# Seasonal variation of Saturn’s Lyman-$\alpha$ brightness

P. Stephenson  Lunar and Planetary Laboratory, University of Arizona, Tucson,
AZ, USA  T. T. Koskinen  Lunar and Planetary Laboratory, University of
Arizona, Tucson, AZ, USA  Z. Brown  Lunar and Planetary Laboratory, University
of Arizona, Tucson, AZ, USA  E. Quémerais  LATMOS/IPSL, Université de
Versailles Saint-Quentin, Guyancourt,France  P. Lavvas  GSMA, Université de
Reims, Reims, France  J. I. Moses  Space Science Institute, Boulder, CO, USA
B. Sandel  Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ,
USA  R. Yelle  Lunar and Planetary Laboratory, University of Arizona, Tucson,
AZ, USA

###### Abstract

We examine Saturn’s non-auroral (dayglow) emissions at Lyman-$\alpha$ observed
by the Cassini/UVIS instrument from 2004 until 2016, to constrain meridional
and seasonal trends in the upper atmosphere. We separate viewing geometry
effects from trends driven by atmospheric properties, by applying a multi-
variate regression to the observed emissions. The Lyman-$\alpha$ dayglow
brightnesses depend on the incident solar flux, solar incidence angle,
emission angle, and observed latitude. The emissions across latitudes and
seasons show a strong dependence with solar incidence angle, typical of
resonantly scattered solar flux and consistent with no significant internal
source. We observe a bulge in Ly-$\alpha$ brightness that shifts with the
summer season from the southern to the northern hemisphere. We estimate atomic
hydrogen optical depths above the methane homopause level for dayside disk
observations (2004-2016) by comparing observed Lyman-$\alpha$ emissions to a
radiative transfer model. We model emissions from resonantly scattered solar
flux and a smaller but significant contribution by scattered photons from the
interplanetary hydrogen (IPH) background. During northern summer, inferred
hydrogen optical depths steeply decrease with latitude towards the winter
hemisphere from a northern hemisphere bulge, as predicted by a 2D seasonal
photochemical model. The southern hemisphere mirrors this trend during its
summer. However, inferred optical depths show substantially more temporal
variation between 2004 and 2016 than predicted by the photochemical model.

Planetary atmospheres (1244) — Saturn (1426) — Ultraviolet astronomy (1736) —
Upper atmosphere (1748) — Atmospheric variability (2119)

##  1 Introduction

Lyman-$\alpha$ is the brightest ultraviolet emission line in the solar system
and has been observed at Saturn since 1976, initially with sounding rockets
and the Copernicus satellite (Weiser et al., 1977; Barker et al., 1980).
Subsequent observations of Lyman-$\alpha$ emissions from Saturn were made by
the Interplanetary Ultraviolet Explorer (IUE, Clarke et al., 1981; McGrath &
Clarke, 1992) and during the Voyager flybys with the Ultraviolet Spectrometer
(UVS, Broadfoot et al., 1981; Sandel et al., 1982; Ben-Jaffel et al., 1995).
Over its 13-year mission, Cassini/Huygens (Matson et al., 2002) orbited
Saturn, compiling an extensive dataset of Lyman-$\alpha$ emissions with the
Ultraviolet Imaging Spectrograph (UVIS, Esposito et al., 2005).

Recently, Ben-Jaffel et al. (2023) identified a bulge in Lyman-$\alpha$
emissions from Saturn’s thermosphere in the northern hemisphere between
latitudes of 5 and 35∘ N, with observations from the Hubble Space Telescope’s
Space Telescope Imaging Spectrograph (HST/STIS) and Cassini/UVIS. They also
identified the same bulge in Voyager/UVS observations that probed Saturn’s
atmosphere close to the northern spring equinox 35 years earlier (Yelle et
al., 1986). They concluded that the observed northern hemisphere bulge is a
permanent feature of the thermosphere. We note that this bulge shows no
longitudinal variation and therefore differs significantly from the
Lyman-$\alpha$ bulge that has been observed on Jupiter, which is fixed in
system III longitude (Sandel et al., 1980; Clarke et al., 1980; Dessler et
al., 1981; Skinner et al., 1988). Ben-Jaffel et al. (2023) proposed two
primary mechanisms to drive the emission bulge: variation of the temperature
profile in the lower thermosphere and upper stratosphere or a previously
unidentified suprathermal atomic hydrogen population at high altitudes, both
of which could vary seasonally. They proposed that the suprathermal population
could be created, for example, by a significant influx of material from the
rings or Enceladus into the upper atmosphere. In this study, we examine
Lyman-$\alpha$ emissions over the duration of the Cassini mission, in order to
examine the cause of the bulge, whether it is a permanent feature of Saturn’s
thermosphere, and the source of the increased emissions.

At Lyman-$\alpha$, Cassini/UVIS consistently observed much lower disk
brightnesses compared to the observations of Voyager/UVS, with peak
brightnesses of about 1 kR outside the auroral oval compared to 3-4 kR during
the Voyager flybys (Ben-Jaffel et al., 1995; Gustin et al., 2010; Shemansky et
al., 2009; Koskinen et al., 2020). Gustin et al. (2010) suggested that the
disparity was a result of ring-reflected light during the Voyager
observations, while Shemansky et al. (2009) suggested strong electroglow
emissions could reconcile the differences. The Voyager brightnesses have since
been questioned and revised downward by Quémerais et al. (2013a), who
concluded that the sensitivity of the Voyager/UVS instruments were
underestimated by a factor of 1.5-2.5. The revised Voyager Lyman-$\alpha$
brightnesses are roughly consistent with those observed by Cassini/UVIS
(Koskinen et al., 2020). Using HST observations from Earth orbit as a
calibration standard, Ben-Jaffel et al. (2023) challenged the downward
revision of the Voyager brightnesses and proposed instead a recalibration of
the Cassini/UVIS instrument at Lyman-$\alpha$ that would increase the observed
brightnesses by 70%. After scaling with the solar Lyman-$\alpha$ flux at
different times, they compared several observations by Cassini/UVIS (in 2007,
2013 and 2014), HST/STIS (in 2017) and HST’s Goddard High Resolution
Spectrograph (in 1996) to arrive at this conclusion.

Cross-calibration of UV instruments between missions at Lyman-$\alpha$ remains
difficult. Observations of Lyman-$\alpha$ emissions from interplanetary
background hydrogen provide one method to facilitate it. Observations of the
IPH Lyman-$\alpha$ by the Voyager/UVS instruments (Katushkina et al., 2016,
2017), New Horizons Alice (Gladstone et al., 2018, 2021) and future
observations by PHEBUS on Bepi/Colombo (Quémerais et al., 2020), have and will
continue to constrain models of the IPH background (e.g. Quémerais &
Izmodenov, 2002; Quémerais et al., 2013b; Izmodenov et al., 2013; Pryor et
al., 2022). Models of the interaction between the local interstellar medium
and the solar wind (Quémerais et al., 2006; Izmodenov et al., 2001, 2013) are
dependent on the hydrogen density of the local interstellar medium (LISM) and
at the terminator shock near 90 au, with estimates of the LISM Hydrogen
density varying from 0.12 to 0.195 cm-3 (Dialynas et al., 2019; Swaczyna et
al., 2020). In addition to scattered solar flux, a galactic contribution to
the background of 40 R has been identified at large heliocentric distances
(Gladstone et al., 2021; Pryor et al., 2022). While there remains uncertainty
on the density of the LISM, we compare Cassini/UVIS Lyman $\alpha$
observations to the model of (Quémerais et al., 2013a) and find good agreement
between Cassini/UVIS observed and modelled brightnesses, without the proposed
recalibration by a factor 1.7 (Ben-Jaffel et al., 2023; Pryor et al., 2024).
The situation with regard to calibration, however, is clearly confusing and
based on this, we cannot entirely discount the possibility of a significant
uncertainty in Saturn’s Lyman-$\alpha$ brightness.

Limb observations and solar occultations, in conjunction with photochemical
models, can provide further constraints on the atomic hydrogen columns and
Lyman-$\alpha$ emissions.

In addition to the Lyman-$\alpha$ bulge, Saturn’s thermosphere also exhibits
latitudinal variation in temperature (Brown et al., 2020) and exobase altitude
that are likely to be seasonally variable (Koskinen et al., 2021). Stellar
occultations by Cassini/UVIS have constrained the temperature of Saturn’s
thermosphere, allowing the retrieval of density profiles of upper atmospheric
constituents, including H2, He and CH4 (Koskinen et al., 2013, 2015, 2016;
Koskinen & Guerlet, 2018; Shemansky & Liu, 2012; Brown et al., 2020, 2022).
Stellar occultations, when stable, are essentially self-calibrating and
independent of instrument calibration. We note, however, that stellar
occultations cannot be used to retrieve the density of H.

Latitudinal and seasonal trends in the upper atmosphere have been predicted by
photochemical models (Moses et al., 2000a; Moses & Bass, 2000; Moses &
Greathouse, 2005; Hue et al., 2015, 2016). This is because, for example,
hydrocarbons in the stratosphere are influenced by meridionally-varying
insolation, including changes due to the motion of the ring shadow across
Saturn’s disk (Moses & Greathouse, 2005). The variation of methane in the
stratosphere also has substantial impact on atomic hydrogen through both the
production of H through photolysis (Moses et al., 2000a) and the location of
the homopause. Comparison of photochemical models with Lyman-$\alpha$
observations from Cassini/UVIS can constrain the hydrogen column above the
methane homopause, with methane a strong absorber at Lyman-$\alpha$. This can
subsequently constrain eddy mixing and circulation near the homopause level
(Atreya, 1982; Sandel et al., 1982; Atreya et al., 1984; Emerich et al., 1993;
Moses et al., 2000a; Moses & Greathouse, 2005).

The emissions of Lyman-$\alpha$ from Saturn’s disk, observed by Cassini/UVIS,
provide an extensive dataset over 13 years, with coverage across all
latitudes. Koskinen et al. (2020) examined one observational sequence of
Lyman-$\alpha$ from the Saturn disk in 2007. They found the brightnesses were
consistent with resonance scattering of solar flux by a hydrogen column of
$3\times 10^{16}$ cm-3, in agreement with columns calculated with a
photochemical model (Moses & Bass, 2000; Moses et al., 2000b; Moses &
Greathouse, 2005). Several other case studies have been examined (Mitchell et
al., 2009; Shemansky et al., 2009; Gustin et al., 2010; Koskinen et al., 2020;
Ben-Jaffel et al., 2023), but the full Lyman-$\alpha$ emission dataset has not
yet been fully explored.

By comparing results from a radiative transfer model with Cassini UVIS
observations, we can estimate the effective optical depth of the atomic
hydrogen layer. We follow the approach of Yelle et al. (1989) who modelled
resonance scattering of a deep atmosphere by the iterative doubling and adding
of thin layers, using angle-averaged partial frequency redistribution. We
examine the variation of the effective hydrogen optical depths with latitude
and season, under the assumption that emissions are dominated by resonant
scattering by the ambient, thermal hydrogen population. In addition, we
directly compare the inferred effective optical depths to the results of a
seasonal photochemical model, to identify processes not included in the
photochemical model, and to identify discrepancies that might indicate the
presence of suprathermal atoms or internal emissions generated by
photoelectron or energetic particle impact.

In this study, we examine the extensive dataset of Lyman-$\alpha$ emissions
from Saturn’s dayside disk collected by Cassini UVIS from 2004 to the end of
2016. We consider the Lyman-$\alpha$ observations through three approaches,
with the methods used outlined in Section 2. In section 3.1, we compare
observation of the IPH Lyman-$\alpha$ background to the model of Quémerais et
al. (2013b). In Section 3.2, we employ a multi-variate analysis of the
Lyman-$\alpha$ observations to confirm that resonance scattering of solar flux
is a dominant source of the emissions from Saturn’s non-auroral, dayside disk.
Finally, in Section 3.3, we compare the radiative transfer model, based on
doubling and adding of thin layers, to the Lyman-$\alpha$ observations, and we
retrieve the H optical depth above the methane homopause across the mission,
to determine seasonal variation of Saturn’s thermosphere. We discuss the
results in Section 4, in particular with relation to the nature of the
Lyman-$\alpha$ bulge. We also compare a seasonal photochemical model to the
effective H optical depths retrieved in Section 3.3.

##  2 Methods

###  2.1 Cassini UVIS data

Figure 1: Number of observation pixels on Saturn’s dayside disk, outside the
auroral regions ($\lvert\phi_{lat}\rvert<65^{\circ}$) by year and observation
type (see text for an explanation of the types).

We consider Cassini/UVIS observations from 2003 until 2016 during Cassini’s
orbits of Saturn, including MOSAIC, EUVFUV and COMPSIT/CIRS scans identified
in the Planetary Data System. This first survey of the emission data focuses
on nadir observations. Limb scans will not be considered in this work and are
instead the subject of a future paper. The Cassini/UVIS instrument comprised
64 spatial bins along the slit and 1024 spectral bins with a resolution of
0.78 Å. We focus on Lyman-$\alpha$ emissions from Saturn’s disk using the FUV
channel of UVIS (1115 to 1912 Å), integrating the emission brightness between
1205 and 1225 Å. We use the time-dependent sensitivity and flat field updates
that indicate a degradation of the signal by 30% from launch to the end of the
mission but in our default models, we do not use the proposed recalibration of
the Cassini/UVIS instrument by a factor 1.7 at Lyman-$\alpha$ (Ben-Jaffel et
al., 2023, p24). While additional degradation at Lyman-alpha would not be
surprising, cross-calibrating instruments with different viewing geometries
and the use of IPH models for calibration also includes uncertainties. For
example, our IPH model produces Lyman-$\alpha$ brightnesses consistent with
UVIS observations without the additional calibration factor (see Section 3.1).
The effective optical depths predicted by photochemical models of Saturn’s
atmosphere are also more consistent with UVIS observations with the previous
calibration (see Section 3.3). In addition, we note that a calibration
uncertainty of a factor 2 does not affect our results with respect to both
seasonal and meridional trends observed in the Lyman-$\alpha$ emission.

In order to reduce the downlink data volume and increase spatial coverage, the
wavelength range of observations was sometimes reduced and neighbouring
spectral pixels were combined (typically with a bin width of 4 pixels). In
cases with no spectral binning, we use the Cube Generator from the
Cassini/UVIS team and the pipeline flatfield to process the data. When
spectral binning was applied, the pipeline flatfield does not appropriately
address the presence of ‘evil’ pixels on the detector, which returned much
lower signals than those surrounding them. In the pipeline calibration the
intensity is interpolated to the ‘evil’ pixels, which are assigned the value
NaN in the flatfield. With binned spectra, one evil pixel in the bin results
in a NaN value for the whole bin, losing information from the adjacent pixels.
In these cases, we process the data with the Cube Generator with no flatfield,
applying the derived 2007 flatfield correction of (Koskinen et al., 2020) for
binned data. The only observations that use spectral binning occur between
2004 and 2008, and we do not expect a substantial change in the flatfield
correction over this period.

The three observation classes (MOSAIC, EUVFUV and COMPSIT/CIRS) had different
aims and characteristics. MOSAIC observations were designed to maximise
spatial coverage, observing much or all of the Saturn system (including the
rings). Prior to 2008, spectral binning of width 4 was common allowing
integration times of 25 to 95s. After 2008, the MOSAIC observations did not
use any spectral binning and required longer integration times from 120 to 900
s. During EUVFUV observations, spectra from both the EUV and FUV channels were
retrieved, with no spectral binning. These often capture a substantial part of
the Saturn system, such as in Figure 2, using an integration time between 180
and 260s. Finally, the COMPSIT/CIRS observations were paired with observations
by the Cassini/Composite Infrared Spectrometer (CIRS, Flasar et al., 2005).
These do not use any spectral binning and have much smaller spatial coverage,
often only observing along a single line across Saturn’s disk and limb.
However, the long integration times (1200 or 2400s) provide excellent signal-
to-noise ratios for the spectra.

The UVIS dataset comprises 636 observations over 14 years comprising 140,925
scans and 8,032,725 pixels, of which 3,000,000 are on the dayside of Saturn’s
disk. In this study, we focus on airglow emissions and therefore exclude the
auroral regions with latitudes poleward of 60∘. Figure 1 shows the number of
observation pixels by observation type and year. Additionally, over large
periods much of Saturn’s disc was shadowed by the Saturnian rings. We do not
have a good constraint on the solar flux entering the atmosphere after
absorption in the ring atmosphere, so these points are removed from the
dataset. For each observation, the ring shadow region and rings are mapped
onto the surface of Saturn (e.g. see Figure 2).

Figure 2: Observed Lyman-$\alpha$ emission brightness from Saturn on 21 Jun
2005. The solid white lines outline Saturn’s disk and its rings. Dashed lines
show the ring shadow region (at this time in the northern hemisphere), the
terminator, and the position of the exobase.

###  2.2 Multivariate quadratic regression of UVIS data

In this section, we outline a data based approach to the analysis of the
Lyman-$\alpha$ emissions from Saturn, which is independent of radiative
transfer modelling. This helps to identify the key emission source of
Lyman-$\alpha$ at Saturn and to test the assumptions used in the RT model. For
this purpose, we consider emission observations from 2014 until the end of
mission, from MOSAIC, EUVFUV, and COMPSIT/CIRS observations. We primarily
focus on part of the northern hemisphere summer (2014-2016), such that the
meridional trends in the atmosphere and emissions do not change substantially
with time.

We use a multi-variate regression with independent variables of emission
angle, incidence angle and latitude, which are used to predict the observed
Lyman-$\alpha$ brightness. For simplicity, we have removed the dependence on
the solar flux variation by scaling all the brightnesses to the solar flux
applicable on 1 Jan 2016 (see Section 2.3.1). We choose these variables
because resonance scattering is strongly dependent on the solar incidence
angle and the emission angle of the observation, and because previous studies
have identified meridional trends in the Lyman-$\alpha$ emissions. The model
includes a quadratic in emission and incidence angles, in addition to a cubic
in latitude. The full expression used for the regression is:

$B=p_{0}+p_{1}\cdot\theta_{\text{{em}}}+p_{2}\cdot\theta_{\text{{in}}}+p_{3}\cdot\phi_{\text{{lat}}}+p_{4}\cdot\theta_{\text{{em}}}^{2}+\\\
p_{5}\cdot\theta_{\text{{em}}}\cdot\theta_{\text{{in}}}+p_{6}\cdot\theta_{\text{{in}}}^{2}+p_{7}\cdot\phi_{\text{{lat}}}^{2}+p_{8}\cdot\phi_{\text{{lat}}}^{3},$
(1)

with $p_{i}$ the coefficients, $\theta_{\text{{em}}}$ is the emission angle,
$\theta_{\text{{in}}}$ is the solar incidence angle, and $\phi_{\text{{lat}}}$
is the planetocentric latitude. We prepare the data by standardizing the
independent variables
($\theta_{\text{{em}}},\theta_{\text{{in}}},\phi_{\text{{lat}}}$), and then
transforming them into the required polynomial expressions in Eq. 1. The
86,913 pixels in the NH summer were randomly split by 80% to 20% into training
and testing sets, respectively. The model was then trained using a least-
squares regression. Monte-Carlo analysis was used to retrieve confidence
intervals for the coefficients, resampling the dataset 1000 times.

###  2.3 Radiative transfer modeling

In order to constrain the properties of the atmosphere, we model the
brightness of scattered solar and IPH Lyman-$\alpha$ using a radiative
transfer model based on doubling and adding, which includes an angular
dependent frequency redistribution function (Yelle, 1988; Yelle & Wallace,
1989; Wallace et al., 1989). The model is based on assumptions of a plane-
parallel and isothermal atmosphere. The plane-parallel assumption breaks down
for observations close to either the limb or the terminator, as the incidence
or emission angles near 90∘. Consequently, we only apply the model to cases of
$\theta_{em}<60^{\circ}$ and $\theta_{in}<65^{\circ}$. Thin layer
approximations of scattering and transmission functions (outlined in Appendix
B Yelle, 1988) are iteratively doubled in thickness (see Appendix A),
computing new scattering, transmission and extinction functions until the
required optical depth is reached.

The scattering and transmission functions are dependent on the atmospheric
temperature (via the Lyman-$\alpha$ lineshape in the thermosphere) and the
optical depth. We use a thermospheric temperature that varies with latitude,
based on the pole-to-pole map of stellar occultations of Cassini/UVIS
throughout 2017 (see Section 2.3.3; Brown et al., 2020).

Figure 3: (blue) Brightness vs. optical depth for resonance scattering of
Lyman-$\alpha$ by atomic hydrogen for $(\theta_{in},\theta_{em})=(0,0)$. The
optically thin approximation is given by the black dotted line.

Figure 3 shows the variation in brightness as the atomic H optical depth
increases, for the nadir case ($\theta_{in},\theta_{em})=(0,0)$. The
approximation for an optically thin atmosphere is given by the dotted line.
The RT model is identical to the optically thin approximation at small optical
depths, before deviating as it approaches $\tau=1$. Beyond this, the
atmosphere is optically thick and the output brightness varies little as the
column grows. At $\tau=10^{3}$, the brightness then begins to increase once
again, as frequency redistribution becomes more effective and photons in the
Lorentzian wings begin to be multiply scattered. For deep atmospheres, like
Saturn’s, the frequency redistribution is critical to computing the scattered
brightness, due to the scattering of photons initially far from the line
centre.

####  2.3.1 Scattered solar flux

Figure 4: Lyman-$\alpha$ flux at Saturn throughout the Cassini mission,
extrapolated from LISIRD fluxes at 1 AU. This does not include absorption of
the solar flux between 1 AU and Saturn, which is accounted for separately.

For scattered solar flux, the brightness of Lyman-$\alpha$ is dependent on the
magnitude and shape of the solar flux entering the top of the atmosphere. The
RT code is normalised to the solar flux at line center. For the magnitude of
the flux, we use the LISIRD composite Lyman-$\alpha$ database (Machol et al.,
2019), which is based on fluxes measured at 1 AU by various instruments (e.g.
Hinteregger et al., 1981; Barth et al., 1983; Woods et al., 2000; McClintock
et al., 2005). First, we correct the LISIRD date for the flux to match the
solar longitude at Saturn during the UVIS observation, which requires a shift
of up to 15 days depending on the relative positions of Earth and Saturn. This
is converted to a line center flux using the line shape from Lemaire et al.
(2005), imposed at 1 AU (see Figure 5). After this, we extrapolate the flux to
Saturn using an inverse square law with heliocentric distance (see Figure 4).
We include absorption of the solar flux between 1 AU and Saturn by the IPH
background, using the IPH model described in Section 2.3.2. Once normalised,
the modelled brightness based on scattered solar flux from Saturn’s atmosphere
is interpolated to the emission and solar incidence angle of a given
observation, giving brightness as a function of temperature, effective H
column optical depth, and viewing geometry. The temperature is constrained by
a fit with latitude to the results retrieved from stellar occultations of the
thermosphere (see Section 2.3.3).

Figure 5: (black solid line) Lineshape of the solar Lyman $\alpha$ flux
(black) at 1au normalised to 1 at the linecentre. (black dashed line and
shaded region) Lineshape of solar Lyman-$\alpha$ after absorption of solar
flux by the IPH background. (blue) The Voigt profile of Lyman-$\alpha$ in
Saturn’s thermosphere for a temperature of 350K. (red) The lineshape of the
IPH Lyman-$\alpha$ entering the upper atmosphere at $T_{IPH}=10,000$ K. The
dotted vertical lines show the integration limits of $\pm
44\Delta\nu_{D,Sat}$, used in the radiative transfer model.

####  2.3.2 Lyman-$\alpha$ from the interplanetary background

Lyman-$\alpha$ flux emitted by the IPH background is also scattered by
Saturn’s upper atmosphere. Unlike the solar flux, the IPH Lyman-$\alpha$
radiation covers the sky, so we integrate the IPH flux that enters from all
directions (see Appendix C for further details). The flux entering the
atmosphere from each direction is calculated with an IPH background model of
Quémerais et al. (2013a). This model incorporates angular dependent partial
frequency redistribution to treat resonant scattering of solar Lyman-$\alpha$
radiation by the interplanetary hydrogen distribution (Quémerais, 2000;
Quémerais et al., 2003). The distribution is calculated from the interaction
of the local interstellar medium with the solar wind (Izmodenov et al., 2001,
2013), using a hydrogen density of $n_{H,TS}=0.09$ cm-3 and $n_{H}=0.14$ cm-3
in the local interstellar medium (Bzowski et al., 2009; Izmodenov & Alexashov,
2020). We do not include an additional contribution of a 40 R galactic
contribution (Gladstone et al., 2021; Pryor et al., 2022) that is isotropic.
However, at distances of 9-10 au the galactic contribution is small in
comparison to IPH-scattered solar flux. The brightness, line width and
temperature of the IPH Lyman-$\alpha$ is calculated by integrating the
emissions along the line-of-sight.

The brightness of the IPH emission line varies strongly with direction of
observation, being largest close to the sunward direction when at large
heliocentric distances. The position of the observer relative to the flow of
the local interstellar medium also impacts the variation of Lyman-$\alpha$
with line-of-sight, although to a lesser extent. Here, we consider two
positions in the IPH: the position of Saturn in 2006 and 2016. In both cases,
we construct a full sky map of the Lyman-$\alpha$ brightness at the position
of Saturn (e.g. Figure 6a). The IPH Lyman-$\alpha$ brightness can be
parameterised as a function of the angle to the direction of maximum
brightness ($\theta_{Max}$, Fig. 6b). We fit this as a sum of a quadratic and
an exponential with respect to $\cos\theta_{max}$. At heliocentric distances
of 10 AU, the maximum in the IPH density and Lyman-$\alpha$ brightness are
closely aligned with the sunward direction.

Figure 6: (left) Full sky map of IPH brightness at Saturn in 2016, modelled as
described in Section 2.3.2. (right) Modelled IPH Lyman-$\alpha$ brightness
against angle to direction of maximum brightness. The fit to the IPH
brightness is shown in blue.

####  2.3.3 Hydrogen effective optical depths

The radiative transfer model, combining scattering of solar and IPH
Lyman-$\alpha$ photons is used to calculate brightness as a function of the
optical depth of the effective H column in the atmosphere, emission and
incidence angle, and atmospheric temperature
$B_{RT}(\tau_{H},\theta_{em},\theta_{in},T_{Sat})$. By effective optical
depth, we refer to the vertical Lyman-$\alpha$ line center optical depth of
the H column above the methane homopause. It is important to note that we
assume all light at Lyman-$\alpha$ to be absorbed below the homopause. We do
not self-consistently simulate absorption by CH4 because seasonal changes to
the depth of the homopause are not known a priori. This effort represents the
first, zeroth order effort to constrain variations in the effective H
scattering column in the thermosphere. We have therefore chosen to follow this
simplified, retrieval-type approach in our modeling.

We use a variable temperature profile with latitude, based on stellar
occultations by UVIS (Brown et al., 2020) combined with CIRS limb scans (Brown
et al., 2024; Guerlet et al., 2018; Koskinen & Guerlet, 2018). Thermal
structure is also expected to be seasonally variable but the general
characteristics of the temperature distribution, including the gradient
between the auroral regions and the equator, are relatively stable, even on
multi-decadal timescales (Koskinen et al., 2021). We calculate the pressure
averaged temperatures
($\bar{T}_{P}=1/[\ln(p_{1}/p_{0})]\int_{P_{0}}^{P_{1}}T(d\ln P)$ ) above the
methane homopause, which are then fit with a 6th-order polynomial (red, Figure
8). Here, the CH4 homopause is defined as the location where $\tau_{CH_{4}}=$
1 at Lyman-$\alpha$ in the atmospheric structure models fitted to the
occultations and CIRS observations (e.g., Brown et al., 2024; Koskinen &
Guerlet, 2018). As noted above, the homopause location is obviously expected
to change over time but the pressure-averaged temperatures are nevertheless
assumed to remain relatively stable. The polynomial fit is used to convert the
latitude of observation into an atmospheric temperature in the radiative
transfer model.

For each pixel, characterised by incidence angle, emission angle and latitude
(temperature), the brightness against optical depth is interpolated to the
viewing geometry. Using this monotonic relationship, the observed
Lyman-$\alpha$ brightness for each pixel is converted to an effective hydrogen
optical depth for each pixel (Figure 7). We use Monte Carlo error estimation,
allowing for uncertainty in the Lyman-$\alpha$ brightness from the UVIS
instrument, as well as uncertainty in the solar flux normalisation and
absorption of the solar flux between the Sun and Saturn.

Figure 7: Retrieval of optical depth estimates (blue) by comparison of
Cassini/UVIS observations (red) with the radiative transfer model (black) for
a single pixel.  Figure 8: (blue) Pressure averaged temperatures above the
methane homopause, from a 1D photochemical model (see Section 2.4),
constrained by stellar UVIS occultations throughout 2017 (Brown et al., 2020).
(red) The temperatures are fitted with a 6th order polynomial.

####  2.3.4 Other sources of Lyman-$\alpha$

In comparing the radiative transfer model with the observed brightnesses, we
ignore the possibility of internal sources and scattering by suprathermal
atomic H. However, we considered the following potential emission sources that
are likely to be insignificant:

  * •

photoelectron impact excitation

  * •

H+ recombination

  * •

scattering by hot hydrogen in the thermosphere, produced via H_3+
recombination.

Waite et al. (1983) used a two-stream model to examine Lyman-$\alpha$
emissions driven by photoelectrons at Jupiter, finding a total contribution of
26 R. Extrapolating this to Saturn, we expect a factor 3-4 reduction in the
photoelectrons based on the reduced solar flux at Saturn and the emitting
column of H at Saturn is approximately a factor 3 smaller than at Jupiter.
Therefore, we expect photoelectron-induced Lyman $\alpha$ emissions to be
negligible compared to scattered solar flux.

Recombination is a well known astrophysical source of Lyman-$\alpha$
emissions, and a possible source in hot exoplanet atmospheres. In cold
planetary atmospheres, however, it is negligible. For H+ recombination, we
calculate the case-B recombination rate using H+ and e- profiles from the 1D
photochemical model (Moses et al., 2023), to estimate an upper limit for the
production of excited H atoms ($n\geqslant 2$). Integrating this over the
atmospheric column gives a column emission of only $\sim 10^{-3}$ R.

Observations of Lyman-$\alpha$ with high spectral resolution should be able to
identify, or provide an upper limit on, a hot population of hydrogen in the
atmosphere, such as HST/STIS echelle observations (see Fig. 3 in Ben-Jaffel et
al., 2023). However, the uncertainties in the high-spectral resolution HST
observations are significant, partly due to the subtraction of the strong
geocoronal Lyman-$\alpha$ emission. Consequently, retrieving a useful upper
limit on a hot atomic hydrogen population does not seem feasible. Ben-Jaffel
et al. (2023) compare a reference and boosted (with 2-3 times the atomic
hydrogen) photochemical model to the HST/STIS observations. The additional 280
R of emissions in the boosted model could similarly be attained by including a
hot atomic hydrogen population ($\tau_{H,Hot}=1$, $T_{Hot}=25,000$ K), with
minimal impact on the resulting lineshape. However, the mechanism for
producing a significant suprathermal hydrogen population remains unclear.

One potential source of hot hydrogen in the upper atmosphere is recombination
of H3+, producing H + H +H and H + H_2. We constrain this source by using the
photochemical model profiles (Moses et al., 2023). At zeroth order, we assume
local production of hot H (rates listed in Larsson et al., 2008) balanced by
local cooling through elastic collisions of the hot H with thermal H and H_2,
with collision cross sections from Krstić & Schultz (1999). Under this
assumption, the hot H population generated by recombination of H3+ in the
upper atmosphere is negligible, with column densities a factor $10^{7}$
smaller than the ambient atomic hydrogen column density. This hot H population
is too small to generate Lyman-$\alpha$ emissions comparable to scattering by
the thermal H population (a column of hot atoms a factor of $10^{4}$ smaller
than thermal H could generate similar emissions).

$H_{3}^{+}$ rain from the rings has been observed at Saturn (O’Donoghue et
al., 2019). However, the ring rain latitudes show no obvious correlation with
the observed Lyman-$\alpha$ bulge and it is difficult to anticipate a related
mechanism that could generate enough hot atoms to explain the bulge. While we
cannot rule out a hot population generated by another means, the observations
and modelling at present appear most consistent with emissions driven by
resonance scattering of solar flux by thermal atomic hydrogen.

###  2.4 Photochemical models of Saturn’s upper atmosphere

The upper atmospheric hydrogen is a probe of photochemistry deeper in the
atmosphere, as it is primarily produced through photolysis of methane (Moses
et al., 2000a), with only a small contribution from thermospheric chemistry.
We compare the inferred H optical depths (outlined in Section 2.3) to two
photochemical models: (a) 1D models tuned to the results of stellar
occultations observed by Cassini/UVIS in 2016-2017 (Brown et al., 2020, 2024)
and (b) a 2D model identical to Moses & Greathouse (2005), except we adopt
solar-cycle average incident ultraviolet flux (including at Lyman-$alpha$),
rather than tracking the $\sim$11 year solar cycle variation from the recent
era.

##### 1D models

Throughout 2016 and 2017, stellar occultations provided a pole-to-pole map of
the thermosphere (Brown et al., 2020) and were used to retrieve temperature
profiles as well as the densities of many atmospheric constituents (e.g. H2,
CH4 Brown et al., 2024). However, as we noted before, it is not possible to
retrieve the H density profile from stellar occultations, due to absorption of
starlight by the interstellar medium. The H density profile is calculated with
the 1D photochemical model, constrained by the profiles of other molecules.

For this study, we are interested in the hydrogen and methane profiles,
particularly the H column above the methane homopause. We consider optical
depths of $\tau_{CH4}=1$ and $\tau_{CH4}=10$ at Lyman-$\alpha$, using a cross-
section of $1.79\times 10^{-17}$ cm2 (Chen & Wu, 2004). For the purposes of
the comparison between the inferred optical depths and the model, the H
optical depth is integrated above the homopause altitude for each occultation
location, in line with the pressure averaged temperatures above the homopause
(see Figure 8).

Figure 9: Optical depth of H above the methane homopause ($\tau_{CH4}=1$) from
the 2D photochemical model, similar to Moses & Greathouse (2005). The $x$-axis
is solar longitude with the northern solstice occurring at $L_{S}=90^{\circ}$
and southern solstice at $L_{S}=270^{\circ}$.

##### 2D model

We use the seasonally and meridionally varying 2D model of Moses & Greathouse
(2005), which incorporates the time varying insolation to represent seasonal
changes with solar-cycle averaged ultraviolet fluxes. In this model, the
temperature-pressure profile is constant across latitudes and with season. It
is important to note that this model parameterizes vertical transport by using
an eddy diffusion profile that is constant with latitude and does not include
any meridional transport. We integrate the H optical depth above the methane
homopause ($\tau_{CH4}=1$) as a function of latitude and time (with the time
mapped to the solar longitude; see Figure 9). At present, photochemical models
that account for seasonal changes to the temperature structure and mixing
rates in the middle and upper atmosphere do not exist.The differences between
the inferred H column from the Lyman-$\alpha$ observations and the
photochemical model predictions identified in this study represent the first
step to guide the development of such models in the future.

##  3 Results

###  3.1 A comparison of UVIS observations with the IPH model

In this section, we compare observations of the IPH Lyman-$\alpha$ background
throughout 2006 with modelled IPH brightnesses (see Figures 10 and 11). The
set of observations is listed in Table 3. The brightnesses of the IPH
observations have been scaled to the solar flux on Jan 1st 2009, when the
solar activity was at a minimum. This corrects for variation of the IPH that
results from the Sun’s rotation. The scaled brightnesses are also given in
Table 3. The modelled IPH brightnesses (black) agree well with the
observations from Cassini/UVIS (blue), with an $R^{2}=0.64$. The brightness
increases from 250 R in the anti-sunward direction ($\cos\theta_{Max}=-1$) to
650 R near the sunward direction ($\cos\theta_{Max}=0.8$).

Figure 10: (left) Full sky map of the modelled IPH brightness at Saturn in
2006 (see Section 2.3.2). (right) Modelled IPH Lyman-$\alpha$ brightness
against angle to direction of maximum brightness. The fit to the IPH
brightness is shown in blue.

A cluster of 8 points at $\cos\theta_{Max}=0.4$ with $B=250\pm 40$ R is 100 R
smaller than predicted by the model. However, there are several observations
at similar angles with $B=450$ R, suggesting the IPH model is not
overestimating the brightnesses at this angle. Overall, the IPH observations
by Cassini are consistent with the predictions of the IPH model at a
heliocentric distance of 9.1 au.

Figure 11: Comparison of IPH Lyman-$\alpha$ brightness from Cassini/UVIS
observations throughout 2006 (blue) with the modelled brightness from the IPH
background model (black, see Section 2.3.2 and Figure 10). The $x$ axis is the
cosine of the angle between the observation direction and direction of maximum
brightness The black line gives the best fit, with the shaded region depicting
minimum and maximum modelled values.

###  3.2 Emission trends of Saturnian Lyman-$\alpha$ from multi-variate
regression analysis

(a)(b)(c)(d) Figure 12: (a,b) Testing data counts vs predicted brightness from
the MVR. (c) Observed brightnesses and (d) residuals vs predicted brightnesses
for the testing dataset (blue) with the averages and standard deviations shown
in red.

Figure 12 shows the predicted brightnesses vs the actual observed brightnesses
of Lyman-$\alpha$ for the multi-variate regression model outlined in Section
2.2 for the testing dataset (20% of points). The coefficients of the fit to
Eq. 1 are given in Table 1. The fit has an $R^{2}=0.791$ demonstrating that
the observed brightness can be well parameterised by the latitude, incidence
and emission angle once the observed brightness is scaled with the time-
dependent solar flux. The 3-variable regression does not capture the
brightnesses at small values, but above 250 R the model accurately predicts
the observed brightnesses.

(a)(b)(c) Figure 13: Comparison of the observed brightnesses from 2014-2016 in
the testing dataset (blue) to the multi-variate regression predictions
(orange) as a function of (a) emission angle, (b) solar incidence angle and
(c) latitude. The observed data are within the intervals
$\theta_{em}=[27,33]$, $\theta_{in}=[27,33]$ and $\phi_{Lat}=[18,22]$ for the
other two independent variables. In panel (b), the best fit of a cosine to the
observed brightnesses is given (green).

Figure 13 shows the dependence of the brightness on each of the independent
variables around the constant values of
$[\theta_{em},\theta_{in},\phi_{lat}]=[30,30,20]^{\circ}$ with the observed
brightnesses shown for points within 10% of the constant values. The
Lyman-$\alpha$ brightness is most strongly dependent on the incidence angle of
light arriving from the Sun, with brightness decreasing strongly from the
subsolar point ($\theta_{in}=0$) to the terminator plane at $\theta_{in}=90$.
Conversely, the brightness increases with emission angle, with the brightest
regions near the limb of Saturn’s disk. The dependence on the incidence angle
closely fits a $\cos\theta_{in}$ dependence.

Figure 14a shows the radiation field predicted by the trained model at a
latitude of $20^{\circ}$N, with the combined dependence of the brightness on
the incidence and emission angles. It again demonstrates substantial darkening
towards the terminator and brightening of the disk near the limb. The
dependences of the observed brightness on latitude, emission and incidence
angles are consistent when using different multi-variate regression models,
such as support vector regression using radial basis fields.

(a)(b) Figure 14: (a) Predicted brightness as a function of solar incidence
and emission angles from the MVR analysis at a latitude of 20∘N for the
2014-2016 dataset. (b) Modelled brightness as a function of solar incidence
and emission angles from the radiative transfer model (see Section 2.3) for an
optical depth of $\tau_{H}=10000$ and $T_{Sat}=350$ K. The dashed lines
illustrate the region in which we compare observations to the radiative
transfer model in Section 3.3.

In Figure 12b, there is non-linearity to the residuals, with a slight
parabolic shape between 200 and 1000R. Beyond 1000 R, the MVR model
overestimates the observed brightnesses, but the statistics at these values
are much lower ($<150$) compared to the intermediate brightnesses (200-1000
R). The structure in the residuals is not replicated when considering the
behaviour with respect to the independent variables (Figure 13). Each variable
shows no discernable behaviour in the residuals, suggesting higher order terms
for each variable are not required.

Table 1: Coefficients for the fit used in the multi-variate regression (see Eq. 1), for the 2014-2016 observations.  Variable | Coefficient | Mean Value | Confidence Interval
---|---|---|---
Constant | $p_{0}$ | $6.51\times 10^{2}$ | ($6.49\times 10^{2}$, $6.53\times 10^{2}$)
$\theta_{em}$ | $p_{1}$ | $3.73\times 10^{0}$ | ($3.68\times 10^{0}$, $3.77\times 10^{0}$)
$\theta_{in}$ | $p_{2}$ | $-1.01\times 10^{1}$ | ($-1.02\times 10^{1}$, $-1.01\times 10^{1}$)
$\phi_{lat}$ | $p_{3}$ | $-3.54\times 10^{0}$ | ($-3.64\times 10^{0}$, $-3.44\times 10^{0}$)
$\theta_{em}^{2}$ | $p_{4}$ | $7.95\times 10^{-2}$ | ($7.74\times 10^{-2}$, $8.17\times 10^{-2}$)
$\theta_{em}\cdot\theta_{in}$ | $p_{5}$ | $3.05\times 10^{-2}$ | ($2.85\times 10^{-2}$, $3.25\times 10^{-2}$)
$\theta_{in}^{2}$ | $p_{6}$ | $-9.36\times 10^{-2}$ | ($-9.54\times 10^{-2}$, $-9.17\times 10^{-2}$)
$\phi_{lat}^{2}$ | $p_{7}$ | $2.52\times 10^{-2}$ | ($2.25\times 10^{-2}$, $2.80\times 10^{-2}$)
$\phi_{lat}^{3}$ | $p_{8}$ | $2.55\times 10^{-3}$ | ($2.43\times 10^{-3}$, $2.66\times 10^{-3}$)

Figure 14b similarly shows the modelled brightness of Lyman-$\alpha$ from the
radiative transfer model (Section 2.3) at a temperature of 350 K and an atomic
Hydrogen optical depth of $\tau_{H}=10,000$. The trained MVR model shows very
similar behaviour to the brightness from the RT model, with a bright limb and
dark terminator region, indicating that the observed brightnesses can be
mostly explained by resonance scattering of Lyman-$\alpha$ by upper
atmospheric atomic hydrogen. The MVR and RT model differ in two main areas:
the bright limb of the RT model and the increase in brightness at low emission
angles observed in the MVR model. The very bright limb observed in the RT
model is a result of the plane-parallel assumption, which breaks down near the
terminator and near the limb of the disk. With a plane parallel-model, the gas
column the photons travel through is greatly overestimated, inflating the
modelled brightness. The increase in brightness at low emission angles for the
MVR may be a result of the limited data available at small emission angles
(see Figure 15), which do not strongly constrain the observations at low
emission and incidence angles.

Figure 15: Histogram of observation geometries (solar incidence angles and
emission angles) in the training dataset for the multi-variate regression from
2014-2016 (see Figures 13 and 14). Regions with fewer observations are less
well constrained in the MVR model.

The latitudinal variation seen in Figure 13c, has been disentangled from
variation with the geometry of the observations without reliance on a
radiative transfer model. The northern hemisphere bulge at low latitudes is a
result of a meridional structure in the upper atmosphere. Using the same
analysis for the 2004-2006 period, the latitudinal profile shows the reverse
trend, with the bulge appearing near 20∘S (see Figure 16). The brightness also
decreases into the winter hemisphere and towards the summer pole. This shows a
clear reversal with season in the latitudinal behaviour, and shows that the
bulge is not a permanent feature of the northern hemisphere. Instead, it is a
seasonally changing feature that appears in the summer or spring hemisphere.

(a)(b)(c) Figure 16: Comparison of the observed brightnesses in the 2004-2006
testing dataset (blue) to the multi-variate regression predictions (orange) as
a function of (a) emission angle, (b) solar incidence angle and (c) latitude.
The observed data are within the intervals $\theta_{em}=[27,33]$,
$\theta_{in}=[27,33]$ and $\phi_{Lat}=[-22,-18]$ for the other two independent
variables. In panel (b), the best fit of a cosine to the observed brightnesses
is given (green).

The source of the increased brightness at 20∘ in the summer hemisphere could
be driven by either enhanced H column in the region, or additional sources of
emission within the bulge region. We split the dataset into latitude bins of
10∘ and applied the same multi-variate regression (see Figures 29 and 35). At
all latitudes, the same dependence of the brightness on solar incidence and
emission angles was observed, with $R^{2}>0.7$ and $R^{2}>0.6$ throughout
northern and southern summers respectively. Some latitude bins were limited by
the phase space coverage of the data (Figures 15, 36 and 30), with small
emission angles not probed in the winter hemisphere or close to the poles. The
consistency of the relation between the incidence and emission angles with
latitude implies that there is not a substantial additional internal emission
source in the northern hemisphere bulge. Therefore, the variation in
Lyman-$\alpha$ brightness can likely be attributed to variation of the H
column above the methane homopause. The alternative would be to attribute the
variation to a seasonal change in the hot atomic H population, although no
source of hot atoms that would have the required behaviour has been identified
to date.

###  3.3 Seasonal Variation of Saturn’s atmosphere

####  3.3.1 A case study of northern hemisphere summer

(a)(b) Figure 17: (a) Lyman-$\alpha$ brightness in observation
234SA_EUVFUV002, with the ring geometry, limb and terminator of Saturn
highlighted in blue. The considered region for comparison to the radiative
transfer model is bounded by the light blue line. (b) Modelled Lyman-$\alpha$
brightness using $\tau_{H}=10,000$ and the temperature dependence in Fig. 8 in
the RT model (see Section 2.3).

We consider first an observation of Saturn on 25th March 2016, when the
subsolar latitude was 26.0∘, with incidence and emission angles from 0 to 65∘.
We exclude latitudes poleward of 60∘, and in the ring shadow region which
extends from 14∘S to the south pole. This includes 1617 pixels across the
dayside region highlighted in Figure 17, accumulated over 4.5 hours. The
observed brightness is constant between the latitudes of 0 and 20∘N at $596\pm
15$ R, before decreasing to $518\pm 12$ R 26∘N to 60∘N. The observed
brightness also drops sharply to 421 R at $9.4^{\circ}$S towards the ring-
shadowed region (see Figure 18).

The modelled brightness for a constant H Lyman-$\alpha$ line center optical
depth (hereafter, the H optical depth) of 10,000 also shown in Figure 18
agrees well with the brightness between 0 and 25∘, but outside this range the
modelled brightness increases in contrast to the observation. The increased
brightness is a result of larger emission angles for the southern and higher
latitudes. The scattered Lyman-$\alpha$ from the IPH background has a fairly
constant contribution of $78.5\pm 4.4$ R across latitudes, which is 12% of the
brightness of the scattered solar flux.

(a)(b) Figure 18: (a) Observed Lyman-$\alpha$ brightness (black) for
observation 234SA_EUVFUV002 against latitude. The modelled brightness for the
observation geometry using $\tau=10000$ and the temperature variation in Fig.8
is shown with (blue) and without (orange) the contribution of scattered IPH
Lyman-$\alpha$. (b) Optical depth (blue) vs. latitude retrieved from
comparison of Cassini/UVIS observations. The optical depths of the methane
homopause (using $\tau_{CH4}=1$, triangles; and $\tau_{CH4}=10$, crosses)
predicted by 1D photochemical models tuned to 2017 stellar occultations are
shown. (orange) H optical depth from the 2D photochemical model on 26 Mar 2016
at subsolar latitude of 26.0∘ and $L_{S}=81.2^{\circ}$ (see Figure 9).

Figure 18b shows the H optical depths retrieved from comparison of the
observations with the RT model. The close agreement between the modelled and
observed brightness between 0 and 25∘ is reflected by the constant optical
depth of $10,800\pm 300$ R over this range. Northward of 25∘N the optical
depth decreases continuously towards $\tau=5000$ at 60∘N, as the observed
brightness declines relative to the expected values from the RT model based on
constant H optical depth. This decrease in the H optical depth is not
predicted by the 1D photochemical model, which predicts that the H optical
depth is approximately constant at 10,000 up to 60∘N. However, the 2D
photochemical model (orange) does show a slight decrease in the H optical
depth from the peak at 10∘N, but only by 10%, compared to the 50% reduction
retrieved from the UVIS data. Southward of the equator, the H optical depth
sharply decreases between 5.6∘N and $9.4^{\circ}$S latitude towards the ring
shadow, in close agreement with the photochemical model.

####  3.3.2 A case study of southern hemisphere summer

Figure 19: Observed Lyman-$\alpha$ brightness for the 00ASA_COMPSITA012_CIRS
observation on 8 Nov 2004 with a subsolar latitude of -23.5∘. The region
modelled by the radiative transfer code is highlighted by the light blue line,
with Saturn, the rings and ring shadow regions superimposed in dark blue.
(a)(b) Figure 20: (a) Observed Lyman-$\alpha$ brightness (black) for
observation 00ASA_COMPSITA012_CIRS against latitude. The modelled brightness
for the observation geometry using $\tau=10000$ and the temperature variation
in Fig.8 is shown with (blue) and without (orange) the contribution of
scattered IPH Lyman-$\alpha$. (b) Optical depth (blue) vs. latitude retrieved
from comparison of Cassini/UVIS observations with the radiative transfer
model. (orange) H optical depth from the 2D photochemical model on 8 Nov 2004
at subsolar latitude of -23.5∘ and $L_{S}=302^{\circ}$.

Figures 19 and 20 show results for an observation from 8 Nov 2004, during
southern hemisphere summer, with a subsolar latitude of 23.5∘S. The spatial
coverage of Saturn’s disk was much narrower for this case (see Figure 19), so
the variation in the modelled brightnesses is much smaller at each latitude.
The observed brightness decreases sharply from $1030\pm 38$ R at $60^{\circ}$S
to $780\pm 31$ R at $40^{\circ}$S as a result of the decreasing emission
angle. The lower brightness is closely matched by the modelled brightness
based on constant H optical depth, and can be attributed wholly to changes in
the viewing geometry.

The observations and constant H optical depth model, however, diverge between
$40^{\circ}$S and the equator, with the modelled brightness continuing to
decrease to $690\pm 7$ R while the observed brightness increases to $888\pm
36$ R in this region. The retrieved H optical depths (blue, Figure 20b)
therefore increase substantially over this latitude range, peaking at
$\tau_{H}=17,500\pm 2500$ at $13^{\circ}$S. Equatorward of the peak, the H
optical depths decrease sharply towards the ring shadow region, closely
mirroring the behaviour observed in the northern hemisphere summer hemisphere
(see Figure 18), and predicted by the 2D seasonal photochemical model
(orange). The magnitude of the H optical depths in the summer hemisphere are
also broadly consistent with the model, which predicts peak H optical depths
of $\tau_{H}=16,600$ at $3^{\circ}$S.

####  3.3.3 Implications for upper atmospheric hydrogen

Having examined two observations from the NH and SH summers, we now apply the
same analysis to all the dayside disk observations from Cassini/UVIS from 2004
through to 2016. We exclude 2017 as there is little coverage of Lyman-$\alpha$
emissions from the dayside Saturn disk. We have calculated H optical depths
for each pixel (as outlined in Section 2.3) and taken annual averages and
standard deviations. We have separated the SH summer (2004-2006, Figure 21),
equinox period (2007-2010, Figure 22) and NH summer periods (2011-2016, Figure
23).

Much like in the case study (Figure 20), the annual averages exhibit a
substantial latitudinal variation during the southern summer (Figure 21), with
H optical depths increasing from 60∘S towards the hydrogen bulge in the SH
between 10∘S and 25∘S. Northward of the bulge, the H optical depth declines
sharply with latitude. There are few available observations north of 20∘N
throughout 2004-2006, as these latitudes are typically obscured by either the
rings or the ring shadow region. The sharp decrease in H optical depth around
the equator is also observed in the photochemical model. The decrease in the H
optical depth at the equator is shifted southward relative to the
photochemical model, and the gradient shifts southward from 2004 to 2006. This
may result from the movement of the southern boundary of the ring shadow,
which begins at 6.6-17.3∘N in 2004 and reaches 4.4-12.3∘N in 2006. The peak H
optical depth, seen in the southern hemisphere bulge, decreases from 2004 to
2006. There is substantially more variation than expected based on the
photochemical model (Moses & Greathouse, 2005) and is discussed in Section
4.3.

During the equinox period (Figure 22), the H optical depths increase
continuously from the polar regions at $\pm 60^{\circ}$ towards the equator.
They agree well with the modelled depths from the 2D photochemical model
(dash-dotted lines) in both magnitude and meridional structure. The equatorial
latitudes have been excluded as they are obscured by the ring shadow and ring
atmosphere during these years. The observed brightness decreases around the
equator, but we cannot determine whether the H column is smaller due to the
uncertainty in the illumination conditions. The H optical depths throughout
this period show less variation than seen in the summer hemisphere (Figures 21
and 23), particularly in the northern hemisphere where the maximum H optical
depth is $\tau_{H}=17,200\pm 2300$.

In the southern hemisphere during the equinox period, the peak H optical
depths are similar, except in 2009 and 2010, which both exhibit significantly
reduced H optical depths at mid-latitudes. The lower brightnesses compared to
2007 and 2008 are unexpected, much like the substantial variation during the
southern hemisphere summer. The retrieved H optical depths are also 40%
smaller than the photochemical model predictions at 20∘S. We note that the
highest optical depths are not consistently located in the more illuminated
hemisphere. While this is the case in 2007 and 2010, the optical depths in
2008 are largest in the northern hemisphere despite a subsolar latitude of
6∘S.

During the northern summer (Figure 23), the hydrogen bulge has shifted from
the southern hemisphere (Figure 21) to the northern hemisphere, along with the
subsolar latitude. The H optical depth decreases continuously towards 60∘N, as
seen in the case study (Fig. 20). From 2011-2013, the southern hemisphere is
not completely shadowed by Saturn’s rings. The retrieved H optical depths are
roughly constant at $\tau_{H}=5000$ in the winter hemisphere and much smaller
than the northern hemisphere peaks. This is also in good agreement with the 1D
models based on 2017 stellar occultations (see Section 2.4), which predicted H
optical depths of $\tau_{H}=3000$ in the shadowed winter hemisphere. As seen
during the earlier periods, there is substantial variability in the peak H
optical depths over time. Large H optical depths of $\tau_{H}=22,100\pm 6990$
and $\tau_{H}=24,000\pm 6,200$ are observed in 2013 and 2014, while in 2012
and 2015 the peak H optical depths are $\tau_{H}=10,000\pm 4100$. The H
optical retrieved from the observations in 2013 and 2014 are twice as large as
those predicted by the 1D photochemical models and 50% larger than the
predictions of the 2D model.

Figure 21: Yearly averages of the optical depths during the southern
hemisphere summer (2004-2006) (colored circles), with shaded regions showing
the uncertainty (see Section 2.3). The subsolar latitude for each year is
given by the dashed vertical lines. The dash-dotted lines show $\tau_{H}$
predictions from the 2D photochemical model in each year. Figure 22: Yearly
averages of the optical depths around equinox (2007-2010) (colored points),
with shaded regions showing the uncertainty (see Section 2.3). The subsolar
latitude for each year in given by the dashed vertical lines.The dash-dotted
lines show $\tau_{H}$ predictions from the 2D photochemical model in each
year.  Figure 23: Yearly averages of the H optical depths during the northern
hemisphere summer (2011-2016, colored points), with shaded regions showing the
uncertainty (see Section 2.3). The subsolar latitude for each year in given by
the dashed vertical lines, with the corresponding color. (black points) H
optical depths above the methane homopause (at $\tau_{CH4}=1$, triangles; and
$\tau_{CH4}=1$, crosses) from 1D photochemical models based on 2017 UVIS
occultations (see Section 2.4). The dash-dotted lines show $\tau_{H}$
predictions from the 2D photochemical model in each year.

##  4 Discussion

###  4.1 Source of Lyman-$\alpha$ emissions

The multivariate regression analysis of the Lyman-$\alpha$ brightness
observations strongly supports the dominance of resonance scattering as the
source of non-auroral Lyman-$\alpha$ emissions, with scattered solar flux
comprising the majority of the observed emissions. The strong dependence of
the Lyman-$\alpha$ brightness on the solar incidence angle (see Figure 13b)
demonstrates the importance of solar photons in generating the observed
emissions. Scattered photons from the IPH background would have a much weaker
dependence on solar incidence angle, as they enter the atmosphere from all
directions. Potential internal sources of Lyman-$\alpha$ photons would also
not exhibit the variation with solar incidence angle, such as the electroglow
proposed by (Shemansky et al., 2009) and others. While solar flux dominates,
the contribution of scattered IPH Lyman-$\alpha$ photons is important for
estimating of the H optical depth.

The radiation field retrieved through the multivariate regression analysis
remains consistent across latitudes and mission periods. At all latitudes, the
observed brightness can be fit with a quadratic function of incidence and
emission angles, with brightness decreasing strongly towards larger incidence
angles. For internal sources of emissions, we would not expect a strong
dependence on solar incidence angle. The peak in the brightness and optical
depth at latitudes of 10-20∘ in the summer hemisphere is therefore unlikely to
be driven by an internal emission source. Our results, however, do not
discount the possibility of hydrogen or water entering the upper atmosphere
from the rings in the equatorial region, such as that detected by INMS (Waite
et al., 2018; Yelle et al., 2018; Serigano et al., 2020, 2022), and leading to
local increases in the H density, although Cassini/UVIS occultations and
photochemical modelling during the Grand Finale do not suggest that a
substantial fraction of the influx materials vaporise in the thermosphere
(Moses et al., 2023). A smaller suprathermal (25,000 K) hydrogen population
(0.1% of the ambient H) could drive substantial emissions. The existence of
such a population is currently hypothetical but can be better constrained with
detailed modelling of emission profiles from Saturn’s limb. Assuming no hot
hydrogen population, the $\tau_{H}$ variation is similar to that predicted by
the 2D photochemical model, which predicts a sharp decrease in the H column in
the winter hemisphere (see Fig 9). This strongly suggests that the low-
latitude brightness peak is seasonal in nature and origin.

###  4.2 The IPH model and Cassini/UVIS calibration

The brightnesses predicted by our IPH model (Section 2.3.2) agree well in both
viewing direction and magnitude with observations of the IPH Lyman-$\alpha$
emissions by Cassini/UVIS (see Figure 11).

A proposed recalibration of Cassini/UVIS at Lyman-$\alpha$ by a factor 1.7
(Ben-Jaffel et al., 2023; Pryor et al., 2024) would result in much larger
observed brightnesses for the IPH (1255 R at $\cos\theta_{Max}=0.94$) compared
to those predicted by the IPH model (762 R). This would require a significant
increase in the hydrogen density at the termination shock, or galactic
emissions beyond the scale of current estimates (Gladstone et al., 2021). The
angular dependence of the disparity could not be rectified by additional
galactic Lyman-$\alpha$ flux. We note that Ben-Jaffel et al. (2023) obtained
the calibration factor by comparing Cassini/UVIS observations with Saturn disk
brightnesses observed by HST/STIS in 2017 from Earth orbit. There were,
however, no simultaneous dayside observations of Saturn’s disk obtained by
Cassini/UVIS in 2017 and the comparison also depends on viewing geometry as
indicated here. The required corrections for IPH absorption and geocorona also
make it difficult to extract the line shape from HST/STIS observations.

Our IPH model, which is constrained by observations of New Horizons Alice and
SWAN/SOHO (Quémerais et al., 2013a; Izmodenov et al., 2013), predicts the
brightness of the IPH Lyman-$\alpha$ to reach 250 R in the anti-sunward
direction during the 2006 observations and rise to 900 R in the sunward
direction. SWAN/SOHO was cross-calibrated with HST/STIS in 2001, and later
observations in March 2023 show the instruments agree within 15%. (Quémerais
et al., 2013a). The good agreement between the model and UVIS observations
suggests that the calibrations of Cassini/UVIS and the other instruments are
consistent, although we cannot discount the possibility of recalibration by up
to 20%.

###  4.3 Seasonal variability in thermospheric atomic hydrogen

(a)(b) Figure 24: (a) Optical depth at 26.7∘N$\pm 5$ (northern solstice
latitude, black dashed line), throughout the Cassini mission from the (black)
UVIS observations and RT model comparison (see Section 2.3) and (orange) the
2D photochemical model (see Section 2.4). The subsolar latitude is shown in
green. (b) The same as above but for $-26.7^{\circ}$ (southern solstice
latitude ). The large error bars from the UVIS retrieval reflect shorter term
variability in the Lyman-$\alpha$ brightness and atomic hydrogen atmosphere,
within each temporal bin. The trends in the optical depths shown here
demonstrate the long timescale variation of the atmosphere.

Assuming that a population of hot atoms is not significant, the H optical
depths retrieved from the UVIS observations with the RT model (see Section
3.3) agree well with the optical depths predicted by the photochemical model.
The peak in brightness, identified in the northern hemisphere by Ben-Jaffel et
al. (2023), translates to an increase in the atomic hydrogen optical depth
above the methane homopause. However, the bulge at 20∘ N is not a permanent
feature of the northern hemisphere, but shifts with season and is observed in
the southern hemisphere during the southern summer (see Figure 21). A similar
peak in the H optical depth is also predicted by the 2D photochemical model,
which clearly shows a reversal in season (see Figure 9).

The poleward decrease in H optical depth from 20 to 60∘N in northern summer is
also predicted by the 2D photochemical model (Figure 9), although not to the
same extent as estimated by the UVIS/RT model comparison. The poleward
decrease disagrees with the 1D photochemical models, which were tuned to
stellar occultations in 2017 (black points, Figure 23; Brown et al., 2024).
These models do not show a significant decrease in the H optical depth towards
higher latitudes, and also do not exhibit the same bulge around 20∘N. The
magnitude of the H optical depths is consistent with the results of the
photochemical models (both 1D and 2D) with typical optical depths between
$\tau_{H}=5,000-15,000$ (see Figures 9, 20, and 23). The proposed revision of
the Cassini/UVIS sensitivity by Ben-Jaffel et al. (2023), increasing observed
Lyman-$\alpha$ brightnesses by a factor 1.7, would result in more significant
enhancements in the retrieved optical depths. While the latitudinal and
seasonal trends would be unaffected, the non-linear relationship between
optical depth and brightness (see Figure 3) would see an increase of 2.5 to
$3\times$ in the H optical depth compared to the current estimates. As a
result, the retrieved hydrogen column densities would be inconsistent with
either the 1D or 2D photochemical models, albeit at the same time roughly
following the seasonal behaviour predicted by the models.

The proposed calibration factor means that an additional emission source or an
alternative source of H would be required. The strong dependence of brightness
on solar incidence angle, identified in the multi-variate regression (see
Section 3.2), excludes an emission source of the required magnitude other than
resonance scattering of solar flux. While an additional source of hydrogen is
not excluded by the current analysis, the magnitude of atomic H required would
be beyond the estimates of what is possible from equatorial ring inflow (up to
$1.75\times$, Ben-Jaffel et al., 2023), and would be required planet-wide.
Photochemical modelling suggests that ring rain entering the atmosphere as
solid particles is unlikely to vaporize and contribute to the background
gaseous atmosphere, and therefore should not significantly enhance the H
column. A smaller population of hot H in the thermosphere is not excluded by
our analysis but remains unidentified.

The year-to-year variability of the optical depths retrieved, however, is
highly unexpected, with variation by a factor 2 observed from one year to the
next (see Figure 24). The 2D photochemical model (orange), which incorporates
the variation of the subsolar latitude but neglects EUV solar-cycle variation,
predicts minimal short-term variation of the H column above the homopause.
Between solar minimum and maximum, a variation by about a factor of 2 is
expected in the H column (Ben-Jaffel et al., 2023), but on the shorter
timescales of the observed behaviour, the variability of the H optical depth
is far greater than expected.

To illustrate the variability, Figure 24a shows the optical depth at $26.7\pm
5^{\circ}$N and Figure 24b for $26.7\pm 5^{\circ}$S, the subsolar latitudes of
the northern and southern hemisphere solstices. At $26.7\pm 5^{\circ}$S, the
2D photochemical model predicts a continuous and monotonic reduction in the H
optical depth above the methane homopause from 2004 to the end of mission in
2017, while the reverse is true for the northern hemisphere.

The retrieved H optical depths from Cassini/UVIS show the same overall trend
with time for $\phi_{lat}=26.7^{\circ}$S, initially large at values of
$\tau_{H}=18,000$ and subsequently decreasing into the NH summer, when the
ring shadows much of the southern hemisphere. However, the optical depth
appears to oscillate around a mean value similar to the photochemical model.
The trend is repeated in the northern hemisphere, with a sinusoidal time
dependence around the photochemical model average. This indicates that there
may be additional time-dependent behaviour that is modifying the upper
atmospheric hydrogen content with a timescale of several years. The source of
this time-variability is not incorporated into the photochemical model and
requires further investigation.

The solar cycle is a natural candidate for this temporal dependence, which was
at a minimum in 2009 and peaked in 2013 and 2014 (see Figure 4). The variation
of the UV flux could modulate chemistry (e.g. methane photolysis) and
temperature profiles in the thermosphere leading, which in turn would impact
the upper atmospheric hydrogen. While some years could be explained by this
process (e.g. large optical depths in 2014, Figure 23), the variation of the
thermospheric hydrogen often bucks trends with the solar cycle (e.g. 2015 vs
2016). As such, there is likely an alternative or additional driver of changes
in the thermosphere.

One possible source of the variation in the inferred H optical depths could be
changes in the thermospheric temperature. The 2D seasonal photochemical model
does not include temperature changes with either season or latitude.
Occultations and circulation models have shown that thermospheric temperatures
vary both seasonally and meridionally (Müller-Wodarg et al., 2019; Brown et
al., 2020; Koskinen et al., 2021). Increases in thermospheric temperatures
lead to increases in the scale height and the brightness from resonant
scattering of solar flux. The factor of two variation in the inferred optical
depth (see Figure 24) could be reconciled by a 40% temperature change (i.e.,
350 K to 500 K), for the same density of H above the methane homopause. For
example, a temperature variation of the order of 150 K between 2006 and 2017,
with a peak temperature around 2010-2012, has been observed in the middle
thermosphere around the equator in the UV occultation data (Koskinen et al.,
2021). In addition, the homopause depth likely changes over time, due to the
effect of the changing seasons on dynamics in the middle atmosphere. For
example, downwelling near the homopause would drive methane deeper in the
atmosphere and increase optical depths. Disentangling the possible drivers of
atomic hydrogen variability will require a seasonal model in which the
temperature structure and homopause depth also change with time, combined with
an analysis of the occultations, H2 emission data, and limb scans from the
duration of the Cassini orbital mission. We note that the analysis of solar
occultations and limb scans can be used to retrieve vertical profiles of H and
therefore further constrain additional degradation of the Cassini/UVIS
instrument and calibration at Lyman-$\alpha$.

##  5 Conclusion

We have examined the extensive dataset of Lyman-$\alpha$ emissions from the
Saturn disk throughout the Cassini mission, as well as observations of
Lyman-$\alpha$ emitted from the interplanetary hydrogen background. We
compared the IPH Lyman-$\alpha$ observations with the model of Quémerais et
al. (2013a), which is calibrated with observations from SWAN, New Horizons
Alice, and other platforms (Quémerais et al., 2013a; Izmodenov et al., 2013).
The Cassini/UVIS observations and the IPH model agree well with each other,
suggesting that the calibration of Cassini/UVIS at Lyman-$\alpha$ is
consistent with the instruments underlying the IPH model.

We applied a multi-variate regression analysis to the observed brightness
throughout the mission, to disentangle variations as a result of observation
geometry and those from meridional changes in the atmosphere. The emission
brightness from Saturn’s disk is dependent on four key variables: solar flux
at the top of the atmosphere, latitude, and the emission and incidence angles
applicable to the observations. The dependence of the observed Lyman-$\alpha$
brightness on the emission and incidence angles agrees closely with a model of
resonant scattering of solar photons, exhibiting a strong decrease of the
observed brightness with increasing solar incidence angle. We therefore
exclude the possibility of a substantial internal source of Lyman-$\alpha$
emissions outside auroral regions.

We observe a bulge in the Lyman-$\alpha$ emissions in both the northern and
southern hemispheres, during their respective summer seasons. Therefore, we
conclude that the bulge previously reported by Ben-Jaffel et al. (2023) in the
northern hemisphere during spring and summer is a seasonally modulated
feature. Around the equinox, the Lyman-$\alpha$ brightness and effective H
optical depths increase towards the equator (once the ring and ring shadow are
excluded) in both hemispheres, giving the equatorial region an appearance of a
bulge.

We compared observations of Lyman-$\alpha$ emissions taken during the Cassini
orbital mission to a radiative transfer model, using the model to retrieve
estimates of the H optical depth above the strongly-absorbing methane
homopause level. The magnitude and latitudinal variation of the H column agree
well with the predictions of the photochemical model (2D in particular),
albeit with much more substantial temporal variability in the observations. A
comparison of the temporal changes in the southern and northern hemispheres
again show further evidence of seasonal change in the upper atmospheric
hydrogen, with a peak in the effective H optical depths at a latitude of 20∘
in the summer hemisphere. The atomic H column decreases sharply into the
winter hemisphere towards the ring shadow and more gradually toward higher
latitudes ($60^{\circ}$). While the seasonal photochemical model of Moses &
Greathouse (2005) predicts roughly constant peak optical depths of
$\tau_{H}=15,000$ with season, our peak optical depth estimates varied by up
to a factor 2 year-to-year. At latitudes of $\pm 26.7^{\circ}$, the optical
depth appears to show a sinusoidal variation in time around the mean depth of
the photochemical model, especially in the northern hemisphere. The source of
this variation requires further examination.

Observations of emissions from the limb and terminator of the planet have been
avoided in this study due to the plane-parallel assumption used in the
radiative transfer model. Further work is required to analyse the limb and
near-limb observations, which would provide a constraint on the vertical
profile of the atomic hydrogen in the thermosphere. This would provide much
greater constraints on the possibility of a suprathermal hydrogen layer in the
upper atmosphere than nadir scans that probe scattering by the bulk of the H
column. Additionally, the Lyman and Werner bands of H2 and He 584Å line
emission should also be addressed to infer constraints on energy deposition as
well as the H2 and He densities in the upper atmosphere.

## Acknowledgments

PS, TTK and JM acknowledge support by the NASA Cassini Data Analysis Program
grant 80NSSC22K0306. ZB acknowledges support by the NASA/CDAP grant
80NSSC19K0902. PL acknowledges support from the project ATMOHAZE within the
framework of the CNRS-UArizona collaboration initiative Searching for
Habitable Worlds, in the Solar System and Beyond.

##  Appendix A Radiative transfer model

The radiative transfer model is a plane-parallel model based on iterative
doubling and adding of layers. Two layers of the atmosphere, $A$ and $B$ with
$A$ above $B$ (see Figure 25), are combined with

A$D_{A}$$S_{A}$$T_{A}$B$U_{A}$$S^{*}_{A}$$T^{*}_{A}$$S_{B}$$T_{B}$ Figure 25:
Diagram of layers A and B, with optical depths $\tau_{A}$ and $\tau_{B}$.
Upward ($U_{A,B}$) and downward fluxes for each layer are shown, in addition
to the scattering ($S_{A,B}$) and transmission functions ($T_{A,B}$). Starred
variables are oriented opposite to the typical direction and are important for
multiple scattering.

$U_{A}=S_{B}\,\exp\bigg{[}-\tau_{A}\frac{\psi(x_{in})+\bar{\omega}_{a}}{\mu_{in}}\bigg{]}+S_{B}\cdot
D_{A}$ (A1) $D_{A}=T_{A}+S^{*}_{A}\cdot U_{A}$ (A2)
$S_{A+B}=S_{A}+\exp\bigg{[}-\tau_{A}\frac{\psi(x_{em})+\bar{\omega}_{a}}{\mu_{em}}\bigg{]}U_{A}+T^{*}_{A}\cdot
U_{A}$ (A3)

$T_{A+B}=\exp\bigg{[}-\tau_{A}\frac{\psi(x_{in})+\bar{\omega}_{a}}{\mu_{in}}\bigg{]}T_{B}+\\\
D_{A}\exp\bigg{[}-\tau_{B}\frac{\psi(x_{em})+\bar{\omega}_{a}}{\mu_{em}}\bigg{]}+T_{B}\cdot
D_{A}$ (A4)

where

  * •

$U,D=U,D(x_{em},\theta_{em},\phi_{em};x_{in},\theta_{in},\phi_{in})$ are the
upward and downward fluxes at the layer boundaries.

  * •

$x_{em},x_{in}$ are the emitted and incident frequency

  * •

$\theta,\theta_{0}$ are the angles of emission and incidence, with
$\mu=\cos\theta$.

  * •

$\tau$ is the optical depth of a layer

  * •

$S$ and
$T=S,T(\tau;x_{em},\theta_{em},\phi_{em};x_{in},\theta_{in},\phi_{in})$ are
the scattering and transmission functions for each layer with the starred
values indicating they are defined opposite to the typical direction
(scattering is typically upwards, transmission downwards).

  * •

$\psi(x)$ is the Lyman-$\alpha$ lineshape, characterised by a Voigt profile at
the upper atmospheric temperature

  * •

$\bar{\omega}_{a}$ is the absorption albedo, which has been set to 0
throughout

We note that Equations A2 and A3 differ slightly from Equations 8 and 9 in
Yelle et al. (1989), as we more explicitly define the directions of the
scattering function. The final term in Eq. 10 of Yelle et al. (1989) is a typo
and is corrected in Equation A4.

The dot product is defined as

$S_{A}\cdot
U_{A}=\frac{1}{4\pi\mu}\int\limits_{0}^{2\pi}d\phi^{\prime}\int\limits_{0}^{1}d\mu^{\prime}\int\limits_{-\infty}^{\infty}dx^{\prime}\bigg{\\{}\\\
S_{A}(\tau_{A};\bm{x}_{em};\bm{x}^{\prime})U_{A}(\tau_{A};\bm{x}^{\prime};\bm{x}_{in}).\bigg{\\}}$
(A5)

where the vectors $\bm{x}=(x,\mu,\phi)$ for each set.

The brightness of scattered solar flux is given by

$4\pi I_{Sol}(\mu_{em},\mu_{in})[R]=\Delta\nu_{D,Sat}\times 10^{-6}\times\\\
\int_{-\infty}^{\infty}dx_{in}\int_{-\infty}^{\infty}dx_{em}S(\tau;\bm{x}_{em};\bm{x}_{in})F_{Sol}(x_{in})$
(A6)

where $F_{Sol}$ is the solar Lyman-$\alpha$ flux incident on the atmosphere,
with an incidence angle of $\theta_{in}$. The Doppler width in Saturn’s
thermosphere is

$\Delta\nu_{D,Sat}=\frac{\nu_{0}v_{th}}{c}=\frac{\nu_{0}}{c}\sqrt{\frac{2k_{B}T_{Sat}}{m}}.$
(A7)

.

##  Appendix B Thin Layer approximations

From Yelle (1988), we have the equation for the scattering function, $S$:

$\Big{[}\frac{\psi(x_{em})}{\mu_{em}}+\frac{\psi(x_{in})}{\mu_{in}}\Big{]}S(\tau;\bm{x}_{em};\bm{x}_{in})+\frac{\partial
S(\tau;\bm{x}_{em};\bm{x}_{in})}{\partial\tau}=\\\
R(\tau;\bm{x}_{em};\bm{x}_{in})+\frac{1}{4\pi}\int\limits_{0}^{2\pi}\mathrm{d}\phi^{\prime}\int\limits_{0}^{1}\frac{\mathrm{d}\mu^{\prime}}{\mu^{\prime}}\int\limits_{-\infty}^{\infty}\mathrm{d}x^{\prime}\bigg{\\{}\\\
R(\tau;\bm{x}_{em};x^{\prime},\mu^{\prime},\phi^{\prime})S(\tau;x^{\prime},\mu^{\prime},\phi^{\prime};\bm{x}_{in})\bigg{\\}}\\\
+\frac{1}{4\pi}\int\limits_{0}^{2\pi}\mathrm{d}\phi^{\prime}\int\limits_{0}^{1}\frac{\mathrm{d}\mu^{\prime}}{\mu^{\prime}}\int\limits_{-\infty}^{\infty}\mathrm{d}x^{\prime}\bigg{\\{}\\\
S(\tau;\bm{x}_{em};x^{\prime},\mu^{\prime},\phi^{\prime})R(\tau;x^{\prime},\mu^{\prime},\phi^{\prime};\bm{x}_{in})\bigg{\\}}\\\
+\frac{1}{16\pi^{2}}\int\limits_{0}^{2\pi}\mathrm{d}\phi^{\prime}\int\limits_{0}^{1}\frac{\mathrm{d}\mu^{\prime}}{\mu^{\prime}}\int\limits_{-\infty}^{\infty}\mathrm{d}x^{\prime}\int\limits_{0}^{2\pi}\mathrm{d}\phi^{\prime\prime}\int\limits_{0}^{1}\frac{\mathrm{d}\mu^{\prime\prime}}{\mu^{\prime\prime}}\int\limits_{-\infty}^{\infty}\mathrm{d}x^{\prime\prime}\bigg{\\{}\\\
S(\tau;\bm{x}_{em};x^{\prime},\mu^{\prime},\phi^{\prime})\\\ \times
R(\tau;x^{\prime},-\mu^{\prime},\phi^{\prime};x^{\prime\prime},\mu^{\prime\prime},\phi^{\prime\prime})S(\tau;x^{\prime\prime},\mu^{\prime\prime},\phi^{\prime\prime};\bm{x}_{in})\bigg{\\}}$
(B1)

with $\bm{x}_{in}=(x_{in},\mu_{in},\phi_{in})$ and $\bm{x}_{em}$ is the
corresponding vector for the emitted flux $x_{in,em}$ are the incident and
emitted frequencies $\mu_{in,em}=\cos\theta_{in,em}$ with incidence and
emission angles $\theta_{in,em}$ $\phi_{in,em}$ are the azimuthal incidence
and emission angles $\tau$ \- optical depth of the layer at the line centre
$\psi(x)$ \- line shape as a function of frequency $S$ is the scattering
function of the incident flux from $(x_{in},\mu_{in},\phi_{in})$ to
$(x_{em},\mu_{em},\phi_{em})$ by the layer of thickness $\tau$ $R$ is the
angular dependent partial frequency redistribution function $R_{II}$ from
Hummer (1962). Neglect the multiple scattering terms as we consider a thin
layer and set

$\eta=\frac{\psi(x_{em})}{\mu_{em}}+\frac{\psi(x_{in})}{\mu_{in}}$ (B2)

An integration factor of $\exp(\eta\tau)$ gives

$\frac{\partial\big{(}S(\tau,x_{em},\mu_{em},\phi_{em},x_{in},\mu_{in},\phi_{in})\exp(\eta\tau)\big{)}}{\partial\tau}=\\\
R(\tau,x_{em},\mu_{em},x_{in},\mu_{in})\exp(\eta\tau)$ (B3)

Integrating from zero to optical depth $\tau$ and simplifying we get:

$S(\tau;\bm{x}_{em};\bm{x}_{in})=\frac{R(\tau;\bm{x}_{em};\bm{x}_{in})}{\eta}[1-\exp(-\eta\tau)]$
(B4)

This is then expanded using a MacLaurin series and using $a=\eta\tau$ to get

$S(\tau;\bm{x}_{em};\bm{x}_{in})=\frac{R(\tau;\bm{x}_{em};\bm{x}_{in})\tau}{a}\sum\limits_{n=1}^{\infty}\frac{(-a)^{n}}{n!}$
(B5)

Expanding this to 8th order gives the thin layer approximation:

$S(\tau;\bm{x}_{em};\bm{x}_{in})=R(\tau;\bm{x}_{em};\bm{x}_{in})\tau(1-a(1-\\\
\frac{a}{2}(1-\frac{a}{3}(1-\frac{a}{4}(1-\frac{a}{5}(1-\frac{a}{6}(1-\frac{a}{7}(1-\frac{a}{8}))))))))$
(B6)

For the thin layer transmission function, $T$, the equation of transfer is:

$\frac{\psi(x_{in})}{\mu_{in}}T(\tau;\bm{x}_{em};\bm{x}_{in})+\frac{\partial
T(\tau;\bm{x}_{em};\bm{x}_{in})}{\partial\tau}=\\\
\exp(-\tau\frac{\psi(x_{em})}{\mu})R(\bm{x}_{em};\bm{x}_{in})+...$ (B7)

Setting $\alpha=\frac{\psi(x_{in})}{\mu_{in}}$ and
$\beta=\frac{\psi(x_{em})}{\mu_{em}}$ and using an integrating factor of
$\exp(\alpha\tau)$ gives

$T=R\frac{1}{\beta-\alpha}[\exp(-\alpha\tau)-\exp(-\beta\tau)]\quad\text{if
}\alpha\neq\beta$ (B8)

To get the thin layer approximation expand both of the exponentials and use
$a=\alpha\tau$ and $b=\beta\tau$

$\displaystyle T=R\tau(b-a)^{-1}[$ $\displaystyle
1-a+\frac{a^{2}}{2!}-\frac{a^{3}}{3!}+\frac{a^{4}}{4!}+...$ (B9)
$\displaystyle-$ $\displaystyle
1+b-\frac{b^{2}}{2!}+\frac{b^{3}}{3!}-\frac{b^{4}}{4!}+...]$ (B10)

Reorganising these terms gives the expression

$T=R\tau\Big{[}c_{1}-\frac{1}{2}\Big{(}c_{2}-\frac{1}{3}\Big{(}c_{3}-\frac{1}{4}\Big{(}c_{4}-\frac{1}{5}\Big{(}c_{5}-\\\
\frac{1}{6}\Big{(}c_{6}-\frac{1}{7}\Big{(}c_{7}-\frac{c_{8}}{8}\Big{)}\Big{)}\Big{)}\Big{)}\Big{)}\Big{)}\Big{]}$
(B11)

where

$\displaystyle c_{1}$ $\displaystyle=1$ (B12) $\displaystyle c_{2}$
$\displaystyle=a+b$ (B13) $\displaystyle c_{3}$ $\displaystyle=a^{2}+ab+b^{2}$
(B14) $\displaystyle c_{4}$ $\displaystyle=(a^{2}+b^{2})\times c_{2}$ (B15)
$\displaystyle c_{5}$ $\displaystyle=a^{4}+a^{3}b+b^{2}a^{2}+ab^{3}+b^{4}$
(B16) $\displaystyle c_{6}$ $\displaystyle=(a^{3}+b^{3})\times c_{3}$ (B17)
$\displaystyle c_{7}$
$\displaystyle=a^{6}+a^{5}b+a^{4}b^{2}+a^{3}b^{3}+b^{2}a^{4}+ab^{5}+b^{6}$
(B18) $\displaystyle c_{8}$ $\displaystyle=(a^{4}+b^{4})\times c_{4}$ (B19)

If $\alpha=\beta$, then the differential equation simplifies to

$T=R\tau\exp(-\alpha\tau)=R\tau\exp(-a)$ (B21)

Expanding this to 8th order gives

$T=R\tau\Big{[}1-a\Big{(}1-\frac{a}{2}\Big{(}1-\frac{a}{3}\Big{(}1-\frac{a}{4}\Big{(}1-\frac{a}{5}\Big{(}1-\\\
\frac{a}{6}\Big{(}1-\frac{a}{7}\Big{(}1-\frac{a}{8}\Big{)}\Big{)}\Big{)}\Big{)}\Big{)}\Big{)}\Big{)}\Big{]}$
(B22)

##  Appendix C Modelling IPH brightnesses

Photons from the interplanetary hydrogen background enter the atmosphere from
all directions. Consequently, we integrate over all incidence angles when
computing the brightness of IPH scattered by Saturn’s atmosphere, rather than
the delta function in incidence angle used for the solar case. As outlined in
Section 2.3.2, the IPH brightness is fit with respect to $\cos\theta_{max}$,
the angle to the direction of maximum brightness. At 9-10 au, the direction of
maximum IPH brightness is closely aligned with the sun direction.

For each pixel, a grid of points is generated around the surface normal (see
Fig. 26, spanning both incidence angle and azimuthal direction. The angle of
each point to the direction of maximum brightness is computed, and used to
calculate the IPH brightness in that direction. This brightness of the IPH
incident on the Saturn’s atmosphere is scaled with the 28-day averaged solar
flux at the time of observation. We use a Gaussian lineshape at a temperature
of 16,000K (Figure 5) in the radiative transfer model.

Figure 26: (a) Modelled IPH background brightness as a function of incidence
and azimuthal angle for one observation pixel, using the fit in Figure 6. (b)
Azimuthally averaged model IPH brightnesses, which are incorporated into the
RT model in Eq. C1.

The scattered brightness from the scattered IPH Lyman$\alpha$ photons is given
by

$4\pi I_{IPH}(\mu_{em})=\Delta\nu_{D,Sat}\times
10^{-6}\int_{0}^{2\pi}d\phi_{in}\times\\\
\int_{0}^{1}d\mu_{in}\int_{-\infty}^{\infty}dx_{in}\int_{-\infty}^{\infty}dx_{em}S(\tau;\bm{x}_{em};\bm{x}_{in})F_{IPH}(\bm{x}_{in})$
(C1)

where we have integrated over all incidence angles. The incident IPH flux is
direction dependent such that

$F_{IPH}(\bm{x}_{in})=B_{IPH}(\mu_{in},\phi_{in})\times
f_{IPH}(x_{in},\mu_{in},\phi_{in}),$ (C2)

where $f_{IPH}(x_{in},\mu_{in},\phi_{in})$ is the lineshape, which is
normalised to 1. While this does not explicitly depend on the incidence angle,
the direction of the normal vector relative to the sunward direction does
impact the scattered IPH brightness.

We tested the impact of including a variable IPH Lyman-$\alpha$ lineshift and
temperature into the radiative transfer model. We found that this resulted in
scattered brightnesses varying by several rayleighs ($\sim 5$% of the IPH
brightness), which in the context of Saturn is negligible and small compared
to the observational uncertainties. Therefore, we simplify
$f_{IPH}(x_{in},\mu_{in},\phi_{in})=f_{IPH}(x_{in})$ in Eq, C2.

##  Appendix D Regression of the Lyman-$\alpha$ brightnesses

As described in Section 2.2 and presented in Section 3.2, we have analysed the
extensive data set of Lyman-$\alpha$ emission brightnesses throughout the
Cassini mission. In this section, we present supplementary material and
figures from the regression analyses for the different mission periods.

###  D.1 Northern hemisphere summer

For the northern hemisphere summer (here taken as 2014-2016), the observed
brightnesses show correlations with the emission angle, incidence angle and
latitude (see Figure 27). The comparison of the observed and predicted
brightnesses for 2014-2016 are shown in Figure 12. We noted that the residuals
in Figure 12d show quadratic behaviour, being positive at small and large
predicted brightnesses. However, the quadratic behaviour seen in Fig. 28b, is
not seen with respect to each independent variable (Figure 28a-c). Therefore,
the choice of a combination of quadratic and cubic fits with respect to the
independent variables (see Eq. 1) is sufficient to describe the brightnesses.

Figure 29 shows the fits of the multi-variate regression in bins of latitude
during the northern hemisphere summer. The variation of the observed
brightness with incidence and emission angle is consistent across the full
latitude range, and similar to the prediction of the radiative transfer model
(see Fig. 14b). The regions of difference with the radiative transfer model
can be attributed to poor coverage in the latitude bin (see Figure 30), so the
regression model is not well constrained.

Figure 27: Brightness of the training dataset in the late mission period
(2014-2016; see Section 3.2) against emission angle, solar incidence angle and
latitude. The median and standard deviations of the observed brightnesses are
given in bins of each independent variable.  Figure 28: Residuals of the MVR
analysis for the testing dataset against each independent variable used in the
model for the late mission period (2014-2016). The black points give the mean
and standard deviations of the residual brightness. Figure 29: Predicted
Lyman-$\alpha$ brightness vs incidence and emission angles in bins of latitude
during the northern hemisphere summer (2014-2016). The MVR analysis is applied
using a quadratic expression in incidence and emission angles and is trained
independently for each latitude bin. The observation coverages in the training
datasets are shown in Figure 30. Figure 30: Counts of Lyman-$\alpha$
brightness observations vs incidence and emission angles in bins of latitude
during the northern hemisphere summer (2014-2016). The predicted brightnesses
from each dataset are shown in Figure 29.

###  D.2 Southern hemisphere summer

Figure 31: Brightness of the training dataset in the southern hemisphere summer (2004-2007) against emission angle, solar incidence angle and latitude. The median and standard deviations (black) of the observed brightnesses are given in bins of each independent variable. Table 2: Coefficients for the fit used in the multi-variate regression (see Eq. 1), for the 2004-2006 observations. Variable | Coefficient | Mean Value | Confidence Interval
---|---|---|---
Constant | $p_{0}$ | $5.95\times 10^{2}$ | ($5.93\times 10^{2}$, $5.96\times 10^{2}$)
$\theta_{em}$ | $p_{1}$ | $2.61\times 10^{0}$ | ($2.55\times 10^{0}$, $2.66\times 10^{0}$)
$\theta_{in}$ | $p_{2}$ | $-8.85\times 10^{0}$ | ($-8.89\times 10^{0}$, $-8.81\times 10^{0}$)
$\phi_{lat}$ | $p_{3}$ | $1.26\times 10^{0}$ | ($1.15\times 10^{0}$, $1.37\times 10^{0}$)
$\theta_{em}^{2}$ | $p_{4}$ | $9.28\times 10^{-2}$ | ($9.07\times 10^{-2}$, $9.50\times 10^{-2}$)
$\theta_{em}\cdot\theta_{in}$ | $p_{5}$ | $7.08\times 10^{-3}$ | ($4.96\times 10^{-3}$, $9.19\times 10^{-3}$)
$\theta_{in}^{2}$ | $p_{6}$ | $-5.28\times 10^{-2}$ | ($-5.45\times 10^{-2}$, $-5.11\times 10^{-2}$)
$\phi_{lat}^{2}$ | $p_{7}$ | $-1.35\times 10^{-2}$ | ($-1.61\times 10^{-2}$, $-1.09\times 10^{-2}$)
$\phi_{lat}^{3}$ | $p_{8}$ | $-1.35\times 10^{-3}$ | ($-1.44\times 10^{-3}$, $-1.25\times 10^{-3}$)

We apply the multi-variate regression model (see Section 2.2) to the
observations in the southern hemisphere summer (2004-2007). The fit
coefficients are listed in Table 2. Figure 31 shows the observed
Lyman-$\alpha$ brightness during this period, showing similar behaviour with
incidence angle and emission angle to that observed in the northern hemisphere
summer (see Figure 27). The dependence of the Lyman-$\alpha$ brightness on
latitude has reversed, with a peak at $-20^{\circ}$ and a sharp decrease in
brightness in the northern hemisphere. Figure 32 compares the observed
brightnesses with predicted brightnesses and residuals. The model accurately
reproduces the observed brightnesses between 200 R and 900 R, but
overestimates some large brightnesses. There are few counts where the
residuals are large, whereas the statistics are much stronger where the
residuals are small ($<200$ R).

Figure 32: (a,b) Testing data counts vs predicted brightness from the MVR for
2004-2007. (c) Observed brightnesses and (d) residuals vs predicted
brightnesses for the testing dataset (blue) with the averages and standard
deviations shown in red.

Similar to the northern summer, the quadratic nature of the residuals in
Figure 32b are not replicated in the dependences on each independent variable
(see Figure 33). The residuals show little dependence on each of the
independent variables, suggesting higher order terms are not required to
improve the fits.

Figure 33: Residuals of the MVR analysis for the testing dataset for 2004-2007
against each independent variable used in the model. The black points give the
mean and standard deviations of the residual brightness.

The dependence of the predicted brightness on the incidence and emission
angles at $\phi_{lat}=-20^{\circ}$ (see Figure 34) is very similar to that
seen in the northern hemisphere summer, and to the predictions of the
radiative transfer model (see Figure 14).

Figure 34: (left) Predicted brightness as a function of solar incidence and
emission angles from the MVR analysis at a latitude of -20∘N for the 2004-2007
dataset. (right) Observation counts as a function of solar incidence and
emission angles used to train the MVR analysis for the 2004-2007 dataset.

The same structure in the predicted brightness is retrieved for each latitude
bin with a good fit ($R^{2}>0.6$) for all bins in the southern hemisphere,
where more observations were available (see Figure 36). Some bins, such as
$\phi_{lat}=[-60,-49]$, predict large brightnesses at low emission angles,
contrary to the radiative transfer model. In each of these cases, the data
coverage does not extend to low emission angles, so this should not be
interpreted physically.

Figure 35: Predicted Lyman-$\alpha$ brightness vs incidence and emission
angles in bins of latitude during the southern hemisphere summer (2004-2007).
The MVR analysis is applied using a quadratic expression in incidence and
emission angles and is trained independently for each latitude bin. The
observation coverage in the training dataset are shown in Figure 36.  Figure
36: Counts of Lyman-$\alpha$ brightness observations vs incidence and emission
angles in bins of latitude during the southern hemisphere summer (2004-2007).
The predicted brightnesses from each dataset are shown in Figure 35.

##  Appendix E Absorption by the interplanetary Lyman-$\alpha$

Between 1 au, where the Lyman alpha profile has been measured (Lemaire et al.,
2005), and Saturn, the interplanetary hydrogen background absorbs some of the
Lyman $\alpha$, particularly near the line-centre, reducing the line-centre
flux by up to 13%. As the flux at line centre is important for resonance
scattering, we model the absorption of the IPH background using the IPH model
of Quémerais et al. (2013a); Izmodenov et al. (2013), which is discussed in
Section 2.3.2.

We calculate the absorption for 36 radial profiles distributed along Saturn’s
orbit, relative to the flow of the local interstellar medium. The optical
depth of the IPH background is integrated radially to a heliocentric distance
of 9.5 au, including the Doppler shift and width of the absorption profile
resulting from the bulk IPH velocity.

Using the optical depth, we correct the Lyman-$\alpha$ lineshape and average
the absorption profiles around the linecenter. It is assumed that resonance
scattering is symmetric about the linecenter in the radiative transfer model,
so the absorption must also be made symmetric.

Figure 5 shows the corrected Lyman-$\alpha$ lineshape for the mean absorption
profile (dashed black line) with the absorption variability (black shaded
region), compared to the flux shape without IPH absorption.

##  Appendix F IPH observations

The observations of the Lyman-$\alpha$ IPH background we use in Section 3.1
are listed in Table 3.

Table 3: Observations of the IPH Lyman-$\alpha$ background by Cassini/UVIS in 2006, with heliocentric distance ($D_{H}$) and background brightness. The right ascension and declination of the observation and the sunward direction ($\odot$) are given.  Day of year | Lead | $D_{H}$ | Background [R] | Ly-$\alpha$ [R] | Scaled B [R] | RA | DEC | RA⊙ | DEC⊙
---|---|---|---|---|---|---|---|---|---
014 | CIRS | 9.08 | 4.04$\pm$2.20 | 251$\pm$72 | $229\pm 65$ | 101.7 | -2.75 | 309.91 | -19.04
015 | UVIS | 9.08 | 5.50$\pm$1.10 | 557$\pm$61 | $508\pm 55$ | 279.1 | 3.01 | 309.92 | -19.02
| ISS | 9.08 | 5.74$\pm$0.69 | 596$\pm$39 | $544\pm 35$ | 277.4 | 3.18 | 309.93 | -19.02
| CIRS | 9.08 | 5.80$\pm$0.94 | 348$\pm$47 | $317\pm 42$ | 101.7 | -2.75 | 309.91 | -19.03
| CIRS | 9.08 | 5.75$\pm$1.66 | 202$\pm$14 | $184\pm 12$ | 101.2 | -2.81 | 309.92 | -19.03
057 | VIMS | 9.10 | 5.53$\pm$1.22 | 183$\pm$12 | $172\pm 11$ | 134.2 | 0.75 | 311.46 | -18.67
058 | UVIS | 9.10 | 5.45$\pm$1.67 | 619$\pm$52 | $582\pm 48$ | 309.6 | -0.24 | 311.53 | -18.65
| CIRS | 9.10 | 5.40$\pm$0.64 | 673$\pm$45 | $633\pm 42$ | 309.0 | -0.17 | 311.54 | -18.65
| ISS | 9.10 | 5.91$\pm$1.04 | 286$\pm$74 | $269\pm 69$ | 133.2 | 0.64 | 311.49 | -18.66
077 | CIRS | 9.10 | 5.47$\pm$1.44 | 304$\pm$27 | $288\pm 25$ | 73.5 | -5.12 | 312.28 | -18.52
| ISS | 9.10 | 6.24$\pm$2.71 | 341$\pm$36 | $323\pm 34$ | 72.2 | -5.2 | 312.29 | -18.52
078 | CIRS | 9.09 | 5.46$\pm$1.70 | 433$\pm$32 | $411\pm 30$ | 248.8 | 5.42 | 312.30 | -18.51
| VIMS | 9.09 | 4.97$\pm$1.51 | 444$\pm$15 | $421\pm 14$ | 249.5 | 5.38 | 312.30 | -18.51
121 | UVIS | 9.11 | 5.66$\pm$1.33 | 596$\pm$55 | $546\pm 50$ | 282.1 | 2.72 | 313.84 | -18.12
140 | UVIS | 9.11 | 5.27$\pm$1.54 | 274$\pm$37 | $250\pm 33$ | 44.0 | -6.31 | 314.62 | -17.97
183 | UVIS | 9.12 | 5.78$\pm$1.15 | 445$\pm$28 | $416\pm 26$ | 255.3 | 4.92 | 316.14 | -17.58
250 | CIRS | 9.14 | 5.44$\pm$0.92 | 333$\pm$21 | $310\pm 19$ | 19.2 | -15.1 | 318.67 | -16.97
| UVIS | 9.14 | 4.85$\pm$2.06 | 330$\pm$27 | $307\pm 25$ | 20.6 | -15.8 | 318.67 | -16.96
| CIRS | 9.14 | 5.52$\pm$1.15 | 247$\pm$84 | $229\pm 78$ | 25.9 | -18.0 | 318.68 | -16.96
| ISS | 9.14 | 6.08$\pm$1.12 | 249$\pm$43 | $231\pm 40$ | 204.7 | 17.6 | 318.69 | -16.96
| ISS | 9.14 | 5.51$\pm$1.37 | 237$\pm$52 | $220\pm 48$ | 205.3 | 17.8 | 318.67 | -16.96
266 | UVIS | 9.14 | 5.10$\pm$0.94 | 288$\pm$38 | $267\pm 35$ | 27.2 | -20.6 | 319.26 | -16.81
282 | CIRS | 9.14 | 4.73$\pm$1.79 | 300$\pm$8 | $279\pm 7$ | 28.5 | -28.5 | 319.84 | -16.66
| CIRS | 9.14 | 5.40$\pm$1.01 | 337$\pm$15 | $313\pm 13$ | 210.2 | 33.8 | 319.86 | -16.66
283 | CIRS | 9.14 | 5.01$\pm$1.12 | 292$\pm$17 | $271\pm 15$ | 208.0 | 32.3 | 319.89 | -16.64
297 | UVIS | 9.15 | 5.43$\pm$0.88 | $317\pm 11$ | $298\pm 10$ | 31.0 | -35.3 | 320.40 | -16.52
299 | VIMS | 9.15 | 5.73$\pm$1.55 | $315\pm 31$ | $296\pm 29$ | 219.4 | 36.3 | 320.44 | -16.50
346 | UVIS | 9.16 | 5.00$\pm$1.18 | $293\pm 33$ | $271\pm 30$ | 39.6 | -38.7 | 322.15 | -16.04
| CIRS | 9.16 | 5.10$\pm$2.15 | $289\pm 28$ | $267\pm 25$ | 215.1 | 42.9 | 322.18 | -16.02
361 | CIRS | 9.16 | 5.37$\pm$1.01 | $308\pm 12$ | $288\pm 11$ | 34.8 | -44.6 | 322.72 | -15.88
362 | CIRS | 9.16 | 5.21$\pm$0.51 | $300\pm 25$ | $281\pm 23$ | 218.3 | 48.5 | 322.74 | -15.87
| CIRS | 9.16 | 5.21$\pm$0.84 | $286\pm 22$ | $268\pm 20$ | 219.2 | 49.6 | 322.75 | -15.87

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