- fh = 0.1,fe = 1 × 10 -6, a n d f a = 2 x 10 s. tune). The value for Uranus is roughly con- These values for C2H6 and C2H2 are somesistent with results based on stellar occulta- what smaller than those suggested by tion measurements. Elliot et al. (1981) A t r e y a and Ponthieu's (1983) " n o m i n a l " determined an equatorial radius Re = results for photochemical equilibrium. 26,145 _+ 3 0 k m at the n -~ 8 × 1013-cm -3 Since this article is concerned primarily level (which corresponds to P -~ 1 /zb), If with the lower stratosphere, however, the altitude difference between this level these differences will not effect the discusand the 100 mb level is about 505 km (using sion here due to condensation effects. Also, the output from model c presented in Fig. 2 the effective temperatures for these profiles below), and assuming a planetary mass M fall within Te = 58.5 --- 0.5 K. Two calcula= 14.51Mq~ (Nicholson et al., 1981) and a tions with no aerosol heating are shown, rotation period of 16 _+ 1 h (Belton and Ter- designated (a): they have the same stratorile, 1984), one obtains ge(100 mb) = 849 -+ spheric and upper tropospheric profiles, but 6 cm sec -2. Somewhat larger values for g they diverge at lower levels according to would hold at higher latitudes and the ac- the CH4 mixing ratios 0.02 and 0.05, as inditual error bars (everywhere) exceed ---6 cm cated. These values for fm correspond to sec -2 due, for example, to uncertainties in regions below the CH4 condensation levels, the thermal structure (which influence the which o c c u r at P -~ 1.2 and 2.7 bars f o r f ~ = altitude estimate of 505 km). Thus, 830 cm 0.02 and 0.05, respectively. The thermal sec -2 is considered satisfactory for present profiles for these two models would differ purposes and, in any case, changes of less somewhat at higher altitudes, had they not than about 10% have a negligible effect on been constrained (by changing the lower the results. F o r Neptune, 1100 cm sec -2 is boundary temperature at P -~ 3.2 bar) to within about 2% of a value based on recent give essentially the same Te. The temperaestimates (Hubbard et al., 1985) of the ture inversion arises from what little CH4 equatorial radius at 1 bar and the rotation remains above the condensation region: in rate. this case, fm has dropped roughly 3 orders Compositions for all the models will be of magnitude. These runs illustrate a familspecified by molar mixing ratios relative to iar point, that for fixed Te and a convenH2, where fh, fro, fe, and fa correspond to tional model such as (a), the thermal strucHe, CH4, C2H6, and C2H2, respectively. ture at P ~< 1 bar is largely decoupled from With one exception (discussed below), all the CH4 fraction deep in the convective of the models employ global average insola- zone (P ~> 3 bar). In turn, the infrared data tions (as specified in Appleby and Hogan, in Fig. i, or as reproduced in Fig. 3, corre1984). Though a more sophisticated treat- spond to atmospheric levels at P ~< i bar. ment would certainly be o f interest, the The theoretical spectrum for (a) (Fig. 3) is data currently available for constraining roughly consistent with the data iongward such models are largely whole-disk mea- o f about 37 /xm. (To minimize confusion, surements covering only a fraction of the this portion of the model spectrum was annual cycles o f Uranus and Neptune. The omitted.) H o w e v e r , the 32- and 18- to 20Voyager encounters will change this pic- /zm emissions, which originate near the 80-

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Io 100-rob region of the a t m o s p h e r e , are not nearly high enough: for example, T~ (32.1 /zm) = 48.2 K for model a versus 53,1 ± 0.9 K from recent observation (Moseley et al., 1985), as illustrated in Fig. 3. These data indicate that an additional heat source is needed near the tropopause. Aerosol absorption could provide a relatively simple solution, as illustrated by models b and c in Fig. 2. These profiles c o r r e s p o n d to uniform height distributions from the radiative-convective (R-C) boundaries upward. 5 and 15% of the available insolation are a b s o r b e d in (b) and (c), respectively; i.e., y = 0.05 and 0.15, where the total aerosol heat input (integrated o v e r all levels) is ySU4 and S< is the planet's solar constant. The a s s u m e d vertical distribution places m o s t of this energy near and below the tropopause. Thus, in model c, for e x a m p l e , 90% ofySo/4 is a b s o r b e d between 48 mb and the R - C b o u n d a r y at 480 mb. N o t e that the influence of aerosol heating is greatest at m u c h higher levels for these (or similar) aerosol distributions, as (moving upward) radiative control of the atmo-

sphere shifts steadily from pressure-induced to solar absorption. In (c), aerosol heating accounts for 1 and 20%, of the total bolometric flux a b s o r b e d at 480 and 48 mb, respectively, and it reaches a m a x i m u m of 60% at 2.5 rob. The CH4 condensation region is indicated schematically for model c in Fig. 2:0.21 ~< P ~< 1.6 bar. The figure also shows the level ( P = 3.6 mb) at which C2H6 and C2H2 reach saturation as these constituents m o v e d o w n w a r d in the stratosphere, away from the higher levels where they originate. (CH4 photolysis occurs at P ~ 0.02 mb in A t r e y a and Ponthieu's (1983) nominal results, using the altitude, pressure, and density scales for (c).) Ice crystals of C2H6 and C2H2 would be predicted in a conventional picture, since the ambient t e m p e r a t u r e near the 4-mb level is far below the T.P. values for these constituents (about 90 and 191 K, respectively). G a s e o u s C2H6 and C2H2 are essentially absent at P ~> 10 mb in model c. The brightness t e m p e r a t u r e spectra for models b and c are shown in Fig. 3. Although b (y = 0.05) is too cold, reasonably

U R A N U S - N E P T U N E EQUILIBRIUM MODELS good a g r e e m e n t with the data is achieved in (c) (y = 0.15). It should be noted that the U r a n u s models p r e s e n t e d in this article are consistent with the 8- to 13-/am data discussed in O r t o n et al. (1983). Synthetic spectra are automatically calculated throughout z7 _< 1500 cm ], including crude flux estimates in the C2H6 and C2H2 bands (Appleby, 1980; A p p l e b y and Hogan, 1984). H o w e v e r , is y = 0.15 consistent with a determination b a s e d on Eq. (2), and what about the aerosol vertical distribution assumed here? The analysis given in the Appendix suggests 0.01 ~< y ~< 0.15 for aerosol heating a b o v e the 2-bar level. The lower limit for y is based on I U E data and is essentially model-independent. Values from 1 to about 15% are easily reconcilable with Baines and Bergstralh's (1986) study of the geometric albedo s p e c t r u m b e c a u s e y depends sensitively on the choice of a CH4-free and hazefree spectral continuum at h ~> 6000 A. Also, the aerosol distribution a s s u m e d in model c is acceptable: Baines and Bergstralh's results indicate that scattering and absorbing haze material can extend up to the 100-mb level (or higher). On the other hand, m o s t of the haze material could be below the R - C b o u n d a r y , and good constraints are lacking for several key input parameters. Therefore, other r a d i a t i v e - c o n vective models with y < 0.15 should be examined. Figure 4 shows the effects of changes in several basic assumptions. Model b (y = 0.05) is r e p r o d u c e d here for reference. (d) is a localized aerosol heating model: in this case, y = 0.06, half of this energy is deposited uniformly (from the R - C b o u n d a r y upward), as in models b and c, and the other half is consigned to a broad but nonuniform vertical distribution centered near the tropopause. (The top of the profile is slightly colder than (b) due to a minor compositional difference:fe = 2 × 10 -6 andf~ = 5 × 10 -8 were used in models d - f . ) The spectrum for (d) is given in Fig. 3. Emission in the cores o f the Hz S(0) and S(1) lines (at 28

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and 17/am, respectively) is necessarily enhanced relative to (b) to c o m p e n s a t e for larger aerosol heat depositions near the tropopause. Overall, (d) fits the data in Fig. 3 about as well as (c), even though (c) required y = 0.15. Localized heating in the 100-rob region of the a t m o s p h e r e is certainly possible, since aerosols associated with p h o t o c h e m i s t r y and condensation m a y be present. Since no information is available h o w e v e r , and given the aerosol heating uniqueness p r o b l e m s , there is no point in dwelling on these results. T h e y simply illustrate, quantitatively, h o w a reasonable alternative IR s p e c t r u m can be produced using a nominal value for y. Model e in Fig. 4 illustrates the effect of a major perturbation in the bulk composition: this profile c o r r e s p o n d s tofh = [He]/[H2] =

394

JOHN F. APPLEBY

1.0, in contrast to b, which assumes a solarlike ratio (fh = 0.1). The two models are essentially equivalent otherwise. Although the stratospheric profile is significantly warmer in (e), the spectrum, shown in Fig. 3, is inconsistent with the data at 18-20/~m and at 37/~m. Also, the fit from 35-55/~m is relatively poor. (The H2 pressure-induced opacity is so strongly enhanced in (e), that the S(0) and S(I) lines are seen in absorption.) Moreover, a convincing explanation for such a large He-to-H,, enhancement is not available. The separation of these gases by Jeans escape is ruled out by Hunten's (1973; p. 1485) calculations. In later remarks (1974; p. 5), Hunten dismissed two other suggestions: separation of the elements in the solar nebula by ~'unspecified hydromagnetic processes," an idea put forward by Alfven; and an early suggestion by Opik concerning condensation of H2 in the solar nebula. Also, the composition used in e would result in relatively large discrepancies at P ~< 10/.~bar, comparing temperature profiles based on radiative equilibrium models and those from stellar occultation data. Temperatures in the latter case would increase by 36% (the ratio of the mean molecular weights assumed) whereas the models show little sensitivity at these levels (cf. Fig. 4; Appleby, 1980; French et al.. 1983). Model f in Fig. 4 illustrates the effect of removing the diurnal average in the solar heating expressions. Models f and b are essentially equivalent otherwise. A factor of ½ (used in all the other models presented in this article) holds for rapid rotation near equinox, and at midlatitudes away from equinox if the obliquity is small. For Uranus, obviously, there is good reason to test the sensitivity of the results to this parameter, even lacking full knowledge of radiative and dynamical time scales. (For example, there is no information on meridional transport in the middle and lower atmosphere, or on radiative effects such as time-dependent aerosol absorption.) The maximum warming, about 15 K,

occurs near 0.1-1.0 mb, while the tropopause region is warmer by 2-3 K. The spectrum for (f) (shown in Fig. 3) fits the data as well as that for model c. However, the actual amplitude of the variations will be less than indicated by (f) because radiative time constants in the stratosphere are long compared to the duration of seasons. For example, the heating rates in model f indicate t,~d rv 103y~ in the region 1-100 rob. (The calculation of (f) is only a sensitivity test; it would be relevant if radiative adjustment were virtually instantaneous.) Further complexity would arise from advective heat transport if, for instance, there is organized meridional flow in the visible atmosphere (cf. Ingersoll, 1984). In any case, (f) seems consistent with detailed time-dependent models by Wallace (1983) and with preliminary results of Bezard and Gautier (1985). These studies indicate that, despite Uranus' extraordinary obliquity, latitudinal and seasonal variations in temperature have relatively small amplitudes (<10 K). Also, since temperature extremes occur at the equinoxes (one-quarter cycle after the solar heating maxima), the current pole-on orientation of the planet should correspond closely to annual mean conditions. It follows that, for the infrared data under scrutiny here, a global mean insolation (as in models c or d, say) is appropriate. Finally, a variation in the CH4 saturation law was studied, following comments by Cess (1979; personal communication). Normally, the CH4 condensation region is determined by using the lowest equilibrium vapor pressure at the ambient temperature of each level. This assumption amounts to a standard "quasi-equilibrium" approach (e.g., Weidenschilling and Lewis, 1973). For the models discussed above, the resulting partial pressure of CH4 follows the saturation law for vapor-solid equilibrium throughout the condensation regions. The top of the condensation region is near the temperature minimum at around 50 K (depending on the model). The CH4 mixing ratio for all higher altitudes equals the value

URANUS-NEPTUNE EQUILIBRIUM MODELS determined at this level, which, in turn, results from the level-by-level development of the condensation region, beginning at much deeper levels. Therefore, the stratospheric thermal structure is relatively sensitive to the details of the condensation process. This sensitivity was tested by inserting the vapor-liquid saturation law at all levels. Using a model similar to (b) or (c) in Fig. 2, this change warmed the profile by about 10 K at I mb. (Note that the ratio of CH4 vapor-liquid to vapor-solid equilibrium pressures is about 4.4 at 50 K.) In fact, there is good reason to question a model that adheres strictly to the "quasi-equilibrium" approach, as discussed by Weidenschilling and Lewis (1973) and Belton (1974). (See also, Hunten's (1974) summary.) A familiar example is supercooled water droplets, which are commonly observed in terrestrial clouds. Lewis (1974) states, apparently on theoretical grounds, that C H 4 at low temperatures is " . . . almost like a partially ordered liquid," and, indeed, laboratory data corroborate this description.

Neptune Results for Neptune are shown in Figs. 5 and 6. These models correspond tofh = 0. I, f~ = 2 × 10 -5, f a = 5 × 10 -7, and Ze = 59.7 --- 0.2 K. Also, fm = 0.02 at the base of the models ( P = 3.17 bar). a represents a clear column of gas (no aerosol heating), analogous to model a (Uranus) in Figs. 2 and 3. The CH4 mixing ratio in the upper atmosphere is thus constrained by the value obtained at the top of the condensation region, with the result that the profile is much too cold. The 8-/xm flux from this model is nearly three orders of magnitude lower than observed. As illustrated by (b) in Figs. 5 and 6, the addition of aerosol heating in the radiative zone does not solve the problem, as long as y ~< 0.1. A uniform distribution was assumed in (b) (as in (b) and (c) for Uranus), along with y = 0.1, but the outgoing flux at h ~< 33 ~m is still too low. Models with nonuniform aerosol heating

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were also studied in which the maximum heat deposition was shifted progressively to higher altitudes. Localized heating near 1 mb brought the 8-p~m emission into compliance with observation, but only at the expense of additional heating needed at P -~ I0-I00 mb. An example is model c in Figs. 5 and 6, in which the flux at 8 and 18 ~m is satisfactory, yet it is too low near 32/~m. (Moseley and his collaborators (1985; personal communication) have acquired additional data from 28 to 38/~m which confirm the 32-/~m measurement shown in Fig. 6.)

396

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These localized heating models suggest that N e p t u n e ' s thermal inversion is not sustained primarily by aerosol heating, regardless of its vertical distribution, as long as y ~< 0.10 to 0.15. For this type of model to succeed, larger values of y would be required, most of this additional heat must go into levels at P <~ 100 mb, and, apart from its influence at other levels, the aerosol must dominate the total heating (solar plus planetary) at P ~ 0.1-1.0 and 10-100 rob. In model c, for instance, it accounts for more than 98% of the volumetric absorption at 1 mb. Unfortunately, observational data for Neptune do not provide tight constraints on y, nor on the aerosol vertical profile. As indicated under Method, estimates of 3' in the literature are between 5 and 10%. Although geometric albedos for Uranus and Neptune are comparable overall, there are some important differences (see Figs. 7 vs 9 in N e f f et al., 1984). For example, the continuum at 3500 ~< ;~ -< 5500 A is relatively flat in N e p t u n e ' s spectrum and, toward longer wavelengths, the apparent interband regions (between CH4 features) are more

strongly absorbed, with band wings sloping more gradually away from the band centers (a characteristic that has been attributed to increased aerosol scattering; Fink and Larson, 1979). Neptune models currently under study (Bergstralh and Baines, 1986) indicate that the visible spectrum could be influenced significantly by aerosols that extend into the lower troposphere, far beneath the 2-bar pressure level discussed above regarding Uranus models. This type of model may or may not preclude significant absorption at P ~< 100 rob. As in Uranus, C2H6 and C2H~ ice should form high in the inversion region. This material in N e p t u n e could be related to the optically thin clouds inferred from measurements at 1-3 p,m (Pilcher, 1977). In any case, although y = 0.1 is certainly within current uncertainties for Neptune, there is no independent evidence for much larger values or for relatively high heat depositions above the tropopause, and, therefore, alternatives to aerosol heating should be considered. (d) and (e) in Figs. 5 and 6 are models in which CH4 provides the main stratospheric heat source. These calculations employ 5 ~

URANUS-NEPTUNE EQUILIBRIUM MODELS aerosol heating in a uniform vertical distribution. (d), which is included here for comparison with (e), is a CH4 supersaturation model: [CH4]/[H2] = 0.02 at all levels. The CH4 partial pressures in (d) exceed the equilibrium values by factors as large as about 500. Presumably, a uniform CH4 distribution would occur if there were no condensation nuclei, if CH4 does not condense under the actual atmospheric conditions in Neptune, or if the vertical mixing is so rapid that condensation cannot occur before the gas is swept upward, above the condensation region [see discussions by Hunten (1974) and Belton (1974)]. The latter case could be described as a nominal "convective penetration" model. (e) is a variant regarding convective penetration, in which saturation equilibrium is operative. In this case, the atmosphere is mixed to the extent that fm = [CH4]/[H2] = 0.02 at all levels where the CH4 partial pressure does not exceed the saturated vapor pressure. In other words, vertical mass motions are assumed to be vigorous enough to tend toward the establishment of a uniform CH4 distribution at all levels, although, on the average, condensation still occurs. Such a distribution could be realized if, for example, the CH4 condensate, once formed, is swept upward above the condensation region, where, in quasi-equilibrium it eventually returns to the vapor phase (Hunten, 1974; Macy and Trafton, 1975). Alternatively, perhaps convective penetration without condensation occurs within isolated regions on the planet. In this case, condensation equilibrium would hold with respect to a global average and a high CH4 mixing ratio in the inversion region would be sustained planet-wide by strong convective forcing at isolated points. The CH4 condensation region in model e develops normally up to the temperature minimum, where fm - 10-5. Then fm increases rapidly upward, above Train, until 0.02 is reached (near 30 mb), and it remains constant (at 0.02) at all higher levels. The bump in the profile near 30 mb, which re-

397

flects the rapidly changing mixing ratio, is analogous to levels at the base of the CH4 condensation region (P = 1.2 mb in this case), where the mixing ratio also changes very rapidly with altitude. The spectrum for (e) (Fig. 6) exhibits fluxes that are somewhat too high near 18 /zm and slightly too low near 33 tzm. However, when additional (more recent) measurements from 28 to 32/~m are considered (Moseley, 1985; personal communication) the fit in this spectral region is good overall. The 8-/zm flux for e is also acceptable (it is within the error bars given by Gillett and Ricke, 1977). Neptune's internal flux could provide the driving mechanism for relatively large convective currents. Although its fractional internal heat is comparable to that of Jupiter and Saturn, and NH3 does not undergo convective penetration in these atmospheres, the comparison may be misleading, at least for Jupiter. CH4 in Uranus and Neptune has a relatively high mixing ratio, and the condensation levels are far below the R - C boundaries. Also, at least in simple models such as those presented here, the latent heat of condensation strongly influences the thermal structure in the upper troposphere (near the R - C boundaries). By contrast, the NH3 cloud in Jupiter is near or above the R - C boundary and NH3 has a minor effect on temperature. (Perhaps these differences result in comparatively weak dynamical forcing in Jupiter.) What about Uranus? The visible spectra of Uranus and Neptune are similar, as are their bulk emissions. Consistent with the models presented here, their thermal structures should be similar in the pressure range 0.05 ~< P ~< 1.5 bar. Weidenschilling and Lewis' (1973) results for Neptune indicated the formation of a solid argon cloud at around 0.2 bars. The temperature profile which these authors employed, however, is too cold. Tmi, = 51 K is obtained in the models presented here, therefore Ar condensation in Neptune is not expected to occur (see their Fig. 2). Neptune has a mas-

398

J O H N F. A PP LEBY

sive satellite (Triton) in close retrograde orbit, which could give rise to atmospheric tidal interactions. Macy and Trafton (1975). however, found that the maximum effects occur far above the temperature minimum (see also Gillett and Rieke, 19771. As noted above, although Uranus has an extreme axial orientation, the long radiative response times tend to suppress stratospheric seasonal effects. Evidently, spatial and temporal variations in Neptune are even smaller, due to the planet's large internal heat flux (e.g., wave amplitudes in T~ arc < 1 K; Wallace, 1984). Nonetheless, both Uranus and Neptune show atmospheric variability. The microwave brightness of Uranus increased steadily as the north pole turned sunward (Gulkis and de Pater, 1984). For Neptune, there is evidence of atmospheric activity based on observations between 1 and 3 p,m: the reflectivity measured by Joyce et al. (1977) increased significantly over about a I-year period. As noted above, Pilcher (1977) argued that a high-altitude haze (P ~-:~50 rob) could account for these variations. In summary, the CH4 convective penetration hypothesis is at least sell-consistent to the extent that relatively high stratospheric temperatures in Neptune. which will result if the CH4 mole fraction is a few percent at 1 rob, can be linked with a large internal heat source. By contrast, much lower temperatures at I mb in Uranus would correspond to an upwelling internal flux that is ineffective or altogether absent. CONCLUSIONS

Despite their similarities. Uranus and Neptune have emerged as "individuals" in the far outer solar system regarding, for example, their internal energy generation and their stratospheric structure. The results presented here reflect this picture: straightforward model atmosphere calculations lead to Uranus profiles having infrared spectra that are reasonably consistent with the observations. For example, it was found that localized aerosol heating may be

very important in the stratosphere of Uranus. Comparable models for Neptune significantly underestimate the observed infrared emission from 8 to 33 p~m. The total aerosol heat input in the Neptune models was limited to about 10% of the planet's insolation (in accord, roughly, with Neptune's geometric albedos). The results showed that aerosol heating can very readily power the observed 8-t~m emission. However, regardless of the vertical distribution, aerosol heating evidently cannot provide sufficient energy to explain simultaneously the 8- and the 33-p~m data. Alternatively, a CH4 ~'convective penetration" model was produced, in which a relatively large (~-2%) stratospheric concentration of methane is permitted, within the constraint of saturation equilibrium at all levels. The resulting thermal spectrum for this model provides a good overall fit to the data.

APPENDIX

The aerosol heating models presented in this article are trivial extensions of the corresponding clear-column cases ((a) in Figs. 2 or 5), based on long-standing but uncertain estimates of continuum absorption at ;~ ~< 5000 A. This Appendix compares aerosol heat inputs used in the Uranus thermal structure models (Figs. 2-4) with results from a new study by Baines and Bergstralh (1986), who have modeled the planet's reflection spectrum. These two types of modeling efforts can be combined for self-consistency, as discussed by Wallace (1980). Nonetheless, although current uncertainties in several key parameters in both modeling domains (solar and thermal) are shrinking, they are still large enough that overall consistency between the two studies can be readily established in lieu of a combined, iterative approach. For example, note that Baines and Bergstralh (1986) used the Uranus P - T profile c in their models, they examined aerosol (haze) vertical profiles extending up to the 100-mb level (where "additional" heating may be needed), and their best-fit synthetic spectra

URANUS-NEPTUNE EQUILIBRIUM MODELS correspond to CH4 mole fractions which are entirely consistent with the thermal models presented here (see discussion of a in Fig. 2 regarding fm = 0.02 VS 0.05). In turn, their models can be examined more or less directly for constraints on aerosol heat inputs, with the result that no inconsistencies are found: Figures A l a - c show the geometric albedo spectrum of Uranus between 2000 and 8000 .&. The IUE results (), < 3500 A) are averages of several measurements at each wavelength reported by Caldwell et al. (1981). Clearly, in view of the solar spectrum, any estimate of y from Eq. (2) will depend primarily on data at longer wavelengths. The ground-based spectrophotometric measurements by Neff et al. (1984) will be used here, as plotted every 10 ,& in Fig. AI (k -> 3500 .&). Prominent spectral features include the Raman-excited Fraunhofer ghosts at 3750 ~< )~ ~< 4100 ,~ (above the continuum) and, of course, the CH4 absorption bands beginning at ~4416. The authors estimate that systematic errors in these data are ~ 5%. Note that the detailed structure evident in the p~ data (at all wavelengths) will not influence this discussion. Of interest here is the observed continuum, assuming that the aerosol opacity in question varies relatively slowly with wavelength compared to that for molecular absorptions. Figures A l a - c also show several theoretical curves in which, for convenience, CH4 opacity has been removed. The spectra labeled A correspond to an aerosol-free column of 191 km-ama of H2 and 20.8 kmama of He, overlying a semi-infinite, isotropically scattering cloud extending downward from the 2-bar level, with single-scattering albedo O5c. (I thank K. Baines for producing the spectra for cases A and B shown here.) At most wavelengths, reasonably good fits to the observed spectrum can be obtained, based on (A), by adding CH4 opacity, an absorbing haze above the 2-bar level, and using o5c = 0.9985 ()~ -< 6000 ,&) and o5c = 0.9709 ()~ > 6000 ,~). Baines employed the P-T profile

399

for model c (Fig. 2), and the haze particles were assumed to scatter isotropically. Other free parameters in his approach include pressure levels at the top and bottom of the haze region, the haze optical depth versus pressure, and the haze single-scattering albedo. Not unexpectedly, the uniqueness problems which arise are formidable. (A detailed discussion is presented by Baines and Bergstralh (1986).) For example, estimates of y in Eq. (2) depend sensitively on the parameter oSc. The value 0.9709 (see Fig. Alc) corresponds to the use of haze models having almost no extinction at )~ > 6000 A. In other words, adding only CH4 opacity to the case A (0.9709) continuum shown in Fig. Alc, gives a reasonably good fit to that portion of the observed spectrum. Alternatively, as can be inferred from the figure, other models with O5c> 0.9709 and compensatory haze absorption at X > 6000 A cannot be ruled out. Note the much brighter continuum that would result if O5c = 0.9985 were used at these wavelengths. Bergstralh (1985, personal communication) has observed that a discontinuity in o5~ near 6000 could have some basis in fact if sulfurbearing compounds are present in the deep cloud layers of Uranus (cf. Figs. 5 and 7 of Lebofsky and Fegley, 1976). (B) in Fig. A1 is the same as case A (0.9985) except for the addition of " h a z e " extinction. The dashed portion of (B) (X < 3000 ,~) is not part of Baines' model; it was drawn freehand through the IUE data. (The model is somewhat too bright at these wavelengths; however, details in the ultraviolet have little impact on the value of y.) (B) will be used here only as a convenient representation of the data, since it provides a plausible methane-free continuum, at least for ,k ~< 5500 A. Unfortunately, a unique continuum cannot be specified for the longer wavelength data, due to overlapping CH4 bands. Tests indicate that the observations longward of about 3900 A can be matched with haze-free models using wavelengthdependent absorption in the cloud layer,

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6000

FIG. A1. G e o m e t r i c a l b e d o s of U r a n u s from 2 0 0 0 - 8 0 0 0 A. The i n f o r m a t i o n in t h e s e figures is u s e d to c a l c u l a t e a r o u g h u p p e r limit for the total a e r o s o l heat d e p o s i t i o n a b o v e an o p a q u e c l o u d top, w h i c h is a s s u m e d to lie at P = 2 b a r s . The c o n t i n u u m s p e c t r a w e r e v e r y k i n d l y f u r n i s h e d by K. B a i n e s {cases A a n d B) a n d M. T o m a s k o ( c a s e C). C a s e s A and C c o r r e s p o n d to an a e r o s o l - f r e e c o l u m n of H, ( 191 kma m a ) plus H e (20.8 k m - a m a ) a b o v e the 2-bar level, and to i s o t r o p i c a l l y s c a t t e r i n g c l o u d p a r t i c l e s {at P -> 2 bars) w i t h s i n g l e - s c a t t e r i n g a l b e d o &~. C a s e B is u s e d h e r e to r e p r e s e n t the " ' o b s e r v e d " c o n t i n u u m for ~. -< 6000 A. F o r c o n v e n i e n c e , CH4 o p a c i t y w a s e x c l u d e d in t h e s e c a l c u l a t i o n s • The d i f f e r e n c e b e t w e e n c a s e C a n d c a s e B c a n be a t t r i b u t e d to a e r o s o l heating, a l t h o u g h l o n g s t a n d i n g u n i q u e n e s s p r o b l e m s r e g a r d i n g a e r o s o l p r o p e r t i e s a n d v e r t i c a l d i s t r i b u t i o n s m u s t be a d d r e s s e d (see text). The d a t a s h o w n h e r e are t h o s e of C a l d w e l l et al. (1981; 1984) and N e f f el al. (1984).

URANUS-NEPTUNE EQUILIBRIUM MODELS

401

0.8 (el

0.7

............

-~2c

] ~C = 0.9985

0.6 O .,..,1 tu 0.5

=,

0.4 I'-14J

. . . . . . . . . .

=E ,o, 0.3

.

A (~c = 0.8").

.

.

•

.

.

"-'.

.

.. ..

.

-...:".

'

..."

•

•

..'.".""".. ..

.

,

0.2 "..

:

....

.

".'~

.~ ",-o

0•1 0 55O0

,

i

i

i

I

6000

i

i

i

i

J

i

i

i

i

I

6500 7000 WAVELENGTH, ~

L

,

L

~

]

7500

i

i

i

L

8000

FIG. A I - - C o n t i n u e d .

with o3c = o3c(h) < 0.9985, and assuming isotropic scattering in the cloud. Compare the two spectra for case A shown in Fig. A1, which correspond to the (h-independent) values, o3~ = 0.9985 and 0.8, respectively. Note, for example, that haze extinction is not needed near 4000 ,~ ifoSc -~ 0.8 at that wavelength. The sensitivity of px to the cloud singleparticle phase function has also been checked. F o r example, Fig. AI shows px = 0.661 for (A) (o3~ = 0.9985) at 4500 A, based on isotropic scattering. A test run (very kindly furnished by M. Tomasko) showed that this result can be reproduced using o5~ = 0.9995 and a strongly anisotropic phase function (namely, a double H - G expression with g= = 0.8, g2 = - 0 . 7 , a n d f = 0.938; cf. Smith and T o m a s k o ' s (1984) Table V). Regarding the aerosol heating estimate as expressed by Eq (2), this test showed essentially no change in the corresponding phase integral and Bond albedo for the model. The situation is more complex at longer wavelengths. F o r example, test runs at 8000 ,~ show large changes in qa and Ax when anisotropic scattering is assumed. Evidently, the ground-based data (h ->

3500 ,~) c o u l d be matched with a haze-free model using O3c = O3c(h) and anisotropic scattering in the cloud. On the other hand, stratospheric absorbers are needed to reproduce the I U E data since all the hazefree model spectra are too bright. As suggested by the two curves for case A in Fig. A l a , p~ at h ~ 3000 ,~ is insensitive to atmospheric properties at P ~> 2 bar. Haze-free models for O5c = 1.0 and O5c < 0.8 were also computed, with the same result. (The model albedos are too high even for a black cloud (O5c= 0), given existing constraints on the cloud top pressure; see Baines and Bergstralh (1986).) With recent independent confirmation of the 1981 I U E data by Caldwell et al. (1984), the evidence for a " U V a b s o r b e r " is thus very strong. It seems unlikely that the opacity of this material drops off abruptly at )~ = 3500 ,&, and therefore, despite uncertainties regarding the cloud opacity, the integral in Eq. (2) should encompass longer wavelengths. What has been done here, prior to constraining observations by Voyager 2, is to illustrate test cases which include some variations in O)c and h2. In the spectra labeled A in Fig. A1, H2 +

402

JOHN

F. A P P L E B Y

He scattering was modeled according to the (scalar) Rayleigh phase function, with an approximate treatment for Raman scattering from Wallace (1972; his case C). This means that the rotational Raman excitation was represented as nonconservative Rayleigh scattering and the vibrational component as pure absorption. Raman-shifted photons are not reintroduced into the radiation field in this type of model, in contrast to a more rigorous treatment of the source function at all wavelengths, as in the studies by Cochran and his collaborators (e.g., Cochran and Trafton (1978)). Thus, geometric albedos based on Wallace 11972) will be too low, depending on the model in question. For the parameters employed here, test calculations (very kindly provided by W. Cochran) indicate that errors in the pa continua arising from use of the Wallace (1972) approximation are < I% at all wavelengths. Corrections for polarization effects are larger: case C in Figs. A l a - c is the same as case A except that molecular scattering was treated according to the Rayleigh phase matrix. Numerical values are given every 500 in Table AI, including corresponding results for the phase integral qA and the Bond albedo Aa. (These calculations were very kindly performed by M. Tomasko using algorithms described, for example, in Smith and Tomasko 11984).) For the model parameters considered here, polarization effects increase pa by about 7% at 2000 A, 3% at 5500 A, and 1% at 7500 A. Model C assumes that the aerosols in the main cloud deck are nonpolarizing, as might be expected for large (size > h), nonspherical ice crystals. In the stratosphere, photochemically derived particles presumably would be much smaller; thus, adding submicron size material would not significantly alter the polarization corrections obtained in (CI for Rayleigh scatterers. E v e n if larger aerosols are present ( C 2 H 6 and C2H2 ice?) and if they are polarizing, the stratospheric haze would not exert a major influence on the

TABLE AI RESULTS FOR THE URANUS MODEL-C CONTINUUM

_ . . . . . . . .

X (A)

o5~

px

q~

,4,

2000 2500 3000 3500 4000 4500 5000 5500 6000

0.9985 0.9985 0.9985 0.9985 0.9985 0.9985 0.9985 0.9985 I). 9985

I}.593 0.607 0.624 0.652 0.680 0.696 0.697 0.691 0.681

1.235 1.237 1.239 I. 241 1.246 1.259 1.280 1.305 I. 331

I).732 I).75 I 0.773 1).809 1/.847 0.876 0.892 0.91/2 0.906

6000 6500 7000 7500 8000

0.9709 0.9709 0.9709 0.9709 0.9709

0.524 0. 509 0.497 I).487 0.430

1.335 I. 364 1.389 1.410 1.428

0. 7011 0. 694 0.690 0.687 0.614

Note. T h e s e results correspond to model C {Figs. A l a - c ) , which includes polarization effects in the gas. p, q, and A signify geometric albedo, planetary phase integral, and spherical or Bond albedo, respectively, tl thank M. T o m a s k o for performing these calculations. I The model is an aerosol-free c o l u m n of 191 kin-area of Ha and 20.8 kin-area of He, overlying a semi-infinite, isotropically scattering cloud extending downward l'rom the 2-bar level, with single-scattering albedo w~. Note that H2 pressure-induced opacity influences p~ at 8000 A, 1see Fig. A I c L

total polarization if its optical thickness is small compared to that of the gas (as in many of the models examined to date; see Baines and Bergstralh, 1986). At X ~< 5000 in Fig. A1, the optical depths for Rayleigh plus Raman scattering are one to two orders of magnitude larger than those for the haze model used in generating (B). Table AII gives y as determined from Eq. (2) with hi = 2000 A. For h -< 6000 A, geometric albedos for case B (Fig. AI), which represents the " o b s e r v e d " continuum, were subtracted from those for case C, the haze-free continuum (including polarization). The phase integrals calculated for (Ct were also used for the " o b s e r v e d " phase integral spectrum. Tests showed that, at least for the haze model used in generating spectrum B (see discussion above), the

URANUS-NEPTUNE EQUILIBRIUM MODELS TABLE All ESTIMATES OF AEROSOl. HEAT DEPOSITIONS FOR URANUS

;~2 (,~) y

3500 0.010

4000 0.022

5000 0.045

6000 0.055

8000 0.065

No te. T h e s e results for y are based on Eq. {2J with

Xt -

20{M} A.

haze had no effect on qx. Thus, in the integrand in Eq. (2), Ax,clear - Ax,obs was replaced by qx,clear (Px,clear -- PX,obs) = qX,C ( P M c - px.B). The values for qx,c are consistent with V o y a g e r 2 cruise data currently available, which c o v e r p h a s e angles from - 2 5 to - 7 0 °. For example, Pollack et al. (1984) reported q =- 1.2 -+ 0.15 for 3500 ~< ?` ~< 5000 ,~, based in part on these observations. F a r better constraints on qx can be e x p e c t e d from V o y a g e r if the e n c o u n t e r includes m e a s u r e m e n t s at e x t r e m e phase angles and the acquisition of limb-darkening curves. y ~- 0.01 (corresponding to ?,2 = 3500 A) is a firm lower limit: although the o b s e r v e d albedos longward of 3500 i~ couht be m a t c h e d using a model with no stratospheric absorber, the I U E data (?` < 3500 ,i0 are lower than any plausible haze-free continuum. Table A I I indicates y = 0.055 for ?`2 = 6000 ,i,, consistent with the value used (a priori) in model b, Fig. 2. Thus, y -~ 0.05 is easily reconcilable with the data, though it is strongly model-dependent. M o r e o v e r , model b c o r r e s p o n d s to aerosol heating a b o v e the R - C b o u n d a r y whereas the estimates in Table AII could arise primarily f r o m aerosols that reside at deeper levels. An u p p e r limit for y(?`2 = 6000 A) can be obtained using a c o n s e r v a t i v e cloud (aSc = 1.0): such a model was calculated but is not shown in Fig. A1. The corresponding s p e c t r u m , analogous to (C), gave y -~ 0.08. Contributions to y longward of 6000 1~ are v e r y uncertain. F o r 6000 -< ?` -< 8000 ,~, Table A I I indicates Ay = 0.01, which is

403

p r o b a b l y a lower limit. N o t e that px.c PX,obs ~ 0.04 near 6040 and 6330 i~. If this d i s c r e p a n c y is attributed to haze absorption a b o v e the main cloud, and if it is applied (arbitrarily) from 6000 to 8000 ,~, one obtains Ay = 0.01. On the other hand, "Aphaze" could be m u c h larger than 0.04: results at 6000 i~ suggest the upper limit is roughly 0.2, taking Pobs ~ 0 . 5 5 and Pclear = 0.75 (which c o r r e s p o n d s to the use of o3c = 1.0). Ay --~ 0.05 would then result for 6000 -< ?` -< 8000,1~, giving y < 0.08 + 0.05 = 0.13 for ?`2 = 8000 i~. O f course, some haze absorption could o c c u r longward of 8000 A. It could be relatively small, judging from the similarity b e t w e e n observations and a higha b u n d a n c e CH4 laboratory spectrum (e.g., Fink and L a r s o n (1979)). In conclusion, the value y = 0.15 used in Uranus model c (Fig. 2) seems to be close to an u p p e r limit for heating a b o v e the 2-bar level. The models presented by Baines and Bergstralh (1986) imply values that are smaller than 0.15 by factors of roughly 2 to 3, primarily because their continuum levels at ?` ~> 6000 A are relatively dark, as in the e x a m p l e for O3c = 0.9709 in Fig. A l c . Their study does not preclude a brighter continu u m coupled with greater haze absorption at ?` ~> 6000 A. Since their haze distributions e n c o m p a s s 0. I ~< P ~< 3 bars, a model such as (c) (Fig. 2) seems to be acceptable. On the other hand, given the m a n y uncertainties discussed above, other r a d i a t i v e c o n v e c t i v e models with y < 0.15 should be considered. ACKNOWLEDGMENTS It is a pleasure to acknowledge several very helpful d i s c u s s i o n s with G. Orton, M. T o m a s k o , W. Cochran, J. Bergstralh, J. H o g a n , and K. Baines. Part of this work is based on m y doctoral thesis (1980; D e p a r t m e n t of Earth and Space Sciences, S U N Y - S t o n y Brook) and, accordingly, I t h a n k D. Peterson, T. Owen. J. CaldweU, and J. Hardorp. This research was conducted primarily at the Jet Propulsion Laboratory, California Institute of Technology, u n d e r contract with the National A e r o n a u t i c s and Space Administration.

404

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F. A P P L E B Y

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