Polar warming in the middle atmosphere of Mars

We report ground-based laser heterodyne spectroscopy of non-thermal emission in the cores of the 10.33-μmR(8) and 10.72-μmP(32) lines of ¹²C¹⁶O₂, obtained at 23 locations on the disk of Mars during the 1984 opposition, at L_s = 130°. The data were obtained at a sub-Doppler spectral resolution, and the temperature of the middle Martian atmosphere (50–85 km) is derived from the frequency width and intensity of the R(8) emission, and from the total intensity of the P(32) emission. We find that the temperature of the middle Martian atmosphere varies with latitude. Near the subsolar latitude, the average 50- to 85-km temperature is close to the radiative equilibrium value for a CO₂ atmosphere. However, at high latitudes in both the northern (summer) and southern (winter) hemispheres the 50- to 85-km temperature exceeds the CO₂ radiative equilibrium value; a meridional gradient in the range of 0.4 – 0.9°K per degree of latitude is indicated by our data. The highest temperatures are seen at high latitudes in the winter hemisphere, reminiscent of the seasonal effects seen at the Earth's mesopause. As in the terrestrial case, this winter polar warming in the Martian middle atmosphere necessitates departures from radiative equilibrium; dynamical heating of order 4 × 10² ergs g⁻¹ sec⁻¹ is required at the edge of the winter polar night. A comparison with 2-D circulation models shows that the presence of atmospheric dust may enhance this dynamical heating at high winter latitudes, and may also account for heating at high latitudes in the summer hemisphere.     

Titan’s 5-μm window: observations with the Very Large Telescope

We report on mid-resolution (R∼2000) spectroscopic observations of Titan, acquired in November 2000 with the Very Large Telescope and covering the range 4.75-5.07 μm. These observations provide a detailed characterization of the CO (1-0) vibrational band, clearly separating for the first time individual CO lines (P10 to P19 lines of ¹³CO). They indicate that the CO/N₂ mixing ratio in Titan’s troposphere is 32±10 ppm. Comparison with photochemical models indicates that CO is not in a steady state in Titan’s atmosphere. The observations confirm that Titan’s 5-μm continuum geometric albedo is ∼0.06, and further indicates a ∼20% albedo decrease over 4.98-5.07 μm. Nonzero flux is detected at the 0.01 geometric albedo level in the saturated core of the ¹²CO (1-0) band, at 4.75-4.85 μm, providing evidence for backscattering on the stratospheric haze. Finally, emission lines are detected at 4.75-4.835 μm, coinciding in position with lines from the CO(1-0) and/or CO(2-1) bands. Matching them by thermal emission would require Titan’s stratosphere to be much warmer (by ∼ 25 K at 0.1 mbar) than indicated by the methane 7.7-μm emission and the Voyager radio-occultation. We show instead that a nonthermal mechanism, namely solar-excited fluorescence, is a more plausible source for these emissions. Improved observations and laboratory measurements on the vibrational-translational relaxation of CO are needed for further interpretation of these emissions in terms of a CO stratospheric mixing ratio.     

Vertical cloud structure of Jupiter's Equitorial Plumes

A model for the vertical cloud structure of Jupiter's Equitorial Plumes is deduced based on an analysis of Voyager images of the equitorial region in the 6190Å methane band and the 6000-Å continuum, and ground-based 8900-Å methane band images of Jupiter. A computer code that represents scattering and absorption from aerosol and gas layers was applied to a heirarchy of increasingly complex model aerosol structures to match the observations in the three wavelengths. The observations are consistent with a model for the vertical cloud structure of the equitorial region that consists of four aerosol layers. A high-altitude haze layer (HAL) with optical depth τ = 1 uniformly blankets the equitorial region at an altitude between 100 and 250 mbar. Below that, a middle-level cloud layer between 400 and 800 mbar contains the well-known Equatorial Plumes. The Plume clouds are optically thick (τ ≥ 12), bright clouds with single scattering albedo $\text{ω}\text{= 0.997}$. They are probably composed of ammonia ice. The darker ($\text{ω}\text{= 0.990}$) interplume regions contain optically thinner clouds (2 ≤ τ ≤ 5) at the same altitude as the Plumes. An opaque cloud deck between 4000 and 6000 mbar, which is probably composed of water, forms the lowest model layer. In addition to these three layers, a thin forward scattering haze layer above 100 mbar was included in the models for consistency with previous work (Tomasko et al., 1978). We conclude that the vertical structure of the Equatorial Plume clouds is consistent with the hypothesis (Hunt et al., 1981) that the Plumes are caused by upwelling at the ammonia condensation level produced by bouyancy due to latent heat release from the condensation of water clouds nearly three scale heights below the Plumes.     

High-resolution spectroscopy of Saturn at 3 microns: CH4, CH3D, C2H2, C2H6, PH3, clouds, and haze

We report observation and analysis of a high-resolution 2.87-3.54 μm spectrum of the southern temperate region of Saturn obtained with NIRSPEC at Keck II. The spectrum reveals absorption and emission lines of five molecular species as well as spectral features of haze particles. The ν₂+ν₃ band of CH₃D is detected in absorption between 2.87 and 2.92 μm; and we derived from it a mixing ratio approximately consistent with the Infrared Space Observatory result. The ν₃ band of C₂H₂ also is detected in absorption between 2.95 and 3.05 μm; analysis indicates a sudden drop in the C₂H₂ mixing ratio at 15 mbar (130 km above the 1 bar level), probably due to condensation in the low stratosphere. The presence of the ν₃+ν₉+ν₁₁ band of C₂H₆ near 3.07 μm, first reported by Bjoraker et al. [Bjoraker, G.L., Larson, H.P., Fink, U., 1981. Astrophys. J. 248, 856-862], is confirmed, and a C₂H₆ condensation altitude of 10 mbar (140 km) in the low stratosphere is determined. We assign weak emission lines within the 3.3 μm band of CH₄ to the ν₇ band of C₂H₆, and derive a mixing ratio of 9±4×¹⁰⁻⁶ for this species. Most of the C₂H₆ 3.3 μm line emission arises in the altitude range 460-620 km (at ∼μbar pressure levels), much higher than the 160-370 km range where the 12 μm thermal molecular line emission of this species arises. At 2.87-2.90 μm the major absorber is tropospheric PH₃. The cloud level determined here and at 3.22-3.54 is 390-460 mbar (∼30 km), somewhat higher than found by Kim and Geballe [Kim, S.J., Geballe, T.R., 2005. Icarus 179, 449-458] from analysis of a low resolution spectrum. A broad absorption feature at 2.96 μm, which might be due to NH₃ ice particles in saturnian clouds, is also present. The effect of a haze layer at about 125 km (∼12 mbar level) on the 3.20-3.54 μm spectrum, which was not apparent in the low resolution spectrum, is clearly evident in the high resolution data, and the spectral properties of the haze particles suggest that they are composed of hydrocarbons.     

Clouds, haze, and CH4, CH3D, HCN, and C2H2 in the atmosphere of Titan probed via 3 μm spectroscopy

Using synthetic spectra derived from an updated model atmosphere together with a continuum model that includes contributions from haze, cloud and ground, we have re-analyzed the recently published (Geballe et al., 2003, Astrophys. J. 583, L39-L42) high-resolution 3 μm spectrum of Titan which contains newly-detected bands of HCN (in emission) and C₂H₂ and CH₃D (in absorption), in addition to previously detected bands of CH₄. In the 3.10-3.54 μm interval the analysis yields strong evidence for the existence of a cloud deck or optically thick haze layer at about the 10 mbar (∼ 100 km) level. The haze must extend well above this altitude in order to mask the strong CH₄ lines at 3.20-3.50 μm. These cloud and haze components must be transparent at 2.87-2.92 μm, where analysis of the CH₃D spectrum demonstrates that Titan's surface is glimpsed through a second cloud deck at about the 100 mbar (∼ 50 km) level. Through a combination of areal distribution and optical depth this cloud deck has an effective transmittance of ∼ 20%. The spectral shape of Titan's continuum indicates that the higher altitude cloud and haze particles responsible for suppressing the CH₄ absorptions have a largely organic make-up. The rotational temperature of the HCN ranges from 140 to 180 K, indicating that the HCN emission occurs over a wide range of altitudes. This emission, remodeled using an improved collisional deactivation rate, implies mesospheric mixing ratio curves that are consistent with previously predictions. The stratospheric and mesospheric C₂H₂ mixing ratios are ∼¹⁰⁻⁵, considerably less than previous model predictions (Yung et al., 1984), but approximately consistent with recent observational results. Upper limits to mixing ratios of HC₃N and C₄H₂ are derived from non-detections of those species near 3.0 μm.     

Titan: Far-infrared and microwave remote sensing of methane clouds and organic haze

It is shown that Titan's surface and plausible atmospheric thermal opacity sources—gaseous N₂, CH₄, and H₂, CH₄ cloud, and organic haze—are sufficient to match available Earth-based and Voyager observations of Titan's thermal emission spectrum. Dominant sources of thermal emission are the surface for wavelenghts λ ? 1 cm, atmospheric N₂ for 1 cm ? λ ? 200 μm,, condensed and gaseous CH₄ for 200 μm ? λ ? 20 μm, and molecular bands and organic haze for λ ? 20 μm. Matching computed spectra to the observed Voyager IRIS spectra at 7.3 and 52.7° emission angles yields the following abundances and locations of opacity sources: CH₄ clouds: 0.1 g cm^? at a planetocentric radius of 2610–2625 km, 0.3 g cm^?2 at 2590–2610 km, total 0.4 ± 0.1 g cm^–2 above 2590 km; organic haze: $\text{4 ± 2 × 10}{}^{\mathrm{?6}}\text{,}\text{g cm}\text{,}{}^{\mathrm{?2}}$ above 2750 km; tropospheric H₂: 0.3 ± 0.1 mol%. This is the first quantitative estimate of the column density of condensed methane (or CH₄/C₂H₆) on Titan. Maximum transparency in the middle to far IR occurs at 19 μm where the atmospheric vertical absorption optical depth is $\text{?0.6}$ A particle radius $\text{r}\text{? 2 μ}\text{m}$ in the upper portion of the CH₄ cloud is indicated by the apparent absence of scattering effects.     

The three-dimensional structure of Saturn's equatorial jet at cloud level

The three-dimensional structure of Saturn's intense equatorial jet from latitudes 8° N to 20° S is revealed from detailed measurements of the motions and spectral reflectivity of clouds at visible wavelengths on high resolution images obtained by the Cassini Imaging Science Subsystem (ISS) in 2004 and early 2005. Cloud speeds at two altitude levels are measured in the near infrared filters CB2 and CB3 matching the continuum (effective wavelengths 750 and 939 nm) and in the MT2 and MT3 filters matching two methane absorption bands (effective wavelengths 727 and 889 nm). Radiative transfer models in selective filters covering an ample spectral range (250-950 nm) require the existence of two detached aerosol layers in the equator: an uppermost thin stratospheric haze extending between the pressure levels ∼20 and 40 mbar (tropopause level) and below it, a dense tropospheric haze-cloud layer extending between 50 mbar and the base of the ammonia cloud (between ∼1 and 1.4 bar). Individual cloud elements are detected and tracked in the tropospheric dense haze at 50 and 700 mbar (altitude levels separated by 142 km). Between latitudes 5° N and 12° S the winds increase their velocity with depth from 265 m s⁻¹ at the 50 mbar pressure level to 365 m s⁻¹ at 700 mbar. These values are below the high wind speeds of 475 m s⁻¹ measured at these latitudes during the Voyager era in 1980-1981, indicating that the equatorial jet has suffered a significant intensity change between that period and 1996-2005 or that the tracers of the flow used in the Voyager images were rooted at deeper levels than those in Cassini images.     

The abundance of acetylene in the atmosphere of Jupiter

The Q and R branches of the C₂H₂ ν₅ fundamental, observed in emission in an aircraft spectrum of Jupiter near 750 cm^?1, have been analyzed with the help of an improved line listing for this band. The line parameters have been certified in the laboratory with the same interferometer used in the Jovian observations. The maximum mixing ratio of C₂H₂ is found to be between 5 × 10^?8 and 6 × 10^?9, depending on the form of its vertical distribution and the temperature structure assumed for the lower stratosphere. Most consistent with observations of both Q and R branches are: (1) distributions of C₂H₂ with a constant mixing ratio in the stratosphere and a cutoff at a total pressure of 100 mbar or less, and (2) the assumption of a temperature at 10^?2 bar which is near 155°K.     

The 2.9-4.2 micron spectrum of Saturn: Clouds and CH4, PH3, and NH3

We have used synthetic spectra to analyze a medium resolution 2.9-4.2 μm spectrum of Saturn's temperate region observed at UKIRT using CGS4. The synthetic spectra include CH₄, PH₃, and NH₃ lines, for which mixing ratios were adopted from recent Cassini results. The observed absorption features in the spectrum are well accounted for by lines of these molecular species formed 22 +/− 8 km above the 1 bar pressure level at ∼610 mbar. The influence of optically thin haze particles at higher altitudes on the spectrum is not pronounced, with higher spectral resolution probably required to constrain the effects of haze in this wavelength region. Fluorescent line emission by CH₄ in its ν₃ and ν₃+ν₄−ν₄ bands, detected in the 3.2-3.5 μm region, originates between 400 km (∼0.06 mbar) and 800 km (∼0.01 μbar) above the 1 bar level, with peak contributions from the two major contributing bands at 550 km (∼3 μbar) and 700 km (∼0.1 μbar), respectively.     

Saturn's cloud structure and temporal evolution from ten years of Hubble Space Telescope images (1994-2003)

We present a study of the vertical structure of clouds and hazes in the upper atmosphere of Saturn's Southern Hemisphere during 1994-2003, about one third of a Saturn year, based on Hubble Space Telescope images. The photometrically calibrated WFPC2 images cover the spectral region between the near-UV (218-255 nm) and the near-IR (953-1042 nm), including the 890 nm methane band. Using a radiative transfer code, we have reproduced the observed center-to-limb variations in absolute reflectivity at selected latitudes which allowed us to characterize the vertical structure of the entire hemisphere during this period. A model atmosphere with two haze layers has been used to study the variation of hazes with latitude and to characterize their temporal changes. Both hazes are located above a thick cloud, putatively composed of ammonia ice. An upper thin haze in the stratosphere (between 1 and 10 mbar) is found to be persistent and formed by small particles (radii ∼0.2 μm). The lower thicker haze close to the tropopause level shows a strong latitudinal dependence in its optical thickness (typically τ∼20-40 at the equator but τ∼5 at the pole, at 814 nm). This tropospheric haze is blue-absorbent and extends from 50 to 100 mbar to about ∼400 mbar. Both hazes show temporal variability, but at different time-scales. First, there is a tendency for the optical thickness of the stratospheric haze to increase at all latitudes as insolation increases. Second, the tropospheric haze shows mid-term changes (over time scales from months to 1-2 years) in its optical thickness (typically by a factor of 2). Such changes always occur within a rather narrow latitude band (width ∼5-10°), affecting almost all latitudes but at different times. Third, we detected a long-term (∼10 year) decrease in the blue single-scattering albedo of the tropospheric haze particles, most intense in the equatorial and polar areas. Long-term changes follow seasonal insolation variations smoothly without any apparent delay, suggesting photochemical processes that affect the particles optical properties as well as their size. In contrast, mid-term changes are sudden and show various time-scales, pointing to a dynamical origin.     

On the vertical wind shear of Saturn's Equatorial Jet at cloud level

In this work we analyze and compare the vertical cloud structure of Saturn's Equatorial Zone in two different epochs: the first one close to the Voyagers flybys (1979-1981) and the second one in 2004, when the Cassini spacecraft entered its orbit around the planet. Our goal is to retrieve the altitude of cloud features used as zonal wind tracers in both epochs. We reanalyze three different sets of photometrically calibrated published data: ground-based in 1979, Voyager 2 PPS and ISS observations in 1981, and we analyze a new set of Hubble Space Telescope images for 2004. For all situations we reproduced the observed reflectivity by means of a similar vertical model with three layers. The results indicate the presence of a changing tropospheric haze in 1979-1981 (Ptop∼100 mbar, τ∼10) and in 2004 (Ptop∼50 mbar, τ∼15) where the tracers are embedded. According to this model the Voyager 2 ISS images locate cloud tracers moving with zonal velocities of 455 to 465 (±2) m/s at a pressure level of 360 ± 140 mbar. For HST observations, our previous works had showed cloud tracers moving with zonal wind speeds of 280±10 m/s at a pressure level of about 50±10 mbar. All these values are calculated in the same region (3°±2° N). This speed difference, if interpreted as a vertical wind shear, requires a change of per scale height, two times greater than that estimated from temperature observations. We also perform an initial guess on Cassini ISS vertical sounding levels, retrieving values compatible with HST ones and Cassini CIRS derived vertical wind shear, but not with Voyager wind measurements. We conclude that the wind speed velocity differences measured between 1979-1981 and 2004 cannot be explained as a wind shear effect alone and demand dynamical processes.     

Titan's prolific propane: The Cassini CIRS perspective

Although propane gas (C₃H₈) was first detected in the stratosphere of Titan by the Voyager IRIS infrared spectrometer in 1980, obtaining an accurate measurement of its abundance has proved difficult. All existing measurements have been made by modeling the ν₂₆ band at : however, different analyzes over time have yielded quite different results, and it also suffers from confusion with the strong nearby ν₅ band of acetylene. In this paper we select large spectral averages of data from the Cassini Composite Infrared Spectrometer (CIRS) obtained in limb-viewing mode at low latitudes (30^°S-30^°N), greatly increasing the path length and hence signal-to-noise ratio for optically thin trace species such as propane. By modeling and subtracting the emissions of other gas species, we demonstrate that at least six infrared bands of propane are detected by CIRS, including two not previously identified in Titan spectra. Using a new linelist for the range 1300-1400 cm^-1, along with an existing GEISA list, we retrieve propane abundances from two bands at 748 and 1376 cm^-1. At 748 cm^-1 we retrieve 4.2±0.5×10^-7 (1-σ error) at 2 mbar, in good agreement with previous studies, although lack of hotbands in the present spectral atlas remains a problem. We also determine 5.7±0.8×10^-7 at 2 mbar from the 1376 cm^-1 band — a value that is probably affected by systematic errors including continuum gradients due to haze and also an imperfect model of the ν₆ band of ethane. This study clearly shows for the first time the ubiquity of propane's emission bands across the thermal infrared spectrum of Titan, and points to an urgent need for further laboratory spectroscopy work, both to provide the line positions and intensities needed to model these bands, and also to further characterize haze spectral opacity. The present lack of accurate modeling capability for propane is an impediment not only for the measurement of propane itself, but also for the search for the emissions of new molecules in many spectral regions.     

The atmosphere of Saturn: An infrared perspective

A model of the radiative portion of the equatorial atmosphere of Saturn, constrained by the infrared data various observers, has been constructed using a technique which includes the variation of thermal flux with depth. The model has a high-altitude temperature inversion due to the absorption of ultraviolet sunlight. The inversion causes the observed infrared emission peaks at 8 ωm (methane) and 12 ωm (ethane). Mixing ratios of these gases to hydrogen are computed from these emission features. The bottom of the modeled region occurs at the radiative-convective boundary. At this level, an opaque cloud consisting of solid ammonia condensation particles is postulated. Above the cloud is a thin haze, also composed of ammonia particles. The haze is required to match infrared observations near 9.5 ωm and hydrogen quadrupole equivalent widths near 0.64 ωm. Predictions of the model are given for further observational tests.     

Linear size–extended radio luminosity (D–P E ) correlation in BL Lacertae objects: evidence for large scale beaming?

In this paper, we use the distributions of projected linear size (D), core- (P _C ) and extended- (P _E ) radio luminosities, to investigate a consequence of relativistic beaming and radio source orientation scenario for low-luminosity extragalactic radio sources. In this scenario, BL Lacertae objects (BL Lacs) are believed to be Fanaroff-Riley type I (FR I) radio galaxies, but with radio axes aligned close to the line of sight. At this orientation, the core emission is greatly enhanced by relativistic Doppler boosting and linear size foreshortened due to geometrical projection. A simple outcome of this scenario is that the extended luminosity is expected to be orientation invariant, but a D–P _C correlation is envisaged. Results show that both the relative core dominance (R) and linear size are strongly correlated with extended luminosity (r≥ 0.7). Using the R-distribution and R–P _E anti-correlation, we show that the difference in radio core-dominance between FR I radio galaxies and X-ray selected BL Lacs can be accounted for by a bulk Lorentz factor γ～5–13 and viewing angle ?～5–15^°, which can be understood in terms of the scenario, with relativistic beaming persisting at largest scales.     

The upper Jovian atmosphere aerosol content determined from a satellite eclipse observation

The Galilean satellite eclipse technique for measuring the aerosol distribution in the upper Jovian atmosphere is described and applied using 30 color observations of the 13 May 1972 eclipse of Ganymede obtained with the 5-m Hale telescope. This event probes the South Temperate Zone. The observed aerosol lies above the visible cloud tops, is very tenuous and varies with altitude, increasing rapidly with downward passage through the tropopause. The aerosol extinction coefficient, κ_a (λ1.05 μm), is ～1.1 × 10^?9 cm^?1 in the lower stratosphere and ～1.1 × 10^?8 cm^?1 at the tropopause. The 1σ uncertainty in these values does not exceed 50% The observations require some aerosol above the tropopause but do not clearly determine its structure. The present analysis emphasizes an extended haze distribution, but the alternate possibility is not excluded that the stratospheric aerosol resides in a thin layer. The aerosol extinction increases with decreasing wavelength and indicates the particle radius to be ?0.2 μm. Larger radii are impossible. These overall results confirm Axel's (1972) suggestion of a small quantity of dust above the Jovian cloud tops and the optical depths are consistent with those required to explain the low uv albedo.     

Photometry of 433 Eros from 0.65 to 2.2 μm

Lightcurves of 433 Eros are reported for 11 bandpasses ranging from 0.65 to 2.2 μm in wavelength. The relative spectral reflectance, R(λ), was not seen to vary during our observations. Eros has R(1.6 μm) = 1.5 ± 0.1 and R(2.2 μm) = 1.7 ± 0.1, where R(λ) is the spectral reflectance scaled to unity at λ = 0.56 μm. This spectral reflectance is suggestive of a mixture of silicates and material with high infrared reflectance, perhaps a metallic phase such as meteoritic “iron”.     

Infrared properties of haze particles of Venus

The computed variation of the infrared flux and polarization of Venus as a function of phase angle, based upon multiple-scattering calculations for the cloud model of Kawabata et al. (1980) with an internal heat source, precludes the possibility of sulfuric acid as the composition of the haze particles located above the main cloud. Furthermore, our calculations reveal that the hazticle should have a large absorption coefficient at these wavelengths, i.e, $\text{k(}\text{imaginary part of the complex refractive index}\text{) ? 1.3}\text{at a wavelength}\text{λ = 3.4 μ}\text{m}$. The optical thickness of the haze layer must be about 0.15 at λ = 3.4 μm.     

Emission line ratios for C III in the sun

Theoretical electron-density-sensitive C III emission line ratios are presented forR ₁ =I(2s2p ³ P – 2p ^{2 3} P)/I(2s2p ¹ P – 2p ^{2 1} S) =I(1176 Å)/I(1247 Å),R ₂ =I(2s2p ³ P – 2p ^{2 3} P)/I(2s ^{2 1} S – 2s2p ³ P ₁) =I(1176 Å)/I(1908 Å), andR ₃ =I(2s2p ¹ P – 2p ^{2 1} S)/I(2s ^{2 1} S – 2s2p ³ P ₁) =I(1247 Å)/I(1908 Å). These are significantly different from those deduced previously, principally due to the adoption of improved electron impact excitation rates in the present analysis. Electron densities deduced from the present theoretical line ratios, in conjunction with observed values ofR ₁,R ₂, andR ₃ measured from solar spectra obtained by the Naval Research Laboratory's S082B instrument on boardSkylab, are found to be generally compatible. In contrast, previous diagnostic calculations imply electron densities fromR ₁,R ₂, andR ₃ that differ by up to two orders of magnitude. These results provide observational support for the accuracy of the atomic physics adopted in the present calculations, and the methods employed in the derivation of the theoretical line ratios.     

An auroral enhancement of O2λ1.27-μm emission

The ground-level zenith radiance of the atmospheric emission at λ1.27 μm was radiometrically observed to increase by a factor of approximately two with the onset of an IBC III⁺ auroral breakup above Chatanika, Alaska, on 10 March 1975. Time-resolved optical spectra clearly show that the slow component of the enhancement is associated with the (0,0) band of the infrared atmospheric system of O₂. Photometric and incoherent scatter radar data are used to define the energy-deposition profile and the absolute energy flux for the event. The magnitude of the O₂λ1.27-μm enhancement compares favourably with the predictions of an auroral excitation model which includes only secondary-electron excitation of molecular oxygen in the O₂(a¹Δ_g) source term.