Saturn Helium Abundance: A Reanalysis of Voyager Measurements

Icarus 144, 124–134 (2000) doi:10.1006/icar.1999.6265, available online at http://www.idealibrary.com on

Saturn Helium Abundance: A Reanalysis of Voyager Measurements Barney J. Conrath Center for Radiophysics and Space Research, Cornell University, Ithaca, New York 14853 E-mail: [email protected]

and Daniel Gautier DESPA, Observatoire de Paris, 92195 Meudon Cedex, France Received March 16, 1999; accepted October 12, 1999

Thermal emission spectra calculated using Voyager Jupiter radio occultation (RSS) temperature profiles rescaled to the Galileo probe value of the helium abundance do not agree with the spectra measured by the Voyager infrared spectrometer (IRIS). The ∼2 K offset in brightness temperature suggests the possibility of a systematic error source yet to be identified. This raises the question of the validity of the Voyager Saturn helium abundance that was determined using the same RSS–IRIS approach. We address this issue by developing an inversion algorithm for the simultaneous retrieval of the temperature, the para H2 fraction, and the helium abundance from the IRIS spectra alone. This approach can not be successfully applied to Jupiter because of strong gaseous NH3 and cloud opacity near the low-frequency end of the spectrum, but this restriction is less severe at the lower temperatures of Saturn. Applications of the algorithm to Saturn spectra yield a volume mixing ratio He/H2 between 0.11 and 0.16 corresponding to a helium mass fraction relative to the total helium and hydrogen in Saturn’s atmosphere of Y = 0.18–0.25. Although these retrievals depend on subjective filtering of the solutions in the inversion algorithm to reduce the range of non-uniqueness for the helium values, they strongly suggest a value for He/H2 significantly larger than the value of 0.034 ± 0.024 previously obtained by Conrath et al. (Conrath, B. J., D. Gautier, R. A. Hanel, and J. S. Hornstein 1984, Astrophys. J. 282, 807–815) using the RSS–IRIS method. °c 2000 Academic Press Key Words: Saturn; Atmospheres, Composition, Structure.

1. INTRODUCTION

Determination of the helium abundance in the giant planets is of fundamental importance in studies of the evolutionary history of these bodies. Hydrogen and helium were acquired during their formation from the primitive solar nebula. As a consequence, their bulk He/H2 ratio must be equal to that in the nebula (the so-called protosolar ratio). Fractionation processes that occurred during the evolution of Jupiter and Saturn have modified the initially uniform distribution of He/H2 within each planet;

helium is expected to be depleted with respect to the protosolar abundance in the outer molecular envelope and enriched in the deep interior. The relative values of He/H2 in the two regions of Jupiter and Saturn strongly constrain theories of the evolution of these objects. The helium abundances in the atmospheres of the four giant planets have been previously retrieved using a combination of Voyager infrared spectrometer (IRIS) measurements and temperature profiles obtained by the radio occultation experiment (RSS). With this technique, a modest depletion of helium with respect to the protosolar abundance was found for Jupiter. A value of the volume mixing ratio He/H2 = 0.110 ± 0.032 was obtained, corresponding to a mass fraction of Y = 0.18 ± 0.04 (Gautier et al. 1981; Conrath et al. 1984) compared with the protosolar abundance of Y = 0.28 (Profitt 1994). In contrast, a very large depletion was found for Saturn with a value of He/H2 = 0.034 ± 0.024 corresponding to a helium mass fraction Y = 0.06 ± 0.05 (Conrath et al. 1984). More recently, the helium abundance detector (HAD) aboard the Galileo atmospheric probe made precise in situ measurements of the Jupiter He/H2 ratio (von Zahn and Hunten 1996; von Zahn et al. 1998), obtaining He/H2 = 0.157 ± 0.003 (Y = 0.234 ± 0.005, assuming a 1.9% heavy element contribution). This result is supported by measurements from the mass spectrometer also carried on the Galileo probe (Niemann et al. 1996, 1998). Although the difference between the Voyager and Galileo results is not extremely large when the combined error bars are taken into account, it may nevertheless be significant. This has motivated a re-examination of the Voyager result, with the conclusion that the Galileo He/H2 value could be made consistent with the Voyager IRIS and RSS measurements if the published nominal radio occultation profile (Lindal 1992) were made cooler by about 2 K as discussed further in Section 2 below. At the present time, it is unclear whether there may be systematic errors in the Voyager measurements peculiar to the Jupiter encounter or whether similar errors may be present in

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the results obtained for the other giant planets. It is important to re-evaluate and possibly refine the remote sensing results and understand the possible implications for the Voyager measurements of He on Saturn, Uranus, and Neptune. Until such time as probes can be placed into the atmospheres of these three planets, we are forced to rely on remote sensing results. In the present paper, we address a reanalysis of the Voyager Saturn data. Currently, we do not have available a detailed error analysis of the Lindal (1992) radio occultation retrieval, but we have attempted to use a method, initially proposed by Gautier and Grossman (1972), that permits us to constrain the He/H2 ratio from the IRIS spectra alone. Unfortunately, this approach can not be used for Jupiter within the spectral range covered by IRIS because the presence of gaseous NH3 and cloud opacity prevents inversion of radiances at appropriate frequencies to simultaneously constrain the temperature, the H2 ortho–para ratio, and the helium abundance. However, the troposphere of Saturn is cooler than that of Jupiter so that the ammonia abundance is lower, and there is less cloud opacity, permitting better access to a portion of the translational H2 continuum at the low-frequency end of the spectrum. The direct inversion method discussed here takes advantage of this property of the Saturn spectrum, permitting new inferences to be made concerning the Saturn He/H2 ratio. The Voyager Jupiter results are summarized in Section 2 in light of the Galileo results. In Section 3, the retrieval algorithm used is discussed and applied to synthetic data to demonstrate the usefulness and limitations of the method. Selections of IRIS spectra at several latitudes on Saturn are made in Section 4. The results of the retrieval of the He/H2 ratio and of the H2 ortho– para ratio from the selected samples are given in Section 5. The results are discussed in Section 6, and our conclusions are summarized in Section 7. 2. JUPITER

The various determinations of the Jovian He abundance from Voyager measurements are compared in Table I with the Galileo TABLE I Determinations of Jovian Helium Abundance from Voyager and Galileo Measurements Determination Voyager Gautier et al. 1981 IRIS–RSS ingress IRIS–RSS egress IRIS retrieval Conrath et al. 1984 IRIS–RSS egress Weighted mean Galileo von Zahn et al. 1998 Niemann et al. 1998

He/H2

0.143 0.136 ± 0.046 0.115 ± 0.037 0.110 ± 0.032 0.114 ± 0.025 0.157 ± 0.004 0.156 ± 0.006

probe determinations. Here we have expressed all values in terms of He/H2 , the volume mixing ratio of He with respect to molecular hydrogen. This quantity is independent of assumptions concerning the presence of other atmospheric constituents. Two separate approaches were used by Gautier et al. (1981) to retrieve the Jovian He abundance from the Voyager IRIS thermal emission spectra. The first consisted of a direct retrieval from the spectra alone, under the assumption that the H2 ortho–para ratio is the thermal equilibrium value at the local temperature. Since this assumption is now known to be incorrect, the value of He/H2 obtained in this way is no longer considered valid. The second method used by Gautier et al. (1981) made use of a combination of IRIS data and measurements from the Voyager radio occultation experiment. By combining atmospheric radio occultation measurements with measurements of thermal emission spectra acquired near the tangent point of the radio occultation path, a determination of the He abundance can be obtained. The basic quantity retrieved from the measurements of the spacecraft signal as it is occulted by the planetary atmosphere is the atmospheric refractivity as a function of distance from the center of the planet. If the mean molecular weight and mean coefficient of refractivity per molecule are known, then by invoking hydrostatic balance and an equation of state a profile of temperature versus pressure can be obtained (see for example, Lindal 1992). Since the mean molecular weight and coefficient of refractivity are dependent on the atmospheric composition, a temperature profile can be constructed for any specified composition. For an atmosphere with uniformly mixed gaseous constituents, the occultation temperature profile scales as m T = , T0 m0 p mα0 , = p0 m0α

(1) (2)

where m and α are mean molecular weight and mean coefficient of refractivity, respectively, and the subscript 0 indicates reference values. Once a temperature profile is obtained for a given composition, a radiative transfer code can be used to calculate a theoretical thermal emission spectrum. This spectrum is then compared with spectra measured near the occultation point. An attempt is then made to fit the measured spectrum by adjusting the temperature profile using (1) and (2). In deriving their result, Gautier et al. (1981) assumed that only He and H2 contribute significantly to the mean molecular weight and mean coefficient of refractivity. This was later refined by Conrath et al. (1984), who included the contribution of methane. The temperature profile given in Table 1 of Lindal (1992) for the Voyager 1 Jupiter egress occultation is shown in Fig. 1. The atmospheric composition was assumed to consist of hydrogen and helium with He/H2 = 0.11/0.89. It should be noted that Lindal gives the latitude of this profile as 12◦ S, which is in fact the latitude of the ingress profile. Apparently the correct latitude for the profile shown is near the equator (Lindal et al. 1981).

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the shape of the spectrum is affected by the assumed He abundance and the para H2 profile, the overall scaling of the spectrum is controlled by the scaling of the occultation profile through (1) and (2). The results shown in Fig. 2 demonstrate that it is not possible to match the Voyager IRIS measurements using the Jupiter profile of Lindal (1992) scaled to the Galileo value of He/H2 = 0.157; the calculated brightness temperature spectrum is ∼2 K warmer across the 300–700 cm−1 region. Although this difference may lie close to the combined error bars of the various factors, the disagreement is clearly systematic, and the source needs to be understood. Possible sources of systematic error include uncertainties in the collision-induced absorption coefficients, atmospheric opacity sources not included in the theoretical calculations, systematic errors in the calibration of the IRIS spectra, and errors in the radio occultation profiles. It is unlikely that absorption coefficient errors could be entirely responsible for producing the observed offset across a broad portion of the spectrum. For example, in the S(1) line near 600 cm−1 , the measured radiance originates from a region of the atmosphere near the tropopause where the lapse rate is low and there is little sensitivity to the exact atmospheric opacity. This same reasoning suggests that unknown opacity sources are unlikely to be a significant source of systematic error. Cloud opacity has not been included in the calculations and could produce some effect below 320 cm−1 and FIG. 1. Jupiter Voyager 1 egress radio occultation profile. The solid curve is the profile given by Lindal (1992) for He/H2 = 0.11/0.89. The broken curve is obtained by rescaling the original profile to the Galileo value He/H2 = 0.157 using Eqs. (1) and (2) in the text.

Using (1) and (2), the profile of Lindal has been rescaled using the Galileo value of He/H2 = 0.157, and this is also shown in Fig. 1. In carrying out the rescaling, it has been assumed that, in addition to helium and hydrogen, only methane contributes significantly to the mean molecular weight and coefficient of refractivity. A value of CH4 /H2 = 2.18 × 10−3 was adopted. This profile has been used to calculate the theoretical spectrum shown in Fig. 2, which is compared with an average of measured IRIS spectra acquired near the egress occultation point. The portion of the spectrum between 300 and 650 cm−1 is dominated by the collision-induced S(0) and S(1) absorption lines of molecular hydrogen. The collision-induced absorption coefficients were calculated using the codes of A. Borysow (Borysow et al. 1985, 1988, Birnbaum et al. 1996) and include H2 –H2 , H2 –He, and H2 –CH4 contributions, although the effect of the latter is small. The detailed shape of the collision-induced spectrum is dependent on the He abundance and the ortho–para ratio of molecular hydrogen. Various investigations have indicated that the ortho– para ratio is not in thermodynamic equilibrium (see, for example, Conrath and Gierasch 1984, Carlson et al. 1992.). We have used a profile of the para H2 fraction retrieved from zonal mean spectra at the appropriate latitudes (Conrath et al. 1998). While

FIG. 2. Comparison of synthetic and measured Jupiter thermal emission spectra. The solid curve is an average of Voyager IRIS spectra acquired near the Voyager 1 egress occultation point. The broken curve is a synthetic spectrum calculated using the occultation profile rescaled to the Galileo He/H2 value as shown in Fig. 1.

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between 440 and 520 cm−1 , but should not affect the S(0) and S(1) line centers (Carlson et al. 1992). Optically thin particulate layers in regions of the upper stratosphere where temperatures are relatively warm could increase the emission, but this would have the effect of making the discrepancy between measured and calculated spectra even greater. The absolute calibration of the IRIS spectra is dependent on the instrument cavity temperature, which is measured with an absolute accuracy believed to be better than 0.1 K. In addition, an error in the measured instrument temperature would produce a spectrally dependent error in the brightness temperature spectrum, which is not observed. For example, to produce an approximately 2 K offset in brightness temperature at 350 cm−1 , an error of 1 K in the instrument temperature would be required, and this same error would result in a brightness temperature error greater than 6 K at 600 cm−1 . Finally, we note that the C/H ratio in Jupiter obtained using a thermal profile directly retrieved from the IRIS spectra (Gautier et al. 1982), rather than using an RSS profile, is quite close to that measured in situ by the Galileo probe (Niemann et al. 1996, 1998). From these considerations, it seems unlikely that errors in the IRIS absolute calibration could account for the differences observed in Fig. 2. This leaves systematic errors in the radio occultation profiles as the remaining candidate in our list of error sources. A detailed error analysis of these profiles has not been published. However, a reanalysis of the RSS measurements, along with a complete error propagation study, is needed. At this time, it does not seem possible to make further progress in the analysis of errors in the Voyager determination of the Jupiter helium abundance using the RSS–IRIS approach. In addition, direct inversion of Jupiter IRIS spectra to simultaneously obtain temperature, para H2 fraction, and helium abundance is not feasible for the portion of the spectrum available in the IRIS measurements, as discussed further below. However, it does appear to be possible to obtain some constraints on the Saturn helium abundance by direct inversion of IRIS spectra, and the remainder of the paper addresses this topic. 3. RETRIEVAL METHOD

The shape of the Saturn thermal emission spectrum between 200 and 600 cm−1 is sensitive to the upper tropospheric temperature vertical profile, the profile of the para H2 fraction, and the He/H2 ratio. The latter sensitivity results from the contribution of He–H2 interactions to the collision-induced absorption of H2 . In addition to the H2 opacity, weak gaseous NH3 absorption and possibly some cloud opacity may exist near the low-frequency end of this spectral region. Whether useful information on helium can be extracted from this part of the spectrum depends on the degree to which the effects of helium and the ortho–para H2 ratio can be separated with the signal-to-noise ratio available in the Saturn spectra. We explore this question by first developing an inversion algorithm and applying it to synthetic Saturn data.

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The concept of retrieving information on helium by direct inversion of appropriate thermal emission spectra of the giant planets was first proposed by Gautier and Grossman (1972). In their approach, the temperature profile and the He/H2 ratio are simultaneously retrieved under the assumption that the para H2 fraction is known. More recently, Conrath et al. (1998) have developed an algorithm for the simultaneous retrieval of the temperature and para H2 profiles, under the assumption that He/H2 is known. In the present application, we have extended the formulation of Conrath et al. to include He/H2 as an additional parameter to be retrieved. The details of this formulation are given the Appendix. The problem is highly “ill-posed” in the sense that unique solutions can not be directly obtained and arbitrarily small changes in the measurements can produce finite changes in the retrieved parameters. This behavior, which is a consequence of the smoothing properties of the radiative transfer process, can lead to hypersensitivity to measurement errors and non-physical solutions. Even in the case when only the temperature profile is sought and all other parameters are assumed to be known, the number of degrees of freedom required to describe the actual profile greatly exceeds the number of independent pieces of information in the spectrum. As a consequence, only a smoothed (low-pass filtered) approximation of the actual profile can be retrieved (Gautier and Revah 1975). The solution is non-unique since higher spatial frequency components of arbitrary amplitudes can be added to it while still satisfying the measurements to within the noise level. In the present case, it is necessary to impose constraints simultaneously on the temperature profile, the para hydrogen profile, and the He/H2 ratio in order to obtain physically meaningful solutions. General approaches for accomplishing this have been discussed by Craig and Brown (1986). We have chosen to constrain the iterated solutions to lie “near” their initial values, and the temperature and para H2 profiles are strongly low-pass filtered with regard to their vertical structure as described in the Appendix. To investigate the characteristics of this inversion method as summarized by Eqs. (A12), (A13), and (A14), we first apply it to synthetic Saturn spectra to determine how well the known He/H2 ratio can be recovered. The spectral region between 230 and 600 cm−1 is used with a sampling interval of 4.3 cm−1 (the IRIS spectral resolution) for a total of 86 spectral data points. The first synthetic spectrum inverted was calculated assuming He/H2 = 0.034, the very small value inferred for Saturn by Conrath et al. (1984) using the RSS–IRIS method. The inversion results are shown in Fig. 3 where the retrieved values of He/H2 as a function of iteration number are shown along with the corresponding rms radiance residuals. The only noise included in the synthetic data is that due to numerical round-off (∼10−10 W cm−2 ster−1 /cm−1 ). Inversions were made using three different first guess values for He/H2 , spanning a broad range from 0 to 0.2. Results for inversions of a synthetic spectrum with He/H2 = 0.12 are shown in Fig. 4. The rms residuals in each case show an initial rapid decrease with iteration and then a continued

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slow decrease toward the round-off error limit. The solutions show some dependence on the first guess as a consequence of the constraint imposed on He/H2 by the inversion algorithm. The strength of this dependence reflects the degree of correlation of the sensitivities of the spectral shape to para hydrogen and to He/H2 . Variations in the para H2 and in the helium abundance can have partially compensating effects on the spectral shape; even in the presence of strong filtering, some non-uniqueness remains. While it may not be possible to get a highly accurate determination of helium abundance from the IRIS spectra alone, these results suggest that it should be possible to distinguish between the low value obtained by Conrath et al. using the RSS–IRIS approach and significantly larger values of He/H2 . When the algorithm is applied to the accessible spectral range on Jupiter, it is found that the inferred helium abundance remains highly dependent on the first guess, so the method cannot be successfully applied to IRIS Jovian data. In the case of Saturn, the lower temperatures, resulting in reduced gaseous ammonia and cloud opacities at the lower wavenumbers, permit access to

FIG. 4. Same as Fig. 3 except for a “true” He/H2 value of 0.12.

more of the translation portion of the H2 spectrum. In addition, the lower temperatures result in a relative spectral sensitivity of the H2 opacity more favorable to the separation of He and para H2 effects. 4. SELECTION OF SATURN DATA SETS

FIG. 3. Results of a Saturn helium abundance retrieval from synthetic data illustrating the behavior of the inversion algorithm. (Lower panel) Retrieved values of the helium mixing ratio as a function of iteration number for three different first guesses for He/H2 . The “true” value of He/H2 is 0.034. (Upper panel) The rms radiance residuals in W cm−2 ster−1 /cm−1 ).

The algorithm described in the preceding section is dependent on the detection of subtle differential spectral shape effects for the extraction of information on the helium abundance. To achieve the required signal-to-noise ratio, it is necessary to average sets of IRIS Saturn spectra. Care must be taken in selecting spectral ensembles for averaging; inhomogeneities within the data sets due to varying observing geometry or varying atmospheric conditions can introduce spurious spectral shape effects. Four sets of spectra, summarized in Table II, have been chosen for this analysis. Three of these were selected from measurements acquired during the Voyager 1 incoming north–south mapping sequence when the spatial resolution of the IRIS instrument was ∼10◦ of great circle arc on the planet. Data were binned into 10◦ -wide latitude intervals between 30◦ N and 60◦ N where the data coverage is best. The averages represent essentially

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TABLE II Saturn Spectral Data Sets and Retrieved Helium Abundances N latitude (deg)

µ

30–40 40–50 50–60 28–38

0.878 0.793 0.686 0.923

No. of Effective spectra NESRa Residualsa 79 49 61 19

0.79 1.00 0.90 1.60

0.71 0.89 0.79 1.29

He/H2

Yb

0.122–0.145 0.138–0.162 0.118–0.146 0.113–0.151

0.196–0.225 0.216–0.245 0.191–0.226 0.184–0.231

In 10−9 W cm−2 ster−1 /cm−1 . Helium mass fraction relative to total helium and hydrogen in Saturn’s atmosphere. a

b

measured spectrum in Fig. 6. The fit to the spectral shape obtained here is better than those obtained by Conrath et al. (1984). The results of the helium retrievals for the four cases are summarized in the final three columns of Table II. Note that in all cases the rms radiance residuals are slightly below the estimated effective NESR for the average spectra. The range of values given for the volume mixing ratio He/H2 indicates the extremes obtained with first guesses ranging from 0 to 0.2 and does not represent a formal error estimate. In the final column the results are expressed in terms of the helium mass fraction relative to the total hydrogen and helium in the Saturn atmosphere, Y =

◦

zonal means over 360 of longitude with little variation in the emission angle. The fourth set of spectra consists of measurements taken between 28◦ N and 38◦ N with a higher spatial resolution (∼4◦ great circle arc). In each case, the average emission angle cosine µ is given in the table along with the number of spectra in the ensemble and the effective noise equivalent spectral radiance (NESR) associated with the mean spectrum. The latter quantity was estimated by dividing the NESR for individual spectra by the square root of the number of spectra in the ensemble. The value used for the NESR of an individual spectrum is 7 × 10−9 W cm−2 ster−1 /cm−1 , based on space-viewed spectral measurements (Hanel et al. 1982).

[He]/[H2 ] . [He]/[H2 ] + 0.504

(3)

Temperature profiles retrieved from the mean spectra for the three low spatial resolution bins are shown in the left-hand panel of Fig. 7. The temperature retrieval for the higher resolution

5. RESULTS AND DISCUSSION

An example of an application of the retrieval algorithm to the mean spectrum for the 30◦ N to 40◦ N bin of Table II is shown in Fig. 5. The upper panel shows the behavior of the rms radiance residual as a function of iteration number. This quantity is the rms value of the difference between the measured and calculated radiances for the 86 spectral points used in the inversion. The lower panel shows the retrieved values of He/H2 . Three different retrievals are shown for three different first guess values of He/H2 . The behavior shown in Fig. 5 is observed in the other three cases as well. The residuals drop to near the effective measurement noise level by the third iteration and remain essentially constant for subsequent iterations. The retrieved values of He/H2 for the three different first guesses show a behavior similar to that obtained for synthetic data. Some dependence on the first guess is observed, along with a slow drift in retrieved values with repeated iterations. These characteristics reflect the lack of complete uniqueness in retrieving both para H2 and helium abundance simultaneously. As the number of iterations increases, the effect is to reduce the strength of the constraints on the para H2 profile, permitting compensating changes in para hydrogen and helium abundance to occur while maintaining essentially the same level of rms radiance residual. The spectrum calculated using the He/H2 retrieval in Fig. 5 for a first guess of 0.10, along with the corresponding retrieved temperature and para hydrogen profiles, is compared with the

FIG. 5. Saturn helium abundances retrieved by inversion of an average of 79 Voyager IRIS spectra acquired in the latitude range 30◦ N to 40◦ N. (Lower panel) Retrieved values of He/H2 as a function of iteration number for three first guesses. (Upper panel) rms radiance residuals in W cm−2 ster−1 /cm−1 . The effective NESR for this average spectrum is indicated with a horizontal line segment in the upper panel.

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FIG. 6. Comparison of the mean Voyager IRIS spectrum for the latitude range 30◦ N to 40◦ N (broken curve) with a theoretically calculated spectrum (solid curve). The theoretical spectrum was calculated using the retrieved temperature and para hydrogen profiles and the retrieved He/H2 value obtained with a first guess of He/H2 = 0.10 (see Fig. 5). Spectra calculated using retrievals for other first guess values are indistinguishable from the theoretical spectrum shown.

28◦ N to 38◦ N bin (not shown) is indistinguishable from that for the 30◦ N to 40◦ N bin. The range of He/H2 values retrieved is shown for each bin in the right–hand panel. The values have been plotted at the central latitude of each bin. Also shown in the left-hand panel is the radio occultation profile for 36◦ N taken from Lindal (1992) and rescaled to be consistent with the central value of He/H2 = 0.135 obtained here. The original occultation profile (not shown) lies very close to the IRIS retrieved profile for the 30◦ N to 40◦ N bin, as expected, since that profile uses the helium abundance originally derived with the RSS–IRIS method. The offset of the rescaled profile indicates the magnitude of the systematic shift that would be required to bring the occultation results and the IRIS results into agreement if the helium abundance suggested by the inversion results presented here is assumed to be correct. Para hydrogen profiles retrieved for the three lower spatial resolution bins are given in Fig. 8. In each case, two retrievals are shown corresponding to the two extreme He/H2 first guesses. A thermal equilibrium profile is also shown for comparison. This illustrates how differences in the retrieved helium abundance can be compensated by the retrieved para hydrogen profile to yield essentially the same rms radiance residuals. Large changes in the shape of the retrieved para hydrogen profile are not allowed because of the filtering constraints imposed by the inversion algorithm; this limits the range of retrieved values of He/H2 obtainable. Consequently, the range of non-uniqueness of the He/H2 retrieval is dependent on the arbitrary but plausible constraint imposed by the algorithm that the para hydrogen profile

FIG. 7. (Left-hand panel) Retrieved temperature profiles for the three latitude bins indicated. The profile retrieved for the higher spatial resolution bin between 28◦ N and 38◦ N is indistinguishable from the 30◦ N to 40◦ N retrieved profile. Also shown is the radio occultation profile (RSS) for 36◦ N taken from Lindal (1992) and rescaled to a He/H2 value of 0.135. (Right-hand panel) Range of He/H2 values retrieved from average spectra in each of four latitude bins. The results are plotted at the central latitude of each bin. The bars do not represent formal error estimates, but rather the range of values obtained for the various first gusses used, indicating the extent to which the solutions are non-unique.

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131

FIG. 8. Retrieved para hydrogen profiles corresponding to the extremes of the He/H2 values obtained. The results shown are for the latitude bins (a) 30◦ N to 40◦ N, (b) 40◦ N to 50◦ N, and (c) 50◦ N to 60◦ N. The thermal equilibrium para hydrogen profile is shown for comparison in each case.

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should be the smoothest solution obtainable on a vertical scale of about a pressure scale height. The combined RSS–IRIS approach for obtaining the helium abundance used by Conrath et al. (1984) is quite sensitive in principle, but is strongly susceptible to systematic errors. The present direct retrieval method makes use of the IRIS data only and is less sensitive to systematic errors, such as the absolute calibration of the spectra. However, it is less sensitive to the He abundance than the RSS–IRIS approach and is highly sensitive to all factors that affect the shape of the measured or calculated spectra. The propagation of random errors in the measurements is minimized through the use of averages of several tens of spectra and the use of all available data points within the selected spectral range in the retrievals. We are unaware of any instrumental effects that could produce a distortion of the spectral shape. Spectrally dependent opacity sources not included in our radiative transfer calculations could cause uncertainties. Ammonia gas and cloud opacities are the most likely candidates. We have avoided any significant absorption by NH3 by restricting our measurements to frequencies no lower than 230 cm−1 . Tropospheric cloud opacity might be expected to have a spatial dependence as is observed in the visible portion of the spectrum, but there is little significant difference in our He abundance retrievals over the latitude range from 30◦ N to 60◦ N. However, a spatially uniform cloud opacity effect can not be completely ruled out. Differential errors in the H2 absorption coefficients can also introduce spurious effects. It is difficult or impossible to make quantitative estimates for most of these error sources; however, it seems likely that the greatest source of uncertainty is in the non-uniqueness of the retrieved He/H2 values. The values of He/H2 obtained with the direct retrieval method fall in the range 0.11–0.16, which is significantly larger than the value of 0.034 ± 0.024 obtained by Conrath et al. (1984) using the combined RSS–IRIS technique. The corresponding range of Y = 0.18–0.25 from the present determination would indicate considerably less depletion of helium in the outer layers of Saturn. While the results presented here can not be regarded as definitive, they do impose constraints that are inconsistent with the earlier results, suggesting the possible presence of systematic errors in that determination. It is interesting to note that the range of values obtained here for the helium mass fraction of Saturn’s atmosphere is more nearly consistent with recent evolutionary modeling results (Hubbard et al. 1999, Guillot 1999) that suggest values in the range Y = 0.11–0.25. Also, a helium abundance equivalent to He/H2 = 0.11 ± 0.04 was obtained by Orton and Ingersoll (1980) from Pioneer II infrared and radio occultation data. 6. SUMMARY AND CONCLUSIONS

When the Voyager Jupiter radio occultation profiles are rescaled to the Galileo probe value of the helium abundance, the spectra calculated from these profiles do not match the measured IRIS spectra. The nearly constant offset in brightness tempera-

ture of ∼2 K between calculated and measured spectra suggests the presence of a systematic error; the source of such an error has not yet been identified. The experience with Jupiter motivated a re-examination of the determination of the He abundance on Saturn from Voyager data. An inversion algorithm was developed for the simultaneous retrieval of temperature, para hydrogen, and helium abundance from IRIS spectra directly. This algorithm could not be applied successfully to Jupiter IRIS data because of the presence of strong gaseous ammonia and cloud opacity at the low-frequency end of the available spectral range. Because of the lower temperatures on Saturn, this constraint is less severe, and estimates of the helium abundance have been retrieved. Volume mixing ratios He/H2 between 0.11 and 0.16 were obtained, corresponding to a helium mass fraction relative to the total helium and hydrogen in the Saturn atmosphere in the range 0.18–0.25. It must be emphasized that the retrieval of He/H2 is non-unique. The algorithm used imposes a subjective but plausible filtering constraint on the para H2 profile to reduce the range of non-uniqueness in the solutions for helium. However, the results strongly suggest a helium abundance significantly larger than that previously inferred by Conrath et al. (1984) using the RSS–IRIS approach. Further progress on the determination of the Saturn He abundance in the immediate future will presumably depend on remote sensing techniques, since no Saturn entry probe is currently planned. In the near term, reanalysis of the Voyager Jupiter and Saturn radio occultation profiles may provide additional understanding of the combined RSS–IRIS approach. Finally, the Cassini spacecraft en route to the Saturn system will obtain new radio occultations, and the CIRS instrument will provide high–quality thermal emission spectra extending to wavelengths longer than those obtained by Voyager IRIS, providing new opportunities to infer the Saturn helium abundance. APPENDIX: RETRIEVAL ALGORITHM FORMULATION Assume we have spectral measurements at m frequencies within the S(0) and S(1) collision-induced hydrogen absorption lines, and we wish to retrieve the helium mixing ratio X = He/H2 simultaneously with the profiles of para hydrogen fraction f p (z) and temperature T (z) where z = ln p and p is barometric pressure. For this purpose, the algorithm given by Conrath et al. (1998) for the simultaneous retrieval of temperature and para H2 is extended to include the He mixing ratio as a third parameter. To calculate the radiance I (ν), numerical quadrature must be used, and T (z) and f p (z) are defined at n atmospheric levels. We first linearize the radiative transfer equation about the reference values X 0 , f p0 (z), and T 0 (z), 1Ii =

n n X X δ Ii δ Ii δ Ii 1T j + 1 f pj + 1X, δT δ f δ X j pj j=1 j=1

(A1)

where δ Ii /δT j , δ Ii /δ f pj , and δ Ii /δ X are values of the functional derivatives of the radiance at νi with respect to T , f p , and X . The perturbations with respect to the reference profiles are 1T j = T (z j ) − T 0 (z j ), 1 f pj = f p (z j ) − 1X = X − X , 0

f p0 (z j ),

(A2) (A3) (A4)

133

SATURN HELIUM ABUNDANCE 1Ii = I (νi ) − I 0 (νi ),

(A5)

where I 0 (νi ) is the radiance calculated using T 0 , f p0 , and X 0 . The direct solution of (A1) for T (z), f p (z), and X , given measurements of I (νi ), is an “ill-posed” problem, and to obtain physically meaningful solutions it is necessary to introduce constraints. Usually, these take the form of strong low-pass filtering of the solutions. This general approach to inverse problems is reviewed in detail by Craig and Brown (1986). Define the matrices Ki j =

δ Ii , δT j

(A6)

Mi j =

δ Ii , δ f pj

(A7)

δ Ii . δX

(A8)

In the present application, it can be assumed that the random measurement errors at any two points in the spectrum are uncorrelated so E can be taken as a diagonal matrix with the diagonal elements equal to the square of the effective NESR associated with the instrument. When analyzing ensemble averages of spectra, the effective NESR for the average spectrum is assumed to be given by the NESR of an individual spectrum divided by the square root of the number of spectra in the ensemble. The magnitudes of the factors α, β, and γ determine the degree of damping imposed on the solutions and their ratios determine the relative emphasis placed on the temperature, para hydrogen fraction, and He mixing ratio. Their values can most easily be determined through numerical experiment. The values of S and U must also be specified. In the present application, we find it useful to specify S and U as Gaussians of the form Si j = Ui j = expf−(z i − z j )2 /2c2 g,

(A16)

and the vector Ri =

Equation (A1) can then be written in the form 1I = M1fp + K 1T + R1X.

(A9)

In general, the profiles 1fp and 1T can be expressed in terms of sets of basis vectors, i.e., 1T = Fa,

(A9)

1fp = Gb,

(A10)

where F and G are matrices whose columns are the basis vectors, and a and b are column vectors of expansion coefficients. Now consider minimization with respect to T j , f pj , and X of the quadratic form ˆ + R1X )T E −1 (1I − Kˆ a − Mb ˆ + R1X ) Q = (1I − Kˆ a − Mb + λaT a + ηbT b + ξ 1X 2 ,

1T = αS K T (α K S K T + β MU M T + γ RRT + E)−1 1I, 1fp = βU M (α K S K + β MU M + γ RR + E) T

T

T

1X = γ R (α K S K + β MU M + γ RR + E) T

T

Ki j =

∂ B[νi , T (z j )] ∂ τ˜ (νi , z j ) , ∂Tj ∂z j

(A17)

where B(ν, T ) is the Planck radiance, and τ˜ (ν, z) is the transmittance at wavenumber ν from level z to the top of the atmosphere. In this formulation, the temperature dependence of the atmospheric opacity is neglected since it is generally much weaker than the temperature dependence of the Planck function. An analytic expression for the functional derivative of the spectral radiance with respect to f p is derived in Conrath et al. (1998). The functional derivative of the radiance with respect to the helium mixing ratio is calculated numerically at each iterative step.

ACKNOWLEDGMENTS (A11)

where E is the measurement error covariance matrix (usually diagonal), and the superscript T denotes matrix transposition. Here we have introduced the definitions Kˆ = K F and Mˆ = M G. In the case when λ = η = ξ = 0, (A11) reduces to the usual quadratic merit function that is minimized in least-squares fitting. The last three terms in (A11) impose the constraint that T , f p , and X lie close to their reference values; λ, η, and ξ are the weights with which the constraints are imposed relative to the least-squares fitting of the measurements. Carrying out the minimization yields the equations

T

where c is the correlation length in pressure scale heights. This simply provides a convenient means of filtering the solutions, and, based on estimates of the resolution from K and M, a value of c = 0.5 is found to be appropriate. The remaining task in the development of the inversion algorithm is to provide formulations for the rapid calculation of the functional derivatives K , M, and R. The functional derivative with respect to temperature can be directly evaluated from the expression

T

T

−1

−1

1I,

1I,

(A12) (A13) (A14)

where α = 1/λ, β = 1/η, and γ = 1/ξ , and we have introduced the two-point correlation matricies U = GG T ,

(A15)

S = FF .

(A16)

T

The nonlinearity of the problem is taken into account through iterative application of Eqs. (A12)–(A14). Note that only the correlation matrices of the basis vectors appear in the expressions and not the basis vectors themselves.

This research was supported in part by the NASA Planetary Atmospheres Program. The work was begun while B.C. was a visiting scientist at DESPA, Observatoire de Paris—Meudon, and the support and hospitality of that organization are acknowledged.

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