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date: 25 June 2022

# The Atmosphere of Uranus

• Leigh N. FletcherLeigh N. FletcherUniversity of Leicester

### Summary

Uranus provides a unique laboratory to test current understanding of planetary atmospheres under extreme conditions. Multi-spectral observations from Voyager, ground-based observatories, and space telescopes have revealed a delicately banded atmosphere punctuated by storms, waves, and dark vortices, evolving slowly under the seasonal influence of Uranus’s extreme axial tilt. Condensables like methane and hydrogen sulphide play a crucial role in shaping circulation, clouds, and storm phenomena via latent heat release through condensation, strong equator-to-pole gradients suggestive of equatorial upwelling and polar subsidence, and the formation of stabilizing layers that may decouple different circulation and convective regimes as a function of depth. Phase transitions in the watery depths may also decouple Uranus’s atmosphere from motions within the interior. Weak vertical mixing and low atmospheric temperatures associated with Uranus’s negligible internal heat means that stratospheric methane photochemistry occurs in a unique high-pressure regime, decoupled from the influx of external oxygen. The low homopause also allows for the formation of an extensive ionosphere. Finally, the atmosphere provides a window on the bulk composition of Uranus—the ice-to-rock ratio, supersolar elemental and isotopic enrichments inferred from remote sensing, and future in situ measurements—providing key insights into its formation and subsequent migration.

As a cold, hydrogen-dominated, intermediate-sized, slowly rotating, and chemically enriched world, Uranus could be the closest and best example of atmospheric processes on a class of worlds that may dominate the census of planets beyond our own solar system. Future missions to the Uranian system must carry a suite of instrumentation capable of advancing knowledge of the time-variable circulation, composition, meteorology, chemistry, and clouds on this enigmatic “ice giant.”

### Subjects

• Observational and Experimental Techniques
• Planetary Atmospheres and Oceans
• Planetary Interiors
• Planetary Chemistry and Cosmochemistry

### Introduction

Of all the primary planets in the solar system, Uranus could be considered as the most unusual. Knocked onto its side, potentially by a cataclysmic impact during the epoch of planetary formation and migration (Kegerreis et al., 2019; Safronov, 1966; Stevenson, 1986), the 98º axial tilt subjects the atmosphere to extreme seasons, with each pole spending four decades in the darkness of polar winter, and four decades in perpetual sunlight. Unlike the other giants, Uranus lacks any detectable internal heat source emanating from its deep interior (i.e., no appreciable “self-luminosity”; Pearl et al., 1990), which may be related to the apparent dearth of large-scale meteorological activity and atmospheric mixing. The sluggish vertical mixing results in stratospheric chemistry operating in a higher-pressure regime quite unlike that found on other worlds. Compared to the gas giants Jupiter and Saturn, Uranus (and Neptune) possess substantial super-solar enrichments in heavy elements, such that the resulting volatiles (methane, hydrogen sulphide, ammonia, and water) play a crucial role in the stability and energetics of the weather layer. These gaseous species exhibit strong equator-to-pole gradients, hinting at large-scale tropospheric circulation. Combined with the smaller size (4.2 Earth radii), slower rotation (17 hours), and colder tropospheric temperatures (~50K, sufficient for methane to condense) compared to the gas giants, these factors all have substantial implications for Uranus’s banded structure, circulation, meteorology, and photochemistry. Indeed, Uranus’s atmosphere occupies a unique region of dynamical and chemical parameter space not found elsewhere in the solar system.

Uranus therefore provides a planetary-scale laboratory to test current understanding of atmospheric physics and chemistry under extreme environmental conditions. Comparison to Neptune, the archetype for a seasonal ice giant, reveals how planets can evolve down different evolutionary paths despite their common origins, allowing one to contrast the extreme seasons and negligible internal heat flux of tilted Uranus with the ever-changing storms of Neptune. Taken together, the ice giants are the closest and best examples of a class of “intermediate-sized” astrophysical objects that may be common in the universe. Indeed, they are only slightly larger than the sub-Neptunes that dominate the exoplanetary census (Fulton & Petigura, 2018). Exploration of Uranus and Neptune therefore reveals how atmospheric circulation, planetary banding, and seasonal photochemistry operate on cold, intermediate-sized, chemically enriched, slow-rotating, hydrogen-rich worlds (Fletcher, Helled, et al., 2020; Wakeford & Dalba, 2020). Finally, the basic composition of Uranus—its ice-to-rock ratio, supersolar elemental enrichment, and isotopic composition—provides an essential missing piece in the puzzle of the architecture of the solar system, revealing the location and time scale for Uranus’s accretion (Mousis et al., 2018) and providing insights into its subsequent migration and thermal evolution.

More than 35 years have now passed since the only spacecraft to visit Uranus, Voyager 2, flew past in 1986 on its journey out of the solar system. Our knowledge of ice giant atmospheres was reviewed shortly after the Voyager encounters by Lunine (1993). Since then, observations from ground- and space-based observatories have revealed a picture of Uranus’s atmosphere that is complex, perplexing, and altogether unlike that seen on the gas giants. The reader is directed to several recent reviews that investigate aspects of Uranus’s atmosphere in great depth: the bulk composition and implications for planetary origins (Atreya et al., 2020; Mousis et al., 2018), meteorology (Hueso et al., 2020; Hueso & Sánchez Lavega, 2019), global circulation (Fletcher, de Pater, et al., 2020), and electrical influences (Aplin et al., 2020) on Uranus’s troposphere; stratospheric chemistry (Moses et al., 2020); and upper atmospheric structure (Melin, 2020). Looking to the future, Uranus’s atmosphere will be a key target for orbital and in situ exploration, alongside in-depth studies of the interior, magnetosphere, rings, and icy satellites (Arridge et al., 2012; Fletcher, Helled, et al., 2020; Hofstadter et al., 2019; Mousis et al., 2018).

This article attempts to create a synthesis of these works to review the current state of our understanding of this enigmatic world. “Observations of Uranus” provides a brief observational history of Uranus and describes the basic properties of its atmosphere. “Atmospheric Processes” then explores contemporary knowledge of Uranus’s circulation, meteorology, and chemistry, revealing how the atmosphere may be connected to both the deep, hidden, water-rich interior and the charged environment of the magnetosphere. “Conclusions and Outstanding Questions” closes with the significant outstanding questions that must be addressed by future missions and Earth- and space-based telescopic observations that will explore this extreme ice giant.

### Observations of Uranus

#### A Brief Observational History

Although the colorful cloud patterns of Jupiter and Saturn have been intensely studied for centuries, Uranus’s atmosphere remained stubbornly out of reach until the late 20th century. Faint dusky belts were suggested as early as 1883 by Young and in 1884 by the Henrys (Henry & Henry, 1884; Young, 1883), and can be seen in drawings (Alexander, 1965) by G. Fournier (1913) and R. L. Waterfield (1915–1916). As Uranus was at autumn equinox (planetocentric solar longitude, LS = 180º) in 1882, and at spring equinox (LS = 0º) in 1923, both hemispheres would have been visible at these times, but the curious orientations of the sketched bands left their existence in doubt. Using spectroscopy, Slipher (1904) discovered dark absorption bands in the visible spectrum that were later demonstrated to be signatures of methane (Wildt, 1932). Furthermore, spectral features identified by Kuiper (1949) were found to be due to molecular hydrogen (Herzberg, 1952), such that the basic atmospheric composition (a hydrogen atmosphere enriched in methane) was known long before the space age.

The secrets of Uranus’s atmosphere only really started to be revealed by the Voyager 2 encounter on January 24, 1986, as part of its “Grand Tour,” and this 1970s spacecraft remains the only mission to ever provide close-up views (Lindal et al., 1987; Smith et al., 1986; Tyler et al., 1986). At the time of the flyby, Uranus was just past northern winter solstice (Ls = 271.4º), meaning that the southern polar region was pointing almost directly at the Sun. Visible light images from the Voyager cameras (Smith et al., 1986) in Figure 1 revealed a greenish-blue world because of the high abundance of red-absorbing methane, with faint banding associated with methane condensation clouds, but very few discrete features—only eight features were available to understand the pattern of zonal winds. Early Hubble observations of Uranus in 1994–1995, during northern winter (Ls = 310º) (Karkoschka, 1997) were similarly bland, although near-IR images began to provide more contrast in 1997 (Ls = 319º) (Karkoschka, 1998, 2001a, 2001b). Modern image processing of the Voyager 2 data by Karkoschka (2015) revealed many more southern hemisphere features (Figure 1c), but the perception of Uranus as a featureless planet had already started to take root, albeit unfairly. Subsequent Earth-based near-IR imaging (Figure 1d) exploited strong methane and hydrogen absorption bands in the 1–2.2 $μ$m range to improve contrast, sensing the aerosol distribution (condensate clouds and photochemical hazes) as a function of altitude (e.g., Fry et al., 2012; Sromovsky et al., 2015; and others). Uranus passed northern spring equinox (Ls = 0º) in December 2007 (Figure 2), permitting the first views of the north pole as it emerged into sunlight (de Pater et al., 2011; Sromovsky et al., 2009, 2012). Today, modern equipment and image-processing techniques now allow amateur observers to track prominent storms and bands on Uranus (de Pater et al., 2015). These visible and near-IR observations, summarized by Hueso and Sánchez-Lavega (2019) and described in “Atmospheric Processes,” started to reveal Uranus’s finely banded structure, bright clouds, vortices, and ephemeral storms in exquisite detail (Figure 2).

Interpreting aerosol distributions requires an understanding of atmospheric temperature and composition, which requires observations in the mid-IR, far-IR, sub-millimeter, and radio (Figure 2). The extreme cold and distance has severely limited the ability to explore this wavelength domain. Although Voyager 2 carried a capable thermal-IR spectrometer, the radiance was so low that only the hydrogen–helium continuum in the 25–50 μm range could be used, revealing tropospheric temperature contrasts between cold midlatitudes, associated with upwelling, and the warm equator and pole, potentially due to subsidence (Conrath et al., 1998). Space telescopes like the Infrared Space Observatory (ISO) (Encrenaz et al., 2000), Herschel (Feuchtgruber et al., 2013), and Spitzer (Orton, Fletcher, et al., 2014) provided high-quality spectra from the infrared to the sub-millimeter (Figure 3), but these were disc-integrated and unable to resolve contrasts across Uranus’s atmosphere. The development of ground-based observatories with $>8$ m primary mirrors (Keck, VLT, Gemini, and Subaru) now permits spatially resolved imaging of Uranus’s thermal emission in the 7–25 μm range, sensing both tropospheric (0.1–1.0 bar) and stratospheric (0.1–10 mbar) temperatures (Orton et al., 2015; Roman et al., 2020), albeit with a low signal-to-noise ratio, as shown in Figure 2. Such temperature measurements are supported by occultations of stars in the UV (Bishop et al., 1990). At longer wavelengths, centimeter and millimeter-wavelength arrays like the Karl G. Jansky Very Large Array (VLA) and Atacama Large Millimeter Array (ALMA) (de Pater et al., 2014; de Pater & Gulkis, 1988; Hofstadter & Butler, 2003) provide maps of gaseous contrasts at greater depths (Figure 2), with spatial resolutions now approaching those of the reflected-sunlight images (de Pater et al., 2018; Molter et al., 2020). These microwave observations sense thermal emission down to tens of bars, modulated by the pressure-broadened wings of NH3 and H2S, along with other potential contributions from CH4, CO, H2O, and PH3, although uniquely distinguishing between these contributions remains a challenge (see Figure 3d, de Pater, 1991). Finally, observations of thermospheric emission from H2 and $H3+$ enable studies of the temperatures and circulation of the upper atmosphere (ionosphere and thermosphere).

Taken together, the observations of reflected sunlight and thermal emission in Figure 3 reveal Uranus’s atmosphere in three dimensions, and how it evolves over time, as discussed in “Vertical Structure and Composition.”

#### Vertical Structure and Composition

Before delving into the details of the processes shaping Uranus’s atmosphere, this section first reviews the basic properties of the troposphere and stratosphere, shown in Figure 4. Based on Uranus’s expected formation region beyond the snow line, the bulk of the planet is thought to be made up of 10%–20% hydrogen and helium, plus 80%–90% heavier elements (Hubbard et al., 1995; Podolak et al., 2019). The proportion of rocky refractory materials versus ice-rich materials remains unclear, but the most abundant condensable species are expected to be CH4, H2O, NH3, and H2S. Of these, only CH4 and H2S (Irwin et al., 2018) have been directly measured: methane was first estimated to comprise around 1.6%–2.3% from Voyager radio occultations (Lindal et al., 1987) and ground-based visible quadrupole measurements (Baines et al., 1995), but later estimates are in the 2.7%–3.5% range (Karkoschka & Tomasko, 2009; Sromovsky et al., 2019). Based on these chemical abundances, and the Clausius–Clapeyron equation (Sánchez-Lavega et al., 2004; Weidenschilling & Lewis, 1973), Uranus’s topmost clouds (Figure 5) are expected to be comprised of a thin layer of CH4 ice with a base near 1.3 bars and approximately 80 K, where latent heat released by condensation was observed to modify the temperature lapse rate observed in Voyager radio occultations (Lindal et al., 1987). Cloud sounding via visible and near-IR spectroscopy (Figure 3a) confirms the presence of optically thin hazes in the upper troposphere and stratosphere, above a thin methane ice cloud near 1.3 bars and a thicker H2S cloud in the 2–4 bar range (de Kleer et al., 2015; Irwin et al., 2009; Tice et al., 2013).

H2 and He become more enriched with height as the volatile gases condense out to form clouds, such that the observable upper troposphere is more “dilute” and lacks signatures of H2O and NH3 and is unlikely to be representative of the bulk. The helium mass fraction Y = 0.262 ± 0.048 was reported by Conrath et al. (1987) using Voyager data, which is just slightly less than the protosolar value (0.278) (Lodders et al., 2009).

The methane deep volume mixing ratios of Sromovsky et al. (2019) suggest a bulk carbon enrichment of ~50–85 times the protosolar abundance (Asplund et al., 2009), which is some ten times larger than that seen on the gas giants. However, there remains the possibility that the elements are not uniformly enriched. Radio observations (de Pater, 1991; Gulkis et al., 1978) suggest a significant superabundance of H2S ($37−6+13×$ protosolar) compared to NH3 ($1.4−0.3+0.5×$ solar) (Molter et al., 2020), which is quite different from Jupiter’s approximately solar N/S ratio, and the low abundance of sulphur in a solar-composition mixture. NH3 and H2S are expected to combine to form a solid NH4SH cloud layer in the 30–50 bar region (Atreya & Wong, 2005; Sánchez-Lavega et al., 2004; Weidenschilling & Lewis, 1973), leaving only the most abundant species—H2S—at lower pressures (DeBoer & Steffes, 1994), as shown in Figure 5. The superabundance of H2S has been supported by measurements of microwave opacity of H2S in the lab (DeBoer & Steffes, 1996) and by the direct detection of H2S vapor above the clouds in the H‑band (Figure 3a) near 1.57–1.58 μm (Irwin et al., 2018). Condensation of H2S ice therefore forms an important cloud deck near 2–4 bars, and possibly deeper. But does this sulphur excess imply a formation scenario where H2S accretion was more efficient, potentially via trapping in clathrates (Hersant et al., 2004), or is chemistry in the deeper layers responsible, where some NH3 and H2S will be dissolved in a deep water (solution) cloud, a liquid water ocean at tens of kilobars, and potentially an ionic or superionic ocean at hundreds of kilobars (Atreya et al., 2020)? Even sampling the well-mixed abundances of NH3 and H2S below the NH4SH cloud will be challenging given that NH3 is not well mixed beneath the foreseen cloud base on Jupiter (Bolton et al., 2017).

Observers use tropospheric measurements in the CH4 (~1.3 bar) and H2S (2–4 bars) cloud regions to infer the properties of Uranus’s deeper interior. The upper troposphere is effectively desiccated of other volatiles, so accessing Uranus’s bulk water abundance, and thus the O/H ratio, remains a significant challenge. Both planetary interior models (Guillot, 1995) and measurements of CO (Cavalié et al., 2017; Venot et al., 2020) suggest that the H2O abundance is potentially much larger, placing the water cloud base at hundreds of bars in Figure 5. Not only is this inaccessible to an entry probe, but it is also a challenge for microwave sounding because of the lack of knowledge of the deep temperature structure, as measured thermal profiles from Voyager probe no deeper than 2.3 bars (Lindal et al., 1987). Indeed, the potentially water-rich layers would cause lapse rates to depart substantially from adiabatic behavior (Leconte et al., 2017), lending further complexity to Uranus’s deep temperature profile assumed in Figure 5.

Estimating Uranus’s water content therefore relies on indirect approaches such as using the chemistry of disequilibrium species. Thermochemical equilibrium controls composition in the deeper atmosphere (Fegley & Prinn, 1986), but when the rate of vertical transport is faster than the rate of chemical destruction, the composition can be frozen in at abundances representative of the “quench” point. Thus disequilibrium species can be studied at higher, colder altitudes, but none of the potential species studied by Fegley and Prinn (1986) have been definitively detected. CO, which is present in the stratosphere from external sources (Cavalié et al., 2014), has only an upper limit in the troposphere (Teanby & Irwin, 2013), so Teanby et al. (2020) pointed out that there is no need to bring CO upward from an oxygen-rich interior to explain the measurements. Nevertheless, if CO is present, then Cavalié et al. (2017) estimated that O/H is <160 times the solar value in Uranus, but these estimates have been revised downward to <45 times solar (Cavalié et al., 2020), using a revised chemical and transport scheme (Venot et al., 2020), thus highlighting the potential model dependency associated with this approach. Vertical transport, parameterized through an eddy diffusion coefficient in these chemical schemes, may also depend on latitude, further confusing the picture. Ethane (Fegley & Lodders, 1994) and PH3 could also be used to constrain O/H, but neither have been detected in the troposphere, and only upper limits are available for PH3 from millimeter wavelengths (Moreno et al., 2009; Teanby et al., 2019). And just as CO could aid in measuring the O/H abundance, molecular N2 could aid in the determination of the deep N/H abundance: N2 contributes to the molecular weight and collision-induced continua in the infrared, and although this has been studied on Neptune (Burgdorf et al., 2003; Conrath et al., 1993), it has not been assessed on Uranus, in part because of the expected weak vertical mixing. Uranus’s bulk O/H, N/H, and C/O ratios therefore remain extremely uncertain.

Further insights into the bulk composition come from isotopic ratios. The supersolar D/H ratio in H2, derived from Herschel measurements (Feuchtgruber et al., 2013), suggests accretion of a substantial water-ice-rich component, perhaps in addition to CO-rich ices (Ali-Dib et al., 2014) and rocky materials. Furthermore, if Uranus accreted from amorphous ices, then each element (C, N, and S, and the noble gases Ar, Kr, and Xe) should show similar enrichments with respect to protosolar, as the trapping efficiencies in amorphous ices are similar at low temperatures (Owen et al., 1999). Alternatively, if the volatiles were originally trapped in clathrates, then the trapping efficiency can vary substantially from molecule to molecule (Mousis et al., 2018). Direct or indirect measurements of the elemental abundances in Uranus’s atmosphere are therefore needed to understand which primordial icy and rocky reservoirs contributed to Uranus during its formation (Mousis et al., 2018).

### Atmospheric Processes

Having discussed the basic composition and vertical structure of Uranus’s atmosphere, this section explores the various physical and chemical processes that reshape the temperatures, winds, clouds, and composition as a function of time and location.

#### Atmospheric Circulation

##### Planetary Banding

The zonal, meridional, and vertical motions of Uranus’s troposphere and stratosphere may be studied via observations of zonal winds, temperature, composition, and aerosols, and how these parameters change over time. Contrast-enhanced images in reflected sunlight, such as those in Figures 1 and 2 (Fry et al., 2012; Karkoschka, 2015; Sromovsky et al., 2015), reveal a banded atmosphere, with fine-scale 5º–20º latitudinal lanes of different reflectivity. These contrasts are probably due to variations in aerosol optical depth with latitude, although Karkoschka (2015) suggested that at least one of the bands could have been caused by variations in aerosol absorption, suggestive of different materials in some bands. Unlike Jupiter and Saturn, this banding appears to have no strong correlation with the zonal “Winds and Temperatures,” which means that canonical ice giant “belts” and “zones” cannot be defined in quite the same way (Fletcher, de Pater, et al., 2020). Furthermore, zonal medians of the reflectivity maps (Sromovsky et al., 2015) indicate that these zonal contrasts are not static, but change from observation to observation, potentially because of obscuration of the banded structure by discrete features. However, millimeter (1.3–3.1 mm) observations of Uranus from ALMA in 2017–2018 and centimeter (0.9–10 cm) observations from VLA in 2015 (Figure 2, Molter et al., 2020) suggest subtle thermal banding in the 1–50 bar range that is qualitatively similar to that observed in reflected sunlight in the 1–4 bar range. This suggests that Uranus’s albedo contrasts could be related to thermal and/or compositional contrasts in the troposphere, although the long-term stability and location of the zonal bands remain to be rigorously assessed.

##### Winds and Temperatures

Tropospheric zonal winds are constrained by tracking of discrete cloud features by Voyager 2 (Karkoschka, 2015; Smith et al., 1986), the Hubble Space Telescope (Hammel et al., 2001; Karkoschka, 1998), and ground-based facilities like Keck and Gemini (Hammel et al., 2005b; Sromovsky & Fry, 2005; Sromovsky et al., 2009, 2015). These sources were used to develop a “canonical wind profile” by fitting Legendre polynomials to the measured drift of features (Figure 6c, Sromovsky et al., 2015), although there remain some dispersion and potential temporal variations around these measurements. Most of the tracked cloud features on Uranus are near the 1.3-bar methane condensation level or in the deeper 2–4 bar main clouds of H2S ice (Irwin et al., 2018), but some can reach the 250–600 mbar region in the upper troposphere (Roman et al., 2018; Sromovsky et al., 2007, 2012). Some of the smaller-scale clouds evolve quite rapidly (Irwin et al., 2017), making it difficult to track them over multiple rotations. Nevertheless, Figure 6 shows that the cloud tracking has revealed a single, broad retrograde jet at Uranus’s equator, and a single high-latitude ~260 m/s prograde jet in each hemisphere near 50º–60º, quite unlike the multi-jet circulations of Jupiter and Saturn (see the reviews by Sánchez-Lavega et al., 2019). Note that it is common practice to define Uranus’s westward winds as positive (prograde) and the eastward equatorial jet as negative (retrograde), unlike on the other giants. Furthermore, reconstruction of the gravity field measured by Voyager 2 (Hubbard et al., 1991) out to the fourth-order harmonic (Kaspi et al., 2013) suggests that these tropospheric winds are confined to a layer approximately 1,000 km deep (i.e., down to 2 kbar).

Geostrophic balance suggests a correlation between horizontal temperature gradients and the vertical shear on the zonal winds (Holton, 2004). The thermal banding observed in the microwave in Figure 2 (Molter et al., 2020) is too fine scale to be related to the broad structure of the zonal flow (Sromovsky et al., 2015), but thermal-IR observations by Voyager 2 (Conrath et al., 1998; Flasar et al., 1987) and ground-based facilities (Figure 2, Orton et al., 2015; Roman et al., 2020) reveal larger-scale structure, with cool midlatitudes in the 80–800 mbar range, contrasted with a warmer equator and poles. The tropospheric temperature patterns in Figure 4b appear to be unchanging in the time since Voyager 2 and imply maximum positive windshears near ±15º–30º latitude on the flanks of the equatorial retrograde jet, and maximum negative windshear near the high-latitude prograde jets at ±60º–70º (Fletcher, de Pater, et al., 2020). The windshear is minimal (i.e., close to barotropic, with no wind variability with height) in the ±30º–50º latitude range associated with the midlatitude temperature minima, which is also where the most notable storm activity occurs (see “Uranian Meterology”). Taken together, this suggests that the cloud-top prograde jets of Sromovsky et al. (2015) should decay with altitude, potentially because of frictional damping related to the breaking of vertically propagating waves in the upper troposphere (Flasar et al., 1987). The equatorial winds would be more complicated, with the most rapid decay of the retrograde flow with height occurring away from the equator. Furthermore, existing measurements of latitudinal temperature gradients and zonal winds have relatively poor spatial resolution, so it is reasonable to assume that finer-scale temperature or wind bands exist throughout the troposphere, as suggested by the reflectivity (Figure 6a) and microwave emission maps. Future missions with multi-spectral mapping capabilities are needed to resolve this.

Finally, there are hints in Figure 6 that both the zonal winds (Sromovsky et al., 2015) and the tropospheric temperatures (Conrath et al., 1998; Orton et al., 2015) exhibit a north–south asymmetry and that this asymmetry might be weakening with time. Southern summertime winds at 50ºS–90ºS (Karkoschka, 2015) appear to be very different to corresponding northern latitudes 60ºN–83ºN during spring (Sromovsky et al., 2015), where zonal drift rates adhere to solid-body rotation. Long-term tracking of the winds will be required to understand whether this asymmetry is seasonal or a permanent feature.

##### Composition Contrasts

Besides temperatures, winds, and aerosols, planetary banding is observed in the gaseous composition in Figure 7. Fine-scale banding in microwave emission in Figure 7c (Molter et al., 2020) has already been discussed and can be interpreted in terms of variations in H2S abundance as a function of latitude. The collision-induced continuum measured by Voyager 2 also reveals the latitudinal contrasts in para-hydrogen, the even spin isomer of H2 (Conrath et al., 1998; Flasar et al., 1987; Fletcher, de Pater, et al., 2020). The para-H2 fraction is quenched at 25% in the warmer deep troposphere, and chemical equilibration is sufficiently slow that upward motion can bring this low-para-H2 air upward into the cooler upper troposphere, where the equilibrium para-H2 abundance should be much larger. Sub-equilibrium para-H2 in Figure 7a is therefore indicative of upwelling, which is seen at midlatitudes in the 80–800 mbar region. Conversely, super-equilibrium para-H2 is observed at the equator and poles, which is taken to be an indicator of subsidence from the cold tropopause. Caution is required, though, as poorly understood catalysis processes on aerosol surfaces could cause the equilibration time scale to vary with location (Fouchet et al., 2003; Massie & Hunten, 1982).

Small-scale contrasts in Figure 7b–c are superimposed onto a significant equator-to-pole decline in the volatile abundances CH4 and H2S. Uranus’s south polar depletion of methane was first detected via Hubble 300–1,000 nm spectroscopy in 2002 (Karkoschka & Tomasko, 2009), when the north polar region was still largely hidden from view. After equinox in 2007, further Hubble observations in 2012 (Sromovsky et al., 2014) and 2015 (Sromovsky et al., 2019) revealed that the north polar region exhibited a similar CH4 depletion and that the equator-to-pole gradient from ~4% near the equator to ~2% at high latitudes appeared to be relatively stable over time. Furthermore, this gradient exhibited a steplike structure, with a transition in abundance near the ±30º–40º latitude region, although degeneracies with the assumed aerosol distribution can modify the derived gradients. Uranus’s polar methane depletion was also observed independently from ground-based facilities in the near-infrared, including the IRTF (Tice et al., 2013), Keck (de Kleer et al., 2015), Palomar (Roman et al., 2018), and the VLT and Gemini (Irwin et al., 2018; Toledo et al., 2018). Sromovsky et al. (2019) reproduced the spectra with a methane distribution that was latitudinally uniform for p > 5 bar (in the 2.7%–3.5% range, depending on aerosol scattering properties), and only latitudinally variable at lower pressures in the upper troposphere, with a factor of three decrease in the upper tropospheric CH4 from 30ºN to 70ºN.

However, the implied equator-to-pole circulation, with rising methane-rich air at low latitudes, moving poleward as CH4 is removed by condensation, precipitation, and sedimentation, before methane-poor air sinks at high latitudes, may in fact extend much deeper. VLA centimeter observations of Uranus between 1982 and 1994 (de Pater, 1991; de Pater & Gulkis, 1988; Hofstadter & Butler, 2003), probing down to 50 bars, revealed Uranus’s microwave-bright south polar region due to a strong depletion of absorbers, with a strong brightness gradient near 45ºS (Figure 2). VLA observations since 2003 show that the north polar region was similarly bright (de Pater et al., 2018; Hofstadter et al., 2004; Molter et al., 2020), and this high brightness can be primarily explained by the absence of H2S down to ~35 bars (Molter et al., 2020). Whether this polar depletion of deep H2S is caused by the same circulation and subsidence responsible for the shallow CH4 depletion is an open question (Fletcher, de Pater, et al., 2020; Sromovsky et al., 2014). Furthermore, neither the methane nor the H2S appear to be tracing the upper tropospheric circulation revealed by the para-H2 and temperature distribution, which would suggest midlatitude upwelling rather than equatorial upwelling. Indeed, polar subsidence would tend to inhibit convection and CH4 cloud formation (Sromovsky et al., 2012), and yet discrete clouds were clearly visible at Uranus’s north pole during early northern spring, as seen in Figure 1 (Sromovsky et al., 2014).

#### Meridional Overturning

Uranus’s tropospheric circulation has yet to be explored via detailed numerical simulation, but the differences between (a) the broad contrasts in upper tropospheric temperatures, winds, and para-H2; (b) the fine-scale banding observed in the aerosols and deep thermal emission; and (c) the strong equator-to-pole contrasts in CH4 and H2S imply that circulation cells might be restricted to two or more discrete tropospheric layers (Fletcher, de Pater, et al., 2020; Sromovsky et al., 2015). Similar “stacked circulation cells” have been proposed for Jupiter and Saturn (Fletcher, Kaspi, et al., 2020; Ingersoll et al., 2000; Showman & de Pater, 2005), and it is possible that the stabilizing influence of molecular weight gradients where clouds condense might help in separating these layers, creating stable regions where convection is inhibited (Friedson & Gonzales, 2017; Guillot, 1995; Leconte et al., 2017) and possibly preventing vertically extended convection and circulation entirely. Indeed, thin-layered convection was proposed to explain Voyager observations of Uranus’s lapse rate and para-H2 distribution by (Gierasch & Conrath, 1987). The CH4 and H2S condensation levels may be good candidates for transitional layers, decoupling motions above and below (Hueso & Sánchez-Lavega, 2019). Such differences in atmospheric circulation as a function of depth remain to be proven, but the transition from a deep domain where eddies provide momentum to the zonal jets, to a shallow domain where zonal jets decay with altitude because of unidentified frictional drag (i.e., a radiative- or wave-driven circulation regime) remains a plausible explanation for the multi-spectral observations available today.

Finally, the tropospheric circulation may also have implications for Uranus’s stratosphere. Stratospheric chemistry is discussed in the section “Chemistry,” but photolysis of stratospheric methane produces acetylene, C2H2, which is the only stratospheric chemical to have been mapped on Uranus to date (Roman et al., 2020). Using images at 13.7 μm, Roman et al. (2020) revealed excess C2H2 emission at midlatitudes, counter to the expectations of seasonal photochemical models (Moses et al., 2018). They proposed two equally plausible mechanisms: either tropospheric midlatitude upwelling was transporting methane-rich air through the cold trap and into the stratosphere, where it was subsequently photolyzed to form C2H2; or the stratosphere exhibited subsidence and adiabatic warming at midlatitudes. The first case can be considered as an upward extension of the upper tropospheric circulation, whereas the second case is a circulation where the midlatitude stratospheric subsidence opposes the midlatitude tropospheric upwelling, yet another tier of a stacked circulation system. Distinguishing between these two possibilities is a topic of ongoing research.

#### Uranian Meteorology

##### Convective Storms

Discrete meteorological features, from small bright clouds to planet-encircling waves, giant storms, and dark vortices, are superimposed on and intricately connected to the large-scale “Atmospheric Circulation” described herein. Some of the small-scale bright clouds, presumably comprised of methane ice reaching pressures of 300–600 mbar, high above the main H2S cloud deck at 2–4 bars, may be convective in origin, driven by a combination of latent heat release as CH4 condenses (Hueso et al., 2020; Stoker & Toon, 1989) and potentially by the energy released during the interconversion between para- and ortho-H2, which have different specific heats (Smith & Gierasch, 1995). However, Uranus shares the same problem as the other giant planets, in that air saturated with condensates will be heavier than the surrounding “dry” H2–He mixture (Friedson & Gonzales, 2017; Guillot, 1995; Leconte et al., 2017). As described in “Atmospheric Circulation,” the strong vertical gradient in molecular weight prevents convection extending over tremendous heights, instead limiting it to vertically thin layers (Gierasch & Conrath, 1987). Strong initial perturbations are therefore required to counteract this static stability (effectively a negative buoyancy) to drive self-sustaining CH4-rich updrafts, but this has yet to be verified observationally (Hueso et al., 2020; Hueso & Sánchez-Lavega, 2019). Indeed, Guillot (2019) pointed out that this moist CH4-driven convection is occurring at shallow pressures (0.1–1.5 bars) and low optical depths compared to the deep, hidden H2O clouds of Jupiter and Saturn, meaning that Uranus (and Neptune) provide ideal destinations to investigate how convection operates on hydrogen-rich worlds where it is inhibited by the weight of the condensables. Furthermore, Uranus’s moist convection might be rather different from Jupiter’s, as H2O will play a very limited role in the observable upper troposphere, and H2S condensation provides only a limited source of latent heat to drive convection.

Hueso et al. (2020) defined Uranus’s convective features to be storm features that show divergence above the clouds over short time scales, but they point out that the frequency and distribution of such storms is not well known, and much of the observed cloud activity might not be convective. Instead, some bright features may be orographic structures associated with pressure perturbations at deeper levels (e.g., those associated with dark, hidden vortices), as on Neptune (Stratman et al., 2001). A particularly long-lived feature at southern midlatitudes, some 5,000–10,000 km wide, was known as the “Berg” and had bright features rising to the 550–750 mbar level, but with the main parts of the structure near 1.7–3.5 bars (de Pater et al., 2011). The Berg was captured by HST and Keck between 1994 and 2009 (de Pater et al., 2011, 2015; Hammel et al., 2005b; Sromovsky et al., 2009, 2015), and may even have been observed by Voyager in 1986 (Sromovsky & Fry, 2005). It oscillated in latitude (around 35.2ºS) and longitude, migrating toward the equator after 2005, reaching 27ºS in 2007, and disintegrating when it reached 5ºS in late 2009 (de Pater et al., 2011; Sromovsky et al., 2009). The Berg exhibited intense brightenings in 2004 and 2007, suggestive of convective storms within the feature (de Pater et al., 2015), but there was no evidence that it was related to a deeper unobserved vortex (Hueso & Sánchez-Lavega, 2019).

Discussion of strong convection naturally raises questions about the potential for lightning. Methane is non-polar, but H2S, NH3, NH4SH, and H2O provide the mixed-phase materials that could produce lightning (Aplin et al., 2020). Microphysical modeling suggests that the radio emissions discovered by Voyager (Zarka & Pedersen, 1986) are more likely related to the NH4SH cloud than the deep water layers (Aplin et al., 2020).

##### Discrete Cloud Activity

Uranus has exhibited a range of discrete cloud phenomena since the early Voyager and Hubble observations. Eight features were observed between 35ºS and 70ºS by Voyager (Smith et al., 1986) near northern winter solstice. As winter proceeded in the mid-1990s, notable bright cloud features were observed at mid-northern latitudes (Karkoschka, 1998) as they were emerging into sunlight. Since 2000, ground-based facilities like Keck were frequently detecting cloud features (de Pater et al., 2002; Hammel et al., 2001, 2005b), including some in 2004 that reached sufficiently high altitude to be detected in the strong CH4 band near 2.12 μm (Hammel et al., 2005a). Northern midlatitudes 28ºN–42ºN appear to be intrinsically active (Sromovsky & Fry, 2007), particularly in the years following northern spring equinox (Ls = 0º, 2007), with clouds reaching 300–500 mbar, and new outbreaks observed in 2004–2006 and 2011 (Sromovsky et al., 2007, 2009, 2012). A record-breaking bright storm occurred at 15ºN in 2014, as shown in Figure 2 (de Pater et al., 2015), thought to be formed from a complex of smaller storms. And another at 32ºN possessed a longitudinally elongated tail in the 1–2 bar range, qualitatively similar to storm tails observed on Jupiter and Saturn (de Pater et al., 2015; Irwin et al., 2016, 2017; Sromovsky et al., 2015).

It is natural to ask whether Uranian storms exhibit a temporal dependency given that some of the brightest activity appeared to occur in the hemisphere emerging from the darkness of winter. Alternatively, maybe Uranian storms occur episodically because of powerful outbursts followed by long periods of quiescence as CAPE (convective available potential energy) accumulates, as suggested for Saturn’s annual storms (Li & Ingersoll, 2015). However, storm statistics are currently insufficient, and the observations are biased to the sunlit hemisphere, so robust assessment of the destabilizing influence of increased insolation remains to be performed. The one exception is in the polar domain in Figure 1, where the south pole displayed no discrete activity during southern summer (Karkoschka, 2015; Smith et al., 1986), whereas clusters of bright spots with 600–800 km diameters were observed in the north polar region after it emerged into sunlight in northern spring (Sromovsky et al., 2012, 2015). This notable asymmetry suggests some form of convective inhibition (or obscuration by overlying hazes) developing as spring turns to summer, something that will be testable as Uranus approaches northern summer solstice (Ls = 90º) in 2030.

##### Vortices and Waves

The vast majority of discrete features tracked on Uranus have been bright cloud features, rising above the main cloud decks. However, Uranus also infrequently exhibits dark anticyclonic ovals, which can form and dissipate on the time scales of years. Hubble was the first to observe a dark vortex in 2006 at 28ºN (Hammel et al., 2009), and since that time several more have been observed (e.g., Sromovsky et al., 2015). These are typically smaller than those observed on Neptune, and whether they are accompanied by bright (and more readily visible) companion clouds due to air being forced up and over the vortex (orographic clouds, Stratman et al., 2001) remains a topic of ongoing exploration. It is not clear whether the dearth of anticyclonic ovals on Uranus with respect to Neptune is an observational bias (e.g., because of different overlying aerosols and gas absorption) or a real difference between the two worlds.

Finally, Uranus exhibits wave phenomena that manifest as longitudinal reflectivity contrasts in the troposphere. A chain of diffuse bright features just north of the equator, first suggested by Keck images in 2003 (Hammel et al., 2005b), were observed by Keck in 2012 (Sromovsky et al., 2015) to be spaced every 30º–40º longitude. A second wave was captured by the 2012–2014 Keck observations, using adaptive optics and derotation techniques (Figure 6a, Fry et al., 2012; Sromovsky et al., 2015), revealing a transverse “scalloped” equatorial wave pattern just south of the equator, with diffuse bright features every 19º–21º longitude (wavenumbers 17–19) and a latitudinal amplitude of 2.4º–2.9º. Sromovsky et al. (2015) suggested that Kelvin waves or mixed internal gravity-Rossby waves may be at work, but this cannot be properly characterized until the dispersion relation (phase speed versus wavelength) is determined.

Wave phenomena are unlikely to be restricted to Uranus’s troposphere and may also be present in the stratosphere, modulating emission from stratospheric hydrocarbons (Roman et al., 2020) and generating rotational variability in disc-averaged mid-IR spectra observed by Spitzer (Rowe-Gurney et al., 2021).

#### Chemistry

Although the thermochemistry and condensation chemistry of Uranus’s troposphere were described in “Observations of Uranus,” the influence of UV photolysis, coupled with the “Atmospheric Circulation,” can contribute to the sources, sinks, and spatiotemporal distributions of chemicals detected in Uranus’s upper troposphere and stratosphere. Although much of this section is devoted to stratospheric chemistry, photolysis of the yet-to-be detected PH3 (Moreno et al., 2009; Teanby et al., 2019) and H2S lofted to high altitude may contribute to upper tropospheric hazes, and potentially more so than on Jupiter and Saturn because of the absence of NH3 to act as a UV shield (Moses et al., 2020; Visscher & Fegley, 2005).

##### Methane Photochemistry

Methane photolysis products dominate the stratospheric composition of both ice giants (Atreya et al., 1991), but the strength of vertical mixing by eddy diffusion (weak on Uranus, strong on Neptune) has stark consequences for the distribution of species, with Uranian methane photochemistry operating in a higher-pressure regime than on any other giant planet (Moses et al., 2018). The lower CH4 homopause on Uranus (Herbert et al., 1987) has the surprising consequence that seasonal contrasts are more muted than on Neptune, and that CH4 chemistry is not as well coupled to exogenic oxygenated materials deposited at lower pressures (see “Coupling to External Oxygen”). Nevertheless, photochemistry on Uranus results in a complicated mix of hydrocarbons (Atreya & Ponthieu, 1983; Dobrijevic et al., 2010; Moses et al., 2005, 2018.) that can be investigated via mid-infrared remote sensing (Encrenaz et al., 1998; Feuchtgruber et al., 1997; Orton et al., 1987; Orton, Fletcher, et al., 2014; Roman et al., 2020; Rowe-Gurney et al., 2021) and UV occultations (Bishop et al., 1990; Herbert et al., 1987). Acetylene (C2H2) was first discovered by the Infrared Space Observatory (ISO) (Encrenaz et al., 1998); ethane (C2H6), methyl-acetylene (C3H4), and diacetylene (C4H2) were observed by Spitzer (Burgdorf et al., 2006; Orton, Moses, et al., 2014). Some products, such as ethylene, propane, benzene, and methyl, have not yet been detected (Moses et al., 2020). The effects of the high-pressure photochemistry are apparent in mid-IR observations: Uranus’s hydrocarbons are confined to altitudes below the ~0.1 mbar level in Figure 4, and the ratio of ethane to acetylene is very different on Uranus compared to all the other giants (Orton, Moses, et al., 2014).

The latitudinal distribution of these species is expected to vary due to stratospheric circulation and seasonal photochemistry (Figure 8). Although photochemistry occurs primarily in UV sunlight (145 nm), and thus photolytic rates will be largest at polar latitudes that receive a higher annual average solar insolation than the equator, small amounts of solar Lyman alpha scattered from hydrogen in the local interstellar medium can provide a secondary photolysis source during the darkness of winter. Chemical abundances respond quickly at low pressures to changes in the UV flux, but Figure 8 shows that contrasts should decrease at higher pressures, where diffusion and chemical time scales increase, producing subtle hemispheric asymmetries at p < 1 mbar, with phase lags in response to insolation changes (Moses et al., 2018). Thus short-lived species like acetylene, methyl-acetylene, and diacetylene are expected to have maximum abundances at the summer or autumn poles, whereas long-lived species like ethane are expected to peak near the equator because photolytic destruction effectively competes with production in the high-latitude summer. However, these predictions assume uniform distributions of stratospheric CH4 as the source material, which may not be the case given the strong equator-to-pole CH4 gradients in the troposphere (see “Radiative Balance”). Furthermore, there remains some disagreement about what the stratospheric CH4 abundance actually is, with a factor of six in stratospheric abundance between Spitzer results in 2007 (Orton, Moses, et al., 2014) and Herschel results in 2011 (Lellouch et al., 2015). Both inferences were from disc-integrated spectra, so this could be a sign of changing viewing geometry, latitudinal temperature gradients, changes in efficiency of vertical mixing, temporal variability, or some combination of the above.

Of these species, only emission from C2H2 has been mapped across Uranus, showing stark discrepancies from the photochemical model predictions (Moses et al., 2018; Roman et al., 2020), and implying a key role for circulation: either strong midlatitude upwelling of CH4-rich air from the troposphere; or midlatitude subsidence in the stratosphere (see “Atmospheric Circulation”). The ground-based VLT maps of stratospheric C2H2 emission in Figure 2 suggest a distinct equatorial minimum at latitudes <±25º (Roman et al., 2020), counter to photochemical expectations, and the complete opposite of what is observed in the tropospheric temperatures.

The products of CH4 photochemistry, such as benzene, acetylene, ethane, and propane (in addition to exogenic H2O and CO2) can condense to form thin haze layers in the 0.1–30 mbar range (Figures 4 and 5), some of which are visible in high-phase imaging from Voyager (Moses & Poppe, 2017; Rages et al., 1991; Romani et al., 1993; Toledo et al., 2018), and may be modulated by vertically propagating waves. The condensed hydrocarbons could also sediment downward to coat tropospheric hazes or serve as cloud-condensation nuclei for CH4 and H2S condensation at higher pressures.

##### Coupling to External Oxygen

Species that contain oxygen, like CO, CO2, and H2O, are present in the upper stratospheres of each of the giant planets (Feuchtgruber et al., 1997; Moses & Poppe, 2017), originating from cometary impacts, satellite debris, and ablation of interplanetary dust and ring particles. What makes Uranian photochemistry particularly interesting is that this oxygen deposition is separated from the methane photochemistry because of the sluggish vertical mixing, making it an intriguing counterpoint to the other giants (Moses & Poppe, 2017). Uranus’s stratospheric water was detected by ISO (Feuchtgruber et al., 1997); CO from the fluorescent emission in the infrared (Encrenaz et al., 2004) and sub-millimeter emission (Cavalié et al., 2014); and CO2 from Spitzer (Burgdorf et al., 2006; Orton, Fletcher, et al., 2014). Photolysis of CO and CO2 can lead to secondary peaks of hydrocarbon production at high altitudes above the levels where the CH4 abundance is dropping away (Moses et al., 2018), whereas H2O will condense to form a 10 mbar ice haze. An interplanetary dust particle source is sufficient to explain the observed amount of CO, CO2, and H2O in Uranus’s stratosphere (Moses & Poppe, 2017), although cometary impacts could also contribute (Lara et al., 2019).

The vertical distribution of hydrocarbons, particularly CH4, C2H2, and C2H6, determine the radiative heating and cooling in Uranus’s middle atmosphere. For this reason, the presence of thin hazes and secondary peaks of hydrocarbon production can have a significant influence. Uranus’s upper troposphere and stratosphere are warmed by shortwave absorption by CH4 and aerosols, and cooled by longwave emission from C2H2 and C2H6 in the stratosphere and from the collision-induced H2–He continuum in the troposphere (Conrath et al., 1998; Li et al., 2018). However, it has long been noted that absorption of weak sunlight by CH4 alone cannot balance the efficient cooling from H2 and, to a lesser extent, the hydrocarbons, so the middle atmosphere is warmer (Orton, Fletcher, et al., 2014) than expected. Additional sources of heat from vertically propagating gravity waves or the radiative contribution of hazes have been invoked to explain the discrepancy between observations and predictions (Li et al., 2018).

Uranus’s extreme obliquity, resulting in a larger irradiance at the poles than the equator, could also yield seasonal temperature asymmetries. However, Uranus’s radiative time constant in the troposphere is longer than the Uranian year (Conrath et al., 1990; Friedson & Ingersoll, 1987), such that tropospheric temperatures are not expected to change significantly with time, and to track the annual average equilibrium values (Lunine, 1993). This is consistent with the lack of observed tropospheric temperature variation between Voyager (Figure 6) and the present day (Orton et al., 2015; Roman et al., 2020). Stratospheric temperature asymmetries have never been measured because of the extreme cold and resulting low mid-IR radiance, but Uranus’s stratospheric radiative time constant is actually longer than that on Neptune, due to the low CH4 abundance and cold temperatures, such that stratospheric temperature asymmetries might be weaker on Uranus and again simply track the annual average (i.e., warmer at the poles and cooler at the equator). Future missions capable of measuring temperature contrasts between the summer and winter hemisphere are needed to address this question.

#### Temporal Change

A theme running through this discussion of Uranus’s circulation (see “Atmospheric Circulation”), meteorology (see “Uranian Meteorology”), and chemistry (see “Chemistry”) is that one can learn more about a system by observing how it evolves with time. Given that resolved observations of Uranus’s atmosphere spans almost four decades, and photometric observations span even longer, Uranus is the ideal test bed for studying variability on an ice giant. Previous sections have described three time-variable aspects of Uranus’s atmosphere. First, predictions of radiative-climate models and diffusive photochemistry models suggest that upper tropospheric and stratospheric temperatures and hydrocarbon distributions might show subtle hemispheric asymmetries that reverse as seasons progress, but that the relevant time constants are so long that such contrasts are expected to be small. Nevertheless, subtle midlatitude asymmetries in tropospheric temperatures (Conrath et al., 1998) and winds (Sromovsky et al., 2015) have been observed. Second, high-latitude winds were found to be different between northern spring in 2012–2014 (where a broad region of solid-body rotation from 62ºN–83ºN was identified, Sromovsky et al., 2015) and southern summer in 1986 (with a much smaller region of solid-body rotation, Karkoschka, 2015). Whether this is a permanent asymmetry or something that will change as northern summer solstice approaches in 2030 remains to be seen. Third, there is evidence that small-scale cloud and storm activity varies over time, potentially as a result of the changing atmospheric stability as the seasons change, and this is particularly notable in the polar domain (Sromovsky et al., 2012).

These three time-variable processes are dwarfed by long-term variations associated with Uranus’s seasonal polar cap and collars (Figure 9), which modulate the disc-integrated photometry as seen from Earth (Lockwood, 2019). Voyager observations during southern summer revealed a reflective polar cap poleward of 45ºS, originally thought to be due to an increase in the optical depth of the CH4 clouds near 1.2–1.3 bars (Rages et al., 1991). In the years after Voyager, Hubble observations between 1994 and 2002 revealed a darkening of the south pole (Figure 9), the formation of a bright ring near 70ºS and a south polar collar at 45ºS (Rages et al., 2004). As the northern hemisphere came into view after 2007, the south polar collar diminished (Irwin et al., 2010; Roman et al., 2018), and a north polar collar developed at 45ºN (Irwin et al., 2012), with a bright “north polar cap” observed after 2014 (de Pater et al., 2015; Sromovsky et al., 2015; Toledo et al., 2018). By computing photometric brightness from Hubble imaging between August 1994 and October 2015, Karkoschka (2016) showed that both the changing view of Uranus in Figure 9 and the real physical changes were modulating the brightness: darkening of high southern latitudes, and brightening of high northern latitudes.

What could be causing these changes at high latitudes? CH4 is strongly depleted at the poles, so there is less absorption (see “Composition Contrasts”), and although Toledo et al. (2018) suggested that temporal evolution of this “methane hole” could be responsible, Sromovsky et al. (2019) revealed that the polar CH4 was relatively stable over time, and suggested that increased scattering from seasonally changing aerosols is required. There are problems with this, as the polar aerosols appear to be changing over short time scales in Figure 9, so cannot be explained via long radiative (Conrath et al., 1990), photochemical (Moses et al., 2018), or microphysical processes (aerosol accumulation and sedimentation are slow processes that would produce a substantial seasonal lag, Toledo et al., 2019). It seems likely that tropospheric and stratospheric circulation must be playing a role (Fletcher, de Pater, et al., 2020), again highlighting the complexity of the coupled circulation, chemistry, and clouds of Uranus’s atmosphere.

Long-term monitoring of visible albedo (472 nm and 551 nm) between 1972 and 2016 (Hammel et al., 2007; Lockwood, 2019; Lockwood & Thompson, 1999) and broadband B and V photometry from 1950 to 1966 (Lockwood & Jerzykiewicz, 2006) are consistent with the geometric effects as the bright polar caps and collars come in and out of view, but also suggest more subtle, secular variations in brightness. In particular, Uranus’s reflectivity is related to the 11-year solar cycle, both through changing levels of UV (which could be modulating the colors of aerosols via a “tanning” process, Baines & Smith, 1990), and also through the ion-induced nucleation generated by galactic cosmic rays (Aplin & Fischer, 2017; Aplin et al., 2020). These studies highlight the value of long-term Earth-based observations to provide temporal context to short-lived spacecraft missions.

#### Connections

##### Connection to the Interior

Although this article is focused on Uranus’s atmosphere, it cannot be considered in isolation from the deeper interior, and the external charged particle environment. It is not immediately clear where an ice giant atmosphere (molecular envelope) ends and its interior or icy mantle begins. Differential rotation due to zonal winds appears to be restricted to the outermost 1,000 km of Uranus’s 25,559 km equatorial radius (i.e., the outer 4%, down to ~2 kbar, Kaspi et al., 2013). Deeper down at ~20 kbar and 1,200 K, it is possible that H2O is insoluble in H2 (Bali et al., 2013), leading to the formation of a liquid water ocean at tens of kilobars (Bailey & Stevenson, 2015). This immiscibility can lead to sharp interfaces, or an ocean surface, within the interior (Bailey & Stevenson, 2021). This ocean would transition from being non-conducting to ionic or superionic at hundreds of kilobars and could be responsible for removing NH3 by dissolution to explain the low N/S ratio detected in the atmosphere (see “Observations of Uranus”). The generation of Uranus’s internal dynamo (Ness et al., 1986) may be associated with convection in the partially dissociated fluid water layers (Redmer et al., 2011; Soderlund et al., 2013), and appreciable conductivity can be achieved at 20%–30% of the radii below the surface (Soderlund & Stanley, 2020). Dynamo simulations predict large-scale circulation in this watery layer, with equatorial upwelling and, depending on the thickness of the convecting layer, polar circulation cells (Soderlund et al., 2013), although it is unclear how this might relate to the circulation in the molecular envelope in “Atmospheric Circulation.” At even greater depths, high-pressure laboratory experiments suggest that water could form a thick superionic “icy mantle,” creating a solid-phase viscous lattice of oxygen ions surrounded by a sea of free hydrogen (Millot et al., 2018, 2019; Wilson et al., 2013).

These substantial transitions between exotic phases of matter in Uranus’s interior could lead to inhibition of convection and retention of interior heat (e.g., Hubbard et al., 1995), significantly influencing the thermal evolution of Uranus (Fortney et al., 2011). Indeed, different rates of hydrogen–water demixing, due to H2O immiscibility and the formation of a water ocean, could explain the very different heat flow on Uranus compared to Neptune (Bailey & Stevenson, 2021). Furthermore, the presence of these transitions could successfully decouple motions in the observable “dry” atmosphere from those in the deeper, water-rich interior oceans. However, Helled et al. (2020) provided a cautionary note that the bulk oxygen content of Uranus is still unknown, to the extent that it might instead be dominated by rocky materials rather than water, and implying that the oft-used term “ice giant” may be misleading.

##### Connection to the Exterior

The circulation and chemistry of the middle atmosphere (e.g., the stratosphere) will also be intricately connected to processes shaping the upper atmosphere (e.g., ionosphere or thermosphere), above the homopause where hydrogen dominates. Weak solar heating alone cannot explain the high temperatures of Uranus’s thermosphere (Herbert et al., 1987), a deficit known as the “energy crisis,” and contributions from auroral heating and potentially the breaking of vertically propagating waves have been invoked to close the gap between observations and expectations. Extreme UV radiation, or impact ionization from the aurora, produces the $H3+$ ion, which was first detected on Uranus in 1992 (Trafton et al., 1993) and has been monitored ever since, revealing a long-term cooling of Uranus’s thermosphere over three decades (Melin et al., 2019). That $H3+$ production appears to be efficient on Uranus, but not Neptune, may be related to the weak atmospheric mixing and low homopause height. The thermospheric cooling trend appears to be decoupled from Uranus’s geometric season (i.e., it did not change after the 2007 equinox) but might be explained by a “magnetic season:” differing amounts of Joule heating at different points in the planet’s orbit, changes to the homopause height with time, or a hemispherically asymmetric homopause that changes the $H3+$ density on the hemisphere visible from Earth (see Melin, 2020, for a review). Whether this redistribution of energy in Uranus’s upper atmosphere has any implications for the energy balance, circulation, and chemistry of the stratosphere remains to be seen.

Auroral emission has been observed via excitation of H Lyman-α and H2 bands due to precipitating energetic particles (Lamy, 2020). These are located at magnetic poles tilted from the rotational axis by ~60º, as observed by Voyager UVS (Herbert et al., 1987) and later Hubble (Lamy et al., 2012, 2017). Auroral brightening has also been tentatively detected in the infrared via $H3+$ emission (Thomas et al., 2020). On Jupiter ion-neutral chemistry related to the aurora can lead to a unique balance of stratospheric chemicals and aerosols in the polar domain, so future observations could test whether this is also true on Uranus, given the midlatitude location of the magnetic poles.

### Conclusions and Outstanding Questions

This article should dispel any remaining myths about Uranus’s atmosphere held over from the first Voyager images, revealing it to be an excellent laboratory for testing our understanding of the processes shaping planetary atmospheres. Indeed, Uranus’s atmosphere occupies a unique regime in parameter space:

It is an intermediate-sized world rotating at a rate slower than Jupiter’s but faster than Earth’s, with implications for the banded structure (see “Atmospheric Circulation”).

It has a hydrogen-rich molecular envelope with a bulk that is substantially enriched in heavy materials (volatiles and rocks) and strong equator-to-pole gradients in condensables (see “Observations of Uranus” and “Atmospheric Circulation”).

It has unusual seasonal effects due to both the extreme axial tilt, weak sunlight, and negligible internal heat, perhaps sequestered by strong layering associated with phase transitions in watery oceans and icy mantles at tens and hundreds of kilobars (see “Temporal Change” and “Connections”).

Its low temperatures and weak vertical mixing mean that photochemistry occurs in a high-pressure, low-temperature regime, and a low homopause permits the formation of an extensive ionosphere (see “Chemistry”).

The small-scale cloud phenomena, episodic outbursts, equatorial waves, and moist convective activity operate at shallow and accessible depths, potentially driven by CH4 condensation and ortho/para-H2 interconversion (see “Uranian Meteorology”).

The stabilizing effects of phase transitions imply layered tiers of circulation patterns and convection, potentially decoupled from one another as a function of depth (see “Atmospheric Circulation” and “Uranian Meteorology”).

The above list is not intended to be exhaustive but provides numerous compelling reasons for future exploration of Uranus, both as a fascinating object in its own right but also as one of the closest and best examples of a class of worlds that might be commonplace, being only slightly larger than the “sub-Neptunes” that appear to dominate the census of exoplanets (Fulton & Petigura, 2018). Uranus and Neptune will be the last remaining class of solar system planets to have a dedicated orbital explorer, and international efforts are underway to develop an ambitious mission to Uranus in the coming decades, combining an orbital tour with in situ descent probes (Arridge et al., 2012; Fletcher, Helled, et al., 2020; Fletcher, Simon, et al., 2020; Hofstadter et al., 2019; Mousis et al., 2018; Simon et al., 2020).

Such a mission would be transformative for our understanding of Uranus, in the same way as Cassini, Galileo, and Juno transformed our understanding of the gas giants. Key questions to be addressed by such a mission, supported by Earth- and space-based multi-spectral remote sensing, are as follows:

What is the bulk composition of Uranus’s interior, particularly the ice-to-rock ratio, elemental abundances (including noble gases), and isotopic ratios? How well does the atmospheric composition represent the bulk, and how do water-rich oceans or mantles influence the observed composition? These will provide crucial constraints on Uranus’s formation and migration by revealing which reservoirs were available to the forming Uranus.

What are the dynamical, meteorological, and chemical impacts of the negligible planetary luminosity, weak vertical mixing, and Uranus’s extreme seasons, and how do atmospheric phenomena differ between Uranus and Neptune?

What is the large-scale circulation of Uranus’s atmosphere; how deep does it penetrate into the interior; is it coupled to interior motions; and do stable layers produce decoupled tiers of circulation and convection?

What is the role of moist convection and precipitation in Uranus’s hydrogen-dominated atmosphere, and how do updrafts overcome the static stability of the cloud bases?

Does Uranus really have no appreciable internal heat, or is this merely trapped or time variable? If so, what does this imply about Uranus’s thermal evolution, and why does it differ from Neptune’s?

What are the sources of energy responsible for heating the middle and upper atmosphere of Uranus, and how does the ionosphere couple the atmosphere to the external magnetosphere?

How does Uranus’s atmosphere evolve with time? Is the banded structure relatively stable, or do the winds, cloud features, and albedo patterns shift over time? How do vortices form, migrate, and interact in Uranus’s atmosphere?

To date, progress in addressing these questions has been hampered by the extreme challenge of observing Uranus in the decades since Voyager. Nevertheless, this review demonstrates how far science has come in developing an understanding of this unusual world, and whets one’s appetite for exciting new discoveries in the coming decades.

### Acknowledgments

Fletcher is supported by a European Research Council Consolidator Grant (under the European Union’s Horizon 2020 research and innovation program, grant agreement No. 723890) at the University of Leicester. I would like to extend my thanks to Michael Roman for providing the artwork for Figure 4, Ricardo Hueso for Figure 5, Julianne Moses for providing the model data plotted in Figure 8, Glenn Orton for providing guidance for spectroscopy in Figure 3, and Helmut Feuchtgruber and Martin Burgdorf for allowing me to show the ISO observations in Figure 3.


text:
# Revision history [back]

### Customizing the wireframe / strides of a 3D plot

I am experimenting with 3D plotting to see how I can customize the output. In particular, I am trying to obtain something more similar to how Mathematica or Wolfram Alpha do things.

For example, I plotted Taubin's heart surface:

$$\left(x^2+\frac{9 y^2}{4} + z^2 - 1\right)^3-x^2 z^3 - \frac{9 y^2 z^3}{80} = 0$$

Using the following code:

var("x y z")

f = (x^2 + 2.25*y^2 + z^2 - 1)^3 - (x^2)*(z^3) - 0.1125*(y^2)*(z^3)
xint = (x, -1.5, 1.5)
yint = (y, -1.5, 1.5)
zint = (z, -1.5, 1.5)

my_plot = implicit_plot3d(f, xint, yint, zint, opacity=.7, mesh=2)
my_plot.show()

http_i.imgur .com/DbLGA50.png

The mesh corresponds to the triangulation of points that Sage performs on the data calculated with implicit_plot3d. However, Wolfram Alpha gives a different result (https_www.wolframalpha .com/input/?i=heart+surface), because it does not use all the edges of the triangulation:

http_i.imgur .com/VZtyWql.gif

What I think they are doing is they are using a high number of points to draw the surface, however they are "slicing" the object with planes parallel to the three axes (at fixed distances) to draw the contours.

How can I replicate these lines?

Matplotlib calls them "strides" as documented for Axes3d.plot_surface (http_matplotlib .org/mpl_toolkits/mplot3d/tutorial.html#surface-plots). My final goal is exporting 3D vector drawings which I managed to do with Matplotlib thanks to the numpy-stl package (https_github .com/WoLpH/numpy-stl):

my_plot.triangulate()
my_plot.save("/tmp/test.stl")
from mpl_toolkits import mplot3d
from matplotlib import pyplot

# Create a new plot
figure = pyplot.figure()
axes = mplot3d.Axes3D(figure)

# Load the STL files and add the vectors to the plot
your_mesh = mesh.Mesh.from_file('/tmp/test.stl')

# Auto scale to the mesh size
scale = your_mesh.points.flatten(-1)
axes.auto_scale_xyz(scale, scale, scale)

# Show the plot to the screen
pyplot.show()
pyplot.savefig("/home/andrea/Scrivania/wow.pdf")

In this case the presence of too much grid lines looks even worse:

http_i.imgur .com/s0LniGy.png

 2 No.2 Revision calc314 4161 ●21 ●48 ●112

### Customizing the wireframe / strides of a 3D plot

I am experimenting with 3D plotting to see how I can customize the output. In particular, I am trying to obtain something more similar to how Mathematica or Wolfram Alpha do things.

For example, I plotted Taubin's heart surface:

$$\left(x^2+\frac{9 y^2}{4} + z^2 - 1\right)^3-x^2 z^3 - \frac{9 y^2 z^3}{80} = 0$$

Using the following code:

var("x y z")

f = (x^2 + 2.25*y^2 + z^2 - 1)^3 - (x^2)*(z^3) - 0.1125*(y^2)*(z^3)
xint = (x, -1.5, 1.5)
yint = (y, -1.5, 1.5)
zint = (z, -1.5, 1.5)

my_plot = implicit_plot3d(f, xint, yint, zint, opacity=.7, mesh=2)
my_plot.show()

http_i.imgur .com/DbLGA50.png

The mesh corresponds to the triangulation of points that Sage performs on the data calculated with implicit_plot3d. However, Wolfram Alpha gives a different result (https_www.wolframalpha .com/input/?i=heart+surface), (https://www.wolframalpha.com/input/?i=heart+surface), because it does not use all the edges of the triangulation:

http_i.imgur http://i.imgur .com/VZtyWql.gif

What I think they are doing is they are using a high number of points to draw the surface, however they are "slicing" the object with planes parallel to the three axes (at fixed distances) to draw the contours.

How can I replicate these lines?

Matplotlib calls them "strides" as documented for Axes3d.plot_surface (http_matplotlib .org/mpl_toolkits/mplot3d/tutorial.html#surface-plots). (http://matplotlib.org/mpl_toolkits/mplot3d/tutorial.html#surface-plots). My final goal is exporting 3D vector drawings which I managed to do with Matplotlib thanks to the numpy-stl package (https_github .com/WoLpH/numpy-stl):(https://github.com/WoLpH/numpy-stl):

my_plot.triangulate()
my_plot.save("/tmp/test.stl")
from mpl_toolkits import mplot3d
from matplotlib import pyplot

# Create a new plot
figure = pyplot.figure()
axes = mplot3d.Axes3D(figure)

# Load the STL files and add the vectors to the plot
your_mesh = mesh.Mesh.from_file('/tmp/test.stl')

# Auto scale to the mesh size
scale = your_mesh.points.flatten(-1)
axes.auto_scale_xyz(scale, scale, scale)

# Show the plot to the screen
pyplot.show()
pyplot.savefig("/home/andrea/Scrivania/wow.pdf")

In this case the presence of too much grid lines looks even worse:

http_i.imgur http://i.imgur .com/s0LniGy.png

### Customizing the wireframe / strides of a 3D plot

I am experimenting with 3D plotting to see how I can customize the output. In particular, I am trying to obtain something more similar to how Mathematica or Wolfram Alpha do things.

For example, I plotted Taubin's heart surface:

$$\left(x^2+\frac{9 y^2}{4} + z^2 - 1\right)^3-x^2 z^3 - \frac{9 y^2 z^3}{80} = 0$$

Using the following code:

var("x y z")

f = (x^2 + 2.25*y^2 + z^2 - 1)^3 - (x^2)*(z^3) - 0.1125*(y^2)*(z^3)
xint = (x, -1.5, 1.5)
yint = (y, -1.5, 1.5)
zint = (z, -1.5, 1.5)

my_plot = implicit_plot3d(f, xint, yint, zint, opacity=.7, mesh=2)
my_plot.show()

http_i.imgur .com/DbLGA50.png

The mesh corresponds to the triangulation of points that Sage performs on the data calculated with implicit_plot3d. However, Wolfram Alpha gives a different result (https://www.wolframalpha.com/input/?i=heart+surface), because it does not use all the edges of the triangulation:

http://i.imgur .com/VZtyWql.gif

What I think they are doing is they are using a high number of points to draw the surface, however they are "slicing" the object with planes parallel to the three axes (at fixed distances) to draw the contours.

How can I replicate these lines?

Matplotlib calls them "strides" as documented for Axes3d.plot_surface (http://matplotlib.org/mpl_toolkits/mplot3d/tutorial.html#surface-plots). My final goal is exporting 3D vector drawings which I managed to do with Matplotlib thanks to the numpy-stl package (https://github.com/WoLpH/numpy-stl):

my_plot.triangulate()
my_plot.save("/tmp/test.stl")
from mpl_toolkits import mplot3d
from matplotlib import pyplot

# Create a new plot
figure = pyplot.figure()
axes = mplot3d.Axes3D(figure)

# Load the STL files and add the vectors to the plot
your_mesh = mesh.Mesh.from_file('/tmp/test.stl')

# Auto scale to the mesh size
scale = your_mesh.points.flatten(-1)
axes.auto_scale_xyz(scale, scale, scale)

# Show the plot to the screen
pyplot.show()
pyplot.savefig("/home/andrea/Scrivania/wow.pdf")

In this case the presence of too much grid lines looks even worse:

http://i.imgur .com/s0LniGy.png

format: plain


text:
## Example 1: Securing an Unmanned Ground Vehicle

This is an example of using Imhotep-SMT in estimating the state of an Unmanned Ground Vehicle (UGV) while one of its sensors is under attack.

## Example 2: Active Cruise Control

In this example, we show how Imhotep-SMT can be used to securely estimating the state of a driving car while performing active cruise control. The simulation consists of two leader cars followed by two followers cars. The follower cars are under sensor attacks. One of the cars (blue) uese Imhotep-SMT to estimate the state while the other follower car (red) uses standard least squares estimator. The simulation uses a 3D virtual reality simulator to show the perfromance of the two estimators. The simulation is currently restricted to the case when the cars are driving in straight lines.

## Example 3: IEEE 14-bus Power Grid

This is an example of using Imhotep-SMT in estimating the state of the IEEE 14-bus power network. The bus composed of 5 synchronous generators and 14 buses. The state of each generator includes rotor angle and frequency. The overall system has 35 sensors: 14 sensors measure the real power injections at every bus, 20 sensors measure the real power flows along every branch, and one sensor measures the rotor angle of generator number 1. In this simulation, the attacker is assumed to pick a random set of 14 sensors. We compare the efficiency of Imohtep-SMT by comparing its estimation error with respect to the estimation error of a classical least squares estimator.

To assess the scalability of Imhotep-SMT, we randomly generated a set of matrices for systems of increasing size. For each of these systems, we configured Imohtep-SMT, provided it with the system inputs and outputs for state estimation, and recorded the execution time. In the first experiment, we fix the number of sensors $p = 20$ and increase the number of system states from $n = 10$ to $n = 150$. In the second experiment, we fix the number of states n=50 and increase instead the number of sensors from $p = 3$ to $p = 150$. In both cases, half of the sensors are attacked. As a result, increasing n has a small effect on the overall execution time, which reflects the fact that the number of constraints to be satisfied does not depend on $n$. Conversely, as the number of sensors increases, the number of constraints increases, hence the execution time of Imhotep-SMT.


text:
## Algebra 1

Published by Prentice Hall

# Chapter 1 - Foundations for Algebra - 1-4 Properties of Real Numbers - Practice and Problem-Solving Exercises: 47

The first plan

#### Work Step by Step

The first plan is Hannah buying each friend the same gift for $22$ dollars. If she goes through with this she will spend $66$ dollars. This is under her budget so this will work. In the second plan, she will spend $29.26+23.99+23.74= 76.99$ dollars. This is more than her budget, so only the first plan works.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations

Abstract : We study the boundary behaviour of the solutions of (E) $\;-\Gd_p u+|\nabla u|^q=0$ in a domain $\Gw \sbs \BBR^N$, when $N\geq p> q>p-1$. We show the existence of a critical exponent \$q_*
Keywords :
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-01095162
Contributor : Laurent Veron <>
Submitted on : Thursday, February 12, 2015 - 11:24:56 AM
Last modification on : Tuesday, August 13, 2019 - 2:30:11 PM
Document(s) archivé(s) le : Wednesday, May 13, 2015 - 10:16:16 AM

### Files

Bound15.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-01095162, version 2
• ARXIV : 1412.4613

### Citation

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Véron. Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations. 2015. ⟨hal-01095162v2⟩

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text:
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# In the Given Figure , Two Circles with Centres O and P Are Touching Internally at Point A. If Bq = 9, De = 5, Complete the Following Activity to Find the Radii of the Circles. - Geometry

#### Question

In the given figure, two circles with centres O and P are touching internally at point A. If BQ = 9, DE = 5, complete the following activity to find the radii of the circles.

#### Solution

Let the radius of the bigger circle be R and that of a smaller circle be r.
OA, OB, OC and OD are the radii of the bigger circle.
∴ OA = OB = OC = OD = R
PQ = PA = r
OQ = OB − BQ = R - 9

OE = OD − DE = R - 5

As the chords QA and EF of the circle with centre P intersect in the interior of the circle, so by the property of internal division of two chords of a circle,
OQ × OA = OE × OF
R - 9 x r = R - 5 x R - 5             .....(∵ OE = OF)
R2 − 9R = R2 − 10R + 25
R = 25

AQ = 2r = AB − BQ                      (∵AB = 2R)
2r = 50 − 9 = 41
r = $\frac{41}{2}$ = 20.5

Is there an error in this question or solution?


text:
# A voltage source and two resistors are connected in parallel as in the given circuit. Suppose that vs = 150 V, R1  = 50 ohm and R2 = 25ohm .Find the currents i1 and i2 in each resistor.

A voltage source and two resistors are connected in parallel as in the given circuit. Suppose that vs = 150 V, R1  = 50 ohm and R2 = 25ohm .Find the currents i1 and i2 in each resistor.

### Right Answer is: i1 = 3A , i2 = -6A

#### SOLUTION

Given Vs = 150 V, R1 = 50 Ω, R2 = 25 Ω

Since voltage parameter in parallel connected elements is same.

Vs = V1 = V2 = 150 V

∴ i1 = V1 / R1 = 150 / 50 = 3 A

Since the direction of the arrow mentioned is in opposite direction to the actual current flow i2 is considered as negative.

i2 = -V2 / R2 = -150 / 25 = – 6 A

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text:
# If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above?

STATEMENT OF THE PROBLEM

If $q^k n^2$ is an odd perfect number with Euler prime $q$, is $\sigma(q^k)/n + \sigma(n)/q^k$ bounded from above?

MOTIVATION

Let $\sigma=\sigma_{1}$ denote the classical sum-of-divisors function, and denote the abundancy index of $x \in \mathbb{N}$ by $I(x)=\sigma(x)/x$.

It is known that the inequality $$I(q^k) + I(n) < \frac{\sigma(q^k)}{n}+\frac{\sigma(n)}{q^k}$$ holds if and only if the biconditional $$q^k < n \iff \sigma(q^k) < \sigma(n)$$ is true. This biconditional is true if $\sigma(q^k)<n$, or if $\sigma(n) \leq q^k$. (I currently am not aware of any other conditions for which the biconditional holds.) Edit (August 11 2017): The biconditional is also true when $q^k < n$. (This follows from $I(q^k)<I(n)$.)

Note that if $\sigma(q^k)/n + \sigma(n)/q^k < C$ for some absolute constant $C$, then $$\sqrt{\frac{8}{5}}\frac{n}{C} < q^k < Cn,$$ so that $C > \sqrt[4]{8/5}$.

However, I know that $C > \sqrt[4]{8/5}$ is far from the truth, as I have recently been able to verify that either $$\frac{\sigma(q^k)}{n} < \sqrt{2} < \frac{\sigma(n)}{q^k}$$ or $$\frac{\sigma(n)}{q^k} < \sqrt{2} < \frac{\sigma(q^k)}{n}$$ is true. In the first case, $q^k < n\sqrt{2}$, while in the second case, we have $n < q^k$.

Of course, trivially we have $$I(q^k) + I(n) < I(q^k) + I(n^2) < 3.$$

Suppose to the contrary that $$\frac{\sigma(q^k)}{n}+\frac{\sigma(n)}{q^k}$$ is bounded from above.
Since $x < \sigma(x)$ for all $x > 1$, and $\sigma(y) < 2y$ for deficient $y$, $$\frac{q^k}{n}+\frac{n}{q^k}<\frac{\sigma(q^k)}{n}+\frac{\sigma(n)}{q^k}<2\cdot\bigg(\frac{q^k}{n}+\frac{n}{q^k}\bigg),$$ so that the biconditional $$\frac{\sigma(q^k)}{n}+\frac{\sigma(n)}{q^k} \text{ is bounded from above } \iff \frac{q^k}{n}+\frac{n}{q^k} \text{ is bounded from above }$$ holds.
But in general, we know that the function $$f(z) = z + \frac{1}{z}$$ is not bounded from above. (It suffices to let $z$ approach $0$ [from the right] and to let $z$ approach $\infty$.)
Consequently, the sum $$\frac{\sigma(q^k)}{n}+\frac{\sigma(n)}{q^k}$$ may not be bounded from above.
• This only shows that there is no trivial, i.e. purely analytic, argument proving that this expression is unbounded. But there might be only finitely many odd perfect numbers, or there might be some relation between $n$ and $q$, which implies that the ratio is bounded. – Jan-Christoph Schlage-Puchta Dec 9 '17 at 18:33


text:
### Interference 2D centered square lattice

Scattering from cylinders distributed along a squared centered lattice.

• The particles are cylinders with radii and heights of $3$ nm.
• Their spatial distribution is composed of two square lattices (lattice length $l$), shifted by half a lattice length in both directions:
• The first square lattice is centered at the origin, with a lattice length of $25$ nm.
• The second one, with the same lattice spacing and the same type of particles at its nodes is initialized at $x = y = l/2 = 12.5$ nm.
• The lattices’ base vectors are parallel to the axes of the reference cartesian frame.
• The wavelength is equal to $1$ $\unicode{x212B}$.
• The incident angles are $\alpha_i = 0.2 ^{\circ}$ and $\phi_i = 0^{\circ}$.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73  """ 2D lattice with disorder, centered square lattice """ import numpy import bornagain as ba from bornagain import deg, angstrom, nm def get_sample(): """ Returns a sample with cylinders on a substrate, forming a 2D centered square lattice """ # defining materials m_ambience = ba.HomogeneousMaterial("Air", 0.0, 0.0) m_substrate = ba.HomogeneousMaterial("Substrate", 6e-6, 2e-8) m_particle = ba.HomogeneousMaterial("Particle", 6e-4, 2e-8) # collection of particles interference = ba.InterferenceFunction2DLattice.createSquare(25.0*nm) pdf = ba.FTDecayFunction2DCauchy(300.0*nm/2.0/numpy.pi, 100.0*nm/2.0/numpy.pi) interference.setDecayFunction(pdf) particle_layout = ba.ParticleLayout() position1 = ba.kvector_t(0.0, 0.0, 0.0) position2 = ba.kvector_t(12.5*nm, 12.5*nm, 0.0) cylinder_ff = ba.FormFactorCylinder(3.*nm, 3.*nm) cylinder = ba.Particle(m_particle, cylinder_ff) basis = ba.ParticleComposition() basis.addParticles(cylinder, [position1, position2]) particle_layout.addParticle(basis) particle_layout.setInterferenceFunction(interference) # assembling the sample air_layer = ba.Layer(m_ambience) air_layer.addLayout(particle_layout) substrate_layer = ba.Layer(m_substrate) multi_layer = ba.MultiLayer() multi_layer.addLayer(air_layer) multi_layer.addLayer(substrate_layer) print(multi_layer.treeToString()) return multi_layer def get_simulation(): """ Create and return GISAS simulation with beam and detector defined """ simulation = ba.GISASSimulation() simulation.setDetectorParameters(200, -2.0*deg, 2.0*deg, 200, 0.0*deg, 2.0*deg) simulation.setBeamParameters(1.0*angstrom, 0.2*deg, 0.0*deg) return simulation def run_simulation(): """ Runs simulation and returns intensity map. """ simulation = get_simulation() simulation.setSample(get_sample()) simulation.runSimulation() return simulation.result() if __name__ == '__main__': result = run_simulation() ba.plot_simulation_result(result)
Interference2DCenteredSquareLattice.py