# Latitudinal variation of methane mole fraction above clouds in Neptune’s
atmosphere from VLT/MUSE-NFM: Limb-darkening reanalysis

###### Abstract

We present a reanalysis of visible/near-infrared (480 – 930 nm) observations
of Neptune, made in 2018 with the Multi Unit Spectroscopic Explorer (MUSE)
instrument at the Very Large Telescope (VLT) in Narrow Field Adaptive Optics
mode, reported by Irwin et al., Icarus, 311, 2019. We find that the inferred
variation of methane abundance with latitude in our previous analysis, which
was based on central meridian observations only, underestimated the retrieval
errors when compared with a more complete assessment of Neptune’s limb
darkening. In addition, our previous analysis introduced spurious latitudinal
variability of both the abundance and its uncertainty, which we reassess here.
Our reanalysis of these data incorporates the effects of limb-darkening based
upon the Minnaert approximation model, which provides a much stronger
constraint on the cloud structure and methane mole fraction, makes better use
of the available data and is also more computationally efficient. We find that
away from discrete cloud features, the observed reflectivity spectrum from 800
– 900 nm is very well approximated by a background cloud model that is
latitudinally varying, but zonally symmetric, consisting of a H2S cloud layer,
based at 3.6 – 4.7 bar with variable opacity and scale height, and a
stratospheric haze. The background cloud model matches the observed limb
darkening seen at all wavelengths and latitudes and we find that the mole
fraction of methane at 2–4 bar, above the H2S cloud, but below the methane
condensation level, varies from 4–6% at the equator to 2–4% at south polar
latitudes, consistent with previous analyses, with a equator/pole ratio of
$1.9\pm 0.2$ for our assumed cloud/methane vertical distribution model. The
spectra of discrete cloudy regions are fitted, to a very good approximation,
by the addition of a single vertically thin methane ice cloud with opacity
ranging from 0 – 0.75 and pressure less than $\sim 0.4$ bar.

Icarus

Department of Physics, University of Oxford, Parks Rd, Oxford, OX1 3PU, UK
Instituto Nacional de Técnica Aeroespacial (INTA), 28850, Torrejón de Ardoz
(Madrid), Spain. School of Earth Sciences, University of Bristol, Wills
Memorial Building, Queens Road, Bristol, BS8 1RJ, UK School of Physics &
Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, UK
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA University of the Basque Country UPV/EHU, 48013
Bilbao, Spain

Patrick<EMAIL_ADDRESS>
Minnaert limb-darkening analysis improves modelling of Neptune’s reflectivity
spectrum in visible/near-IR.

General cloud distribution can be modelled with zonally-symmetric H2S cloud
and stratospheric haze.

Mole fraction of methane at 2–4 bar (above H2S cloud) found to decrease from
4–6% at the equator to 2–4% at the south pole.

Discrete cloud features can be fitted with an additional methane ice cloud at
pressures less than $\sim$ 0.4 bar.

##  1 Introduction

The visible and near-infrared spectrum of Neptune is formed by the reflection
of sunlight from the atmosphere, modulated primarily by the absorption of
gaseous methane, but also to a lesser extent H2S [Irwin . (2018)]. Measured
spectra can thus be inverted to determine the cloud structure as a function of
location and altitude, providing we know the vertical and latitudinal
distribution of methane. Although for some years the vertical profiles of
methane determined from Voyager 2 radio-occultation observations were used at
all latitudes, HST/STIS observations of Uranus recorded in 2002 [Karkoschka
Tomasko (2009)] and similar observations of Neptune recorded in 2003
[Karkoschka Tomasko (2011)] both showed that the tropospheric cloud-top (i.e.,
above the H2S cloud) methane mole fraction varies significantly with latitude
on both planets, later confirmed for Uranus by several follow-up studies
[Sromovsky . (2011), Sromovsky . (2014), Sromovsky . (2019)]. These HST/STIS
observations used the collision-induced absorption (CIA) bands of H2–H2 and
H2–He near 825 nm, which allow variations of CH4 mole fraction to be
differentiated from cloud-top pressure variations of the H2S cloud.
kark09,kark11 found that the methane mole fraction above the main observable
H2S cloud tops at 2–4 bar varies from $\sim 4$% at equatorial latitudes to
$\sim 2$% polewards of $\sim 40^{\circ}$ N,S for both planets.

More recently, an analysis of VLT/MUSE Narrow-Field Mode (NFM) observations
(770 – 930 nm) along Neptune’s central meridian [Irwin . (2019)] found a
similar latitudinal variation of cloud-top methane mole fraction, with values
of 4–5% reported at equatorial latitudes, reducing to 3–4% at polar latitudes,
but with considerable pixel-to-pixel variation that was not understood at the
time. In this study we reanalyse these data using a new limb-darkening
approximation model, which makes much better use of all the data from a given
latitude, observed at many different zenith angles, and which we find
considerably improves our methane mole fraction determinations. We also find
that having constrained the smooth latitudinal variation of opacity of the
tropospheric cloud and stratospheric haze, we are able to efficiently retrieve
the additional opacity of discrete upper tropospheric (0.1 - 0.5 bar) methane
clouds seen in our observations.

##  2 MUSE Observations

As reported by irwin19, commissioning-mode observations of Neptune were made
on 19th June 2018 with the Multi Unit Spectroscopic Explorer (MUSE) instrument
[Bacon . (2010)] at ESO’s Very Large Telescope (VLT) in Chile, in Narrow-Field
Mode (NFM). MUSE is an integral-field spectrograph, which records 300 $\times$
300 pixel ‘cubes’, where each ‘spaxel’ contains a complete visible/near-
infrared spectrum (480 – 930 nm) with a spectral resolving power of 2000 –
4000. MUSE’s Narrow-Field Mode has a field of view of 7.5” $\times$ 7.5”,
giving a spaxel size of 0.025”, and uses Adaptive Optics to achieve a spatial
resolution less than 0.1”. These commissioning observations are summarised in
Table 1 of irwin19. The spatial resolution was estimated to have a full-width-
half-maximum of 0.06” at 800 nm. The observed spectra were smoothed to the
resolution of the IRTF/SpeX instrument, which has a triangular instrument
function with FWHM = 2 nm, sampled at 1 nm, in order to increase the signal-
to-noise ratio without losing the essential shape of the observed spectra.
This resolution was also more consistent with the spectral resolution of the
methane gaseous absorption data used, which are described in section 3.2.

In our previous analysis of these data [Irwin . (2019)], spectra recorded from
single pixels along the central meridian of one of the longer integration time
observations (120s) were fitted with our NEMESIS retrieval model [Irwin .
(2008)] to determine latitudinal variations of methane and cloud structure.
The cloud-top (i.e., immediately above the H2S cloud) methane mole fractions
were found to be consistent with HST/STIS observations [Karkoschka Tomasko
(2011)], but were not very well constrained with significant latitudinal
variation that we attributed at the time to the random noise from single-pixel
retrievals. However, they also did not make full use of the limb-darkening
behaviour visible in these IFU observations, although we verified that our
cloud parameterization reproduced the observed limb-darkening well at 5 –
$10^{\circ}$S.

Since making our initial report on the MUSE-NFM Neptune observations, an
analysis of HST/WFC3 observations for Jupiter has been conducted by perez20,
which makes much better use of the limb-darkening information content of
multi-spectral observations using a Minnaert limb darkening approximation
scheme. We have adapted this technique for use with our Neptune MUSE-NFM
observations and find that it greatly improves the quality of our fits and our
estimates of the latitudinal variation of cloud-top methane mole fraction at
2–4 bar in Neptune’s atmosphere. This reanalysis has also highlighted an
erroneous retrieval artefact in our previous work [Irwin . (2019)] at some
locations, which can now be explained.

##  3 Reanalysis

###  3.1 Minnaert Limb-darkening analysis

The dependence of the observed reflectivity from a location on a planet on the
incidence and emission angles can be well approximated using an empirical law
first introduced by minnaert41. For an observation at a particular wavelength,
the observed reflectivity111$I/F$ is $\pi R/F$, where $R$ is the reflected
radiance (W cm-2 sr-1 $\mu$m-1) and $F$ is the incident solar irradiance at
the planet (W cm-2 $\mu$m-1).  $I/F$ can be approximated as:

$\frac{I}{F}=\left(\frac{I}{F}\right)_{0}\mu_{0}^{k}\mu^{k-1}$ (1)

where $(I/F)_{0}$ is the nadir-viewing reflectivity, $k$ is the limb-darkening
parameter, and $\mu$ and $\mu_{0}$ are, respectively, the cosines of the
emission and solar incidence angles. With this model a value of $k>0.5$
indicates limb-darkening, while $k<0.5$ indicates limb-brightening. Taking
logarithms, Eq. 1 can be re-expressed as:

$\ln\left(\mu\frac{I}{F}\right)=\ln\left(\frac{I}{F}\right)_{0}+k\ln(\mu\mu_{0})$
(2)

and we can see that it is possible to fit the Minnaert parameters $(I/F)_{0}$
and $k$ if we perform a least-squares fit on a set of measurements of $\ln(\mu
I/F)$ as a function of $\ln(\mu\mu_{0})$.

Figure 1: Appearance of Neptune on June 19th 2018 at 09:43:21UT (Observation
‘3’ of irwin19) at 830 nm (top row) and 840 nm (bottom row). The left-hand
column shows the observed images and the middle column shows the images
reconstructed following a Minnaert limb-darkening analysis. Here, for each
location with latitude $\phi$ and cos(zenith) angles $\mu_{0}$ and $\mu$ the
reflectivity was calculated as $I/F=R(\phi)\mu_{0}^{k(\phi)}\mu^{k(\phi)-1}$
where $R(\phi)$ and $k(\phi)$ are the interpolated values of Minnaert
parameters $\left(I/F\right)_{0}$ and $k$ at that latitude. The right-hand
column shows the difference (reconstructed - observed). The equator and
$60^{\circ}$S latitude circles are indicated by the red-dashed and blue-dashed
lines, respectively. The dark spot seen at continuum wavelengths near the
equator is an artefact of the reduction pipeline of unknown origin. We found
it to have negligible effect on our analysis. Figure 2: Appearance of Neptune
on June 19th 2008 at 09:43:21UT (Observation ‘3’ of irwin19) at 840 nm,
showing regions masked to fit the Minnaert curves (top) and the areas selected
for later additional discrete cloud fitting (bottom).

We analysed the same ‘cube’ of Neptune as was studied by irwin19, namely
Observation ‘3’, recorded by VLT/MUSE at 09:43:21(UT) on 19th June 2018. We
analysed the spectra in this cube in the wavelength range 800 – 900 nm and
Fig. 1 shows the observed appearance of the planet at 830 and 840 nm, which
are wavelengths of weak and strong methane absorption, respectively. The limb-
darkening behaviour of the observed spectra were analysed in latitude bands of
width $10^{\circ}$, spaced every $5^{\circ}$ to achieve Nyquist sampling. For
each latitude band, the observed reflectivities were used to construct plots
of $\ln(\mu I/F)$ against $\ln(\mu\mu_{0})$ and straight lines fitted to
deduce $(I/F)_{0}$ and $k$ for each wavelength. Locations on the disc where
there were bright clouds were masked out (Fig. 2) and examples of the fits at
830 and 840 nm for latitude bands centred on the equator and $60^{\circ}$S are
shown in Fig. 3. Here it can be seen that the Minnaert empirical law provides
a very accurate approximation of the observed dependence of reflectivity with
viewing zenith angles. Although all the measurements are plotted, only those
measurements with $\mu\mu_{0}\geq 0.09$ (i.e., $\mu$, $\mu_{0}>\sim 0.3$) were
used to fit $(I/F)_{0}$ and $k$ to make sure that the fitting procedure was
not overly affected by points measured near the disc edge and thus potentially
more ‘diluted’ with space. Also plotted in Fig. 3 are the reflectivities
calculated with our radiative transfer and retrieval model from our best-fit
retrieved cloud and methane mole fractions at these latitudes, reported in
section 3.3. It can be seen that there is very good agreement between the
reflectivities calculated with our multiple-scattering matrix operator model
and the Minnaert limb-darkening approximation to the observations for zenith
angles less than $\sim 70^{\circ}$.

Extending this analysis to all wavelengths under consideration, Fig. 4 shows a
contour plot of the fitted values of $(I/F)_{0}$ and $k$ for all wavelengths
and latitude bands. It can be seen that at wavelengths near 830 nm, the fitted
$k$ values are greater than 0.5, indicating limb darkening, while at longer
wavelengths, values of $k$ less than 0.5 are fitted, indicating limb
brightening. It can also just be seen in Fig. 4 that the width of the
reflectance peak of $(I/F)_{0}$ is noticeably wider at latitudes southwards of
20 – $40^{\circ}$S, a trend that is also just discernible in the fitted $k$
values near the reflectance peak.

Figure 3: Minnaert analysis of observed limb-darkening curves at 830 nm (top
row) and 840 nm (bottom row). The left-hand column shows all observations,
while the right-hand column is limited to points with $\mu\mu_{0}>0.1$. The
red points are the measurements at the equator, while those coloured blue are
at $60^{\circ}$S. The red and blue solid lines are the lines fitted with the
Minnaert analysis for points with $\mu\mu_{0}\geq 0.09$. The vertical blue
dotted lines indicate the values of $\mu\mu_{0}$ corresponding to our 5-point
quadrature ordinates (with the vertical blue dashed line indicating the
smallest value of $\mu\mu_{0}$ used in our retrieval analysis). The blue
diamonds and red squares are simulated reflectivities from our best-fit cloud
structure and methane mole fractions at these latitudes, reported in section
3.3, showing excellent consistency between the Minnaert approximation and the
reflectivities simulated with our matrix operator multiple-scattering model
for $\mu\mu_{0}>0.1$. Figure 4: Minnaert analysis of observed limb-darkening
curves from 800 to 900 nm and at all latitudes visible on Neptune’s disc. The
left-hand column shows the fitted values of $(I/F)_{0}$ and $k$ at the equator
(red) and $60^{\circ}$S (blue), while the right-hand column shows a contour
plot of these fitted parameters at all latitudes. The horizontal lines in the
left hand column indicates the value of the white colour in the contour plots
in the right hand column, and for the $k$ contour plot indicates the
transition from limb darkening $(k>0.5)$ to limb-brightening $(k<0.5)$. The
horizontal lines in the right hand column indicate the equator (red) and
$60^{\circ}$S (blue), respectively.

Having fitted values of $(I/F)_{0}$ and $k$ for all wavelengths and latitude
bands, it is then possible to reconstruct the apparent image of the planet at
any observation geometry. Using the measured observation values of $\mu$ and
$\mu_{0}$ we reconstructed the images of Neptune at 830 and 840 nm, where for
each location with latitude $\phi$ and cos(zenith) angles $\mu_{0}$ and $\mu$
the reflectivity is calculated as
$I/F=R(\phi)\mu_{0}^{k(\phi)}\mu^{k(\phi)-1}$, where $R(\phi)$ and $k(\phi)$
are the interpolated values of Minnaert parameters $\left(I/F\right)_{0}$ and
$k$ at that latitude. We compare these reconstructed images with the observed
images in Fig. 1, which also shows the differences between the observed and
reconstructed images. As can be seen the observed general dependence of
reflectivity with latitude and position on disc is well reproduced compared
with the original MUSE observations and the differences are very small, except
at: 1) locations of known artefacts in the reduced data; 2) locations of the
small discrete clouds, which were masked out when fitting the zonally-averaged
Minnaert limb-darkening curves; and 3) off-disc, where the observed images are
not corrected for the instrument point-spread function (PSF). We will return
to these discrete clouds in section 3.5.

###  3.2 Retrieval model

Having applied the Minnaert model to the observations we then used the fitted
$(I/F)_{0}$ and $k$ parameters to reconstruct synthetic spectra of Neptune for
all visible latitude bands and fitted these as synthetic ‘observations’ using
our radiative transfer and retrieval model, NEMESIS [Irwin . (2008)]. There
are two main advantages in doing this: 1) the spectra reconstructed using the
fitted Minnaert parameters have smaller random error values as they have been
reconstructed from values fitted to a combination of all the points in a
latitude band; and 2) we can reconstruct the apparent spectrum of Neptune at
any set of angles that is convenient for modelling, which can greatly reduce
computation time. In our previous approach [Irwin . (2019)], where we did not
assume to know the zenith-angle dependence, we tried to fit simultaneously to
the observations at several different zenith angles near the equator. For
modelling near-infrared reflectivity observations, NEMESIS employs a plane-
parallel matrix operator multiple-scattering model [Plass . (1973)]. In this
model, integration over zenith angle is done with a Gauss-Lobatto quadrature
scheme, while the azimuth integration is done with Fourier decomposition. For
most calculations, not too near to the disc edge, we have found that five
zenith angles are usually sufficient and the reflectivity at a particular
zenith angle is linearly interpolated between calculations done at the two
closest zenith angles. Although this provides a general purpose functionality,
this approach has some drawbacks: 1) it requires two sets of calculations at
two different solar zenith angles for each location; 2) the linear
interpolation can lead to interpolation errors at larger zenith angles; and 3)
for points near the disc edge, the number of Fourier components needed to
fully resolve the azimuth dependence increases, which can greatly increase
computation time. By reconstructing spectra using the fitted Minnaert
$(I/F)_{0}$ and $k$ parameters we can simulate spectra measured as if they
exactly coincided with the angles in our quadrature scheme, thus avoiding
interpolation error. In addition, if we assume the Minnaert approximation to
be true, which has a linear dependence in logarithmic space, we only need to
fit spectra calculated at two different angles, not several, and can test how
well the linear approximation applies in post-processing. Hence, in the
retrievals presented here we reconstructed two spectra for each latitude band
with viewing zenith angle $\theta_{V}$, solar zenith angle $\theta_{S}$ and
azimuth angle $\phi$ values of ($0^{\circ}$, $0^{\circ}$, $180^{\circ}$) and
($61.45^{\circ}$, $61.45^{\circ}$, $180^{\circ}$), respectively. Here,
$\phi=180^{\circ}$ indicates back-scattering, while $\theta=61.45^{\circ}$ is
the second zenith angle in our five-point Gauss-Lobatto scheme, summarised in
Table LABEL:tab:quadrature, and $\theta=0^{\circ}$ is the fifth. This second
zenith angle is sufficiently high to probe limb-darkening or limb-brightening,
but is not too high that we need an excessive number of Fourier components in
the azimuth angle decomposition to properly model it, which would make the
computation excessively slow. For each latitude band the two synthetic
observations at $0^{\circ}$ and $61.45^{\circ}$ zenith angle were then fitted
simultaneously to determine the vertical cloud structure and tropospheric
methane mole fraction.

Errors in the fitted $(I/F)_{0}$ and $k$ parameters were propagated into the
errors on these reconstructed spectra at $0^{\circ}$ and $61.45^{\circ}$
zenith angle as normal, and were seen to increase towards the poles where the
curves were less well sampled. However, even then, because the synthetic
spectra are derived from linear fits to a large number of data points the
random error is very small and we found that we were unable to fit the
synthetic observations to within these error. We attribute this to ‘forward-
modelling’ systematic errors due to deficiencies in the methane absorption
coefficients of kark11 and also in our chosen cloud and methane
parameterization schemes, described below. In order to achieve final
$\chi^{2}/n$ fits of $\sim 1$ at all latitudes in our retrievals (necessary to
derive representative error values on the retrieved parameters) we found it
necessary to multiply these errors by a factor of $\sim 15$. Although this may
appear to be an alarmingly high factor, we will see later in Fig. 11 that this
leads to error bars on the synthetic $I/F$ reflectivity spectra of only 0.5 –
1.0 %, which is perfectly reasonable given the likely accuracy of the
absorption coefficients used and also the simplicity of our retrieval scheme.
We also decided that this approach was more appropriate than our usual
procedure of simply adding a forward modelling error (which here would have
been 0.5 – 1.0% at all wavelengths and locations) since this would miss the
fact that the Minnaert-fitting errors are dependent on both wavelength and
latitude.

Table 1: Five-point Gauss-Lobatto Quadrature Scheme used in this study Index | $\mu$ |  $\theta$ (∘) | Weight
---|---|---|---
1 | 0.1652790 | 80.4866 | 0.3275398
2 | 0.4779249 | 61.4500 | 0.2920427
3 | 0.7387739 | 42.3729 | 0.2248893
4 | 0.9195339 | 23.1420 | 0.1333060
5 | 1.0000000 | 0.00000 | 0.0222222

As with our previous analysis [Irwin . (2019)], we modelled the atmosphere of
Neptune using 39 layers spaced equally in log pressure between $\sim 10$ and
$0.001$ bar. We ran NEMESIS in correlated-k mode and for methane absorption
used a methane k-table generated from the band model of kark10. The collision-
induced absorption of H2-H2 and H2-He near 825 nm was modelled with the
coefficients of borysow89a,borysow89b,borysow00, assuming a thermally-
equilibriated ortho:para hydrogen ratio. Rayleigh scattering was included as
described in irwin19 and the effects of polarization and Raman scattering were
again justifiably neglected at these wavelengths. We used the solar spectrum
of chance10, smoothed with a triangular line shape of FWHM = 2 nm and took
Neptune’s distance from the Sun on the date of observation to be 29.94 AU. The
reference temperature and mole fraction profile is the same as that used by
irwin19 and is based on the ‘N’ profile determined by Voyager-2 radio-
occultation measurements [Lindal (1992)], with He:H2 = 0.177 (15:85),
including 0.3% mole fraction of N2.

For the methane profile, we adopted a simple model with a variable deep mole
fraction, limited to 100% relative humidity above the condensation level and
further limited to a maximum stratospheric mole fraction of $1.5\times
10^{-3}$ [Lellouch . (2010)] as shown in Fig. 5. Several authors [Karkoschka
Tomasko (2011), Sromovsky . (2019), e.g.,] have pointed out that such a simple
“step” model is not physically well based for either Neptune or Uranus when
extended to great depths and in particular tollefson19 notes that such strong
deep methane latitudinal gradients would induce humidity winds (additional to
the thermal wind equation), which are not seen. Instead, sromovsky19 favours a
“descended profile” model, where downwards motion suppresses the methane mole
fraction in the 2–4 bar region, but which then recovers to a uniform deep mole
fraction at depth. This profile is compared with our “step” model in Fig. 5.
Although this profile is smoother and may be more physically plausible than
the “step” model, we do not have the vertical resolution in the MUSE data to
be able to discriminate between the two and we also cannot see clearly through
the H2S cloud to deeper pressures. This can be seen in Fig. 5, where we have
also plotted the two-way vertical transmission to space through the cloud
only, which shows the fitted cloud to be nearly opaque. In addition, we have
also plotted in Fig. 5 the functional derivatives with respect to methane
abundance, i.e., the rate of change of the calculated radiance spectrum with
respect to the methane abundance at each level if we were to assume a
continuous profile. Here we can see that we are only significantly sensitive
to the methane abundance in the 1–4 bar region. In fact, we see that with the
MUSE data we are only really sensitive to the column abundance of methane
above the H2S cloud and this column abundance will depend on the vertical
distribution of both the methane mole fraction and the cloud; since we do not
have precise constraints on either we thus have a degeneracy. Hence, in this
study we decided used the simpler “step” model for methane, which has the
added advantage of returning a mean value for the methane mole fraction in the
2–4 bar region, which is easy to understand, interpret and compare with
previous studies. It is worth noting that tollefson19, who were able to probe
to slightly deeper pressures than we were, also adopted a simple “step” model
of methane.

Figure 5: Left hand panel: Methane profiles considered in this analysis. Our
simple “step” model is shown for two different values of the ‘deep’ methane
mole fraction (3% and 5%, respectively) and compared with the “descended”
methane profile of sromovsky19 for a deep mole fraction of 5%, a deep pressure
of $P_{d}=5$ bar and scaling coefficient, $vx=3.0$ (see Eq. 3 of sromovsky19
for details). The shaded region is the cloud opacity/bar of our nominal
equatorial cloud distribution model, which has a base pressure of 4.66 bar.
Note that both models have been limited to 100% relative humidity at all
pressures and the stratospheric mole fraction is limited to not exceed
$1.5\times 10^{-3}$. Also plotted for reference is the two-way transmission
from space to each level through the main cloud only, showing the cloud to be
mostly opaque. The haze distribution has been omitted from this figure for
clarity. Right-hand panel: sensitivity of calculated radiances on the
abundance of methane at each pressure level, showing that our main sensitivity
is from 1–4 bar.

For clouds/hazes we again adopted the parameterized model used by irwin16 to
model VLT/SINFONI and Gemini/NIFS H-band observations of Neptune, which was
found to provide good limb-darkening/limb-brightening behaviour. In this model
particles in the troposphere are modelled with a cloud near the H2S
condensation level (and which is thus presumed to be rich in H2S ice [Irwin .
(2019)]), with a variable base pressure ($\sim 3.6$ – 4.7 bar) and a scale
height retrieved as as a fraction of the pressure scale height (called the
fractional scale height). Scattering from haze particles is modelled with a
second layer with base pressure fixed at $0.03$ bar and fixed fractional scale
height of 0.1. Although the base pressure of the stratospheric haze may in
reality vary with latitude, we found that the precise pressure level did not
significantly affect the calculated spectra at these wavelengths (since the
transmission to space is close to unity at the tropopause level from 800 to
900 nm) and so fixed it to a typically representative value stated. The
scattering properties of the cloud were calculated using Mie scattering and a
retrievable imaginary refractive index spectrum. For cases where we allow the
imaginary refractive index to vary with wavelength (as in our previous report)
we use a Kramers-Kronig analysis to construct the real part of the refractive
index spectrum, assuming $n_{real}=1.4$ at 800 nm. Here, however, for
simplicity we forced the imaginary refractive indices to be the same at all
wavelengths across the 800 – 900 nm range considered, and hence the real
refractive index was fixed to 1.4 over the whole range also. The Mie-
calculated phase functions were again approximated with combined Henyey-
Greenstein functions for computational simplicity and also to smooth over
features peculiar to spherical particles, such as the back-scattering ‘glory’.

Table 2: Preset grid of cloud parameterization values used in test retrievals of cloud opacity, cloud fractional scale height, stratospheric haze opacity and cloud-top methane mole fraction at the equator and $60^{\circ}$S Parameter | Number of values | Values
---|---|---
Mean cloud radius | 4 | 0.05, 0.1, 0.5, 1.0 $\mu$m
Cloud radius variance | 2 | 0.05, 0.3
Cloud $n_{imag}$ | 3 | 0.001, 0.01, 0.1
Mean haze radius | 4 | 0.05, 0.1, 0.5, 1.0 $\mu$m
Haze radius variance | 2 | 0.05, 0.3
Haze $n_{imag}$ | 3 | 0.001, 0.01, 0.1
$p_{base}$ | 3 | 3.65, 4.15, 4.66 bar

###  3.3 Retrieval analysis

To ‘tune’ our retrieval model we first concentrated on the latitude bands at
the equator and $60^{\circ}$S. We were aware from our previous study that
there was likely to be a high degree of degeneracy in our best-fit solutions
with respect to assumed particle sizes and other parameters in the 800 – 900
nm range. Hence, we first analysed these two latitude bands for a grid of
preset values of: 1) mean cloud particle radius; 2) variance of cloud radius
distribution; 3) cloud imaginary refractive index; 4) mean haze particle
radius; 5) variance of haze radius distribution; 6) haze imaginary refractive
index; and 7) cloud base pressure, described in Table LABEL:tab:parameter.
From Table LABEL:tab:parameter we can see that the number of grid values is
$4\times 2\times 3\times 4\times 2\times 3\times 3=1728$ setups for each
latitude band. For each setup we retrieved simultaneously four variables from
the synthetic spectra reconstructed at 0∘ and 61.45∘ emission angle: 1) cloud
opacity; 2) cloud fractional scale height; 3) haze opacity; and 4) the cloud-
top methane mole fraction. As explained earlier, we assumed that the set
imaginary refractive indices applied at all wavelengths simultaneously. After
fitting we plotted the $\chi^{2}$ of the fits as a function of the grid
parameters, together with the retrieved cloud-top methane mole fraction, which
we show in Fig. 6 for the equator and Fig. 7 for the $60^{\circ}$S. Here we
can see that the goodness of fit depends little on the assumed deep pressure
of the cloud, nor on the variance of the size distribution of the cloud and
haze particles. However, we can see that there is a strong preference for
solutions with a haze particle mean radius of 0.05 – 0.1 $\mu$m and high
imaginary refractive index of 0.1. For the cloud, it can be seen that the
preference is for a low imaginary refractive index of 0.001. Retrievals where
the cloud imaginary refractive index was set to 0.1 had $\chi^{2}$ in excess
of 1000 and so are not visible in Figs. 6 and 7, but the constraint on the
cloud mean radius is not strong with values of 0.05 – 0.1 $\mu$m slightly
favoured over 1.0 $\mu$m. A cloud mean radius of 0.5 $\mu$m is least favoured.
In addition, we checked to see if the limb-darkening curves modelled with our
radiative transfer model at all zenith angles were consistent with the
Minnaert law and found very good correspondence for zenith angles less than
$\sim 70^{\circ}$ (noted in Fig. 3) for schemes using both five and nine
zenith angles, adding confidence to our approach.

Figure 6: Variation of the goodness-of-fit of our retrievals ($\chi^{2}$) in
the equatorial band (variable cloud fractional scale height, stratospheric
haze opacity and cloud-top methane mole fraction) as a function of the fixed
grid values of the other cloud and haze properties defined in Table.
LABEL:tab:parameter. The first three panels of the top row show the $\chi^{2}$
values of all the fits for different fixed values of the mean cloud particle
radius, cloud particle imaginary refractive index, and variance of the cloud
particle radius distribution, while the first three panels of the bottom row
show the $\chi^{2}$ values for the corresponding haze particle properties. The
top right panel shows the $\chi^{2}$ values for different set values of the
cloud base pressure, while the bottom right panel shows the fitted values of
the cloud-top methane mole fraction, with the cloud base pressure of each case
colour-coded as indicated in the panel above to show that there is no simple
correlation between retrieved methane mole fraction and set cloud base
pressure. As we are fitting simultaneously to $2\times 101=202$ points in
total (i.e., two spectra at $0^{\circ}$ and $61.45^{\circ}$ zenith angle,
respectively), $\chi^{2}=\ \sim 200$ indicates a good fit.  Figure 7: As Fig.
6, but for the $60^{\circ}$S latitude band.

Although there are a wide range of best-fit $\chi^{2}$ values it is apparent
that the best fits are achieved for a methane cloud-top mole fraction of
$\sim$ 4–6% at the equator and $\sim$ 2–4 % at $60^{\circ}$S. However,
although we clearly retrieve lower methane mole fractions near the south pole
than at the equator it can be seen that there are a wide range of possible
cloud solutions that give equally good fits to the data, but rather different
methane abundances. Hence, although it would appear that the polar methane
cloud-top mole fraction is $\sim 0.5$ times that at the equator we can be less
certain of the absolute cloud-top methane mole fraction at the equator and
pole.

Having surveyed the range of cloud properties that best match the observed
limb darkening at the equator and $60^{\circ}$S, we then took one of the best-
fit setup cases and applied this to all latitudes. We chose to fix
$p_{base}=4.66$ bar, $r_{cloud}=0.1$ $\mu$m with 0.05 variance and imaginary
refractive index $n_{imag}=0.001$. For the haze we chose to fix $r_{haze}=0.1$
$\mu$m with 0.3 variance and imaginary refractive index $n_{imag}=0.1$. We
then fitted to the synthetic spectra generated from our fitted Minnaert limb-
darkening coefficients for all the latitude bands sampled by the Neptune MUSE
observations and fitted once more for 1) cloud opacity; 2) cloud fractional
scale height; 3) haze opacity; and 4) the cloud-top methane mole fraction. The
resulting fitted methane cloud-top mole fractions as a function of latitude
are shown in Fig. 8, where we also show the methane mole fraction variation
derived in our previous analysis [Irwin . (2019)] and that derived by kark11.
In Fig. 8 we can see that our derived latitudinal methane distribution for our
default model (indicated as Model 1) varies much more smoothly with latitude
than our previous analysis [Irwin . (2019)] and has more smoothly-varying
error bars. In addition, it can be seen that our new retrieved methane
variation more closely resembles that determined by kark11. The greatest
discrepancy occurs at 20 – 40∘S and we found here that our fits had the
highest $\chi^{2}/n$ values. To introduce additional flexibility into our
model, we ran our retrievals a second time, but additionally allowed the model
to vary the imaginary refractive indices of the cloud and haze particles
(Model 2), where the imaginary refractive indices were still assumed to be
wavelength invariant. It can be seen that Model 2 retrieves lower methane mole
fractions at 20 – 40∘S and even more closely resembles the results of kark11.

Figure 8: Fitted methane mole fractions as a function of latitude. The results
from our previous work [Irwin . (2019)] are shown for reference and compared
with our new model: 1) where the imaginary refractive indices of the haze and
cloud are fixed to 0.1 and 0.001, respectively; and 2) where the imaginary
refractive indices of the haze and cloud are allowed to vary (keeping constant
with wavelength). Also shown are the methane mole fractions estimated by
kark11, scaled to match our estimates, and recalculations with Model 2 where
the base cloud pressure has been reduced to 4.15 and 3.65 bar, respectively.
The difference is not large for $p_{base}=4.15$ bar, but for $p_{base}=3.65$
bar it can be seen that the methane retrieval becomes unstable, for the
reasons described in the text.

Fig. 9 shows the latitudinal variation of the wavelength-invariant values of
$n_{imag}$ retrieved by Model 2 for the cloud and haze, and also shows the
retrieved 2-D (i.e., latitude – altitude) cloud structure. We can see that
$n_{imag}$ for the cloud is poorly constrained, but that of the haze is well
estimated and it would appear that to best match the observations at 20 – 40∘S
the haze particles are required to have slightly lower $n_{imag}$ values than
those found at other latitudes. This is easily understood looking at Fig. 1
where we can see that this latitude has numerous bright, high, discrete
clouds. Although we masked the observations to focus the retrievals on the
background smooth latitudinal variation, to mask completely the brighter
clouds at these latitudes would have left us with no data to analyse at all
(Fig.2). Hence, we would expect Model 2, which allows the cloud/haze particle
reflectivity to vary, to better incorporate the additional reflectivity from
these upper tropospheric methane clouds and so fit the observations more
accurately and also retrieve a more reliable latitudinal variation in cloud-
top methane mole fraction. Please note that the cloud opacity plot in Fig. 9
shows opacity below the 4.66-bar cloud base pressure for two reasons: 1) we
assume the opacity to diminish with a scale height of 1 km below the
condensation level rather than cutting off sharply; and 2) we show here the
opacity in the 39 model atmospheric layers, which are split equally between
$\sim 10$ and $0.001$ bar and so do not coincide exactly with the base
pressures of the cloud and haze.

Figure 9: Fitted latitudinal variation of cloud opacity/bar (at 800 nm) and
imaginary refractive indices of the cloud and haze particles. The top panel
shows a contour plot of the fitted cloud opacity/bar profiles (darker regions
indicate greater cloud density), while the bottom panel shows the latitudinal
variation of the retrieved imaginary refractive indices of the cloud particles
(red) and haze particles (blue), together with error range (grey). For the
imaginary refractive indices it can be seen that this is poorly constrained
for the cloud (large error bars) and we just need the particles to be highly
scattering. However, the imaginary refractive index of the haze is well
constrained, and shows these particles are more scattering at 20–40∘S. The red
line in the top plot indicates the cloud top pressure (i.e., level where
overlying cloud opacity at 800 nm is unity). The cloud contour map indicates
the main cloud top to lie at similar pressure levels at all latitudes and has
a cloud-top pressure of $\sim$ 3–4 bar. We can also see a very slight increase
in stratospheric haze opacity at 20 – 40∘S, associated with the cloudy zone
and then clearing slightly towards the north and south.

In addition to providing a better constrained retrieval of cloud-top methane
mole fraction, showing its mole fraction to decrease from equator to south
pole, our new retrieval scheme appears to detect noticeably lower mole
fractions of methane near 60∘S. This is more easily seen in Fig. 10, which
shows the spatial variation of tropospheric cloud opacity, tropospheric cloud
fractional scale height, stratospheric cloud opacity and cloud-top methane
mole fraction projected onto the disc of Neptune as seen by VLT/MUSE. It is
difficult to be certain if this is a real feature as we have much less
geometrical coverage of the limb-darkening curves as we approach the south
pole. If it is a real feature then it is possible it might perhaps be related
to the South Polar Feature (SPF), e.g., tollefson19. We will return to this
question in the next section.

Figure 10: Fitted tropospheric cloud opacity (Cloud Opac.), tropospheric cloud
fractional scale height (Cloud FSH), stratospheric cloud opacity (Haze Opac.)
and cloud-top methane mole fraction (Methane VMR) projected on to the
Neptune’s disc as seen by VLT/MUSE. Also plotted are the opacity (Meth. Cloud
Opac.) and mean pressure level (Meth. Cloud Press.) of additional methane
clouds used to fit the discrete cloud regions. The methane mole fraction plot
clearly shows lower values of methane polewards of $30-40^{\circ}$S and an
apparent possible local minimum at $60^{\circ}$S.

Finally, in Fig.11 we show the fitted spectra from Model 2 at $0^{\circ}$ and
$61.45^{\circ}$ zenith angle, compared with the synthetic ‘observed’ spectra
at the equator and $60^{\circ}$S. The error bars of the synthetic observations
are shown as the lighter colour-shaded regions and it is apparent here why the
errors in the synthetic observations had to be inflated to enable the
retrieval model to fit to an accuracy of $\chi^{2}/n\sim 1$: even when
inflated the reflectivity errors are still small ($\sim 0.5$ %) compared with
the likely accuracy of the gaseous absorption coefficients used and the
simplicity of our cloud parameterization scheme.

Figure 11: Spectra fitted by NEMESIS to synthetic observations at $0^{\circ}$
(red) and $61.45^{\circ}$ (blue) zenith angle generated from the Minnaert
limb-darkening analysis at $60^{\circ}$S and the equator. The Minnaert-
modelled synthetic observed spectra and error limits (as described in the
text) are shown as the light shaded regions, while the spectra fitted by our
retrieval model are shown as the solid, darker lines.

###  3.4 Comparison with previous retrievals

The difference between our new methane retrievals and our previous estimates
[Irwin . (2019)] are for some locations greater than 3-$\sigma$, although it
should be remembered that these are random errors only, and do not account for
systematic errors arising from differences in the assumed methane/cloud
models. These differences mostly occur in the cloud belt near 20 – 40∘, where
we have unaccounted upper tropospheric methane ice clouds, but we wondered
whether there might be other effects that might explain the sharper
latitudinal changes in methane abundance and estimated errors of [Irwin .
(2019)]. Our previous retrievals assumed a base cloud pressure of
$p_{base}=4.23$ bar, rather than $p_{base}=4.66$ bar as we assumed here.
Hence, we re-ran our retrievals using the two other base pressures listed in
Table LABEL:tab:parameter of 4.15 and 3.65 bar, respectively.

The results for Model 2 (where we also fit for $n_{imag}$ of both cloud and
haze) for all three cloud base pressures are shown in Fig. 8. As can be seen,
the results for $p_{base}=4.15$ bar are very similar to those for
$p_{base}=4.66$ bar, but those for $p_{base}=3.65$ are very different and
appear, in terms of scatter and inflated error bars, more like our previous
results [Irwin . (2019)]. We believe this to be caused by an artefact of our
retrieval model, where we have assumed a single cloud with fixed base pressure
and variable scale height combined with our simple “step” methane model. For
Model 2 with $p_{base}=4.66$ bar it can be seen in Fig. 9 that the retrieved
level of unit cloud optical depth is in the range 3–4 bar, depending on
latitude, comfortably greater than the methane condensation pressure. However,
when the cloud base pressure is lowered to $p_{base}=3.65$ bar the cloud
opacity has to be greater at lower pressure levels in order to give enough
overall reflectivity. This pushes the level of unit optical depth to lower
pressures and, depending on the deep methane abundance, can at some latitudes
become similar to the methane condensation pressure. In such circumstances the
sensitivity of the calculated reflectivity to the deep methane mole fraction
is reduced and the retrieved mole fraction may need to be greatly increased
(and have greater error bars) to give enough methane absorption, exactly as we
see. The retrieved pressure levels of unit optical depth from irwin19 are
shown in Fig. 9 of that paper to be in the range 1.8–3 bar, which is indeed
rather close to the methane condensation pressure level and so would
unfortunately have suffered from this same systematic artefact. However,
irwin19 assumed $p_{base}=4.23$ bar, a value that gave consistent results in
our new retrievals, which indicates that there must be an additional
difference between the two analyses that caused irwin19 to retrieve unit
optical depth values near the methane condensation level. We have identified
this difference to be that rather than using a priori values of
$n_{imag}=0.001$ and 0.1 for the cloud and haze respectively, as used in this
study, irwin19 assumed the imaginary refractive index spectra to be fixed at
all latitudes to those retrieved from their limb-darkening analysis at 5 –
10∘S. Figure 6 of irwin19 shows that the haze particles were found at the
equator to be rather dark ($n_{imag}$ in the range 0.076 to 0.159, depending
on wavelength). These darker, wavelength-dependent haze particles, combined
with the extended wavelength region of 770 – 930 nm (compared with 800 – 900
nm considered here) led to larger retrieved haze opacities and consequently
required larger cloud opacities to match the peak reflectivity at continuum
wavelengths. This then led to the retrieved unit cloud optical depth levels
approaching the methane condensation pressure.

To demonstrate this effect we repeated the analysis of the central median
observations of irwin19, using exactly the same setup as used in this previous
study, but substituting the a priori cloud scattering properties to be those
used by our new Model 1 (fixed $n_{imag}$) and Model 2 (variable $n_{imag}$).
Our re-fitted methane mole fractions are shown in Fig. 12. Here, we can see
that when reprocessed in this way the set of spectra along Neptune’s meridian
return a latitudinal variation in deep methane mole fraction that is much more
consistent with our new analysis and also with the HST/STIS determinations
[Karkoschka Tomasko (2011)]. The apparently small retrieved errors of the
reanalysis towards the south pole should be viewed with caution. Such
latitudes are only seen over a very small range of zenith angles and so we
cannot say anything about the limb-darkening here. Hence, we are much more
dependent at these latitudes on the assumed cloud and methane profile, and so
our methane estimates are prone to larger systematic error. In addition, for
the central meridional analysis we used the MUSE pipeline radiance errors
(scaled to give $\chi^{2}/n\sim 1$ for fits to spectra near the equator),
which are smaller near the pole giving smaller apparent methane mole fraction
errors. In contrast our new limb-darkening analysis has fewer points to define
the limb-darkening and so assigns larger error bars to the reconstructed
spectra near the pole. Hence, at these latitudes the retrieved deep mole
fraction is retrieved with larger error.

Returning to the question of the apparent minimum of methane at 60∘S in our
new analysis, with larger retrieval errors towards the south pole the solution
might be expected to partially relax back to the a priori value of ($4\pm
4$)%222Note that this parameter is treated logarithmically within NEMESIS, and
hence it is the fractional error, i.e., 1.0, that is used in the covariance
matrix.. However, when we repeated the retrievals using a lower methane mole
fraction of ($2\pm 2$)% (i.e., same fractional error), the same latitudinal
behaviour was determined as can be seen in Fig. 12 (Model 2A), so this cannot
be the cause. Instead, we believe this apparent methane feature may arise from
the limited range of zenith angles sampled to fit the Minnaert parameters at
these latitudes, since they will appear only at higher zenith angles and our
analysis further omits observations with $\mu>0.3$ ($\mu$ is the cosine of the
zenith angle), to avoid locations too near the disc edge. Figure 4 shows
(bottom right panel) that at these latitudes the Minnaert limb-darkening
parameter, $k$, appears to tend to $\sim 0.5$ at $\sim 80^{\circ}$S at all
methane-absorbing wavelengths, but there is no clear difference in the
appearance of Neptune at this latitude at any wavelength. Hence, we believe
this effect to be a geometrical artefact of our limb-darkening analysis, in
the same way that the central meridional analysis shows a continuing decrease
towards the pole as we view the locations at higher and higher zenith angle.
Only observations recorded with even higher spatial resolution would be able
to better constrain the latitudinal variability of methane at such high
latitudes. In the meantime, our new determinations of methane mole fraction at
polar latitudes have larger retrieval errors properly indicating this greater
uncertainty.

Figure 12: As Fig. 8, but comparing the results from our previous work [Irwin
. (2019)] with the results of a reprocessing of the central meridian spectra
considered by irwin19, where the cloud/haze scattering properties have been
replaced by those used in our new analysis and either fixed at all latitudes
(Reproc. Model 1), or allowed to vary with latitude (Reproc. Model 2). In
addition, we show the results of our limb-darkening analysis with Model 2
(previously shown in Fig. 8 and which has an a priori methane mole fraction of
4%) and a revised version of Model 2, where the a priori methane mole fraction
has been reduced to 2% (Model 2A).

Finally, we return to the question of assumed methane and cloud profile
parameterizations and the absolute accuracy of our methane retrievals. As
noted earlier, with spectral observations such as these in this wavelength
region what we are actually sensitive to is the column abundance of methane
above the cloud top. Sadly, the vertical resolution of such nadir/near-nadir
observations cannot physically be less than $\sim$ one scale height, which
means that it is very difficult to discriminate between the “step” methane
model used here and more sophisticated models such as the “descended profile”
model favoured by, e.g., kark11 and sromovsky19. This is especially the case
when considering that we do not have a good ab initio model for the vertical
cloud structure either. It is apparent that for models with higher cloud
opacity at lower pressures, the mole fraction of methane will need to be
higher to give the same column abundance and so it can be seen that there
exists a wide range of possible cloud and methane vertical distribution models
that could fit our observations equally well and give the same methane column
abundance for a given latitude. For the nominal Model 2 retrievals presented
here, with cloud base at 4.66 bar we see a clear reduction in the column
abundance of methane towards the pole, which we interpret here in terms of a
deep mole fraction varying from ($5.1\pm 0.3$)% at the equator to ($2.6\pm
0.2$)% at $60^{\circ}$S, i.e., a reduction factor of $1.9\pm 0.2$. However, in
terms of the latitudinal dependence of the mole fraction of methane this
depends on the methane profile, cloud profile and the reference pressure level
and so the cloud-top methane mole fraction could conceivably vary by as much
as $\sim\pm 1$%. Hence, here we estimate the equatorial deep mole fraction of
methane to be in the range 4–6%, with the abundance at polar latitudes reduced
from those at equatorial latitudes by a factor of $1.9\pm 0.2$.

###  3.5 Extension to discrete cloud retrievals

Having greatly improved our fit to the background atmospheric state, we
wondered if it might be possible to retrieve, in addition, the cloud profiles
for the discrete cloud regions masked out in our analysis so far. Discrete
clouds such as these are known to exist at pressures from 0.5 – 0.1 bar [Irwin
. (2011), e.g.] and as such are almost certainly clouds of methane ice. It can
be seen from Fig. 1 that in our observations these clouds are mostly
restricted to latitudes 30 – 40∘S and are of highly variable reflectivity and
only cover a small range of central meridian longitudes. Hence, our Minnaert
limb-darkening analysis, which assumes that the clouds at a particular
latitude are zonally-symmetric and do not vary with central meridian
longitude, was not appropriate for analysing these discrete clouds. Instead,
assuming that the cloud properties of the background tropospheric and
stratospheric clouds would not be different from their zonally-retrieved
Minnaert values, we looked to see what opacity and pressure level additional
discrete methane ice clouds might need to have to best match the observed
spectra in the previously masked pixels. We assumed that the methane ice
particles at such low pressures were likely small and assumed a size
distribution with mean radius 0.1 $\mu$m and variance 0.3. Methane ice is
highly scattering at short wavelengths [Martonchik Orton (1994), Grundy .
(2002)] and so the complex refractive index was set to $1.3+0.00001i$, with
scattering properties calculated via Mie theory and phase functions again
approximated by combined Henyey-Greenstein functions. For vertical location,
we assumed that the cloud had a Gaussian dependence of specific density
(particles/gram) with altitude and had a variable peak pressure and opacity.
The a priori pressure level of the opacity peak was set to 0.3 bar. We
reanalysed the areas that had previously been masked and extended the area
slightly to capture some of the thinner discrete clouds seen (Fig. 2). Then,
for each pixel in this extended area, we set the tropospheric cloud,
stratospheric haze and cloud-top methane mole fraction to that determined from
the zonally-averaged Minnaert analysis for that latitude and retrieved the
opacity and peak pressure of an additional methane ice cloud.

Our retrieved methane ice cloud properties can be seen in Fig. 10, where we
retrieve opacities of up to 0.75. Although there are clearly some regions of
thick methane ice cloud, the median value of this additional opacity is found
to be only 0.0063 and so it only makes a significant difference to the
observed radiances in the small discrete regions seen. Figure 10 also shows an
apparent variation in mean pressure of these discrete clouds, remaining near
the a priori pressure of $\sim 0.3$ bar for the thinnest clouds, but reducing
to as low as 0.15 bar for the thickest clouds. However, we find this pressure
variation to be insignificant compared with the retrieval errors. Reviewing
the two-way transmission-to-space within the 800 – 900 nm wavelength region
examined here, we find that we are only weakly sensitive to the actual
pressure level of detached methane clouds in the 0.5 – 0.1 bar region. Our
chosen wavelength band includes the strong methane absorption band at 887 nm,
but even here the two-way transmission to space only reduces to 0.5 at the
0.35 bar level for nominal cloud/haze conditions. To really probe the
altitudes of such clouds we need to use observations in the much stronger
methane bands at 1.7 $\mu$m in the H-band, as has been done by numerous
previous authors [Irwin . (2011), Irwin . (2016), Luszcz-Cook . (2016),
e.g.,], but which is not observable by MUSE. We examined the raw MUSE
observations near 727 nm and 887 nm, but could not see any clear difference in
the brightness of the discrete clouds with wavelength for either weak or
bright detached clouds. Hence, all we can really say with the MUSE
observations is that the discrete clouds (for all opacities) must lie
somewhere at pressures less than $\sim 0.4$ bar.

With the addition of discrete methane clouds our forward-modelled
reconstructed images of Neptune at 830 and 840 nm are compared with the MUSE
observations in Fig. 13. Comparing with Fig. 1 it can be seen that we achieve
a very good fit at all locations on Neptune’s disc. Although we have only
shown the fit at two wavelengths here, the fit was found to be very good for
all wavelengths in the 800 – 900 nm range and Fig. 14 shows the root-mean-
square (RMS) reflectivity differences of our fits at all wavelengths, showing
that we match the observed spectra to an RMS of typically 0.05%, increasing to
only 0.15% in the brightest methane ice clouds. At these locations it may be
that model is struggling with the fact that the stratospheric haze opacity was
set and fixed to that derived from zonally-averaged fits where the discrete
clouds had not been entirely masked.

Figure 13: As Fig. 1, but here the reconstructed images are generated from
NEMESIS forward-modelling calculations from the fitted cloud and methane
profiles, including fitting of discrete cloud regions using an additional thin
methane ice cloud layer. As can be seen the residuals are reduced greatly and
are small at all locations on the planet’s disc. Figure 14: Root-mean-square
reflectivity (I/F) differences of our final fit to the observations of Neptune
over the entire visible disc from 800 – 900 nm.

##  4 Conclusions

In this work, we have reanalysed VLT/MUSE-NFM observations of Neptune, made in
June 2018 [Irwin . (2019)], with a Minnaert limb-darkening analysis recently
developed for Jupiter studies [Pérez-Hoyos . (2020)]. We find that the new
scheme allows us to use the observations much more effectively than simply
analysing along the central meridian, as we had previously done, as it also
accounts for the observed limb-darkening/limb-brightening at different
wavelengths and latitudes. Once having fitted the general latitudinal
variation of cloud and haze with this analysis we are then able to fit for the
properties of discrete methane clouds seen in our observations, allowing us to
fit all locations on the visible disc to a reflectivity (I/F) precision of 0.5
– 1.0% (RMS $<0.15$%). Our main conclusions are:

  * •

We find that we are able to fit the background reflectivity spectrum of
Neptune from 800 – 900 nm with a simple two-cloud zonally symmetric model
comprising a deep cloud based at 4.66 bar, with variable fractional scale
height and a layer of stratospheric haze based near 0.03 bar.

  * •

The cloud-top mole fraction of methane at 2–4 bar (i.e., above the H2S cloud,
based at 4.66 bar) is found to decrease by a factor of $1.9\pm 0.2$ from
equator to pole, from 4–6% at the equator to 2–4% at the south pole.

  * •

While this latitudinal decrease in methane mole fraction is well defined, the
absolute mole fractions at different latitudes depends on the precise choice
of cloud parameterization (and indeed methane parameterization), for which a
wide range of setups give similar goodness of fit.

  * •

The previous retrievals along the central meridian of these data reported by
irwin19 appear to have suffered from an unfortunate retrieval artefact at some
locations due to a less sophisticated incorporation of limb-darkening. Our new
methane retrievals are more robust, vary more smoothly with latitude, and give
a clearer and conservative estimate of the likely error limits.

  * •

The opacity of both the tropospheric cloud and stratospheric haze is found to
be maximum at 20 – $40^{\circ}$S and 20 – $40^{\circ}$N, which are also the
latitudes of the discrete methane clouds seen. While this may be a real
feature, it may also be that the limb-darkening curves analysed by our
Minnaert scheme were contaminated by thin discrete clouds at these latitudes.

  * •

Adding localised methane clouds to our zonal model allows us to additionally
retrieve the properties of the discrete cloud locations; we find these clouds
must lie at pressures less than 0.4 bar and have opacities of up to 0.75.

  * •

The latitudinal variation of cloud-top methane mole fraction with latitude
observed here in 2018 seems little changed from that seen by HST/STIS in 2003
[Karkoschka Tomasko (2011)], indicating that this apparent latitudinal
distribution of cloud-top methane has not varied significantly in the
intervening fifteen years.

Having now developed a scheme that matches the observed spectra of Neptune
from 800 – 900 nm, the next step will be to reproduce the appearance of
Neptune at other wavelengths to see if our fitted cloud/methane model is more
generally applicable. There will be two main challenges to this:

  1. 1.

Firstly, extending to shorter wavelengths the contribution of Rayleigh
scattering becomes more important, which limits our ability to see to deeper
cloud layers. However, more importantly for the MUSE-NFM observations is the
fact that the full-width-half-maximum (FWHM) of the point-spread-function
(PSF) becomes significantly worse. It can be seen in Fig. 1 that even at
800–900 nm, there is a considerable residual off-disc signal due to the PSF.
We could have tried to model this in our fitting procedure, but concluded that
it was simpler to limit ourselves to locations not too near the disc edge in
the Minnaert analysis. However, this would become more difficult to justify at
shorter wavelengths. Instead, in future work we hope to take our existing
model, calculate the appearance at shorter wavelengths, and convolve with a
PSF model, which still needs to be developed. By thus simulating the shortwave
observations we will be able to see if we can discount more of the possible
cloud/haze paramaterisation setups, which will help refine our methane
retrievals.

  2. 2.

Extending to longer wavelengths, the contribution of Rayleigh scattering
becomes less and the increasing strength of the methane absorption bands means
that it is possible to probe the vertical extent and location of clouds more
precisely. We would need to extend to J, H and K-band observations to see
which of our cloud/haze setups can be discounted, using the fact that the
opacity of small cloud particles will fall more quickly with wavelength than
large particles. However, this will mean trying to analyse observations taken
at different apparitions and with different instruments, which will mean that
the cloud distribution will not be the same and that systematic errors in the
photometric calibration and PSF characterisation will arise. In addition, the
complex refractive indices of the particles will be different from those we
have derived at these wavelengths.

Although these difficulties are not insurmountable, they will require careful
evaluation and effort to overcome, which is why we leave them as future work.
However, it is clear from this study that splitting the atmosphere of Neptune
into a background zonal part (that can be fitted with a the Minneart limb-
darkening scheme) and an additional spatially-varying part due to discrete
methane clouds greatly simplifies the retrieval process and allows us to
efficiently reconstruct the entire 3-D structure of Neptune’s clouds and
methane cloud-top mole fraction. Our model is then simultaneously consistent
with the observations at all wavelengths under consideration and all locations
on Neptune’s disc. The approach could also be applied to observations of
Neptune at other wavelengths and also Uranus and Saturn, which to a first
approximation appear zonally symmetric with additional discrete cloud features
at visible/near-infrared wavelengths. It could also be further applied to
Jupiter observations, building upon the work of perez20, at wavelengths and
latitudes that appear relatively homogeneous.

###### Acknowledgements.

We are grateful to the United Kingdom Science and Technology Facilities
Council for funding this research. Glenn Orton was supported by NASA funding
to the Jet Propulsion Laboratory, California Institute of Technology. Leigh
Fletcher was supported by a Royal Society Research Fellowship and European
Research Council Consolidator Grant (under the European Union’s Horizon 2020
research and innovation programme, grant agreement No 723890) at the
University of Leicester. The observations reported in this paper have the ESO
ID: 60.A-9100(K).

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#
NIMA POST-PROCESS BANNER TO BE REMOVED AFTER FINAL ACCEPTANCE

Lightweight Thermal Management Strategies for the Silicon Detectors of CBM at
FAIR

Franz Matejcek111Joint first authors with equal contribution.
<EMAIL_ADDRESS>Kshitij Agarwal 222Joint first authors with
equal contribution. for the CBM Collaboration

###### Abstract

Lightweight thermal management is central to the design of the all-silicon
Inner Tracker of the Compressed Baryonic Matter Experiment (CBM) at the
Facility for Antiproton and Ion Research (FAIR). This experiment aims to study
strongly interacting matter at neutron star core densities through fixed-
target Au-Au collisions ($\sqrt{s_{NN}}=2.9-4.9$ GeV; up to 107 beam-target
interactions per second; polar angular acceptance of
$2.5^{\circ}\leq\Theta\leq 25^{\circ}$). Studies with thermal demonstrators
for both the pixel-based Micro Vertex Detector and the strip-based Silicon
Tracking System have validated their thermal management strategies, preparing
them for series production now and global commissioning in 2028.

###### keywords:

Thermal Management, Silicon Detectors, Heavy Ion Collisions, 3MTM NovecTM 649,
Air Cooling

[1]organization=Goethe-Universität Frankfurt, addressline=Institut für
Kernphysik, Max-von-Laue-Straße 1, postcode=60438, city=Frankfurt am Main,
country=Germany [2]organization=Eberhard-Karls-Universität Tübingen,
addressline=Physikalisches Institut, Auf der Morgenstelle 14, postcode=72076,
city=Tübingen, country=Germany

##  1 Introduction

The all-silicon Inner Tracker of the CBM [1, 2], comprising the Micro Vertex
Detector (MVD) [3] and Silicon Tracking System (STS) [4], is tasked to
accurately determine secondary vertices and reconstruct tracks in a wide
momentum range down to a a few 100 MeV/c in a high track-density environment
resulting from unprecedentedly high beam-target interaction rates (see Tab.
1). This is expected to result in maximum accumulated non-ionising damage of
up to $10^{14}\,n_{eq}/\textrm{cm}^{2}$ over the detector’s lifetime.
Lightweight thermal management strategies are crucial to mitigate radiation-
induced effects, such as the rise of leakage currents over the detector’s
lifetime, while also minimizing the material budget to guarantee the required
precision in momentum determination.

|  | MVD
---
| STS
---
| Number of
---
Layers
| 4 (8 – 20 cm
---
Downstream)
In vacuum
| 8 (30 – 100 cm
---
Downstream)
In air
| Active Area and
---
Granularity
|  0.15 m2 (288 Sensors)
---
150M Pixels
|  4 m2 (876 Sensors)
---
1.8M Strips
| Sensor
---
Technology
| CMOS MAPS
---
(MIMOSIS)
| Double-Side Strips
---
(Hamamatsu Sensors)
| Material per Layer
---
|  0.3% – 0.5% X0
---
|  0.3% – 2% X0
---
|  Spat. Res. $\sigma_{xy}$
---
Time Res. $\sigma_{t}$
|  $\approx$ 5 µm
---
5 µs (Frame Length)
|  $\approx$ 10 µm
---
5 ns
| Power
---
Dissipation
|  $P_{\text{Sensor}}~{}\lesssim$ 75 mW/cm2
---
$T_{\text{Sensor}}~{}\lesssim 0\,^{\circ}$C
|  $P_{\text{Sensor}}~{}\approx$ 50 mW/cm2
---
$P_{\text{FEE}}~{}\approx$ 40 kW
$T_{\text{Sensor}}~{}\lesssim 10\,^{\circ}$C
Rate |  | 80 MHz/cm²
---
0.1 MHz Reactions
| 10 MHz/cm²
---
10 MHz Reactions
Table 1: Summary of major features of MVD and STS.

##  2 Thermal Management Concept

MVD: Due to the vacuum operation of the MVD, the power dissipated by the
sensors (MIMOSIS, $\lesssim$ 75 mW/cm2, $\approx$ 120 W total) is transported
conductively through Thermal Pyrolytic Graphite (TPG) sheets. It is guided
into the peripherally located aluminium heat sinks which are actively cooled
with 3MTM NovecTM 649 (see Fig. 1(a)). This minimises the material budget in
the geometric acceptance and allows for a homogeneous temperature distribution
across the sensors ($<$10 K over the TPG), ensuring a high detection
efficiency and low fake hit rate.

STS: The material budget requirement of the STS is maintained by holding the
sensors on carbon fibre ladders and placing the electronics outside the
geometric acceptance. The highly irradiated sensors around the beam-pipe
($\leq$ 50 mW/cm2 at 10 ∘C) are cooled by impinging air-jets via perforated
carbon-fibre tubes while natural air convection is deemed sufficient for the
remaining less irradiated sensors. This is to avoid their thermal runaway by
minimizing the rise of stable operating temperature caused at the end-of-life
fluence and minimise the material budget. Furthermore, the electronics power
dissipation of $\approx$ 40 kW is neutralised by custom-made aluminium heat
sinks actively cooled with 3MTM NovecTM 649 to minimise the residual heat-
transfer to the silicon sensors (see Fig. 1(b)). Both detector subsystems
foresee to use a common 3MTM NovecTM 649 cooling plant with $-20\,^{\circ}$C
nominal operating temperature.

(a)

(b)

Figure 1: Illustrations showing the thermal path of (a) MVD, and (b) STS.
Drawings are not to scale.

##  3 Experimental Validation

Realistic demonstrators with pre-series detector components were built for
both subsystems to validate the respective thermal management strategies,
explore safety margins, and determine optimal operational parameters. The MVD
demonstrator module, comprising an L-shaped heat sink and a TPG carrier in a
vacuum chamber, represents the entire detector with its 16 thermally
independent quadrants. Similarly, the STS demonstrator comprises one thermally
active half-layer with dummy silicon sensor and electronics heating elements
in realistic mechanical boundary conditions to accurately represent the
interplay between the heating elements.

Fig. 2(a) shows the radiation-corrected maximum temperature gradient across
the TPG bulk of a MVD quadrant ($\Delta T_{\textrm{TPG~{}Bulk}}$) for varying
sensor power and coolant temperatures. The data shows the $\Delta
T_{\textrm{TPG~{}Bulk}}<$ 10 K at nominal operational conditions with large
safety margins. The thermal runaway performance of the STS thermal
demonstrator’s half-station at nominal operating conditions is shown in Fig.
2(b). The stable temperature rise ($\Delta T_{\textrm{Stable}}$) observed
after mimicking sensor heating corresponding to end-of-lifetime fluence (EOL)
distribution exhibits importance of the air sensor cooling both forced air
convection via impinging air jets (central ladders; LT201-202; $\Delta
T_{\textrm{Stable}}<$ 7.4 K) and natural convection (peripheral ladders;
LT203-206; $\Delta T_{\textrm{Stable}}\approx$ 0 K), while ensuring sufficient
neutralisation of FEE power dissipation.

(a)

(b)

Figure 2: (a) Measured maximum temperature gradient across the TPG carrier as
a function of the sensor’s power dissipation and the coolant temperature. (b)
Temperature rise of all half-station’s sensors before and after power
dissipation mimicking the EOL fluence distribution.

##  4 Acknowledgements and Funding

FM and KA acknowledge the support from the BMBF Project-ID 05P21RFFC2 and
05P19VTFC1, respectively.

## References

  * Ablyazimov et al. [2017] T. Ablyazimov, et al. (CBM),  Challenges in QCD matter physics –The scientific programme of the Compressed Baryonic Matter experiment at FAIR,  Eur. Phys. J. A 53 (2017) 60. doi:10.1140/epja/i2017-12248-y. arXiv:1607.01487.
  * Agarwal [2023] K. Agarwal (CBM),  The compressed baryonic matter (CBM) experiment at FAIR—physics, status and prospects,  Phys. Scripta 98 (2023) 034006. doi:10.1088/1402-4896/acbca7.
  * Stroth et al. [2022] J. Stroth, et al., Technical Design Report for the CBM: Micro Vertex Detector (MVD), GSI, Darmstadt, 2022.
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