A Wideband Self-consistent Disk-averaged Spectrum of Jupiter Near 30 GHz and Its Implications for NH₃ Saturation in the Upper Troposphere

Reduction of these data converts a time series of flux densities to a single brightness temperature on channel-by-channel basis. The process also isolates and corrects for a variety of systematics throughout the process. The result is 15 time- and disk-averaged measurements of Jupiter's brightness near the 1.3 cm ammonia absorption band.

The effects of Jupiter's distance from Earth during the observational period are removed through distance normalization to the nominal value of 4.04 au, "flattening" each channel's measurement as is made evident between the two panels in Figure 1. This step facilitates examination of antenna gain error, indicated by the variance in the normalized points as well as the cross-channel behavior during individual observations. Jackknife testing, discussed in more detail in Section 2.3, reveals an additional artifact at this stage. The series of flux densities for each channel show a positive linear correlation with Jupiter's elevation in the sky at the time of observation, indicating some potential issue with airmass calibration in the original measurements. The slopes of each channel's flux density are positive and linear with frequency and therefore relatively easy to correct for, as demonstrated in Figure 2; we assumed in this process that the measurements taken at higher elevations through thinner layers of atmosphere are more accurate.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The flux densities are averaged across time with weights corresponding inversely to the thermal error on each measurement. Averages are calculated as

$$\begin{eqnarray}&&{F}_{\nu ,\mathrm{meas}}=\langle {F}_{\nu }\rangle =\displaystyle \frac{{\displaystyle \sum }_{i}{w}_{\nu ,i}{F}_{\nu ,i}}{{\displaystyle \sum }_{i}{w}_{\nu ,i}}\end{eqnarray} \tag{ 1 }$$

where

$$\begin{eqnarray}&&{w}_{\nu ,i}=1/{\sigma }_{\nu ,i}^{2}\end{eqnarray} \tag{ 2 }$$

such that ${\sigma }_{\nu ,i}$ is the stated thermal error in Janskys on the ith measurement for a given channel. Averaging the flux densities themselves is fairly straightforward; the rest of this section will discuss the application of meaningful error estimates.

2.1.1. Absolute (σ_A) and Relative (σ_R) Uncertainty

We define two distinct error measurements for the data set: absolute uncertainty σ_A and relative uncertainty σ_R. Absolute uncertainty quantifies our estimate of the overall calibration accuracy of the SZA, estimating how close the data are to the actual values. This is determined by the systematics of the instrument. An upper limit of <5% on this calibration error is quoted in COPSS, which incorporates results from Sharp et al. (2010) in this estimate.

Relative uncertainty quantifies the stability of each channel with respect to the others, estimating internal consistency rather than absolute accuracy. CARMA has strong spectral stability and can observe this feature in the strongly correlated cross-channel behavior displayed in Figures 1 and 2, which suggests that the σ_R measurement, while allowing some independent uncertainty on points, is expected to be smaller than σ_A and so will not have as great an effect on the overall position of the ensemble. σ_R is useful in defending the use of the relative structure of the ensemble as a comparably reliable tool to the absolute position as well as quantifying a model's match to this relative structure during the model-comparison process. σ_A, by contrast, serves as a more conventional uncertainty estimate.

We approach the absolute uncertainty σ_A estimate using statistical properties of the flux density ensemble along with their stated thermal errors. The absolute uncertainty is the quadrature sum of the weighted average of the individual data and the thermal noise of each measurement and may be written

$$\begin{eqnarray}&&{\sigma }_{A}^{2}(\hat{\sigma },\hat{n})={\hat{\sigma }}^{2}+{\hat{n}}^{2}.\end{eqnarray} \tag{ 3 }$$

The first term is

$$\begin{eqnarray}&&{\hat{\sigma }}^{2}=\displaystyle \frac{{\displaystyle \sum }_{i}{w}_{i}{({F}_{\nu ,i}-\langle {F}_{\nu }\rangle )}^{2}}{{\displaystyle \sum }_{i}{w}_{i}}.\end{eqnarray} \tag{ 4 }$$

The second term, the thermal contribution $\hat{n}$, is calculated as the reciprocal sum of thermal uncertainties for a given channel. With w_i defined as in Equation (2), $\hat{n}$ is

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\hat{n}}^{2}}=\displaystyle \sum _{i}{w}_{i}.\end{eqnarray} \tag{ 5 }$$

In this way, we merge statistical 1σ error on a channel's flux ensemble with the thermal uncertainty of the points themselves and produce the value σ_A, our estimate of uncertainty on the absolute calibration of the instrument. This we find to be closer to ∼2%, which falls under the upper limit stated in COPSS.

Relative uncertainty σ_R comprises both thermal contribution as well as the tendency of each channel to deviate from the others. A perfectly stable instrument should exhibit a consistent cross-channel response to small, day-to-day variation in observed flux; unexpected behavior should be reflected in σ_R. We compare observations across channels by normalizing each channel with its average.⁵ The normalized measurements are denoted f_ν,i.

$$\begin{eqnarray}&&{f}_{\nu ,i}=\displaystyle \frac{{F}_{\nu ,i}}{\langle {F}_{\nu ,i}{\rangle }_{\nu }}.\end{eqnarray} \tag{ 6 }$$

This fraction is averaged across each observation to find the daily mean deviation from each channel's respective average. A stable instrument's channel would exhibit minimal spread around this daily mean deviation, and any spread should be uncorrelated with channel. We isolate the residuals δ_ν,i from this daily average by subtracting the daily mean deviation from each observation set of normalized measurements, so as to exclude the daily deviation itself and examine each channel's deviation from this deviation. These residuals are defractionalized by multiplying by the channel average.

$$\begin{eqnarray}&&{\delta }_{\nu ,i}=({f}_{\nu ,i}-\langle {f}_{\nu ,i}{\rangle }_{i})\cdot \langle {F}_{\nu ,i}{\rangle }_{\nu }.\end{eqnarray} \tag{ 7 }$$

Each channel's behavior is now quantified by its mean deviation from the daily average as well as the spread of these deviations, given by their variance across each channel. During this examination, it was noted that the channels had a tendency to vary together in time; they often would pivot around the 31.188 GHz measurement, which remained stable to within 0.5% of the measurement value. The frequencies at either extreme (27.688 GHz and 34.688 GHz) varied by no more than 6%. It is unclear what causes this linear variation, but it is worth mentioning that even though this estimate describes the uncertainty of these points relative to each other, it is an overestimate. There remains some unidentifiable cross-channel dependency.

We introduce a thermal component as well, calculated by fractionalizing each thermal error by its accompanying measurement, averaging across each channel, and denormalizing with the channel average.

$$\begin{eqnarray}&&{\sigma }_{\mathrm{th}}={\left\langle \displaystyle \frac{{\sigma }_{i}}{{F}_{\nu ,i}}\right\rangle }_{\nu }\langle {F}_{\nu ,i}{\rangle }_{\nu }.\end{eqnarray} \tag{ 8 }$$

The final relative uncertainty measurement contains the thermal component σ_th as well as the mean $\langle {\delta }_{\nu ,i}{\rangle }_{\nu }$ and standard deviation ${\langle {({\delta }_{\nu ,i}-\langle {\delta }_{\nu ,i}{\rangle }_{\nu })}^{2}\rangle }_{\nu }^{1/2}$ of the daily deviations by channel. The third term, the variance of the channel's daily deviations, dominates the entire measurement and gives each channel a relative uncertainty on the order of a jansky.

$$\begin{eqnarray}&&{\sigma }_{R}^{2}={\sigma }_{\mathrm{th}}^{2}+\langle {\delta }_{\nu ,i}{\rangle }_{\nu }^{2}+{\langle {({\delta }_{\nu ,i}-\langle {\delta }_{\nu ,i}{\rangle }_{\nu })}^{2}\rangle }_{\nu }.\end{eqnarray} \tag{ 9 }$$

2.1.2. Correction for the Synchrotron Radiation

The relatively large SZA beam (${\theta }_{B}\approx 11^{\prime}$ full width at half max) contains flux from both thermal and non-thermal components. In this frequency regime, synchrotron radiation dominates the non-thermal emission. As we are interested only in the thermal component, we must subtract out the synchrotron radiation from the total observed flux density.

Jupiter's dynamic synchrotron spectrum has been a subject of discussion since the 1970s when a series of observations suggested time variability in the low-frequency⁶ spectrum (Klein 1976). A survey of that spectrum from 74 MHz up to 8 GHz, described by de Pater et al. (2003), suggests that the synchrotron contribution to the planet's radio spectrum drops off above 2 GHz, leading us to believe that, with frequencies around 30 GHz, our measurements contain negligible synchrotron-induced variability. Klein (1976) observes that fluctuations on the order of days did not exceed 10% and explores variability on timescales of several years using 1–3 month averages, which implies that our 5 month average should capture an approximately constant period of synchrotron activity. Under this assumption, we use a simplified and time-independent model to determine the contribution of synchrotron radiation on Jupiter's thermal spectrum. The correction is purely arithmetic and thus does not propagate into uncertainties.

JG uses a value of 1.5 Jy for the synchrotron contribution to a 28.5 GHz measurement of the thermal spectrum, which has a value of about 145 Jy, based on work done by de Pater & Dunn (2003). In order to adjust extant data in the same frequency regime, JG adopts a relationship of ${F}_{\nu ,\mathrm{synch}}\sim {\nu }^{-0.4}$, leading to the local model

$$\begin{eqnarray}&&{F}_{\nu ,\mathrm{synch}}=(1.5\ {Jy}){\left(\displaystyle \frac{\nu }{28.5\mathrm{GHz}}\right)}^{-0.4},\end{eqnarray} \tag{ 10 }$$

which we apply across our small frequency domain.

The synchrotron model (10) is subtracted from the time-averaged flux density at each channel (1), yielding thermal-only flux density measurements:

$$\begin{eqnarray}&&{F}_{\nu ,\mathrm{thermal}}={F}_{\nu ,\mathrm{meas}}-{F}_{\nu ,\mathrm{synch}}.\end{eqnarray} \tag{ 11 }$$

The thermal radiation flux density, F_ν,thermal, is converted to brightness temperature, T_b, via the Planck function. The resulting T_b,meas from a direct conversion is not yet indicative of the true temperature of the emitter—it is the contrast between the emitter and the microwave background. Correction for this is made during conversion, following a similar adjustment made by de Pater et al. (2014). Observed thermal flux density is set equal to a combination of thermal brightness temperature and CMB contribution by the Planck function, as below, allowing T_cmb = 2.725 K.

$$\begin{eqnarray}&&{F}_{\nu }=\displaystyle \frac{2h{\nu }^{3}}{{c}^{2}}\left(\displaystyle \frac{1}{{e}^{h\nu /{{kT}}_{b}}-1}-\displaystyle \frac{1}{{e}^{h\nu /{{kT}}_{\mathrm{cmb}}}-1}\right)\displaystyle \frac{\pi {R}_{\mathrm{eq}}{R}_{p}^{\prime} }{{D}^{2}}\end{eqnarray} \tag{ 12 }$$

where apparent polar radius is given by ${R}_{p}^{\prime} \,=\sqrt{{R}_{\mathrm{eq}}^{2}{\sin }^{2}\phi +{R}_{p}^{2}{\cos }^{2}\phi }$. Subearth latitude used here is ϕ = 015, which is the average over a tight cluster of small, similar ϕ values over the four-month observation interval according to the JPL Horizons interface.

Working values at each major step as well as final measurements and associated error estimates are laid out in Table 1. The T_b values, along with some combination of the σ_A and σ_R uncertainties, are appropriate for reproduction in future work.

In order to investigate whether identifiable systematics are present, we conducted a set of jackknife tests during which we run arbitrarily selected halves of the time-series data through the analysis pipeline and observe the average resulting change in brightness temperatures. The data are halved both on meaningless criteria as well as by criteria with more systematic potential. We applied arbitrary-parameter tests with the data split along odd versus even index, first versus second half, and first and last quarters versus central half. We applied the more meaningful test of sorting along Jupiter's horizontal elevation at the time of observation.

It was mentioned earlier that what may be an airmass calibration error in the raw data was detected by one of these jackknife tests, specifically one using Jupiter's elevation. After this correction, all subsequent jackknife tests amount to nothing more than noise at less than 2% variation from the full-range values, indicating that we have identified all major systematics about which we have information. The consistency of our measurements with the Gibson and WMAP points corroborates this, or at least suggests that we all suffer from the same unknown systematics.

Our calculations use the radiative transfer code most recently used by de Pater et al. (2016a); this code is based upon an atmosphere in thermochemical equilibrium, as described in detail by de Pater et al. (2005). As in de Pater et al. (2016a), we assume for our nominal model that all constituents (NH₃, H₂S, CH₄, and H₂O) are enhanced by a factor of 4.5 above solar in the deep atmosphere (P > 8 bar), as observed by the Galileo probe for NH₃, H₂S, and CH₄ (Folkner et al. 1998; Sromovsky et al. 1998; Mahaffy et al. 1999; Wong et al. 2004). At higher altitudes, NH₃ will be partially dissolved in the water cloud (∼7.3 bar), will form the NH₄SH cloud (∼2.5 bar, de Pater 1990), and at P < ∼0.8 bar will condense into its own ice cloud and follow the saturated vapor curve. In our nominal model, we thus assume an NH₃ abundance of $5.72\times {10}^{-4}$ in Jupiter's deep atmosphere, which is diminished at altitudes at which clouds form. We adopt a 100% humidity within and above the NH₃ ice cloud in our nominal model. This results in a constant abundance of $1.20\times {10}^{-7}$ near and above the tropopause. This NH₃ profile is shown by the blue curve in Figure 3(b), and the resulting spectrum is shown by the blue curves in Figures 4 and 5.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Note from the weighting functions in Figure 3(a) that our data are sensitive to P < ∼3 bar. While our results should, strictly speaking, only apply down to a "deep atmosphere" cutoff defined by this sensitivity, we extend our results down to P < 8 bar in order to remain consistent with de Pater et al. (2016a) under the assumption that NH₃ abundance should remain roughly constant between 3 < P < 8 bar—it should only increase (with increasing pressure) at the clouds described above. Below P > 8 bar, we adopt the values as measured by the Galileo spacecraft (Wong et al. 2004).

Starting with the nominal NH₃ abundance profile described above, we apply to it small, unity-order adjustments, which in turn gives us the ability to generate a range of theoretical spectra. These spectra are compared to the available measurements in order to isolate a NH₃ abundance profile in maximal agreement with observations. The spectra are generated using the radiative transfer software, pyplanet, described by de Pater et al. (2014) for its use on Neptune's atmosphere and most recently updated with the NH₃ line profile from Bellotti et al. (2016). The software ignores potential opacities from clouds.

We begin this process by separating the atmosphere into regions of altitude based on their radiative contribution to our measurements, according to the nominal NH₃ abundance model. These contribution functions, shown in Figure 3(a), are most prevalent between 0.5 < P < 0.8 bar. This layer coincides with the NH₃ ice cloud-forming region, over which altitude range NH₃ follows the saturated vapor curve, as indicated by the nominal abundance profile (blue curve in Figure 3(b)). We apply to this region, defined by its plummeting NH₃ abundance, a constant humidity multiplier RH. Through this parameter, we will tune humidity in the NH₃ cloud-forming region to fit observations.

We similarly treat the regions of constant NH₃ abundance above and below this layer, granting them their own modifying constants. The sub-cloud region, defined as the region of approximately constant NH₃ abundance from the bottom of the NH₃ cloud down to P ∼ 8 bar, is modified by the parameter α_d. As we consider our model not to apply below P > 8 bar, we jump the abundance below this point back to the nominal model, which is motivated by Galileo measurements sensitive to this deeper region. de Pater et al. (2016a) applies this same practice. The region of constant abundance above the tropopause is modified by the parameter α_h.

These parameters, RH, α_d, and α_h, form a three-parameter grid across which we create slightly modified versions of the nominal abundance model. Each element of the grid is run through the radiative transfer software to generate a spectrum, and each of these spectra is compared to the observations such that each point on the three-dimensional grid is associated with a χ² value. We search the parameter space, bounded by a physically reasonable range of unity-order constants, for a global minimum. This minimum is explored to within 0.5% resolution of the nominal value. The abundance profile associated with this minimum is taken to be the profile in maximal agreement with observational evidence. Uncertainty for each parameter is approximated as the region in which the local χ² value is less than 2× the minimum χ² value.

We examine the NH₃ abundance below the cloud-forming region relative to the nominal abundance value of $5.72\times {10}^{-4}$, and find a value of $2.40\times {10}^{-4}$ just above P ∼ 8 bar, with uncertainty bounding it between $[2.26,2.57]\times {10}^{-4}$. Accounting for reductions at clouds, the P ∼ 0.8 bar abundance is $1.89\times {10}^{-4}$. Figure 5 demonstrates the individual contribution of α_d (dashed cyan line) to the final model (red line). This contribution dominates far from the band center, where it fits especially well to the high-frequency WMAP and lower-frequency VLA points, both sensitive to the deeper atmosphere.

Zoom In Zoom Out Reset image size

Considering that ours is a disk-averaged result, these measurements are consistent with results recently presented in Li et al. (2017) using data from the Microwave Radiometer (MWR) experiment on the Juno spacecraft. Latitudinally resolved abundance measurements from the MWR, shown in Figures 3 and 4 of Li et al. (2017), at all latitudes except the more ammonia-rich Equatorial Zone tend toward this same 2–$2.4\times {10}^{-4}$ from the NH₃ cloud at P ∼ 0.8 bar down to about 10 bar, a regime consistent with our sub-cloud partition.

Humidity is examined relative to the saturation curve region in the nominal model, roughly between 0.3 < P < 0.8 bar. The nominal model follows 100% humidity, so our results will be considered relative to a fully saturated model. Best-fit results suggest a humidity of 56.5%, bounded between [50.0%, 63.5%]. The dotted cyan line in Figure 5 shows the individual contribution of RH, which dominates closer to the band center and produces model spectra that are consistent with the lower-frequency CARMA points and the Gibson point that are most sensitive to this pressure range.

JG states that the NH₃ abundance is, on average, sub-saturated by at least a factor of two at P < 0.6 bar. Our results corroborate this factor-of-two sub-saturation within our own pressure regime stated above but add a tighter bound based on three independent sets of measurements.

The high-altitude atmosphere abundance is examined relative to the nominal (saturated vapor curve) value of $1.2\times {10}^{-7}$. Our model fits suggest a value of about $2.8\times {10}^{-8}$, nearly one fifth of the original, but yield an upper bound of $2.4\times {10}^{-7}$, twice the original value. As we are not very sensitive to these pressures, we are not able to provide a lower bound.

This result relies primarily on the lowest frequency WMAP measurement but has some input from the lower-frequency CARMA measurements. It is difficult to place any reasonable bound on the high-altitude atmosphere abundance due to the lack of data near the band center; the Klein & Gulkis measurements were deemed too uncertain for our comparison, but tend toward temperatures higher than the WMAP point. Higher brightness temperatures at the band center would indicate a lower pressure departure from the saturation curve and consequently a smaller high-altitude atmosphere abundance. The precise number found in our analysis is likely meaningless—nevertheless, it is quite possible that the high atmosphere abundance should be smaller than its value in the nominal model.

Our final model is shown by the red dashed lines in Figures 4 and 5, with upper and lower bounds. In Figure 5, as mentioned above, we also show the spectra resulting from α_d and RH only (cyan dashed and dotted lines, respectively).