the method of Wang et al. (2020), as the empirical covariance matrices cannot
be incorporated in species, other than for GRAVITY data. Fitting for the
covariance parameters was universally favoured by the Bayes factor, and thus
we only present the full fits. Consistent with the expectations of Greco &
Brandt (2016) and Nasedkin et al. (2023), including these parameters also
tended to broaden the posterior distributions, though posterior widths remain
far narrower than the variation between the models.

In addition to the Bayesian fits, we perform a simple $\chi^{2}$ minimisation
over each grid to avoid potential issues with interpolating the spectra along
the different parameter axes. Using this framework, we identify the single
best-fit spectra, as presented in Table 9.

##  5 Results

07

Figure 6: Best fit temperature profiles and spectra for the group $A\cap B$
retrievals. From top to bottom are HR 8799 b, c, d and e.

Figure 7: [M/H] posterior distributions for all retrievals in group $A\cap B$.
From left to right are the distributions for b, c, d, and e. The vertical line
indicates solar metallicity. Model keys are as in tables 14–17, and are sorted
by the Bayes factor from top to bottom.

Figure 8: C/O posterior distributions for all retrievals in group $A\cap B$.
From left to right are the distributions for b, c, d, and e. The vertical line
indicates stellar C/O. Model keys are as in tables 14–17, and are sorted by
the Bayes factor.

Figure 9: Teff posterior distributions for all retrievals in groups $A\cap B$.
From left to right are the distributions for b, c, d, and e. The vertical line
indicates 1000 K. Model keys are as in tables 14–17, and are sorted by the
Bayes factor. Figure 10: Grid fits from Exo-REM, Sonora Diamondback, ATMO and
petitCODE. ALES data for HR8799d is scaled by the overall best-fit scaling
parameter (Exo-REM, 0.57). Fits are the single best-fit $\chi^{2}$ model from
the grid. Table 8: Grid-retrieval results

Planet | Model | $\bm{\Delta\log}_{\mathbf{10}}\bm{\mathcal{Z}}$ | $\bm{T_{\rm eff}}$ | $\bm{\log g}$ | [M/H] | $\bm{f_{\rm sed}}$ | C/O | Radius | $\bm{\log L/L_{\odot}}$ | Mass
---|---|---|---|---|---|---|---|---|---|---
|  |  | [K] | [cgs] |  |  |  |  [RJup] |  |  [MJup]
b | ATMO | -63 | ${1020}_{-7}^{+9}$ | ${4.42}_{-0.04}^{+0.04}$ | ${>0.6}$ | … | ${<0.3}$ | ${0.73}_{-0.02}^{+0.02}$ | ${-5.278}_{-0.009}^{+0.008}$ | ${5.6}_{-0.4}^{+0.4}$
| Diamondback | -98 | ${959}_{-8}^{+10}$ | ${4.18}_{-0.03}^{+0.03}$ | ${>0.5}$ | ${<1.0}$ | 0.458 | ${0.95}_{-0.02}^{+0.02}$ | ${-5.155}_{-0.007}^{+0.007}$ | ${5.6}_{-0.4}^{+0.4}$
| Exo-Rem | 0 | ${862}_{-6}^{+21}$ | ${4.0}_{-0.03}^{+0.02}$ | ${0.69}_{-0.03}^{+0.11}$ | … | ${0.50}_{-0.02}^{+0.10}$ | ${1.19}_{-0.03}^{+0.03}$ | ${-5.14}_{-0.02}^{+0.02}$ | ${5.5}_{-0.4}^{+0.4}$
|  petitCODEa | -105 | ${>850}$ | ${3.48}_{-0.02}^{+0.01}$ | ${>1.4}$ | ${2.53}_{-0.02}^{+0.01}$ | 0.55 | ${2.00}_{-0.04}^{+0.02}$ | ${-4.73}_{-0.02}^{+0.01}$ | ${4.8}_{-0.2}^{+0.8}$
c | ATMO | 0 | ${1195}_{-6}^{+5}$ | ${4.34}_{-0.03}^{+0.03}$ | $>0.6$ | … | ${0.425}_{-0.014}^{+0.012}$ | ${1.01}_{-0.01}^{+0.01}$ | ${-4.723}_{-0.003}^{+0.003}$ | ${9.1}_{-0.5}^{+0.5}$
| Diamondback | -245 | ${1237}_{-2}^{+2}$ | ${<3.5}$ | ${>0.5}$ | ${2.32}_{-0.02}^{+0.02}$ | 0.458 | ${0.99}_{-0.01}^{+0.01}$ | ${-4.683}_{-0.002}^{+0.002}$ | ${1.25}_{-0.01}^{+0.01}$
| Exo-Rem | -30 | ${1065}_{-1}^{+1}$ | ${4.15}_{-0.02}^{+0.02}$ | $>1.0$ | … | ${0.501}_{-0.002}^{+0.003}$ | ${1.314}_{-0.005}^{+0.005}$ | ${-4.694}_{-0.002}^{+0.002}$ | ${9.8}_{-0.5}^{+0.5}$
| petitCODE | -622 | ${1400}_{-0.5}^{+0.2}$ | ${4.00}_{-0.01}^{+0.01}$ | ${>0.3}$ | ${>4.5}$ | ${0.55}_{-0.00}^{+0.00}$ | ${0.80}_{-0.00}^{+0.00}$ | ${-4.652}_{-0.002}^{+0.002}$ | ${2.57}_{-0.02}^{+0.02}$
d | ATMO | -38 | ${1300}_{-2}^{+2}$ | ${4.39}_{-0.01}^{+0.01}$ | ${0.07}_{-0.02}^{+0.02}$ | … | ${<0.3}$ | ${0.926}_{-0.004}^{+0.004}$ | ${-4.652}_{-0.003}^{+0.003}$ | ${8.4}_{-0.2}^{+0.2}$
| Diamondback | -76 | ${1234}_{-7}^{+8}$ | ${4.27}_{-0.02}^{+0.02}$ | ${>0.5}$ | ${1.95}_{-0.06}^{+0.04}$ | 0.458 | ${1.05}_{-0.02}^{+0.01}$ | ${-4.632}_{-0.002}^{+0.002}$ | ${8.4}_{-0.2}^{+0.2}$
| Exo-Rem | 0 | ${1155}_{-4}^{+5}$ | ${4.23}_{-0.01}^{+0.01}$ | ${0.78}_{-0.03}^{+0.03}$ | … | ${0.200}_{-0.002}^{+0.002}$ | ${1.168}_{-0.005}^{+0.006}$ | ${-4.656}_{-0.005}^{+0.007}$ | ${9.3}_{-0.2}^{+0.6}$
| petitCODE | -185 | ${1400}_{-1}^{+1}$ | ${4.47}_{-0.01}^{+0.01}$ | $>0.3$ | ${2.99}_{-0.07}^{+0.06}$ | ${0.55}_{-0.000}^{+0.000}$ | ${0.859}_{-0.003}^{+0.003}$ | ${-4.587}_{-0.002}^{+0.003}$ | ${8.9}_{-0.2}^{+0.2}$
e | ATMO | 0 | ${1251}_{-20}^{+22}$ | ${4.38}_{-0.07}^{+0.06}$ | ${>0.6}$ | … | ${0.52}_{-0.03}^{+0.03}$ | ${0.87}_{-0.03}^{+0.03}$ | ${-4.778}_{-0.006}^{+0.006}$ | ${7.2}_{-0.8}^{+0.8}$
| Diamondback | -16 | ${1244}_{-6}^{+6}$ | ${<3.5}$ | ${>0.5}$ | ${2.31}_{-0.05}^{+0.05}$ | 0.458 | ${0.92}_{-0.01}^{+0.01}$ | ${-4.738}_{-0.005}^{+0.005}$ | ${1.12}_{-0.05}^{+0.07}$
| Exo-Rem | -14 | ${1064}_{-3}^{+3}$ | ${4.13}_{-0.04}^{+0.04}$ | ${>1.0}$ | … | ${0.50}_{-0.01}^{+0.01}$ | ${1.24}_{-0.01}^{+0.01}$ | ${-4.747}_{-0.005}^{+0.006}$ | ${8.0}_{-0.8}^{+0.8}$
| petitCODE | -56 | ${1399}_{-4}^{+2}$ | ${4.00}_{-0.03}^{+0.02}$ | $>0.3$ | ${4.48}_{-0.03}^{+0.02}$ | ${0.55}_{-0.00}^{+0.00}$ | ${0.755}_{-0.004}^{+0.005}$ | ${-4.703}_{-0.004}^{+0.004}$ | ${2.3}_{-0.1}^{+0.1}$

  * •

Notes

  * •

All values presented are the median values from the fits, with uncertainties
given as the $\pm 34.1$% percentiles. Values without uncertainties were fixed
during the fit.

  * •

$a$ the petitcode-cool-cloudy grid was used to fit HR 8799 b.

Table 9: Grid-fit $\chi^{2}$ results

Planet | Model | $\bm{\chi^{2}}$ | $\bm{T_{\rm eff}}$ | $\bm{\log g}$ | [M/H] | $\bm{f_{\rm sed}}$ | C/O | $\bm{\gamma_{\rm ad}}$ | Radius
---|---|---|---|---|---|---|---|---|---
|  |  | [K] | [cgs] |  |  |  |  |  [RJup]
b | ATMO | 744 | 1000 | 4.5 | 0.6 | … | 0.3 | 1.05 | 1.00
| Diamondback | 879 | 1100 | 3.5 | 0.5 | 2.0 | 0.458 | … | 1.07
| Exo-REM | 510 | 850 | 3.5 | 0.5 | … | 0.55 | … | 1.05
|  petitCODEa | 968 | 850 | 3.5 | 1.4 | 2.0 | … | … | 0.79
c | ATMO | 2162 | 1200 | 4.5 | 0.6 | … | 0.3 | 1.01 | 0.97
| Diamondback | 3900 | 1200 | 3.5 | 0.5 | 2.0 | 0.458 | … | 1.08
| Exo-REM | 1868 | 1100 | 3.5 | 1.0 | … | 0.8 | … | 1.22
| petitCODE | 7374 | 1200 | 3.5 | 0.3 | 3.0 | 0.55 | … | 1.09
db | ATMO | 943 | 1300 | 3.0 | 0.6 | … | 0.55 | 1.01 | 0.94
| Diamondback | 1336 | 1200 | 3.5 | 0.5 | 1.0 | 0.458 | … | 1.12
| Exo-REM | 936 | 1200 | 3.0 | 1.0 | … | 0.55 | … | 1.12
| petitCODE | 2254 | 1200 | 4.0 | 0.0 | 1.5 | 0.55 | … | 1.21
e | ATMO | 669 | 1300 | 4.0 | 0.6 | … | 0.55 | 1.03 | 0.81
| Diamondback | 819 | 1200 | 3.5 | 0.5 | 2.0 | 0.458 | … | 1.02
| Exo-REM | 623 | 1100 | 3.5 | 1.0 | … | 0.8 | … | 1.15
| petitCODE | 1136 | 1400 | 4.0 | 0.3 | 4.5 | 0.55 | … | 0.74

  * •

Notes

  * •

All values presented are the single best-fit value according to the
$\chi^{2}$. All of the models share a similar number of parameters, which are
listed in Table 7. Additional parameters for data scaling and covariance
fitting are shared between all models.

  * •

a for HR 8799 b, the petitcode-cool-cloudy grid was used rather than the
petitcode-hot-cloudy grid.

  * •

b the fits for HR 8799 d included a scaling parameter for the ALES data.

Based on both the atmospheric retrievals and the self consistent grid fits,
the HR 8799 planets share enriched atmospheres, with stellar-to-superstellar
C/O ratios. The properties of each atmosphere for subsets of the ensemble of
retrievals are summarised in tables 1–4 for planets b–e, respectively. These
tables also contain ranges of plausible parameter values for each planet based
on the aggregate of the self-consistent models, while Fig. 5 shows the
distributions for a subset of the key parameters. These estimates are
synthesised from the results of the Bayesian fits and the $\chi^{2}$
minimisation, rejecting solutions with unphysical masses and radii. The full
results of the grid fits are found in Table 8 for the Bayesian fits and Table
9 for the single-best $\chi^{2}$ fits. In these tables an index is assigned to
each retrieval, with the format planet.group.index, which will serve as the
retrieval identifier throughout the text.

Nearly 100 retrievals were performed for this analysis across the four
companions in order to derive robust constraints on primary planetary
properties. The aggregate results of the Bayesian model average of group
$A\cap B$ retrievals are presented in Fig. 5, with the best fit models for the
same sample of retrievals compared to the data in Fig. 6. Overall we find
consistently good fits for all four planets, with best-fit $\chi^{2}/\nu<2$
for each planet. The self-consistent grid-fits incorporate additional physical
processes and require fewer free parameters than the retrievals, making them
less flexible. The additional processes, such as radiative-convective
equilibrium, chemistry and cloud physics act as narrow priors on the bulk
atmospheric parameters. While the lack of flexibility leads to worse fits when
compared to retrievals, the fits of multiple self-consistent grids still
result in mutually consistent results. Notably, petitCODE gives consistent
parameter estimates even though it is strongly disfavoured by the goodness-of-
fit metrics.

Both the pRT retrievals and the self-consistent grid fits show that all of the
atmospheres are strongly enriched in metals, finding median [M/H]$\gtrsim$1.0
for each planet. This is driven by the carbon and oxygen abundances as
measured through H2O and CO, and particularly through the CO absorption
feature at 2.3 $\upmu$m. The C/O ratios decrease with decreasing separation
from $0.78\pm 0.04$ in HR 8799 b to $0.60\pm 0.05$ for HR 8799 d, with HR 8799
e breaking the trend with carbon rich C/O ratio of $0.87\pm 0.02$. In addition
to water and CO, HCN is confidently detected in HR 8799 c and e, while CH4 is
detected in HR 8799 c.

We find that cloudy atmospheres are universally favoured over clear
atmospheres. Extended silicate clouds, together with an iron cloud deck are
the preferred models for the three inner planets, with the cooler HR 8799 b
showing evidence for Na2S clouds.

The retrievals found masses, radii and surface gravities consistent with
evolutionary models. By construction, these form Group A of our retrievals. In
general, retrieving the planet radius is challenging: the radius of directly
imaged exoplanets is commonly underestimated by both retrievals and grid fits
(e.g. Bonnefoy et al., 2016). These unphysical solutions are found in several
of the grid fits, while nearly every retrieval finds plausible values of the
mass and radius. The small radius is often compensated for by adjusting the
temperature, metallicity, or cloud properties. Nevertheless, in all cases the
most favoured retrieval produced masses consistent with dynamical mass
measurements and radii consistent with evolutionary models. In the group
$A\cap B$ retrievals that used dynamical mass priors, the retrieved mass
estimate is within $1\sigma$ of the dynamical mass for b, d and e, with the
posterior width consistent with the prior width which. For HR 8799 c the
retrieved mass is moderately higher than the dynamical mass estimate at
$8.5\pm 0.5$ M${}_{\textrm{\tiny Jup}}$ compared to the dynamical mass of
$7.7\pm 0.7$ M${}_{\textrm{\tiny Jup}}$. However, if the dynamical mass is not
used as a constraint and $\log g$ is freely retrieved, the resulting mass
estimate is found to be much larger than the dynamical mass estimate,
highlighting the importance of including additional constraints in a retrieval
framework. The estimate of the effective temperature of each planet is
consistent with previous findings, with e, c, and d sharing temperatures
around 1100 K, and b being cooler at around 950 K. While we do not perform a
comparison to spectral templates due to the inferred high metallicity, low
surface gravity and potential complications from the viewing angle of the
planets, we find that the spectral types found in Bonnefoy et al. (2016)
remain a good description of the four planets. HR 8799 d and e are similar to
late-L type brown dwarfs, consistent with their inferred effective
temperature. HR 8799 c is more likely fit by an early-T spectral type as it is
a few kelvin cooler than e and d and is beginning to show spectral features
from CH4. At 950 K HR 8799 b is solidly in the T dwarf regime.

In this section we will present the measured properties for each of the four
planets individually. Following that, Section 5.5 will discuss in depth the
results and challenges of comparing different thermal structures, chemistry
and cloud parameterisations, and data inclusion.

###  5.1 HR 8799 b

For HR 8799 b we include the GRAVITY and OSIRIS spectra, together with the
full set of photometry, allowing the OSIRIS data to float as the published
data are not flux calibrated. For the grid fits we include additional
parameters to describe the covariance of the OSIRIS data.

HR 8799 b is the coldest and lowest mass planet in the HR 8799 system. The
best estimate of these parameters via Bayesian averaging of group $A\cap B$
retrievals finds an effective temperature of $942^{+12}_{-16}$ K and a mass of
$6.0^{+0.4}_{-0.3}$ MJ, driven by the use of the dynamical mass estimate of
Zurlo et al. (2022) as a prior. The radius is slightly inflated compared to
jupiter, with ($R_{pl}=1.10^{+0.03}_{-0.03}$); combining the mass and radius
estimates leads to $\log g=4.10^{+0.03}_{-0.04}$. This is consistent with the
Bayesian grid fits, though the single-best $\chi^{2}$ fits found both lower
($\log g=3.5$) and higher ($\log g$=4.5) solutions. The Bayesian averaged
results (for group $A\cap B$) are driven by a single retrieval, with a
$\Delta\log_{10}\mathcal{Z}$ of 4 relative to the next best retrieval. This
single best retrieval uses the Z23 temperature profile and free chemistry,
finding a metallicity of $0.96^{+0.08}_{-0.08}$ and a C/O of
$0.78^{+0.03}_{-0.04}$.

The grid fits find temperatures between 850 K$-$1100 K, with the single best
fit, found using Exo-REM, finding $T_{\rm eff}=850$ K. Using the Bayesian fit,
Exo-REM is again the most favoured model by the Bayes factor, and finds
$T_{\rm eff}=862^{+21}_{-6}$ K and a radius of ${1.19}_{-0.03}^{+0.03}R_{\rm
Jup}$, consistent with expectations from evolutionary models (e.g. Marley et
al., 2012). The ATMO and Sonora Diamondback models finds a somewhat higher
temperature and a smaller radius, but are disfavoured by the Bayes factor.
petitCODE is divided into cool-cloudy and hot-cloudy grids, with the cool grid
extending up to 850 K, and the hot grid beginning at 1000 K. As the effective
temperature of HR 8799 b likely falls between these grids, it poorly fit the
data, although the remaining parameters of $\log g$ and $f_{\rm sed}$ are
compatible with the other self-consistent models.

Figure 7 shows the variation of the metallicity across the different models
considered. We see that strong enrichment solution is almost always found,
particularly by models preferred by the Bayes factor. The degree of enrichment
does not systematically vary between disequilibrium and free chemistry
retrievals. All of the self-consistent models favour high metallicity
solutions, reaching the upper bounds of the grid in all cases.

Similarly to the metallicity, the C/O ratio is shown for the different
retrievals in Fig. 8 and $T_{\rm eff}$ in 9. The C/O ratio is generally
constrained to between 0.6 and 0.8. However, for free chemistry retrievals,
the inferred C/O ratio only indicates the gas-phase composition. Additional
oxygen is sequestered in the silicate clouds: accounting for this
sequestration would result in a lower C/O ratio. Among the grid fits, shown in
Fig. 10, the C/O ratio shows more variation, ranging from the lower bound of
the ATMO grid at 0.3 to 0.55 from Exo-REM. The best fit models from Exo-REM
are consistent with the stellar value of 0.54, and additional data covering
the near infrared water features is likely necessary to improve these
constraints. From Fig. 11 we find that both the free retrieval and
disequilibrium retrievals display similar trends in the retrieved chemical
abundances, finding that nearly 10% of the atmosphere is CO by mass, while
water has a lower abundance of around 1% by mass. The best-fit free retrieval
finds systematically lower abundances for both of these species compared to
the best-fit disequilibrium retrieval. H2S and CH4 are found to be the next
most abundant species in both the disequilibrium and free chemistry
retrievals. However, even though their abundances are constrained by the
posterior distribution, there is no evidence for their detection when
comparing between the full free chemistry retrieval (b.AB.1) and retrievals
without these species (b.A.0 and b.A.2). The b.AB.1 retrieval uses the Z23
temperature profile, free chemistry, and clouds condensing at their
equilibrium position. This same setup is used for b.A.0 and b.A.2, apart from
the exclusion of CH4 and H2S respectively. While the inferred CH4 abundance of
$\log X_{{\rm CH}_{4}}=-5.0\pm 0.4$ is compatible with Barman et al. (2015),
there is no statistical evidence to support the detection.

In order to compare the cloud composition for HR 8799 b, we consider a set of
retrievals using the same temperature structure (M20) and cloud
parameterisation, and vary the cloud optical properties and condensation
curve. We find that Na2S clouds are preferred over silicate, iron, and KCl
clouds ($\Delta\log_{10}\mathcal{Z}\geq 2)$. Patchy clouds were also explored,
but no evidence was found for patchiness, regardless of cloud composition. At
the temperature of HR 8799 b silicate clouds are expected to occur below the
photosphere, with Na2S or KCl clouds becoming the primary aerosol opacity
source. This result is likely driven by the condensation temperature of Na2S
rather than the optical properties; unlike silicate clouds which have strong
absorption features in the mid-infrared, crystalline Na2S is featureless out
to 15 $\upmu$m, apart from a characteristic scattering slope.

Figure 11: Mass fraction abundance profiles as a function of pressure in the
atmospheres of HR 8799 b, c, d and e. The solid lines indicate the median
values of the most favoured disequilibrium retrieval for each planet, with the
shaded region indicating the 90% confidence interval. The circular markers
show the median values for each species from the most favoured free chemistry
retrieval, with the error bars indicating the 90% confidence interval. The
position along the pressure axis is arbitrary. The minimum mass fraction
allowed in the free retrievals was $10^{-7}$.

###  5.2 HR 8799 c

In our standard retrieval setup, we include data from SPHERE, GPI, CHARIS,
GRAVITY and ALES for HR 8799 c. We omit the OSIRIS data, as it overlaps nearly
completely with the GRAVITY data, is not flux calibrated, and requires either
binning to lower resolution to be fit with the c-k opacity tables, or the use
of the higher-resolution line-by-line opacities to fit the full resolution
data, dramatically increasing computation time. Nevertheless, we perform
several retrievals incorporating the OSIRIS data rather than the GRAVITY data
to determine how this choice impacts the retrieved chemistry and clouds, and
to determine if we can reproduce the findings of Konopacky et al. (2013).
Overall, HR 8799 c proved challenging to fit: many retrieval setups either
required far more model computations before convergence than the other three
planets, or failed to converge entirely. The grid-fits for HR 8799 c also
displayed the greatest variation between models.

Like HR 8799 b, the group $A\cap B$ retrievals of HR 8799 c are dominated by a
single retrieval, with $\Delta\log_{10}\mathcal{Z}=2$ compared to the next
best retrieval. This retrieval uses the Z23 temperature profile and free
chemistry, and requires high-altitude silicate clouds ($\log P_{\rm
base}=-3.4\pm 1.8$ bar). The inferred effective temperature of
$1159^{+11}_{-12}$ K is consistent with the range of temperatures found by the
self-consistent grids, which spans from 1100 K to 1200 K, with the Bayesian
fits averaging around $\sim 1200$ K. The retrieved mass
($8.5^{+0.4}_{-0.5}\leavevmode\nobreak\ M_{\rm Jup}$) is slightly higher than
the dynamical mass estimate, though radius ($1.10\pm 0.01\leavevmode\nobreak\
R_{\rm Jup}$) is compatible with evolutionary models, and from these we derive
a $\log g$ of $4.26^{+0.02}_{-0.03}$. The grid-fits also find plausible radii,
favouring values slightly larger than 1 ${R}_{\textrm{\tiny Jup}}$. All of the
self-consistent models find temperatures of 1100$-$1200 K, and $\log g$
between 3.5 and 4.5. From the single-best $\chi^{2}$ fits, the ATMO model
finds a $\log g$ of 4.5, while the remaining models find a lower solution of
3.5.

As with the other HR 8799 planets the retrievals favour highly enriched
solutions, finding [M/H] = $1.27^{+0.05}_{-0.05}$. The disequilibrium
chemistry retrievals find slightly lower metallicities than the free chemistry
retrievals, with the most-favoured disequilibrium retrieval finding a
metallicity of $1.05\pm 0.04$. The data for HR 8799 c are highly discrepant in
the H band (Figure 3), with the CHARIS data and photometry being around 50%
brighter than the SPHERE and GPI data. As the metallicity is highly sensitive
to the amplitude of the J, H and K band peaks, such discrepancies need to be
resolved in the data to ensure reliable measurement of this parameter. All of
the grid-fit solutions require high metallicity and are limited by the grid
boundaries. The C/O ratios between the Bayesian fits and the $\chi^{2}$
minimisation are consistent, typically favouring stellar to slightly sub-
stellar C/O/ The retrievals present a more consistent picture, with most
retrievals favouring a mildly super-stellar C/O ratio, with the average group
$A\cap B$ C/O of $0.60^{+0.02}_{-0.01}$. The most favoured free chemistry
retrieval for HR 8799 c finds consistent water and CO abundances with most
favoured disequilibrium retrieval. The free retrieval also found an extremely
high HCN mass fraction of $\log X_{\rm HCN}=-2.54\pm 0.05$, orders of
magnitude higher than the predictions from equilibrium chemistry. This finding
is strongly favoured by the Bayes factor, with $\log_{10}\mathcal{Z}=30$ in
favour of including HCN (c.AB.3 over c.A.8), with both retrievals using the
Z23 temperature profile and clouds condensing at their equilibrium locations.
The detection of HCN is largely driven by the ALES data; the wavelength
dependence of the detection is discussed further in Section 4.2. If HCN is
excluded, the overall metallicity is also increased, mostly due to a 3 dex
increase in the H2S abundance to $\log X_{\rm H_{2}S}=-2.38\pm 0.06$. While
this solution is disfavoured, the higher H2S abundance is more compatible with
the equilibrium chemistry predictions. High resolution spectroscopy is likely
required to precisely characterise the sulphur and nitrogen elemental
abundances, and determine reliable abundances for these trace species.
Although the CH4 abundance is relatively low, with $\log X_{\rm
CH_{4}}=-4.3\pm 0.06$, it is precisely constrained, and detected with high
confidence, $\Delta\log_{10}\mathcal{Z}=11.5$ (c.AB.3 over c.A.6).

HR 8799 c is host to a highly cloudy atmosphere. The most favoured retrieval
finds high altitude ($\log P_{\rm MgSiO_{3}}=-3.4\pm 1.8$ bar) MgSiO3 cloud
with a mass fraction of $\log X_{\rm MgSiO_{3}}=-4.7\pm 1.2$, together with a
deep iron deck. The vertical mixing strength for the clouds is $\log K_{\rm
zz}=8.0\pm 0.9$, while the $f_{\rm sed}$ for both the silicate and iron clouds
are compatible, with values between 5-6. Cloud composition could not be
robustly determined for HR 8799 c, due to difficulties in retrieval
convergence. However, we find that crystalline MgSiO3 clouds (c.AB.5) provide
a better fit by the $\chi^{2}$, and are favoured by the Bayes factor over
patchy amorphous MgSiO3 (c.AB.8), both using the M20 profile and
disequilibrium chemistry. The use of patchy cloud layers may improve this fit,
allowing individual layers to impact the spectrum independently, but this
would come at the cost of substantially increasing the number of parameters to
fit the continuum shape. The crystalline morphology provides a marginally
better fit to the MIRI photometric data, but spectroscopic characterisation of
the silicate feature is likely necessary to robustly distinguish these cases.

We ran a disequilibrium and free chemistry retrieval using the OSIRIS data in
place of the GRAVITY data to check for consistency and to determine if we
could reproduce the findings of Konopacky et al. (2013). In order to use the
correlated-k method of pRT, we binned the OSIRIS data by a factor of 4, to a
spectral resolving power of $R\sim 1000$. We find that a high metallicity
solution is still found using these data, with effective temperatures, and
surface gravities consistent with the GRAVITY retrievals. For the
disequilibrium retrieval, the C/O ratio is found to be significantly higher
than any of the GRAVITY retrievals, as well as the free chemistry OSIRIS
retrieval. While the free retrievals using the GRAVITY data find a slightly
higher metallicity overall, the OSIRIS data finds a slightly higher CH4
abundance of $-3.81\pm 0.09$, as well as a much higher H2S abundance. Overall,
we find that the main findings of metal-rich atmospheres are reproducible
regardless of whether we use the GRAVITY or OSIRIS data, and confirm the
detection of water and CO in the atmosphere of HR 8799 c.

###  5.3 HR 8799 d

For HR 8799 d we include all available spectroscopic data as described in
Section 3, but include a scaling factor for the ALES data set as otherwise it
is incompatible with NACO photometric observations. This shifts the mean
L-band flux of HR 8799 d to a similar magnitude as e and c.

Unlike HR 8799 b or c, d is well fit by a broad selection of models, and no
single retrieval dominates the Bayesian average of figure 5. All of the
preferred models $(\Delta\log_{10}\mathcal{Z}<2)$ use disequilibrium chemistry
and the M20 or Z23 temperature profiles. Using the Bayesian average, the
effective temperature is retrieved to be $1179^{+31}_{-28}$ K, compatible with
the ranges found by the grid fits, which find $T_{\rm eff}$ from 1150 K – 1300
K. The planet mass ($9.2\pm 0.1\leavevmode\nobreak\ M_{\rm Jup}$) was tightly
constrained by the dynamical mass prior, which in turn allowed for precise
measurement of the planet radius ($1.26\pm 0.07\leavevmode\nobreak\ R_{\rm
Jup}$) and $\log g$ ($4.18\pm 0.05$). This surface gravity is consistent with
the estimates from the Bayesian grid fits, but is significantly higher than
the 3.0-4.0 range found by the $\chi^{2}$ fits. The self-consistent fits find
marginally smaller radii than the retrievals, generally between 1.1 and 1.2
${R}_{\textrm{\tiny Jup}}$.

The metallicity of HR 8799 d is consistent with HR 8799 b and c, with [M/H] =
$1.2\pm 0.2$. While most grid-fit solutions also favour high-metallicity
atmospheres, the bayesian fit with the ATMO model finds a solution consistent
with solar metallicity, radius inconsistent with evolutionary models
($0.926\pm 0.004$ ${R}_{\textrm{\tiny Jup}}$). The best fit Exo-Rem model
finds a metallicity of 0.78$\pm 0.03$, a radius of $1.168\pm 0.006$
${R}_{\textrm{\tiny Jup}}$ and an effective temperature of 1155$\pm 5$ K. The
C/O ratio is always found to be consistent with the stellar value, with
retrievals finding C/O = $0.60^{+0.04}_{-0.06}$. The grid-fits are also
typically consistent with stellar, though the best fit Exo-Rem model finds a
substellar C/O ratio of 0.2.

Patchy clouds are marginally disfavoured by the Bayes factor, and the
patchiness is poorly constrained, finding $f_{\rm cloud}=0.45\pm 0.3$. The
most favoured solutions require either amorphous or crystalline MgSiO3 or
crystalline MgSiO3, with a marginal preference for the amorphous structure.
Each of these compositions displays slightly different near infrared slopes,
shown in Figure 16. However, such a slope can be induced by various sources of
continuum opacity that may not be fully accounted for in the retrieval. Thus
mid infrared observations of the silicate absorption features will be
necessary to robustly constrain the composition and particle geometry. There
is no preference for a free cloud base pressure compared to the equilibrium
position, suggesting that the AM01 model is sufficient to describe the clouds
in this atmosphere.

While the disequilibrium retrievals are favoured over the free chemistry
retrievals, there is excellent agreement between the freely retrieved
abundances and the disequilibrium chemical profiles. Water and CO are both
highly abundant, and the freely retrieved abundances agree with the
disequilibrium profiles to within $1\sigma$. No other species are both highly
abundant and well constrained, so we do not perform leave-one-out retrievals
to test for their presence. However, even at low abundances the freely
retrieved CH4 abundance is compatible with the disequilibrium profile.

###  5.4 HR 8799 e

The measurements of HR 8799 e largely reinforce existing literature values.
The single most favoured retrieval, which also dominates the group $A\cap B$,
uses free chemistry and the Z23 temperature profile, together with a cloud
base calculated using equilibrium condensation. This leads to similar results
as for HR 8799 c and d, and is consistent with the results of the grid fits.
An effective temperature of $1138^{+30}_{-22}$ K is retrieved, compatible to
within uncertainties of the grid-fits, which find a temperature range of 1100
K to 1200 K. The mass posterior is determined by the dynamical mass prior, as
is the planet radius, finding $M_{pl}=7.5^{+0.6}_{-0.6}$ and
$R_{pl}=1.13^{+0.05}_{-0.05}$ respectively. This leads to a $\log g$ of
$4.18^{+0.06}_{-0.05}$, slightly higher than the self-consistent estimates of
3.5–4.0. The overall best self-consistent model by the $\chi^{2}$ is Exo-Rem,
which finds and effective temperature of 1100K, a radius of 1.15
${R}_{\textrm{\tiny Jup}}$, and a somewhat low log $g$ of 3.5. ATMO is the
most favoured self-consistent model when using the Bayesian framework, though
it finds an unphysically small radius and higher temperature than other
models. HR 8799 e is the only companion for which the MIRI photometry is not
convincingly fit, as seen below in figure 20. Every model underestimates the
flux beyond 10 $\upmu$m relative to the measurements, though this may be due
to contamination from the host star or inner disk (Boccaletti et al., 2023).

Compared to the other three planets, HR 8799 e is found to have an even more
metal rich atmosphere, with [M/H]=$1.9^{+0.1}_{-0.2}$. Metallicities $>$1 are
a universal feature of the retrievals for e. This is reinforced by the grid-
fits, which uniformly find strong enrichment, running into the upper grid
boundaries. Free chemistry retrievals are always preferred over the
disequilibrium retrievals; from these we find the group $A\cap B$ C/O ratio is
$0.87^{+0.02}_{-0.02}$. As the free chemistry retrieval C/O ratio only
accounts for the gas-phase abundances, there is additional oxygen sequestered
in the silicate clouds that could reduce the C/O ratio. However, the most
favoured disequilibrium retrieval finds a similar value of 0.83$\pm 0.02$,
suggesting that HR 8799 e is somewhat of an outlier compared to the other
three planets. Using a similar setup to Mollière et al. (2020) (e.AB.11), we
find C/O=$0.78^{+0.03}_{-0.03}$. The grid fits tend to find C/O ratios
compatible with the stellar value, though the overall best fit Exo-Rem model
also finds a higher value of 0.8.

In addition to the water and CO rich atmosphere, HCN is found to be highly
abundant, with $\log X_{\rm HCN}=-2.26\pm 0.11$. This detection is strongly
favoured by the Bayes factor, with $\log_{10}\mathcal{Z}=7.5$. As with HR 8799
c and shown in Fig. 25, this detection is driven by using the HCN opacity to
fit the ALES spectrum, though changes to the shape in the H and K-bands also
provide a slightly better fit as well. This is slightly enriched compared to
equilibrium chemistry predictions, but is expected for a metal rich planet
with a relatively high C/O ratio (Giacobbe et al., 2021), and can be produced
through photochemical reactions (Moses et al., 2013). In contrast, the
presence of CH4 is poorly constrained.

As with HR 8799 c and d, the most favoured retrieval for e favours silicate
clouds with a deeper iron deck, both condensing at their equilibrium
locations, with no preference for patchy clouds. A broad range of
disequilibrium retrievals (e.AB.7–11) find consistent cloud properties, with
condensation at the equilibrium location preferred, with a silicate abundance
lower than predicted by equilibrium chemistry (between 10$\times$ to
100$\times$ less than equilibrium), and iron abundances consistent with
equilibrium. Individual $f_{\rm sed}$ parameters for the silicate and iron
clouds are not required, with variations in the Bayes factor driven more by
the choice of patchiness and temperature profile.

###  5.5 Impacts of modelling choices

While we have examined the primary atmospheric characteristics of each of the
four HR 8799 planets, it is crucial to understand how the choice of model
impacts these measurements. As described in Section 4, we performed retrievals
using a broad selection of thermal structures, chemical parameterisations and
cloud models, each of which we will examine in detail below.

####  5.5.1 Thermal structure

Figure 12: Temperature profiles for HR 8799 e. In blue are temperature
profiles from disequilibrium retrievals, while in red are free chemistry
retrievals. The shaded regions indicate 90% confidence intervals. Also
included are the temperature profiles from the best fit self-consistent
models.  Figure 13: Metallicities and elemental number ratios for the HR 8799
planets derived from Baysian averaged group $A\cap B$ free chemistry
retrievals.

For each planet we performed retrievals using four different temperature
profile parameterisations. The Z23 profiles is used in the most favoured
retrieval for three of the four planets, while for HR 8799 d there is equal
evidence for the Z23 and M20 profiles. The Guillot profile is found to be the
second most preferred profile for HR 8799 c and e, while the spline profile
using six nodes is strongly disfavoured by the Bayes factor. In Figure 12 we
compare the retrieved and self-consistent temperature profiles for HR 8799 e.
We find that there is excellent agreement between nearly all of the retrievals
in the photosphere region, as well as with the best-fit Exo-Rem temperature
profile. There is little variation in the photosphere region between the
disequilibrium and free chemistry models. Only one model - the spline profile
with free chemistry - found a profile more similar to that of the clear ATMO
profile, though it is strongly disfavoured by the Bayes factor compared to the
other free chemistry retrievals. The spline profile is the only profile that
does not explicitly assume an adiabat deep in the atmosphere or rely on
assumptions from self-consistent models, and so we cannot fully rule out the
diabatic profiles of Tremblin et al. (2015). The bulk atmospheric properties
are also reasonably consistent across the different temperature
parameterisations. While there are statistically significant variations in the
C/O ratio between the different parameterisations, they remain broadly
consistent between 0.7 and 0.9.

Although the spline profile was disfavoured by the retrievals, it is a useful
parameterisation to determine the amount of flexibility required by the model,
and to explore the known degeneracies between the atmospheric thermal
structure and clouds Tremblin et al. (2015, 2016). We performed a series of
retrievals on HR 8799 b, varying the number of nodes in the spline profile and
observing how the retrieved profile changes with the increased flexibility. We
repeated this test for both a clear atmosphere model and an model with clouds
condensing at the equilibrium base pressure. We find no significant
differences in the temperature profiles between the clear and cloudy
atmospheres. For HR 8799 b, the Bayes factor favours retrievals with three or
four nodes in the spline profile. Fewer nodes mean the profile cannot be
accurately modelled, while more nodes add additional parameters without
improving the fit to the spectra.

####  5.5.2 Chemistry

For all four planets we perform retrievals using a grid derived from an
equilibrium chemistry solver with disequilibrium H2O-CO-CH4 quenching, as well
as free chemistry retrievals where we directly retrieve the mass-fraction
abundance of various species. Fig. 11 shows the abundance profiles from the
best fit disequilibrium and free chemistry retrievals for each planet. Both
types of retrievals produce consistent metallicities and C/O ratios for each
planet: overall there is excellent agreement in the water and CO abundances,
which are the primary opacity sources in these atmospheres. Only HR 8799 b
shows statistically significant discrepancy between the two methods for these
species, with the free retrieval finds a slightly lower metallicity than the
disequilibrium retrieval. Even for trace species, the free retrievals and
disequilibrium retrievals are largely compatible, though only a few species
have statistically significant detections in the free retrievals.

The strongest trace species detections are HCN in HR 8799 c and e, at
abundances far higher than predicted by the equilibrium model. HR 8799 b also
has a well constrained H2S measurement, though it is not statistically
significant. CH4 is significantly detected in the atmosphere of HR 8799 c,
demonstrating that with sufficient $S/N$ and wavelength coverage, it is
possible to constrain abundances at below $10^{-4}$ by mass. The free
chemistry detection of CH4 is at a moderately higher abundance than in the
best fit disequilibrium model. While it is likely also present in the cooler
atmosphere of b, additional wavelength coverage or higher $S/N$ observations
are required for a significant detection. For HR 8799 c, we include in Figures
24 and 25 in the appendix comparisons between the HR 8799 c data and models
both with and without the contribution of CH4 and HCN opacity, demonstrating
the impact of these species on the spectral shape. While the HCN detection is
driven primarily by the low flux of the ALES data, there is a significant
change in the H-band shape, as well as a slight change in the peak amplitude
of the K-band. As the ALES data are relatively low $S/N$, additional H and L
band data should be obtained to confirm this detection. However, the CH4
detection is driven by modest improvements in the fit throughout the K-band.
Several abundant species predicted by the equilibrium network are not
confidently detected by the free chemistry retrievals, such as CO2, NH3 and
H2S. Additional wavelength coverage or higher spectral resolution may allow
for the characterisation of such species.

If we take the averaged free retrieval results at face value, we can derive
elemental abundance ratios for each of the four planets, using a similar
method to calculating the metallicity. Taking the volume mixing ratios of each
molecular species, we can count the total number of C, N, O and S atoms, and
calculate the ratio relative to the planetary hydrogen abundance. Thus for
example

$\mathrm{C/H}=\frac{\rm X_{CO}+X_{\rm CO_{2}}+X_{\rm CH_{4}}+X_{\rm
HCN}}{X_{\rm H_{2}}+2X_{\rm H_{2}O}+2X_{\rm H_{2}S}+X_{\rm HCN}+4X_{\rm
CH_{4}}+3X_{\rm NH_{3}}},$ (16)

where all abundances are measured in number fraction. These ratios are then
normalised to the solar values from Asplund et al. (2009).

In Fig. 13, we present the elemental abundance ratios for each of the four
planets. We find that most elements are enhanced relative to solar for all
four planets. HR 8799 b appears depleted in nitrogen relative to the other
planets, likely due to the non-detection of NH3, which will require
observations of the 10 $\upmu$m feature to characterise. The HCN detections in
HR 8799 c and e tightly constrain the nitrogen enhancement, though these
planets still appear less enriched in nitrogen than in carbon or oxygen,
though this is again likely due to additional nitrogen stored in N2 and NH3,
whose opacities are inaccessible at these wavelengths. The sulphur elemental
ratio is poorly constrained for all of the planets apart from HR 8799 b, which
has a precise - though not statistically significant - constraint on the H2S
abundance. HR 8799 b appears sulphur rich, while the remaining planets appear
consistent with the solar value, or slightly depleted in sulphur, though this
is largely due to a lack of measured chemical species.

The C/O ratio is a key consideration for planetary atmospheres. However,
measuring the atmospheric C/O ratio and linking it to the bulk planet
composition is far from trivial. Lodders & Fegley (2002) and Lodders (2003)
explore the chemistry and condensation of substellar atmospheres, identifying
the condensation sequence of refractory species throughout these atmospheres,
finding that at typical L-dwarf temperatures there will be silicate clouds in
the photosphere region. Fonte et al. (2023) demonstrate how oxygen is
sequestered in silicate clouds and other refractory species. This was followed
by the recent work from Calamari et al. (2024), who calculate the bulk planet
C/O ratio from the atmospheric ratio, finding that the median sequestration of
oxygen due to this condensation is $17.8^{+1.7}_{-2.3}$%. They also identify a
relation between the bulk and observed C/O ratio:

$\left(\mathrm{C/O}\right)_{\rm
obs}\approx\frac{\left(\mathrm{C/O}\right)_{\rm
bulk}}{1-0.371\left(\mathrm{C/O}\right)_{\rm bulk}}.$ (17)

Solving for $\left(\mathrm{C/O}\right)_{\rm bulk}$, we find that for HR 8799
e, with an observed C/O ratio of $0.88^{+0.02}_{-0.02}$, should have a bulk
C/O ratio of 0.66, much closer to the stellar value of 0.54. Likewise, HR 8799
d has the lowest observed C/O ratio of $0.61^{+0.03}_{-0.04}$, which
translates to a modestly sub-stellar bulk C/O ratio of 0.50. In general, this
relation reduces the variation between the four planets, and brings the
planetary C/O ratio more in line with the known stellar value.

While both chemistry models are compatible, they also share similar biases.
The free chemistry model measures the gas phase abundance in the photosphere,
and is primarily impacted by the atmosphere above the silicate clouds.
Conversely, the underlying equilibrium model does remove oxygen from the gas
phase due to condensation, though the additional flexibility in the cloud
parameterisation means that it is only exactly correct for $f_{\rm sed}$ $=1$.
By parameterising disequilibrium via fixing the chemical abundances above a
quench point, the model may lose this sensitivity, and therefore measures the
abundances of CO, H2O and CH4 in a similar fashion to the free chemistry
model. Thus the C/O ratio as inferred by both models will be strongly impacted
by the oxygen-depleted region above the silicate clouds, leading to over-
estimates of the C/O ratio. Throughout this work we present these
measurements, but we note that the adjustment introduced by Calamari et al.
(2024) is likely a more accurate estimate of the bulk planet composition.

In addition to the elemental ratios, we also compute $Z_{\rm pl}/Z_{\star}$,
which allows us to directly compare our metallicities to literature values,
such as those of Thorngren et al. (2016). We convert the metallicity [M/H] of
each atmosphere to $Z_{\rm pl}$ using the methods of Thorngren & Fortney
(2019), adapting for our own notation:

$10^{[\mathrm{M/H}]}=\frac{1+Y/X}{\left(Z_{\rm
pl}^{-1}-1\right)\left(\frac{\mu_{Z}}{\mu_{H}}\right)},$ (18)

where $X$, $Y$ and $Z$ are the solar hydrogen, helium and metal mass fractions
and $\mu$ is the mean molecular weight of the metal content of the atmosphere.
Rearranging and substituting in the measured atmospheric metallicity [M/H], we
find:

$Z_{\rm
pl}=\left(1+\frac{1+Y/X}{10^{[\mathrm{M/H}]}\left(\frac{\mu_{Z}}{\mu_{H}}\right)(Z/H_{\odot})}\right)^{-1}.$
(19)

We take the same assumptions as Thorngren & Fortney (2019), taking $\mu_{Z}$
to be 18, assuming most of the metal content is in water, $\mu_{H}$ to be 1
for atomic hydrogen, and Y/X to be 0.3383 as in Asplund et al. (2009). As the
metallicity of HR 8799 A is near solar, $Z_{\star}/H$ is taken to be the solar
value of $Z/H_{\odot}=1.04\times 10^{-3}$. To normalise to the stellar
metallicity we follow Thorngren et al. (2016) and calculate the $Z_{\star}$ as

$Z_{\star}=0.014\times 10^{\mathrm{Fe/H}}.$ (20)

For HR 8799, we use solar metallicity to calculate Fe/H, but refer to the
discussion in Section 2.1.

Disequilibrium chemistry has long been thought to play a key role in shaping
the composition of the HR 8799 atmospheres (e.g. Marois et al., 2008). With
well-constrained chemical abundances, we can start to place limits on the
strength of vertical mixing that drives this disequilibrium. The quench
pressure we retrieve is defined as the level below which (in pressure) the
abundances of H2O, CO and CH4 become vertically constant. This parameterises
dynamical mixing that homogenises the upper layers of the atmosphere. More
rigorously, the quench point is defined as the point at which the chemical
timescale $t_{\rm chem}$ and the mixing $t_{\rm mix}$ are equal. Following
Zahnle & Marley (2014), the mixing timescale is defined through the ratio of
the local atmospheric scale height $H$ to the vertical eddy diffusion
coefficient, $K_{\rm zz}$:

$t_{\rm mix}=\frac{H^{2}}{K_{\rm zz}}.$ (21)

The chemical timescale depends on the reaction rates involved. Considering the
CO–CH4 reaction chain, Zahnle & Marley (2014) derive a timescale at the quench
point ($t_{q}$) for CO. For strong mixing, pulling from material at depths
below the point where the atmosphere is 1000 K, they find the timescale well
described by an Arrhenius relation for quench pressure $p$ in bar, metallicity
$m$, where $m=10^{\rm[M/H]}$, and temperature $T$in kelvin:

$t_{\rm q1}=1.5\times 10^{-6}p^{-1}m^{-0.7}\exp{\left(42000/T\right)}\ {\rm
s}.$ (22)

For weak mixing, thus drawing from low temperatures with little CO, the
timescale is found to be

$t_{\rm q2}=40p^{-2}\exp{\left(25000/T\right)}\ {\rm s},$ (23)

Combining the two, the total chemical timescale is defined as:

$t_{\rm CO}=\left(\frac{1}{t_{\rm q1}}+\frac{1}{t_{\rm q2}}\right)^{-1},$ (24)

which will favour the lower of the two values $t_{\rm q1}$ and $t_{\rm q2}$.
Equating the mixing and CO reaction timescales, we can infer the strength of
vertical mixing in the atmospheres of the HR 8799 planets:

$K_{\rm zz}=\frac{H^{2}}{t_{\rm CO}},$ (25)

where the scale height is defined as

$H=\frac{k_{\rm B}T}{\mu g},$ (26)

for temperature $T$, surface gravity $g$, mean molecular mass $\mu$ and the
Boltzmann constant $k_{\rm B}$. In order to calculate these quantities for the
HR 8799 planets, we take $T_{\rm eff}$ as a representative temperature to
calculate the timescales and scale height, $g$ as the measured surface gravity
and $\mu$ as the average mean molecular weight of the atmosphere. As $K_{\rm
zz}$ is exponential in temperature, the choice of what temperature to use
strongly influences the measured value. Using the temperature at the quench
pressure, typically deep in the atmosphere, results in unphysically strong
vertical mixing, with $\log K_{\rm zz}\approx 20$. A more thorough analysis
could try to measure the vertical mixing as a function of temperature
throughout the atmosphere, but the current data quality is of insufficient
resolution or $S/N$ for such measurements. Thus we treat $T_{\rm eff}$ as a
representative temperature with which to determine the vertical mixing
strength. We include the results of these calculations, together with the
retrieved $K_{\rm zz}$ used to parameterise the AM01 clouds in Table 10.

Table 10: Quench pressures, vertical mixing parameters, and sedimentation
fractions for the HR 8799 planets.

Planet | $\bm{\log}\mathbf{P_{\rm\textbf{q}}}$ | $\mathbf{K_{\rm\textbf{zz\,q}}}$ | $\mathbf{K_{\rm\textbf{zz,\,\cite[citealias]{\@@bibref{Phrase1}{ackerman_precipitating_2001}{\@@citephrase{AM01}}{}}}}}$ | $\bm{f_{\rm sed}}$
---|---|---|---|---
| [bar] | [cm2/s] | [cm2/s] | MgSiO3 | Fe
b | $1.7^{+0.2}_{-0.2}$ | $2.9^{+0.6}_{-0.7}$ | $8.6^{+0.8}_{-1.0}$ | $1.1^{+0.3}_{-0.3}$ | $4^{+2}_{-1}$
c | $2.3^{+0.1}_{-0.1}$ | $6.3^{+0.3}_{-0.5}$ | $9.2^{+0.4}_{-0.4}$ | $3.3^{+0.3}_{-0.3}$ | $6^{+2}_{-2}$
d | $1.0^{+0.6}_{-0.7}$ | $4.8^{+0.9}_{-1.3}$ | $9.2^{+0.7}_{-0.9}$ | $2^{+4}_{-1}$ | $6^{+2}_{-2}$
e | $2.1^{+0.4}_{-1.2}$ | $5.9^{+0.9}_{-0.6}$ | $8.6^{+0.7}_{-0.8}$ | $1.4^{+0.4}_{-0.3}$ | $6^{+2}_{-3}$

  * •

Notes

  * •

Measured from the group $A\cap B$ retrievals, providing median and $\pm 34$%
confidence regions.

Figure 14: Cloud properties for HR 8799 e, showing the optical depth due to
clouds as a function of pressure (colour map), with the $\tau=0.3$ and
$\tau=1.0$ contours highlighted by the dashed lines. The solid lines indicate
the mass fraction abundance of the MgSiO3 and Fe clouds in blue and red
respectively. Left: Cloud properties e.AB.13, using a freely retrieved cloud
base pressure and abundances. Right: The same, but for Case e.AB.2, which uses
the equilibrium condensation to determine the location of the cloud base. The
abundances are determined using equilibrium chemistry, and retrieving a
scaling factor, ($\log S_{\rm Fe}=0.0\pm 1.1$, $\log S_{\rm MgSiO_{3}}=-0.8\pm
1.0$).

The quench pressure was well constrained in the Bayesian average of group
$A\cap B$ retrievals for all four planets; the values of which are listed in
Table 10. All of the planets quench below the photosphere, with d quenching at
the highest altitude, around 10 bar. Including only the disequilibrium
retrievals in the Bayesian average of group $A\cap B$, we can derive $K_{\rm
zz}$ from the quench pressure. We can then turn to Soni & Acharyya (2023) for
a comparison, who provide predictions for CH4 and CO abundances for varying
$K_{\rm zz}$, $T_{\rm eff}$, and log $g$ across a range of metallicities. For
our measured $T_{\rm eff}$ and CH4 abundances, we should expect $\log K_{\rm
zz}$ of around 6 for the warmer three companions, regardless of whether we use
the measured CO or CH4 abundance. For HR 8799 b a much lower value (less than
$\sim 2$) is expected, assuming a 10$\times$ solar metallicity. Our inferred
$K_{\rm zz}$ values for c, d and e are compatible with this prediction,
finding $\log$$K_{\rm zz}$ between 5 and 6. For HR 8799 b we also measure weak
mixing, with $\log K_{\rm zz}=2.9^{+0.6}_{-0.7}$, which is again compatible
with the predictions of Soni & Acharyya (2023). These measurements from the
quench pressure are also inconsistent with the parameter used in the AM01
clouds, which require stronger vertical mixing of $\log K_{\rm zz}\sim 9$.
This discrepancy is perhaps not surprising: 3D modelling predicts that $K_{\rm
zz}$ should vary with altitude throughout the atmosphere, and the larger cloud
particles likely respond to the atmospheric motion differently than the gas
phase constituents. Ultimately, more precise constraints on the thermal
structure and chemical abundances, as well as trace species detections are
necessary to derive a more precise vertical mixing strength. Further modelling
work is also necessary to provide a more physically motivated transport model
than a vertically constant eddy diffusion coefficient.

####  5.5.3 Clouds

For the inner three planets, we find the most favoured solution is an
optically thin silicate cloud lying above a deeper, optically thick iron cloud
deck. In our standard setup, the clouds are parameterised as in AM01, with the
clouds condensing at the intersection of their condensation curve and the
temperature profile, with their extent determined by $f_{\rm sed}$. The cloud
mass fraction is allowed to scale from equilibrium. In totally free
retrievals, the cloud abundances, locations and vertical extents are all free
parameters of the model. This decouples the clouds from both the chemistry and
the atmospheric thermal structure, allowing them to fit the spectral shape,
but in potentially non-physical configurations. In this framework the cloud
extent is then parameterised as in AM01, determined by $f_{\rm sed}$ and
$K_{\rm zz}$. Using this setup we find an optically thin silicate cloud lying
above a compact iron cloud, while the AM01 setup finds an iron cloud that
extends high above the silicate cloud. Depending on the choice of other
parameters, either the free cloud base or the equilibrium base can be
preferred by the Bayes factor. HR 8799 e free chemistry retrievals strongly
favour the equilibrium condensed clouds, while the disequilibrium retrievals
favour the free cloud base setup. However, in general the clouds condensing at
equilibrium are the most favoured setup for each planet. Without broad
wavelength coverage and high spectral resolution to probe a high dynamic range
in pressure, it is difficult to robustly distinguish between the different
potential cloud structures.

There is a marginal preference for clouds parameterised using a Hansen (1971)
particle size distribution over a log-normal distribution (e.AB.20 over
e.AB.25, $\Delta\log_{10}\mathcal{Z}=0.7$), though this was only compared for
HR 8799 e using the Z23 profile, and assuming the clouds condense at their
equilibrium saturation location. In this case $a_{h}$ is calculated from
$f_{\rm sed}$ and $K_{\rm zz}$, and a lower $f_{\rm sed}$ for the MgSiO3 cloud
was retrieved than with the log normal distribution ($f_{\rm
SED,\,Hansen}=1.18\pm 0.20$). The effective distribution width parameter,
$b_{h}$ was found to be 0.016, which is narrower than the distributions found
by Burningham et al. (2021).

In general our clouds are comparable to those of Burningham et al. (2021) and
other similar studies (e.g. Mollière et al., 2020; Vos et al., 2023; Balmer et
al., 2023). Burningham et al. (2021) find a combination of MgSiO3, SiO2 and Fe
clouds provides the best-fit model to an ultracool field dwarf, 2MASSW
J2224438–015852. The MgSiO3 clouds in their model are located at $10^{-3}$ bar
with a maximum optical depth of $\tau=0.3$ at 1 $\upmu$m and an effective
particle radius of $\sim 0.04\leavevmode\nobreak\ \upmu$m. The SiO2 clouds are
slightly deeper, at $10^{-2}$ bar, and are optically thick at 1 $\upmu$m. This
is the same location where we find MgSiO3 clouds condensing in the atmospheres
of HR 8799 c, d and e. The iron cloud is found to be deep in the atmosphere,
though with an extended structure, with some contribution at the same altitude
as the silicate clouds. While we did not fit for a three cloud model, the
similar locations and optical depths of these clouds suggests similar
structures between the objects, even though they differ in effective
temperature by hundreds of kelvin.

Figure 15 highlights how the difference in particle radius contribute to the
difference in the wavelength dependence of the cloud optical depth. The cloud
particle radius in the AM01 model is a function of many atmospheric factors,
including the temperature, mixing strength, $f_{\rm sed}$, and particle number
density. We see that changes in the particle radius are correlated with
changes in both the temperature and the particle density. Luna & Morley (2021)
explore the impact of particle size and composition on the spectral signatures
of clouds in young brown dwarfs in the mid infrared, finding that small
particle sizes will lead to visible features in the planetary spectrum.
However, we see in Figure 15 that the particle sizes in regions of the
atmospheres with significant cloud mass fraction tend to be larger
($>1\leavevmode\nobreak\ \upmu$m), and that there are no indications of deep
cloud absorption features in the mid-infrared, even though silicate clouds are
present in the atmosphere. Small silicate particles should produce deeper
absorption features, which are not observed in the mid-infrared, suggesting
that the the impact of the small mean particle size in the upper atmosphere
does not contribute strongly to the cloud opacity.

Due to the different slopes in the near-infrared opacity as a function of
wavelength, as shown in Figure 16, there is some sensitivity to different
compositions and particle geometries. This can explain the mild preference for
amorphous MgSiO3 clouds in certain like-for-like retrieval comparisons. In
reverse, the lack of features leads to a preference for Na2S clouds in HR 8799
b. Performing additional tests on HR 8799 e, we find that Mg2SiO4 (e.AB.14)
are mildly disfavoured by the retrievals, and Al2O3 (e.AB.24) clouds are
strongly disfavoured, though this is again more likely from their condensation
location rather than from the impact of aerosol spectral features.
Observations of the silicate absorption features at 10 $\upmu$m would allow
more precise measurement of this wavelength dependence, and in turn place
better constraints on the cloud structure and composition.

Figure 15: Effective particle radii as a function of altitude for silicate
(blue) and iron (red) clouds. The solid lines indicate the radii for Case
e.AB.13, which used a free cloud base pressure and abundance, while the dashed
lines are for Case e.AB.2, which used equilibrium condensation and scaled
equilibrium abundances. The horizontal lines indicate the cloud base pressure.
The green line indicates the temperature profile, with the free cloud profile
in dark green and the equilibrium cloud profile in light green. The shaded
regions indicate unphysical particle sizes where the opacity contribution is
set to 0.

##  6 Discussion

###  6.1 Highly enriched atmospheres

While some enrichment is expected in giant planets formed through core
accretion, metallicities of nearly 100$\times$ the stellar value are far
beyond the expectations for planets with masses larger than that of Jupiter.
Having transformed the atmospheric metallicities from the retrievals to
$Z_{\rm pl}/Z_{\star}$ using equations 19 and 20, we can compare the HR 8799
planets to the broader population. Figure 17 shows how the inferred
metallicities of the HR 8799 system compare to those of other directly imaged
planets, and to a fit from Thorngren et al. (2016) derived from a sample of
transiting exoplanets. The HR 8799 planets are clear outliers amongst the
directly imaged planets, whose metallicities were taken from the literature.
We again use equations 19 and 20 to convert from a measurement of [M/H] to
$Z_{\rm pl}/Z_{\star}$. Where stellar metallicities are not available, we
assume a metallicity [Fe/H]=0. Only the 2 M${}_{\textrm{\tiny Jup}}$ planet Af
Lep b has a comparable degree of enrichment to the HR 8799 planets (Zhang et
al., 2023). However, they are comparable to hot jupiters observed in
transmission, such as WASP 39 b (Rustamkulov et al., 2023). Looper et al.
(2008) and Stephens et al. (2009) demonstrate how high metallicity atmospheres
facilitate condensation, in turn leading to the strong reddening seen in a
subset of L dwarfs. The retrieval of highly metal rich and very cloudy
atmospheres is consistent with this picture from brown dwarfs. Further
supporting the high [M/H] retrievals are their consistency with the self-
consistent grids, which always favour their upper limits.

To validate these findings, we performed a series of test retrievals for HR
8799 e, fixing the metallicity to solar composition, and comparing to cases
where [M/H] is fixed to values between 0.0 and 2.0 in steps of 0.5 dex. Figure
18 shows the best fit spectra from each of these retrievals, showing the clear
differences in the J, H and K bands between the different metallicity cases
that are unable to be compensated for by varying other atmospheric parameters.
We find that the cases of [M/H] = 1.5 and 2.0 are strongly favoured over the
cases between 0.0 and 1.0, with solar composition disfavoured at
$\Delta\log\mathcal{Z}>10$. The remaining atmospheric parameters also
significantly varied between the different retrievals: the C/O ratio increases
with increasing [M/H], maintaining a relatively constant abundance of H2O in
the atmosphere while allowing the CO abundance to increase. This combines with
the decreasing $f_{\rm sed}$, increasing the cloudiness of the planet to
dampen the stronger molecular features at high metallicity. However, if the
bulk C/O ratio calculated using the methods of Calamari et al. (2024) is
correct, a lower metallicity may also ultimately favoured. A more thorough
treatment of the condensation and chemistry in the retrieval framework are
likely necessary to accurately infer both of these parameters.

Figure 16: Cloud absorption (left) and scattering (right) opacities for
different condensate compositions and structure, for 1-$\upmu$m particles.
Dark blue indicates MgSiO3, light blue is Mg2SiO4. Solid (dashed) lines are
for crystalline (amorphous) substances. The solid red (green) line is for
crystalline iron (Na2S). Figure 17: Metallicities of the exoplanet planet
population. In blue are transiting planets, adapted from Thorngren et al.
(2016) (dark blue). In red are directly imaged planets, with references listed
below. Indicated with stars are the HR 8799 planets as measured in this work.
The grey line indicates the fit by Thorngren et al. (2016).
Notes: $\beta$ Pic b (Gravity Collaboration et al., 2020); PDS 70 b (Wang et
al., 2021); 51 Eri b (Whiteford et al., 2023); VHS 1256 b (Hoch et al., 2022);
HIP 65426 b (Petrus et al., 2021); $\kappa$ And b (Bonnefoy et al., 2014;
Wilcomb et al., 2020); YSES 1 b (Zhang et al., 2021) AB Pic b (Palma-Bifani et
al., 2023); AF Lep b (Zhang et al., 2023); GJ 504 b (Bonnefoy et al., 2018);
HD 95086 b (Desgrange et al., 2022); Ross 458 c (Burgasser et al., 2010); DH
Tau b (Patience et al., 2012); HN Peg b (Leggett et al., 2008; Suárez et al.,
2021); CT Cha b (Schmidt et al., 2008); GQ Lup b (Demars et al., 2023). Figure
18: Comparison retrievals of HR 8799 e where the metallicity is fixed to a
constant value. Black points denote the equivalent retrieval where the
metallicity is a free parameter. The top panel shows how the best fit spectra
differ across the set of retrievals, while the bottom row shows how different
atmospheric parameters vary as a function of metallicity.

The spectra shown in Figure 18 show that the retrieved metallicity is strongly
dependant on both the height of the J, H and K band peaks, as well as the
shape of these features. However, the J bands is only covered by the low-
resolution SPHERE data, with relatively poor $S/N$, and different instruments
measure significantly different flux in the H band. Without compatible
measurements in this spectral regime, robust conclusions about the metallicity
are hard to draw. Additional constraints can be obtained from other wavelength
regions; Lodders & Fegley (2002) find that the CO2 abundance scales
proportionally to [M/H]2. With strong features between 3 and 4 $\upmu$m, as
well as in the mid-infrared, future observations should be able to place
robust constraints on this parameter. Section 6.4 discusses the potential of
JWST to make such observations.

Finally, we perform independent comparisons to the moderate resolution OSIRIS
spectra presented in Ruffio et al. (2021). Following their methods, we use the
parameters of the single best-fit disequilibrium models for HR 8799 b, c and d
to compute a high spectral resolution using line-by-line opacity lists. We
convolve this model to the OSIRIS instrumental spectral resolution, and bin to
the wavelength grid. The resulting spectra is multiplied by the atmospheric
transmission function provided by Ruffio et al. (2021). The continuum is
measured by high-pass filtering the spectrum, and is subsequently subtracted
from the model. We fit a scaling factor between the model and the OSIRIS data
and compute the resulting $\chi^{2}$. This exercise was repeated, setting the
metallicity of the model to solar. For HR 8799 c and b we find that the high
metallicity model provides a better fit to the OSIRIS data than the solar
model, with the caveat that fixing the metallicity during the retrieval may
result in a better fit than setting it a posteriori, without changing other
atmospheric parameters. For HR 8799 d the high metallicity model is only a
marginally better fit than the solar metallicity model. In general, the fits
from the GRAVITY and remaining archival data provide reasonable fits to the R
$\approx 4000$ OSIRIS data, of similar quality to the fits displayed in Ruffio
et al. (2021), and are included in the appendix in Figure 23.

###  6.2 Impacts of data selection

Given the inhomogeneity in the data in terms of spectral resolution, $S/N$,
observing strategy and more, we performed a series of retrievals to examine
the impact of different datasets on the retrieval results. We first performed
retrievals using only the GRAVITY data to determine the constraining power of
this new dataset. We found that using only the GRAVITY data we could rule out
clear atmosphere solutions at $\Delta\log\mathcal{Z}>7$ (e.g. e.A.31 over
e.A.32), using the Z23 profile and disequilibrium chemistry. Using only the
GRAVITY observations for HR 8799 e, we could obtain estimates of the
metallicity ($2\pm 0.3$), effective temperature ($1143^{+38}_{-32}$ K), and
C/O ratio ($0.71^{+0.08}_{-0.2}$), which are broadly consistent with the
results from the combined dataset. We obtain similarly reliable estimates for
b (b.A.32), but using only the GRAVITY data for d (d.A.12) and c (c.A.11),
resulted in significantly lower metallicities. All of these retrievals use the
same setup of the Z23 profile, disequilibrium chemistry and silicate and iron
clouds condensing at their equilibrium saturation point. Given the higher
spectral resolution and $S/N$ of the GRAVITY data, this leads to the
conclusion that solutions to the full retrievals are largely driven by the
fits to the GRAVITY spectra. The C/O ratios for d and b are also incompatible
when using only the GRAVITY data, finding substellar values for both planets.

Conversely, we also performed retrievals that exclude the GRAVITY spectra. For
HR 8799 e (e.A.33) and d (d.A.13), we find that the retrieved parameters again
broadly agree with the retrieval including the GRAVITY data. Thus even if the
retrievals are dominated by the GRAVITY data, the conclusions we draw are
robust even when excluding the GRAVITY spectra.

###  6.3 Formation

The formation mechanism of the HR 8799 planets has seen much debate since
their discovery: simply put, how can one form four such massive planets in the
same system? The C/O ratio and metallicity are the best formation tracers
observed to date in these planets. Nevertheless, neither gravitational
instability nor core accretion scenarios have been ruled out. The high degree
of enrichment in these objects would seem to suggest a core accretion
formation scenario. However, Wang (2023) finds that even for [M/H]=0.5,
approximately 100 earth masses of solids are required to enrich the atmosphere
of HR 8799 e. With the even higher metallicities measured in this work, this
number would increase, and it is unclear if it is possible to have nearly 1000
earth masses of metals available in a protoplanetary disk and accreted with
high efficiency. Wang (2023) additionally find that late accretion of
planetesimals would require less material to result in a similar degree of
enrichment, potentially only requiring a few hundred earth masses of solids to
achieve the high metallicities of all four planets, though this material
should quickly settle out of the atmosphere.

Figure 19: Predictions for NIRSpec/G395H based on most favoured disequilibrium
(solid) and free chemistry (dashed) retrievals, together with the best-fit
self-consistent models from each grid. Figure 20: Predictions for MIRI/MRS
based on the most favoured disequilibrium (solid) and free chemistry (dashed)
retrievals, together with the best-fit self-consistent models from each grid.

All four companions have stellar C/O ratios or higher, and the C/O ratio in
this system varies with separation, decreasing from b to d, before a sharp
increase in the C/O ratio for the innermost planet. Super-stellar C/O ratios
have been tied to core-accretion formation together with pebble drift and
evaporation (Schneider & Bitsch, 2021b, a; Mollière et al., 2022). A pathway
to significant metal enrichment was found by Bitsch & Mah (2023), though it
predicts that the CO rich pebbles should evaporate near the CO iceline, which
is outside the radius of even HR 8799 b. Planetesimal accretion cannot be
ruled out either: if large amounts of solids, with near-stellar composition,
are accreted, more metal-rich planets will have C/O ratios that approach the
stellar value. This is consistent with the trends in metallicity and C/O
between the b, c and d planets, though e remains an exception. Such a
transition could be explained by the outer three planets, particularly d,
trapping water ice and preventing these solids from reaching the innermost
planet. Some combination of these mechanisms could explain both the
atmospheric enrichment and the trends in the C/O ratio: for example, early
enrichment from evaporating pebbles could lead to the high planetary
metallicities, while late accretion of planetesimals could then drive the C/O
ratio down towards the solar value. Alternatively, Chen et al. (2024)
demonstrate that the opening of gaps in a protoplanetary disk can
significantly alter the composition of the gas and ices available to accrete
onto forming planets, and it seems likely that substructure induced by the
four HR 8799 planets would strongly impact their eventual composition.

With effective temperatures well over 1000 K and radii significantly larger
than that of jupiter, the coldest initial condition scenarios of Marley et al.
(2007) can be excluded. Beyond this constraint, the masses, luminosities and
radii seem largely consistent with a broad range of potential evolutionary
tracks (e.g. Baraffe et al., 2003; Saumon & Marley, 2008; Mordasini et al.,
2017). Further work to measure more formation tracers is clearly necessary to
unravel this system. Mid infrared spectroscopic observations could
characterise the NH3 abundance, at least in HR 8799 b, as well as the debris
disk observed in Boccaletti et al. (2023). Higher spectral resolution could
enable the measurement of carbon isotopes (e.g., as well as place better
constraints on the metallicity of each of these atmospheres). Such
measurements, combined with dedicated formation models of giant planets
outside the water iceline, are necessary to determine whether these four
planets share a formation pathway, or whether there were different mechanisms
impacting different regions of the protoplanetary disk.

###  6.4 Predictions for JWST

The high spectral resolution modes of JWST may allow us to verify the
measurements made in this work, particularly through observation of CO2, CO
and CH4 features in the near infrared and the silicate absorption features
near 10 $\upmu$m. To this end, we present the range of model predictions in
these wavelength regions at the spectral resolution of the JWST instruments,
for NIRSpec in figure 19 and for the MIRI/MRS in figure 20. The comparisons to
the MIRI photometry shown in figure 20 demonstrate the typical degree of
compatibility between the models and the data in these mid infrared
wavelengths. In the NIRSpec/G395H wavelength range we see significant
discrepancy between models for the same planet in the amplitude of the CO2
feature at 3.8 $\upmu$m, as well as the CH4 feature at 3.3 $\upmu$m and the CO
lines between 4.5 and 5 $\upmu$m. These observations will also be able to
confirm the presence of HCN in the atmospheres of HR 8799 c and e. Precise
measurement of these features should provide robust constraints on the
metallicity of these objects, verifying the degree of enrichment found via the
ground-based observations. While silicate clouds are preferred in the
retrieval comparison, none of the models show signs of deep silicate
absorption features near 10 $\upmu$m, but spectroscopic observations are
required to validate these models. Mid infrared observations will be
particularly valuable for HR 8799 b, and will allow for the clear detection of
ammonia. If combined with a chemical model to determine the ratios of
NH3:HCN:N2, this will allow for the measurement of the N/O ratio, which can
also be used as a formation diagnostic (Turrini et al., 2021; Pacetti et al.,
2022). Recently Ruffio et al. (2023) demonstrated the potential for high-
contrast imaging with NIRSpec; even without the use of a coronagraph it should
be possible to obtain flux calibrated, moderate resolution spectroscopy of HR
8799 b, c and d using the NIRSpec IFU through a combination of forward
modelling and reference differential imaging. In the case where the planet
signal is unable to be separated from that of the host star, Patapis et al.
(2022) demonstrated that it will be possible to at least identify trace
species through molecular mapping in the mid-infrared, though this will be
unable to characterise the broad wavelength features of the silicate clouds.

##  7 Conclusions

After more than 15 years of study, the HR 8799 planets remain mysterious,
though increasing data quality is allowing us to peer deeper into these
atmospheres than ever before. We presented new K-band spectra from the
VLTI/GRAVITY, which formed the basis of the our atmospheric analysis together
with a large set of archival data. Using petitRADTRANS retrievals and fits to
self-consistent grids, we inferred the atmospheric properties of all four
companions, with reasonable agreement between the two methods. Our results are
broadly consistent with the literature in terms of effective temperature,
mass, surface gravity and radius for all four planets. The use the dynamical
mass as a prior in the retrievals when determining $\log g$ allows us to
reliable retrieve physically reasonable planet radii.

We found that all four planets are strongly enriched in metals, though there
is still discrepancy between different models in constraining the precise
value. This was validated by running retrievals using different temperature
profiles, chemical models, and comparing to self consistent grids. Further
self-consistent modelling is necessary, particularly to extend model grids out
to high metallicities. The C/O ratio is stellar to super-stellar for all four
planets. It decreases from the outermost planet to HR 8799 d, while HR 8799 e
has a higher C/O ratio than the other companions. We confidently detect HCN in
HR 8799 c and e, at abundances far higher than predicted by equilibrium
chemistry, though this detection is largely driven by low $S/N$ data from
LBT/ALES. CH4 is also confidently detected in HR 8799 c for the first time.
From the disequilibrium chemistry retrievals H2S appears to be a highly
abundant species in all of the planets, but will require higher signal to
noise and spectral resolution for a confident detection in a free retrieval
framework. Using our retrieved quench pressure and chemical abundances, we are
able to derive a vertical mixing strength, finding $K_{\rm zz}$ values
compatible with high-metallicity predictions from Soni & Acharyya (2023). The
mixing strength is stronger for the warmer planets, at $\log K_{\rm zz}\approx
6$, and is lower for HR 8799 b with $\log K_{\rm zz}\approx 2$.

All of the planets are highly cloudy. For the inner three planets, these
clouds are composed of silicate clouds lying above the photosphere, and deep,
dense iron clouds forming the base of the photosphere. Cooler than the other
three planets, the most favoured model for HR 8799 b requires Na2S clouds. All
of the planets have effective temperatures consistent with literature values,
with HR 8799 b still unique in its lower temperature and mass compared to its
siblings.

We emphasise the use of robust model comparison in this work: while it may be
difficult to present precise measurements of certain properties, the use of
multiple methods and models allows us to draw a robust portrait of each of
these atmospheres. We also note that our conclusions rely on data with
significant incompatibilities, particularly in the H-band flux. While we
performed extensive analysis to mitigate the influence of any individual
dataset, further observations are required to obtain reliable spectroscopic
measurements in the near infrared. Are they four-of-a-kind? The HR 8799
planets share many similarities, but much like our own solar system there are
differences in their atmospheric properties that will require further study.

###### Acknowledgements.

We would like to thank Quinn Konopacky, Alice Zurlo, Alex Greenbaum, Beth
Biller, Olivier Flasseur, David Doelman and Pengyu Lui for providing the
archival datasets used in this work. We are also grateful to Gilles Loupe and
Malavika Vasist for providing helpful suggestions on how to implement the
spline temperature profile. Thanks as well to Daniel Thorngren for providing
the model for calculating the planet metallicity $Z$. Finally, we are grateful
to our anonymous reviewer for their thorough and insightful report. Software
used: petitRADTRANS, pyKLIP, species, VIP-HCI, pyMultiNest, phot_utils,
Python, numpy, matplotlib, astropy, sympy, Aspro. SL acknowledges the support
of the French Agence Nationale de la Recherche (ANR), under grant
ANR-21-CE31-0017 (project ExoVLTI). J.J.W., A.C., and S.B. acknowledge the
support of NASA XRP award 80NSSC23K0280. G.-D.M. acknowledges the support of
the DFG priority program SPP 1992 “Exploring the Diversity of Extrasolar
Planets” (MA 9185/1) amd from the Swiss National Science Foundation under
grant 200021_204847 “PlanetsInTime”. Parts of this work have been carried out
within the framework of the NCCR PlanetS supported by the Swiss National
Science Foundation. This work is based on observations collected at the
European Southern Observatory under ESO programme 1104.C-0651. This
publication makes use of VOSA, developed under the Spanish Virtual Observatory
project supported by the Spanish MINECO through grant AyA2017-84089. VOSA has
been partially updated by using funding from the European Union’s Horizon 2020
Research and Innovation Programme, under Grant Agreement nº 776403
(EXOPLANETS-A)

## References

  * Ackerman & Marley (2001) Ackerman, A. S., & Marley, M. S. 2001, ApJ, 556, 872
  * Adams et al. (1989) Adams, F. C., Ruden, S. P., & Shu, F. H. 1989, ApJ, 347, 959
  * Allard et al. (2001) Allard, F., Hauschildt, P. H., Alexander, D. R., Tamanai, A., & Schweitzer, A. 2001, ApJ, 556, 357
  * Allard et al. (2012) Allard, F., Homeier, D., & Freytag, B. 2012, Philosophical Transactions of the Royal Society of London Series A, 370, 2765
  * Allard et al. (2016) Allard, N. F., Spiegelman, F., & Kielkopf, J. F. 2016, A&A, 589, A21
  * Allard et al. (2019) Allard, N. F., Spiegelman, F., Leininger, T., & Mollière, P. 2019, A&A, 628, A120
  * Amara & Quanz (2012) Amara, A., & Quanz, S. P. 2012, MNRAS, 427, 948
  * Apai et al. (2016) Apai, D., Kasper, M., Skemer, A., et al. 2016, ApJ, 820, 40
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Azzam et al. (2016) Azzam, A. A. A., Yurchenko, S. N., Tennyson, J., & Naumenko, O. V. 2016, MNRAS, 460, 4063
  * Baines et al. (2012) Baines, E. K., White, R. J., Huber, D., et al. 2012, ApJ, 761, 57
  * Balmer et al. (2023) Balmer, W. O., Pueyo, L., Stolker, T., et al. 2023, ApJ, 956, 99
  * Baraffe et al. (2003) Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. H. 2003, A&A, 402, 701
  * Barber et al. (2014) Barber, R. J., Strange, J. K., Hill, C., et al. 2014, MNRAS, 437, 1828
  * Barman et al. (2015) Barman, T. S., Konopacky, Q. M., Macintosh, B., & Marois, C. 2015, ApJ, 804, 61
  * Barman et al. (2011) Barman, T. S., Macintosh, B., Konopacky, Q. M., & Marois, C. 2011, ApJ, 733, 65
  * Barton et al. (2013) Barton, E. J., Yurchenko, S. N., & Tennyson, J. 2013, MNRAS, 434, 1469
  * Baudino et al. (2015) Baudino, J. L., Bézard, B., Boccaletti, A., et al. 2015, A&A, 582, A83
  * Baudino et al. (2017) Baudino, J.-L., Mollière, P., Venot, O., et al. 2017, ApJ, 850, 150
  * Bayo et al. (2008) Bayo, A., Rodrigo, C., Barrado Y Navascués, D., et al. 2008, A&A, 492, 277
  * Bell et al. (2023) Bell, T. J., Welbanks, L., Schlawin, E., et al. 2023, Nature, 623, 709
  * Benneke & Seager (2013) Benneke, B., & Seager, S. 2013, ApJ, 778, 153
  * Bergfors et al. (2011) Bergfors, C., Brandner, W., Janson, M., Köhler, R., & Henning, T. 2011, A&A, 528, A134
  * Beuzit et al. (2008) Beuzit, J.-L., Feldt, M., Dohlen, K., et al. 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7014, Proc. SPIE, 701418
  * Beuzit et al. (2019) Beuzit, J. L., Vigan, A., Mouillet, D., et al. 2019, A&A, 631, A155
  * Biller et al. (2021) Biller, B. A., Apai, D., Bonnefoy, M., et al. 2021, MNRAS, 503, 743
  * Bitsch & Mah (2023) Bitsch, B., & Mah, J. 2023, A&A, 679, A11
  * Blain et al. (2021) Blain, D., Charnay, B., & Bézard, B. 2021, A&A, 646, A15
  * Boccaletti et al. (2023) Boccaletti, A., M., M., P., B., et al. 2023, arXiv e-prints, arXiv:2310.13414
  * Bodenheimer (1974) Bodenheimer, P. 1974, Icarus, 23, 319
  * Bodenheimer et al. (2000) Bodenheimer, P., Hubickyj, O., & Lissauer, J. J. 2000, Icarus, 143, 2
  * Bonnefoy et al. (2014) Bonnefoy, M., Currie, T., Marleau, G. D., et al. 2014, A&A, 562, A111
  * Bonnefoy et al. (2016) Bonnefoy, M., Zurlo, A., Baudino, J. L., et al. 2016, A&A, 587, A58
  * Bonnefoy et al. (2018) Bonnefoy, M., Perraut, K., Lagrange, A. M., et al. 2018, A&A, 618, A63
  * Boss (1997) Boss, A. P. 1997, Science, 276, 1836
  * Bowler et al. (2010) Bowler, B. P., Liu, M. C., Dupuy, T. J., & Cushing, M. C. 2010, ApJ, 723, 850
  * Brandt et al. (2021) Brandt, G. M., Brandt, T. D., Dupuy, T. J., Michalik, D., & Marleau, G.-D. 2021, ApJ, 915, L16
  * Brown-Sevilla et al. (2023) Brown-Sevilla, S. B., Maire, A. L., Mollière, P., et al. 2023, A&A, 673, A98
  * Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, A&A, 564, A125
  * Burgasser et al. (2010) Burgasser, A. J., Simcoe, R. A., Bochanski, J. J., et al. 2010, ApJ, 725, 1405
  * Burningham et al. (2017) Burningham, B., Marley, M. S., Line, M. R., et al. 2017, MNRAS, 470, 1177
  * Burningham et al. (2021) Burningham, B., Faherty, J. K., Gonzales, E. C., et al. 2021, MNRAS, 506, 1944
  * Burrows et al. (2011) Burrows, A., Heng, K., & Nampaisarn, T. 2011, ApJ, 736, 47
  * Burrows & Sharp (1999) Burrows, A., & Sharp, C. M. 1999, ApJ, 512, 843
  * Burrows et al. (2006) Burrows, A., Sudarsky, D., & Hubeny, I. 2006, ApJ, 640, 1063
  * Calamari et al. (2024) Calamari, E., Faherty, J. K., Visscher, C., et al. 2024, ApJ, 963, 67
  * Cameron (1978) Cameron, A. G. W. 1978, Moon and Planets, 18, 5
  * Cannon & Pickering (1993) Cannon, A. J., & Pickering, E. C. 1993, VizieR Online Data Catalog, III/135A
  * Cantalloube et al. (2015) Cantalloube, F., Mouillet, D., Mugnier, L. M., et al. 2015, A&A, 582, A89
  * Chabrier et al. (2000) Chabrier, G., Baraffe, I., Allard, F., & Hauschildt, P. 2000, ApJ, 542, 464
  * Charnay et al. (2018) Charnay, B., Bézard, B., Baudino, J.-L., et al. 2018, ApJ, 854, 172
  * Chen et al. (2024) Chen, K., Kama, M., Pinilla, P., & Keyte, L. 2024, MNRAS, 527, 2049
  * Chubb et al. (2020) Chubb, K. L., Rocchetto, M., Al-Refaie, A. F., et al. 2020, A&A
  * Coles et al. (2019) Coles, P. A., , Yurchenko, S. N., & Tennyson, J. 2019, MNRAS, 490, 4638
  * Contro et al. (2015) Contro, B., Wittenmyer, R. A., Horner, J., & Marshall, J. P. 2015, Origins of Life and Evolution of the Biosphere, 45, 41
  * Covarrubias et al. (2022) Covarrubias, S., Blunt, S., & Wang, J. J. 2022, Research Notes of the American Astronomical Society, 6, 66
  * Cridland et al. (2019) Cridland, A. J., Eistrup, C., & van Dishoeck, E. F. 2019, A&A, 627, A127
  * Cridland et al. (2020) Cridland, A. J., van Dishoeck, E. F., Alessi, M., & Pudritz, R. E. 2020, A&A, 642, A229
  * Currie et al. (2012) Currie, T., Fukagawa, M., Thalmann, C., Matsumura, S., & Plavchan, P. 2012, ApJ, 755, L34
  * Currie et al. (2011) Currie, T., Burrows, A., Itoh, Y., et al. 2011, ApJ, 729, 128
  * Currie et al. (2014) Currie, T., Burrows, A., Girard, J. H., et al. 2014, apj, 795, 133
  * Cushing et al. (2006) Cushing, M. C., Roellig, T. L., Marley, M. S., et al. 2006, ApJ, 648, 614
  * Cushing et al. (2008) Cushing, M. C., Marley, M. S., Saumon, D., et al. 2008, ApJ, 678, 1372
  * Demars et al. (2023) Demars, D., Bonnefoy, M., Dougados, C., et al. 2023, A&A, 676, A123
  * Desgrange et al. (2022) Desgrange, C., Chauvin, G., Christiaens, V., et al. 2022, A&A, 664, A139
  * Dodson-Robinson et al. (2009) Dodson-Robinson, S. E., Veras, D., Ford, E. B., & Beichman, C. A. 2009, ApJ, 707, 79
  * Doelman et al. (2022) Doelman, D. S., Stone, J. M., Briesemeister, Z. W., et al. 2022, AJ, 163, 217
  * Eddington (1930) Eddington, A. S. 1930, MNRAS, 90, 279
  * Eistrup et al. (2018) Eistrup, C., Walsh, C., & van Dishoeck, E. F. 2018, A&A, 613, A14
  * Esposito et al. (2013) Esposito, S., Mesa, D., Skemer, A., et al. 2013, A&A, 549, A52
  * Faherty (2018) Faherty, J. K. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & J. A. Belmonte (Springer, Cham), 188
  * Faramaz et al. (2021) Faramaz, V., Marino, S., Booth, M., et al. 2021, AJ, 161, 271
  * Fegley & Lodders (1996) Fegley, Bruce, J., & Lodders, K. 1996, ApJ, 472, L37
  * Feroz & Hobson (2008) Feroz, F., & Hobson, M. P. 2008, MNRAS, 384, 449
  * Feroz et al. (2019) Feroz, F., Hobson, M. P., Cameron, E., & Pettitt, A. N. 2019, The Open Journal of Astrophysics, 2, 10
  * Flasseur et al. (2018) Flasseur, O., Denis, L., Thiébaut, E., & Langlois, M. 2018, A&A, 618, A138
  * Flasseur et al. (2020) Flasseur, O., Denis, L., Thiébaut, É., & Langlois, M. 2020, A&A, 637, A9
  * Fonte et al. (2023) Fonte, S., Turrini, D., Pacetti, E., et al. 2023, MNRAS, 520, 4683
  * Fortney et al. (2011) Fortney, J. J., Ikoma, M., Nettelmann, N., Guillot, T., & Marley, M. S. 2011, ApJ, 729, 32
  * Fragoso et al. (2018) Fragoso, T. M., Bertoli, W., & Louzada, F. 2018, International Statistical Review, 86, 1
  * Fukagawa et al. (2009) Fukagawa, M., Itoh, Y., Tamura, M., et al. 2009, ApJ, 696, L1
  * Gaia Collaboration (2020) Gaia Collaboration. 2020, VizieR Online Data Catalog, I/350
  * Galicher et al. (2011) Galicher, R., Marois, C., Macintosh, B., Barman, T., & Konopacky, Q. 2011, ApJ, 739, L41
  * Gehren (1977) Gehren, T. 1977, A&A, 59, 303
  * Geiler et al. (2018) Geiler, F., Krivov, A. V., Booth, M., & Löhne, T. 2018, MNRAS, 483, 332
  * Giacobbe et al. (2021) Giacobbe, P., Brogi, M., Gandhi, S., et al. 2021, Nature, 592, 205
  * Goździewski & Migaszewski (2018) Goździewski, K., & Migaszewski, C. 2018, ApJS, 238, 6
  * Gravity Collaboration et al. (2017) Gravity Collaboration, Abuter, R., Accardo, M., et al. 2017, A&A, 602, A94
  * Gravity Collaboration et al. (2019) Gravity Collaboration, Lacour, S., Nowak, M., et al. 2019, A&A, 623, L11
  * GRAVITY Collaboration et al. (2020) GRAVITY Collaboration, Bauböck, M., Dexter, J., et al. 2020, A&A, 635, A143
  * Gravity Collaboration et al. (2020) Gravity Collaboration, Nowak, M., Lacour, S., et al. 2020, A&A, 633, A110
  * Gray et al. (2003) Gray, R. O., Corbally, C. J., Garrison, R. F., McFadden, M. T., & Robinson, P. E. 2003, AJ, 126, 2048
  * Gray & Kaye (1999) Gray, R. O., & Kaye, A. B. 1999, AJ, 118, 2993
  * Greco & Brandt (2016) Greco, J. P., & Brandt, T. D. 2016, ApJ, 833, 134
  * Greenbaum et al. (2018) Greenbaum, A. Z., Pueyo, L., Ruffio, J.-B., et al. 2018, AJ, 155, 226
  * Groff et al. (2017) Groff, T., Chilcote, J., Brandt, T., et al. 2017, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10400, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. S. Shaklan, 1040016
  * Groff et al. (2015) Groff, T. D., Kasdin, N. J., Limbach, M. A., et al. 2015, in Techniques and Instrumentation for Detection of Exoplanets VII, ed. S. Shaklan, Vol. 9605 (SPIE), 96051C, backup Publisher: International Society for Optics and Photonics
  * Guillot (2005) Guillot, T. 2005, Annual Review of Earth and Planetary Sciences, 33, 493
  * Guillot (2010) Guillot, T. 2010, A&A, 520, A27
  * Hansen (1971) Hansen, J. E. 1971, Journal of Atmospheric Sciences, 28, 1400
  * Helled & Bodenheimer (2010) Helled, R., & Bodenheimer, P. 2010, Icarus, 207, 503
  * Hoch et al. (2022) Hoch, K. K. W., Konopacky, Q. M., Barman, T. S., et al. 2022, AJ, 164, 155
  * Hoch et al. (2023) Hoch, K. K. W., Konopacky, Q. M., Theissen, C. A., et al. 2023, AJ, 166, 85
  * Ingraham et al. (2014) Ingraham, P., Marley, M. S., Saumon, D., et al. 2014, ApJ, 794, L15
  * Janson et al. (2010) Janson, M., Bergfors, C., Goto, M., Brandner, W., & Lafrenière, D. 2010, ApJ, 710, L35
  * Jullion & Lambert (2007) Jullion, A., & Lambert, P. 2007, Computational Statistics & Data Analysis, 51, 2542
  * Karalidi et al. (2021) Karalidi, T., Marley, M., Fortney, J. J., et al. 2021, ApJ, 923, 269
  * Kass & Raftery (1995) Kass, R. E., & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773
  * Kaye et al. (1999) Kaye, A. B., Handler, G., Krisciunas, K., Poretti, E., & Zerbi, F. M. 1999, PASP, 111, 840
  * Kirkpatrick et al. (1999) Kirkpatrick, J. D., Reid, I. N., Liebert, J., et al. 1999, ApJ, 519, 802
  * Kitzmann et al. (2020) Kitzmann, D., Heng, K., Oreshenko, M., et al. 2020, ApJ, 890, 174
  * Konopacky et al. (2013) Konopacky, Q. M., Barman, T. S., Macintosh, B. A., & Marois, C. 2013, Science, 339, 1398
  * Lacour et al. (2020) Lacour, S., Wang, J. J., Nowak, M., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11446, Optical and Infrared Interferometry and Imaging VII, ed. P. G. Tuthill, A. Mérand, & S. Sallum, 114460O
  * Lafrenière et al. (2009) Lafrenière, D., Marois, C., Doyon, R., & Barman, T. 2009, ApJ, 694, L148
  * Lang & Brezger (2004) Lang, S., & Brezger, A. 2004, Journal of Computational and Graphical Statistics, 13, 183
  * Lapeyrère et al. (2014) Lapeyrère, V., Kervella, P., Lacour, S., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9146, Optical and Infrared Interferometry IV, ed. J. K. Rajagopal, M. J. Creech-Eakman, & F. Malbet, 91462D
  * Larkin et al. (2006) Larkin, J., Barczys, M., Krabbe, A., et al. 2006, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6269, Ground-based and Airborne Instrumentation for Astronomy, ed. I. S. McLean & M. Iye, 62691A
  * Laughlin & Rozyczka (1996) Laughlin, G., & Rozyczka, M. 1996, ApJ, 456, 279
  * Lavie et al. (2017) Lavie, B., Mendonça, J. M., Mordasini, C., et al. 2017, AJ, 154, 91
  * Lee et al. (2013) Lee, J.-M., Heng, K., & Irwin, P. G. J. 2013, ApJ, 778, 97
  * Leggett et al. (2008) Leggett, S. K., Saumon, D., Albert, L., et al. 2008, ApJ, 682, 1256
  * Linder et al. (2019) Linder, E. F., Mordasini, C., Mollière, P., et al. 2019, A&A, 623, A85
  * Line et al. (2015) Line, M. R., Teske, J., Burningham, B., Fortney, J. J., & Marley, M. S. 2015, ApJ, 807, 183
  * Liu et al. (2023) Liu, P., Bohn, A. J., Doelman, D. S., et al. 2023, A&A, 674, A115
  * Lodders (2003) Lodders, K. 2003, ApJ, 591, 1220
  * Lodders & Fegley (2002) Lodders, K., & Fegley, B. 2002, Icarus, 155, 393
  * Lodders et al. (2009) Lodders, K., Palme, H., & Gail, H. P. 2009, Landolt Börnstein, 4B, 712
  * Looper et al. (2008) Looper, D. L., Kirkpatrick, J. D., Cutri, R. M., et al. 2008, ApJ, 686, 528
  * Luna & Morley (2021) Luna, J. L., & Morley, C. V. 2021, ApJ, 920, 146
  * Macintosh et al. (2014) Macintosh, B., Graham, J. R., Ingraham, P., et al. 2014, Proceedings of the National Academy of Science, 111, 12661
  * Madhusudhan (2019) Madhusudhan, N. 2019, Annu. Rev. Astron. Astrophys., 57, 617
  * Madhusudhan et al. (2011) Madhusudhan, N., Burrows, A., & Currie, T. 2011, ApJ, 737, 34
  * Madhusudhan et al. (2023) Madhusudhan, N., Sarkar, S., Constantinou, S., et al. 2023, ApJ, 956, L13
  * Madhusudhan & Seager (2009) Madhusudhan, N., & Seager, S. 2009, ApJ, 707, 24
  * Maire et al. (2015) Maire, A. L., Skemer, A. J., Hinz, P. M., et al. 2015, A&A, 576, A133
  * Marleau et al. (2017) Marleau, G.-D., Klahr, H., Kuiper, R., & Mordasini, C. 2017, ApJ, 836, 221
  * Marleau et al. (2019) Marleau, G.-D., Mordasini, C., & Kuiper, R. 2019, ApJ, 881, 144
  * Marley et al. (2007) Marley, M. S., Fortney, J. J., Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2007, ApJ, 655, 541
  * Marley et al. (2012) Marley, M. S., Saumon, D., Cushing, M., et al. 2012, ApJ, 754, 135
  * Marley et al. (2010) Marley, M. S., Saumon, D., & Goldblatt, C. 2010, ApJ, 723, L117
  * Marley et al. (2021) Marley, M. S., Saumon, D., Visscher, C., et al. 2021, ApJ, 920, 85
  * Marocco et al. (2014) Marocco, F., Day-Jones, A. C., Lucas, P. W., et al. 2014, MNRAS, 439, 372
  * Marois et al. (2008) Marois, C., Macintosh, B., Barman, T., et al. 2008, Science, 322, 1348
  * Marois et al. (2010) Marois, C., Zuckerman, B., Konopacky, Q. M., Macintosh, B., & Barman, T. 2010, Nature, 468, 1080
  * Matthews et al. (2014) Matthews, B., Kennedy, G., Sibthorpe, B., et al. 2014, ApJ, 780, 97
  * McCarthy et al. (2024) McCarthy, A. M., Muirhead, P. S., Tamburo, P., et al. 2024, arXiv e-prints, arXiv:2402.15001
  * McKemmish et al. (2019) McKemmish, L. K., Masseron, T., Hoeijmakers, H. J., et al. 2019, MNRAS, 488, 2836
  * McKemmish et al. (2016) McKemmish, L. K., Yurchenko, S. N., & Tennyson, J. 2016, MNRAS, 463, 771
  * Metchev et al. (2009) Metchev, S., Marois, C., & Zuckerman, B. 2009, ApJ, 705, L204
  * Miles et al. (2023) Miles, B. E., Biller, B. A., Patapis, P., et al. 2023, ApJ, 946, L6
  * Mollière et al. (2017) Mollière, P., van Boekel, R., Bouwman, J., et al. 2017, A&A, 600, A10
  * Mollière et al. (2015) Mollière, P., van Boekel, R., Dullemond, C., Henning, T., & Mordasini, C. 2015, ApJ, 813, 47
  * Mollière et al. (2019) Mollière, P., Wardenier, J. P., van Boekel, R., et al. 2019, A&A, 627, A67
  * Mollière et al. (2020) Mollière, P., Stolker, T., Lacour, S., et al. 2020, A&A, 640, A131
  * Mollière et al. (2022) Mollière, P., Molyarova, T., Bitsch, B., et al. 2022, ApJ, 934, 74
  * Mordasini et al. (2012) Mordasini, C., Alibert, Y., Klahr, H., & Henning, T. 2012, A&A, 547, A111
  * Mordasini et al. (2017) Mordasini, C., Marleau, G.-D., & Mollière, P. 2017, A&A, 608, A72
  * Morley et al. (2024) Morley, C. V., Mukherjee, S., Marley, M. S., et al. 2024, arXiv e-prints, arXiv:2402.00758
  * Moses et al. (2013) Moses, J. I., Madhusudhan, N., Visscher, C., & Freedman, R. S. 2013, ApJ, 763, 25
  * Moses et al. (2016) Moses, J. I., Marley, M. S., Zahnle, K., et al. 2016, ApJ, 829, 66
  * Moya et al. (2010) Moya, A., Amado, P. J., Barrado, D., et al. 2010, MNRAS, 405, L81
  * Moya et al. (2010) Moya, A., Amado, P. J., Barrado, D., et al. 2010, MNRAS, 406, 566
  * Nasedkin et al. (2023) Nasedkin, E., Mollière, P., Wang, J., et al. 2023, A&A, 678, A41
  * Nasedkin et al. (2024) Nasedkin, E., Mollière, P., & Blain, D. 2024, Journal of Open Source Software, 9, 5875
  * Nero & Bjorkman (2009) Nero, D., & Bjorkman, J. E. 2009, ApJ, 702, L163
  * Nixon et al. (2023) Nixon, M. C., Welbanks, L., McGill, P., & Kempton, E. M. R. 2023, arXiv e-prints, arXiv:2310.03713
  * Nowak et al. (2020) Nowak, M., Lacour, S., Lagrange, A. M., et al. 2020, A&A, 642, L2
  * Öberg et al. (2011) Öberg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, ApJ, 743, L16
  * Oppenheimer et al. (2013) Oppenheimer, B. R., Baranec, C., Beichman, C., et al. 2013, ApJ, 768, 24
  * Pacetti et al. (2022) Pacetti, E., Turrini, D., Schisano, E., et al. 2022, ApJ, 937, 36
  * Palma-Bifani et al. (2023) Palma-Bifani, P., Chauvin, G., Bonnefoy, M., et al. 2023, A&A, 670, A90
  * Patapis et al. (2022) Patapis, P., Nasedkin, E., Cugno, G., et al. 2022, A&A, 658, A72
  * Patience et al. (2012) Patience, J., King, R. R., De Rosa, R. J., et al. 2012, A&A, 540, A85
  * Perri & Cameron (1974) Perri, F., & Cameron, A. G. W. 1974, Icarus, 22, 416
  * Petit dit de la Roche et al. (2019) Petit dit de la Roche, D. J. M., van den Ancker, M. E., Kissler-Patig, M., Ivanov, V. D., & Fedele, D. 2019, MNRAS, 491, 1795
  * Petrus et al. (2021) Petrus, S., Bonnefoy, M., Chauvin, G., et al. 2021, A&A, 648, A59
  * Petrus et al. (2023) Petrus, S., Chauvin, G., Bonnefoy, M., et al. 2023, A&A, 670, L9
  * Phillips et al. (2020) Phillips, M. W., Tremblin, P., Baraffe, I., et al. 2020, A&A, 637, A38
  * Podolak (2003) Podolak, M. 2003, Icarus, 165, 428
  * Pollack et al. (1996) Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, ıcarus, 124, 62
  * Polyansky et al. (2018) Polyansky, O. L., Kyuberis, A. A., Zobov, N. F., et al. 2018, MNRAS, 480, 2597
  * Pueyo (2016) Pueyo, L. 2016, ApJ, 824, 117
  * Radigan (2014) Radigan, J. 2014, ApJ, 797, 120
  * Radigan et al. (2014) Radigan, J., Lafrenière, D., Jayawardhana, R., & Artigau, E. 2014, ApJ, 793, 75
  * Rahman (2005) Rahman, M. M. 2005, PhD thesis, University of British Columbia
  * Rajan et al. (2015) Rajan, A., Barman, T., Soummer, R., et al. 2015, ApJ, 809, L33
  * Rhee et al. (2007) Rhee, J. H., Song, I., Zuckerman, B., & McElwain, M. 2007, ApJ, 660, 1556
  * Rossow (1978) Rossow, W. B. 1978, Icarus, 36, 1
  * Rothman et al. (2010) Rothman, L. S., Gordon, I. E., Barber, R. J., et al. 2010, J. Quant. Spec. Radiat. Transf., 111, 2139
  * Rowland et al. (2023) Rowland, M. J., Morley, C. V., & Line, M. R. 2023, ApJ, 947, 6
  * Ruffio et al. (2019) Ruffio, J.-B., Macintosh, B., Konopacky, Q. M., et al. 2019, AJ, 158, 200
  * Ruffio et al. (2021) Ruffio, J.-B., Konopacky, Q. M., Barman, T., et al. 2021, AJ, 162, 290
  * Ruffio et al. (2023) Ruffio, J.-B., Perrin, M. D., Hoch, K. K. W., et al. 2023, arXiv e-prints, arXiv:2310.09902
  * Rustamkulov et al. (2023) Rustamkulov, Z., Sing, D. K., Mukherjee, S., et al. 2023, Nature, 614, 659
  * Sadakane (2006) Sadakane, K. 2006, PASJ, 58, 1023
  * Samland et al. (2017) Samland, M., Mollière, P., Bonnefoy, M., et al. 2017, A&A, 603, A57
  * Saumon & Marley (2008) Saumon, D., & Marley, M. S. 2008, ApJ, 689, 1327
  * Schmidt et al. (2008) Schmidt, T. O. B., Neuhäuser, R., Seifahrt, A., et al. 2008, A&A, 491, 311
  * Schneider & Bitsch (2021a) Schneider, A. D., & Bitsch, B. 2021a, A&A, 654, A71
  * Schneider & Bitsch (2021b) Schneider, A. D., & Bitsch, B. 2021b, A&A, 654, A72
  * Sepulveda & Bowler (2022) Sepulveda, A. G., & Bowler, B. P. 2022, AJ, 163, 52
  * Sepulveda et al. (2023) Sepulveda, A. G., Huber, D., Li, G., et al. 2023, Research Notes of the American Astronomical Society, 7, 2
  * Showman & Kaspi (2013) Showman, A. P., & Kaspi, Y. 2013, ApJ, 776, 85
  * Skemer et al. (2012) Skemer, A. J., Hinz, P. M., Esposito, S., et al. 2012, ApJ, 753, 14
  * Skemer et al. (2014) Skemer, A. J., Marley, M. S., Hinz, P. M., et al. 2014, ApJ, 792, 17
  * Skemer et al. (2015) Skemer, A. J., Hinz, P., Montoya, M., et al. 2015, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9605, Techniques and Instrumentation for Detection of Exoplanets VII, ed. S. Shaklan, 96051D
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163
  * Soni & Acharyya (2023) Soni, V., & Acharyya, K. 2023, ApJ, 946, 29
  * Soummer et al. (2011) Soummer, R., Hagan, J. B., Pueyo, L., et al. 2011, ApJ, 741, 55
  * Soummer et al. (2012) Soummer, R., Pueyo, L., & Larkin, J. 2012, ApJ, 755, L28
  * Sousa-Silva et al. (2015) Sousa-Silva, C., Al-Refaie, A. F., Tennyson, J., & Yurchenko, S. N. 2015, MNRAS, 446, 2337
  * Stephens et al. (2009) Stephens, D. C., Leggett, S. K., Cushing, M. C., et al. 2009, ApJ, 702, 154
  * Stolker et al. (2019) Stolker, T., Bonse, M. J., Quanz, S. P., et al. 2019, A&A, 621, A59
  * Stolker et al. (2020) Stolker, T., Quanz, S. P., Todorov, K. O., et al. 2020, A&A, 635, A182
  * Su et al. (2009) Su, K. Y. L., Rieke, G. H., Stapelfeldt, K. R., et al. 2009, ApJ, 705, 314
  * Suárez et al. (2023) Suárez, G., Vos, J. M., Metchev, S., Faherty, J. K., & Cruz, K. 2023, ApJ, 954, L6
  * Suárez & Metchev (2022) Suárez, G., & Metchev, S. 2022, MNRAS, 513, 5701
  * Suárez & Metchev (2023) Suárez, G., & Metchev, S. 2023, MNRAS, 523, 4739
  * Suárez et al. (2021) Suárez, G., Metchev, S., Leggett, S. K., Saumon, D., & Marley, M. S. 2021, ApJ, 920, 99
  * Szulágyi & Mordasini (2017) Szulágyi, J., & Mordasini, C. 2017, MNRAS, 465, L64
  * Tan & Showman (2021a) Tan, X., & Showman, A. P. 2021a, MNRAS, 502, 678
  * Tan & Showman (2021b) Tan, X., & Showman, A. P. 2021b, MNRAS, 502, 2198
  * Tennyson & Yurchenko (2012) Tennyson, J., & Yurchenko, S. N. 2012, MNRAS, 425, 21
  * Thompson et al. (2023) Thompson, W., Marois, C., Do Ó, C. R., et al. 2023, AJ, 165, 29
  * Thorngren & Fortney (2019) Thorngren, D., & Fortney, J. J. 2019, ApJ, 874, L31
  * Thorngren et al. (2016) Thorngren, D. P., Fortney, J. J., Murray-Clay, R. A., & Lopez, E. D. 2016, ApJ, 831, 64
  * Tremblin et al. (2016) Tremblin, P., Amundsen, D. S., Chabrier, G., et al. 2016, ApJ, 817, L19
  * Tremblin et al. (2015) Tremblin, P., Amundsen, D. S., Mourier, P., et al. 2015, ApJ, 804, L17
  * Tremblin et al. (2017) Tremblin, P., Chabrier, G., Baraffe, I., et al. 2017, ApJ, 850, 46
  * Tremblin et al. (2019) Tremblin, P., Padioleau, T., Phillips, M. W., et al. 2019, ApJ, 876, 144
  * Turrini et al. (2021) Turrini, D., Schisano, E., Fonte, S., et al. 2021, ApJ, 909, 40
  * Vos et al. (2019) Vos, J. M., Biller, B. A., Bonavita, M., et al. 2019, MNRAS, 483, 480
  * Vos et al. (2023) Vos, J. M., Burningham, B., Faherty, J. K., et al. 2023, ApJ, 944, 138
  * Wahhaj et al. (2021) Wahhaj, Z., Milli, J., Romero, C., et al. 2021, A&A, 648, A26
  * Waldmann et al. (2015) Waldmann, I. P., Rocchetto, M., Tinetti, G., et al. 2015, apj, 813, 13
  * Wang (2023) Wang, J. 2023, arXiv e-prints, arXiv:2310.00088
  * Wang et al. (2018a) Wang, J., Mawet, D., Fortney, J. J., et al. 2018a, AJ, 156, 272
  * Wang et al. (2020) Wang, J., Wang, J. J., Ma, B., et al. 2020, AJ, 160, 150
  * Wang et al. (2023) Wang, J., Wang, J. J., Ruffio, J.-B., et al. 2023, AJ, 165, 4
  * Wang et al. (2018b) Wang, J. J., Graham, J. R., Dawson, R., et al. 2018b, AJ, 156, 192
  * Wang et al. (2020) Wang, J. J., Ginzburg, S., Ren, B., et al. 2020, AJ, 159, 263
  * Wang et al. (2021) Wang, J. J., Vigan, A., Lacour, S., et al. 2021, AJ, 161, 148
  * Wang et al. (2021) Wang, J. J., Ruffio, J.-B., Morris, E., et al. 2021, AJ, 162, 148
  * Wang et al. (2022) Wang, J. J., Gao, P., Chilcote, J., et al. 2022, AJ, 164, 143
  * Wende et al. (2010) Wende, S., Reiners, A., Seifahrt, A., & Bernath, P. F. 2010, A&A, 523, A58
  * Whiteford et al. (2023) Whiteford, N., Glasse, A., Chubb, K. L., et al. 2023, MNRAS
  * Wilcomb et al. (2020) Wilcomb, K. K., Konopacky, Q. M., Barman, T. S., et al. 2020, AJ, 160, 207
  * Wilner et al. (2018) Wilner, D. J., MacGregor, M. A., Andrews, S. M., et al. 2018, ApJ, 855, 56
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868
  * Yurchenko et al. (2017) Yurchenko, S. N., Amundsen, D. S., Tennyson, J., & Waldmann, I. P. 2017, A&A, 605, A95
  * Yurchenko et al. (2020) Yurchenko, S. N., Mellor, T. M., Freedman, R. S., & Tennyson, J. 2020, MNRAS, 496, 5282
  * Zahnle & Marley (2014) Zahnle, K. J., & Marley, M. S. 2014, ApJ, 797, 41
  * Zerbi et al. (1999) Zerbi, F. M., Rodríguez, E., Garrido, R., et al. 1999, MNRAS, 303, 275
  * Zhang et al. (2021) Zhang, Y., Snellen, I. A. G., Bohn, A. J., et al. 2021, Nature, 595, 370
  * Zhang et al. (2023) Zhang, Z., Mollière, P., Hawkins, K., et al. 2023, AJ, 166, 198
  * Zuckerman et al. (2011) Zuckerman, B., Rhee, J. H., Song, I., & Bessell, M. S. 2011, ApJ, 732, 61
  * Zuckerman & Song (2004) Zuckerman, B., & Song, I. 2004, ARA&A, 42, 685
  * Zurlo et al. (2016) Zurlo, A., Vigan, A., Galicher, R., et al. 2016, A&A, 587, A57
  * Zurlo et al. (2022) Zurlo, A., Goździewski, K., Lazzoni, C., et al. 2022, A&A, 666, A133

##  Appendix A Data logs

Table 11: Near infrared stellar Photometry of HR8799, using apparent flux
normalized to 10 pc, retrieved from the Spanish Virtual Observatory (Bayo et
al., 2008).

Filter | $\bm{\lambda}$ | Flux | Ref.
---|---|---|---
|  [$\upmu$m] |  [erg/s/cm2/Å] |
2MASS J | 1.235 | $2.198\pm 0.055\times 10^{-12}$ | S06
2MASS H | 1.662 | $8.754\pm 0.145\times 10^{-13}$ | S06
2MASS Ks | 2.159 | $3.433\pm 0.057\times 10^{-13}$ | S06
WISE W1 | 3.353 | $6.840\pm 1.367\times 10^{-14}$ | W10
WISE W2 | 4.603 | $2.316\pm 0.166\times 10^{-14}$ | W10

  * •

Notes

  * •

References: S06 Skrutskie et al. (2006); W10 Wright et al. (2010);

Table 12: Photometric data for HR 8799 bcde.

Instrument/Filter | m | Ref.
---|---|---
HR 8799 b
Keck/NIRC2.H | $18.05\pm 0.09$ | C12
Keck/NIRC2.Ks | $17.03\pm 0.08$ | M10
Keck/NIRC2.Ms | $16.05\pm 0.3$ | G11
Paranal/NACO.Lp | $15.52\pm 0.1$ | C14
Paranal/NACO.NB405 | $14.82\pm 0.18$ | C14
Paranal/SPHERE/IRDIS.B_J | $19.78\pm 0.09$ | Z16
Paranal/SPHERE/IRDIS.D_H23_2 | $18.08\pm 0.14$ | Z16
Paranal/SPHERE/IRDIS.D_H23_3 | $17.78\pm 0.1$ | Z16
Paranal/SPHERE/IRDIS.D_K12_1 | $17.15\pm 0.06$ | Z16
Paranal/SPHERE/IRDIS.D_K12_2 | $16.97\pm 0.09$ | Z16
Subaru/CIAO.z | $21.22\pm 0.29$ | C11
JWST/MIRI.F1065C | $13.54\pm 0.04$ | B23
JWST/MIRI.F1140C | $13.64\pm 0.07$ | B23
JWST/MIRI.F1550C | $13.49\pm 0.25$ | B23
HR 8799 c
Keck/NIRC2.H | $17.06\pm 0.13$ | C12
Keck/NIRC2.Ks | $16.11\pm 0.08$ | M10
Keck/NIRC2.Ms | $15.03\pm 0.14$ | G11
Paranal/NACO.Lp | $14.65\pm 0.11$ | C14
Paranal/NACO.NB405 | $13.97\pm 0.11$ | C14
Paranal/SPHERE/IRDIS.B_J | $18.6\pm 0.13$ | Z16
Paranal/SPHERE/IRDIS.D_H23_2 | $17.09\pm 0.12$ | Z16
Paranal/SPHERE/IRDIS.D_H23_3 | $16.78\pm 0.1$ | Z16
Paranal/SPHERE/IRDIS.D_K12_1 | $16.19\pm 0.05$ | Z16
Paranal/SPHERE/IRDIS.D_K12_2 | $15.86\pm 0.07$ | Z16
JWST/MIRI.F1065C | $12.97\pm 0.18$ | B23
JWST/MIRI.F1140C | $13.59\pm 0.26$ | B23
JWST/MIRI.F1550C | $11.88\pm 0.23$ | B23
HR 8799 d
Keck/NIRC2.H | $16.71\pm 0.24$ | C12
Keck/NIRC2.Ks | $16.09\pm 0.12$ | M10
Keck/NIRC2.Ms | $14.65\pm 0.35$ | G11
Paranal/NACO.Lp | $14.55\pm 0.14$ | C14
Paranal/NACO.NB405 | $13.87\pm 0.15$ | C14
Paranal/SPHERE/IRDIS.B_J | $18.59\pm 0.37$ | Z16
Paranal/SPHERE/IRDIS.D_H23_2 | $17.02\pm 0.17$ | Z16
Paranal/SPHERE/IRDIS.D_H23_3 | $16.85\pm 0.16$ | Z16
Paranal/SPHERE/IRDIS.D_K12_1 | $16.2\pm 0.07$ | Z16
Paranal/SPHERE/IRDIS.D_K12_2 | $15.84\pm 0.1$ | Z16
JWST/MIRI.F1065C | $12.98\pm 0.14$ | B23
JWST/MIRI.F1140C | $12.98\pm 0.17$ | B23
JWST/MIRI.F1550C | $11.88\pm 0.23$ | B23
HR 8799 e
Keck/NIRC2.Ks | $15.91\pm 0.22$ | C12
Keck/NACO.Lp | $14.49\pm 0.21$ | M10
Keck/NACO.NB405 | $13.72\pm 0.2$ | C14
Paranal/SPHERE/IRDIS.B_J | $18.4\pm 0.21$ | Z16
Paranal/SPHERE/IRDIS.D_H23_2 | $16.91\pm 0.2$ | Z16
Paranal/SPHERE/IRDIS.D_H23_3 | $16.68\pm 0.21$ | Z16
Paranal/SPHERE/IRDIS.D_K12_1 | $16.12\pm 0.1$ | Z16
Paranal/SPHERE/IRDIS.D_K12_2 | $15.82\pm 0.11$ | Z16
JWST/MIRI.F1065C | $12.52\pm 0.26$ | B23
JWST/MIRI.F1140C | $12.52\pm 0.23$ | B23
JWST/MIRI.F1550C | $11.01\pm 0.42$ | B23

  * •

Notes:

  * •

References: C12 Currie et al. (2012); M10 Marois et al. (2010); G11 Galicher
et al. (2011); C14 Currie et al. (2014); Z16 Zurlo et al. (2016); C11 Currie
et al. (2011); B23 Boccaletti et al. (2023).

Table 13: Spectroscopic Observation Log.

Planet | Instrument | Date | $\bm{\lambda}$ | $\bm{\lambda/\Delta\lambda}$ |  $\mathbf{\Delta}$PA | Seeing | Airmass | DIT | NEXP | Ref.
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  [$\mu$m] | [°] | [as], Med. | Min. | [s] |  |
b | GRAVITY | 2019-11-11 |  2.0$-$2.4 | 500 | … | 0.98 | 1.54 | 8x100 | 3 | …
| GRAVITY | 2021-08-26 |  2.0$-$2.4 | 500 | … | 0.88 | 1.48 | 8x100 | 3 | …
| GRAVITY | 2021-08-27 |  2.0$-$2.4 | 500 | … | 0.86 | 1.45 | 8x100 | 2 | …
| OSIRIS | 2009-07-22 |  2.0$-$2.4 | 60 | … | … | 1.0 | 900 | 30 | B11
| OSIRIS | 2009-07-23 |  1.5$-$1.8 | 60 | … | … | 1.0 | 900 | 30 | B11
| OSIRIS | 2009-07-30 |  1.5$-$1.8 | 60 | … | … | 1.0 | 900 | 30 | B11
| OSIRIS | 2010-07-11 |  2.0$-$2.4 | 60 | … | … | 1.0 | 900 | 30 | B11
| OSIRIS | 2010-07-13 |  1.5$-$1.8 | 60 | … | … | 1.0 | 900 | 30 | B11
c | GRAVITY | 2019-11-11 |  2.0$-$2.4 | 500 | … | 1.01 | 1.62 | 8x100 | 3 | …
| GRAVITY | 2021-08-26 |  2.0$-$2.4 | 500 | … | 1.04 | 1.50 | 8x100 | 3 | …
| GRAVITY | 2021-08-27 |  2.0$-$2.4 | 500 | … | 1.00 | 1.45 | 8x100 | 2 | …
| GRAVITY | 2022-08-19 |  2.0$-$2.4 | 500 | … | 0.64 | 1.54 | 4x100 | 10 | …
| GRAVITY | 2023-07-02 |  2.0$-$2.4 | 500 | … | 0.75 | 1.45 | 4x100 | 6 | …
| SPHERE | 2015-07-04 |  0.9$-$1.6 | 30 | 16.4 | 1.43 | 1.44 | … | 46 | F20
| CHARIS | 2018-09-01 |  1.2$-$2.4 | 19 | 202.24 | 0.47 | 1.0 | 20 | 1201 | W22
| CHARIS | 2018-09-02 |  1.2$-$2.4 | 19 | 206.55 | 0.42 | 1.0 | 20 | 1253 | W22
| GPI | 2016-09-19 |  1.5$-$1.8 | 45 | 20.93 | 0.97 | 1.61 | 60 | 60 | G18
| ALES | 2019-09-18 |  2.8$-$4.2 | 35 | 85.64 | 0.8-1.1 | 1.02 | 3.934 | 1300 | D22
| OSIRIS | 2010-2011 |  1.97$-$2.38 | 4000 | … | … | … | 600 | 33 | K13
d | GRAVITY | 2019-11-09 |  2.0$-$2.4 | 500 | … | 0.85 | 1.63 | 8x60 | 4 | …
| GRAVITY | 2019-11-11 |  2.0$-$2.4 | 500 | … | 1.14 | 1.70 | 8x100 | 3 | …
| GRAVITY | 2021-08-26 |  2.0$-$2.4 | 500 | … | 0.96 | 1.54 | 8x100 | 3 | …
| GRAVITY | 2021-08-27 |  2.0$-$2.4 | 500 | … | 1.18 | 1.48 | 8x100 | 2 | …
| GRAVITY | 2022-09-15 |  2.0$-$2.4 | 500 | … | 0.67 | 1.54 | 4x100 | 12 | …
| SPHERE | 2014-08-12 |  0.9$-$1.6 | 30 | 29.65 | 0.87 | 1.43 | 100 | 32 | Z16
| SPHERE | 2014-08-12 |  0.9$-$1.6 | 30 | 15.37 | 0.87 | 1.43 | 60 | 48 | Z16
| CHARIS | 2018-09-01 |  1.2$-$2.4 | 19 | 202.24 | 0.47 | 1.0 | 20 | 1201 | W22
| CHARIS | 2018-09-02 |  1.2$-$2.4 | 19 | 206.55 | 0.42 | 1.0 | 20 | 1253 | W22
| GPI | 2016-09-19 |  1.5$-$1.8 | 45 | 20.93 | 0.97 | 1.61 | 60 | 60 | G18
| ALES | 2019-09-18 |  2.8$-$4.2 | 35 | 85.64 | 0.8-1.1 | 1.02 | 3.934 | 1300 | D22
e | GRAVITY | 2018-08-28 |  2.0$-$2.4 | 500 | … | 0.67 | 1.44 | 10x100 | 7 | G19
| GRAVITY | 2019-11-09 |  2.0$-$2.4 | 500 | … | 0.84 | 1.55 | 8x60 | 3 | M20
| GRAVITY | 2019-11-11 |  2.0$-$2.4 | 500 | … | 1.15 | 1.47 | 8x100 | 3 | M20
| GRAVITY | 2021-08-26 |  2.0$-$2.4 | 500 | … | 0.66 | 1.50 | 8x100 | 2 | …
| GRAVITY | 2021-08-27 |  2.0$-$2.4 | 500 | … | 0.67 | 1.47 | 8x100 | 2 | …
| GRAVITY | 2021-09-27 |  2.0$-$2.4 | 500 | … | 0.82 | 1.55 | 4x100 | 6 | …
| GRAVITY | 2023-07-02 |  2.0$-$2.4 | 500 | … | 0.75 | 1.46 | 4x100 | 6 | …
| SPHERE | 2014-08-12 |  0.9$-$1.6 | 30 | 29.65 | 0.87 | 1.43 | 100 | 32 | Z16
| SPHERE | 2014-08-12 |  0.9$-$1.6 | 45 | 15.37 | 0.87 | 1.43 | 60 | 48 | Z16
| CHARIS | 2018-09-01 |  1.2$-$2.4 | 19 | 202.24 | 0.47 | 1.0 | 20 | 1201 | W22
| CHARIS | 2018-09-02 |  1.2$-$2.4 | 19 | 206.55 | 0.42 | 1.0 | 20 | 1253 | W22
| GPI | 2016-09-19 |  1.5$-$1.8 | 45 | 20.93 | 0.97 | 1.61 | 60 | 60 | G18
| ALES | 2019-09-18 |  2.8$-$4.2 | 35 | 85.64 | 0.8-1.1 | 1.02 | 3.934 | 1300 | D22

  * •

Notes

  * •

References: B11: Barman et al. (2011); K13: Konopacky et al. (2013); Z16:
Zurlo et al. (2016); G18: Greenbaum et al. (2018); G19: Gravity Collaboration
et al. (2019); F20: Flasseur et al. (2020); M20: Mollière et al. (2020); W22:
Wang et al. (2022)

##  Appendix B Reprocessing SPHERE and GPI datasets

To resolve the known discrepancies in the H-band flux between the archival
SPHERE and GPI datasets, we reprocessed each using KLIP (Soummer et al., 2012;
Pueyo, 2016), ANDROMEDA (Cantalloube et al., 2015) and PynPoint (Amara &
Quanz, 2012; Stolker et al., 2019). We optimised the choice of algorithm
parameters through a series of injection/extraction tests into each dataset,
as in Nasedkin et al. (2023). Using two different goodness-of-fit metrics on
injections representative of the true companion contrast and separation, we
choose the number of principal components used in the PSF subtraction in order
to extract the companion spectra with minimal bias. Figures 21 and 22 shows
the extracted spectra for each of HR 8799 c, d, and e, for SPHERE and GPI
respectively, compared to published literature spectra from Zurlo et al.
(2016), Greenbaum et al. (2018) and Flasseur et al. (2018). As the goodness-
of-fit metrics favoured the KLIP extractions, we used these as the basis of
our retrieval analysis.

Figure 21: KLIP and ANDROMEDA extractions from SPHERE for HR 8799 c, d, and e
compared to the spectra published in Zurlo et al. (2016) and Flasseur et al.
(2018).  Figure 22: KLIP and ANDROMEDA extractions from GPI for HR 8799 c, d,
and e compared to the spectra published in Greenbaum et al. (2018)

##  Appendix C Retrieval validation

Extensive validation of the pRT retrieval module was performed as part of this
work. Following updates described in Nasedkin et al. (2024), we verified that
the results of Mollière et al. (2020) could be reproduced. We independently
tested updates to the c-k mixing implementation, the adaptive mesh refinement
implementation, updated opacity sources for H2O and CO, bug fixes for
convergence on multiple scattering in the clouds, the inclusion of photometric
data, the inclusion of scaling factors on the SPHERE and GPI datasets,
including or excluding the GPI K-band spectra, updates to each of the SPHERE,
GPI and GRAVITY datasets, different prior widths and the number of live points
used in the retrieval. All of the posterior distributions were fully
consistent to within 2$\sigma$ with most falling well within 1$\sigma$ of the
published results, apart from the inclusion of new epochs of GRAVITY data,
which led to a significantly higher retrieved metallicity ([M/H]=$1.1\pm
0.32$) and an $f_{\rm sed}$ of $5\pm 2.6$.

We verified several model assumptions through retrievals that only include the
GRAVITY datasets, or the GRAVITY data and photometry. We find that the GRAVITY
data alone could not distinguish between clear and cloudy models
($\Delta\log_{10}\mathcal{Z}<1$), while cloudy models were strongly favoured
once the broad wavelength coverage of the photometry was included
($\Delta\log_{10}\mathcal{Z}>10$).

Models that use the dynamical mass estimates as priors for calculating the
surface gravity were marginally favoured over those that freely retrieve $\log
g$ and $R_{\rm pl}$, but that the posteriors parameter distributions were
generally consistent, with $T_{\rm int}$ and $R_{\rm pl}$ showing the greatest
discrepancy. Using the dynamical mass as a prior and setting a Gaussian prior
on the radius led to more reasonable estimates of the radius of HR 8799 e
($0.97\pm 0.04$) compared to the free retrieval ($0.79\pm 0.05$). However, the
composition of the planet and the degree of cloudiness did not vary
significantly between the two models.

Using the full dataset for HR 8799 b, we verified that retrievals including
scattering clouds are strongly favoured over those without scattering
($\Delta\log_{10}\mathcal{Z}>10$). Without scattering, both the temperature
and composition ([M/H] and C/O) are significantly discrepant from retrievals
that include scattering clouds.

Figure 23: Comparison of best-fit disequilibrium models to OSIRIS data from
Ruffio et al. (2021). From top to bottom is HR 8799 b, c and d. In blue are
the best-fit disequilibrium models, with the spectra generated using high-
resolution line-by-line opacities, before being convolved, binned, and
normalised for comparison. In orange is the same, but with the metallicity set
to 0. Figure 24: Comparison of best-fit disequilibrium models (black) of HR
8799 c to the data, with residuals shown in the bottom panel. In blue are the
same spectra, but without opacity contributions from CH4 . Figure 25:
Comparison of best-fit disequilibrium models (black) of HR 8799 c to the data,
with residuals shown in the bottom panel. In blue are the same spectra, but
without opacity contributions from HCN.

##  Appendix D Using the Hansen distribution with EDDYSED

The EddySed cloud model from Ackerman & Marley (2001) is implemented in pRT,
and is the most physically motivated model incorporated to date. Typically, it
assumes a log-normal particle size distribution, where the geometric particle
radius will vary throughout the atmosphere as a function of the vertical
diffusion coefficient $K_{\rm zz}$ and the sedimentation fraction $f_{\rm
sed}$. Here, we will substitute the log-normal particle size distribution with
the Hansen distribution, originally introduced in Hansen (1971), and will re-
derive the calculation for the particle radius as a function of $K_{\rm zz}$
and $f_{\rm sed}$.

We begin with a review of the EddySed model: the distribution of the number of
particles as a function of particle radius, $n(r)$ is approximated as a log-
normal distribution with width $\sigma_{g}$ and characteristic geometric
radius $r_{g}$.

$n(r)=\frac{N}{r\sqrt{2\pi}\log\sigma_{g}}\exp\left(-\frac{\log^{2}\left(r/r_{g}\right)}{2\log^{2}\sigma_{g}}\right),$
(27)

$N$ is the total number of cloud particles.

The goal of the EddySed model is to calculate $r_{g}$ for each layer in the
atmosphere, given $K_{\rm zz}$ and $f_{\rm sed}$. It balances the upwards
vertical mixing, parameterised by $K_{\rm zz}$ and the particle settling
velocity, $v_{f}$

$v_{f}=w_{*}\left(\frac{r}{r_{w}}\right)^{\alpha}.$ (28)

Here $w_{*}$ is the convective velocity scale. Note that $r_{w}\neq r_{g}$.
$r_{w}$ is the radius at which the particle settling velocity equals the
convective velocity scale:

$w_{*}=\frac{K_{\rm zz}}{L},$ (29)

where $L$ is the convective mixing length. Since $w_{*}$ is known, and $v_{f}$
can be found analytically as in Ackerman & Marley (2001); Podolak (2003), a
linear fit can be used to find both $\alpha$ and $r_{w}$.

With both of these quantities known, we follow AM01 and define $f_{\rm sed}$
as:

$f_{\rm
sed}=\frac{\int_{0}^{\infty}r^{3+\alpha}n(r)dr}{r_{w}^{\alpha}\int_{0}^{\infty}r^{3}n(r)dr}$
(30)

For the log-normal distribution, one finds:

$\int_{0}^{\infty}r^{\beta}n(r)dr=Nr_{g}^{\beta}\exp\left(\frac{1}{2}\beta^{2}\log^{2}\sigma_{g}\right)$
(31)

Which we can then use to solve for $r_{g}$:

$r_{g}=r_{w}f_{\rm
sed}^{1/\alpha}\exp\left(-\frac{\alpha+6}{2}\log^{2}\sigma_{g}\right)$ (32)

In order to use the Hansen distribution, we must recalculate the total number
of particles $N$, and integrate the distribution for $f_{\rm sed}$. We note
here that the Hansen distribution is parameterised by the effective radius,
$\bar{r}$, rather than the geometric mean radius. In this derivation we do not
correct for this difference in definition, as both act as nuisance parameters
in the context of an atmospheric retrieval.

We start by giving the Hansen distribution in full:

$n_{\rm
Hansen}(r)=\frac{N\left(\bar{r}v_{e}\right)^{\left(2v_{e}-1\right)/v_{e}}}{\Gamma\left[\left(1-2v_{e}\right)/v_{e}\right]}r^{(1-3v_{e})/v_{e}}\exp\left(-\frac{r}{\bar{r}v_{e}}\right)$
(33)

In Hansen (1971), the authors use the parameters $a$ and $b$ to denote the
mean effective radius and effective variance, which we write as $\bar{r}$ and
$v_{e}$ respectively. These differ from the simple mean radius and variance by
weighting them by the particle area, as the cloud particle scatters an amount
of light proportional to its area. Thus:

$\bar{r}=\frac{\int_{0}^{\infty}r\pi r^{2}n(r)dr}{\int_{0}^{\infty}\pi
r^{2}n(r)dr}$ (34)

and

$v_{e}=\frac{\int_{0}^{\infty}\left(r-\bar{r}\right)^{2}r^{2}n(r)dr}{\bar{r}^{2}\int_{0}^{\infty}\pi
r^{2}n(r)dr}$ (35)

As in EddySed, we will fit for the settling velocity, which will provide us
with $\alpha$ and $r_{w}$, which we can use to find $f_{\rm sed}$, as in 30.
However, we must now integrate the Hansen distribution. We find that:

$\int_{0}^{\infty}r^{\beta}n_{\rm
Hansen}(r)dr=\frac{v_{e}^{\beta}\left(v_{e}\beta+2v_{e}+1\right)\left(\frac{1}{\bar{r}}\right)^{-\beta}\Gamma\left(\beta+1+\frac{1}{v_{e}}\right)}{\left(-v_{e}+v_{e}^{\beta+3}+1\right)\Gamma\left(1+\frac{1}{v_{e}}\right)}$
(36)

We can then use Eqns. 30 and 36 to solve for $\bar{r}$:

$\bar{r}=\left(\frac{f_{\rm
sed}r_{w}^{\alpha}v_{e}^{-\alpha}\left(v_{e}^{3+\alpha}-v_{e}+1\right)\Gamma\left(1+\frac{1}{v_{e}}\right)}{\left(v_{e}\alpha+2v_{e}+1\right)\Gamma\left(\alpha+1+\frac{1}{v_{e}}\right)}\right)^{\frac{1}{\alpha}}.$
(37)

Thus for a given $K_{\rm zz}$, $f_{\rm sed}$ and $v_{e}$, we can find the
effective particle radius for every layer in the atmosphere.

However, in order to compute the cloud opacity, we still require the total
particle count. For a volume mixing ratio of a given species, $\chi_{i}$, we
can integrate $n(r)$ to find $N$:

$N=\frac{\chi_{i}}{\left(\bar{r}^{3}v_{e}-1\right)\left(2v_{e}-1\right)}$ (38)

##  Appendix E Complete retrieval results

We include in the text abridged tables that present key parameters of
interest. The complete set of inferred parameters for every retrieval is
available online.

Legend: Chemistry/Profile/Clouds/Data/Info.

  * •

Chemistry: (D)isequilibrium or (F)ree.

  * •

Profile: (M)olliere, (Z)hang, (G)uillot or (S)pline(NNodes).

  * •

Clouds: Clear (CLR); (f)ree or (eq)uilibrium condensation location,
(species)_(cd/am)_(P)atchy_(h)ansen. ‘*’ indicates $f_{\rm sed}$ was retrieved
independently for each cloud species.

  * •

Data: -(not included) or (only included). O indicates OSIRIS data was used in
place of GRAVITY data.

  * •

Info: ‘-’ indicates not included. ‘mr’ indicates mass and radius were used as
parameters instead of $\log g$ and radius.

In the text, models will be referred to as planet.group.index.

Table 14: Abridged retrieval results HR 8799 b

Planet | Index | Model | $\bm{\Delta\log}_{\mathbf{10}}\bm{\mathcal{Z}}$ | $\bm{\chi^{2}/\nu}$ | $\mathbf{T_{\rm Eff}}$ | $\bm{\log g}$ | [M/H] | C/O | Radius | $\bm{\log L/L_{\odot}}$ | Mass
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | [K] | [cgs] |  |  |  [RJup] |  |  [MJup]
b | b.A.0 | F/Z/eq*FeMg_am/ALL/-CH4/ | 0 | 1.46 | $948_{-14}^{+11}$ | $4.11_{-0.04}^{+0.03}$ | $0.94_{-0.08}^{+0.08}$ | $0.78_{-0.04}^{+0.03}$ | $1.1_{-0.03}^{+0.03}$ | $-5.06_{-0.04}^{+0.04}$ | $6_{-0.4}^{+0.3}$
| b.AB.1 | F/Z/eq*FeMg_am/ALL/ | 0 | 1.47 | $942_{-13}^{+12}$ | $4.1_{-0.04}^{+0.03}$ | $0.96_{-0.08}^{+0.08}$ | $0.78_{-0.04}^{+0.03}$ | $1.11_{-0.03}^{+0.03}$ | $-5.08_{-0.04}^{+0.04}$ | $6_{-0.3}^{+0.3}$
| b.A.2 | F/Z/eq*FeMg_am/ALL/-H2S/ | 0 | 1.49 | $958_{-15}^{+13}$ | $4.1_{-0.04}^{+0.03}$ | $1.1_{-0.1}^{+0.2}$ | $0.83_{-0.04}^{+0.03}$ | $1.1_{-0.03}^{+0.03}$ | $-5.05_{-0.04}^{+0.04}$ | $5.9_{-0.4}^{+0.4}$
| b.A.3 | F/S1/eqMg_am/ALL/ | -2 | 1.48 | $977_{-10}^{+11}$ | $4.18_{-0.04}^{+0.04}$ | $1.25_{-0.1}^{+0.1}$ | $0.89_{-0.03}^{+0.02}$ | $1.01_{-0.02}^{+0.02}$ | $-5.07_{-0.03}^{+0.03}$ | $6_{-0.4}^{+0.4}$
| b.A.4 | F/S2/eqMg_am/ALL/ | -4 | 1.50 | $970_{-13}^{+14}$ | $4.17_{-0.04}^{+0.04}$ | $1.2_{-0.1}^{+0.1}$ | $0.87_{-0.03}^{+0.03}$ | $1.02_{-0.02}^{+0.02}$ | $-5.08_{-0.04}^{+0.04}$ | $5.9_{-0.4}^{+0.4}$
| b.A.5 | F/S1/CLR/ALL/ | -4 | 1.53 | $985_{-11}^{+11}$ | $4.18_{-0.04}^{+0.04}$ | $1.43_{-0.09}^{+0.08}$ | $0.92_{-0.02}^{+0.02}$ | $1.01_{-0.02}^{+0.02}$ | $-5.06_{-0.04}^{+0.03}$ | $6_{-0.5}^{+0.5}$
| b.AB.6 | D/M/eqNa_P/ALL/ | -4 | 1.52 | $931_{-23}^{+21}$ | $4.12_{-0.05}^{+0.05}$ | $1.16_{-0.08}^{+0.08}$ | $0.73_{-0.02}^{+0.02}$ | $1.09_{-0.04}^{+0.06}$ | $-5.07_{-0.06}^{+0.06}$ | $6_{-0.3}^{+0.3}$
| b.AB.7 | D/M/eqNa/ALL/ | -4 | 1.50 | $966_{-19}^{+17}$ | $4.19_{-0.04}^{+0.04}$ | $0.95_{-0.07}^{+0.07}$ | $0.78_{-0.03}^{+0.02}$ | $1.0_{-0.04}^{+0.05}$ | $-5.05_{-0.05}^{+0.05}$ | $6_{-0.3}^{+0.3}$
| b.AB.8 | F/Z/f*FeMg_am/ALL/ | -4 | 1.48 | $944_{-10}^{+10}$ | $4.15_{-0.03}^{+0.03}$ | $0.97_{-0.08}^{+0.08}$ | $0.78_{-0.04}^{+0.03}$ | $1.04_{-0.02}^{+0.02}$ | $-5.11_{-0.03}^{+0.03}$ | $5.9_{-0.3}^{+0.4}$
| b.A.9 | F/S3/CLR/ALL/ | -4 | 1.49 | $962_{-12}^{+13}$ | $4.17_{-0.04}^{+0.04}$ | $1.1_{-0.1}^{+0.1}$ | $0.85_{-0.04}^{+0.03}$ | $1.02_{-0.02}^{+0.02}$ | $-5.09_{-0.04}^{+0.04}$ | $6_{-0.5}^{+0.4}$
| b.A.10 | F/S2/CLR/ALL/ | -4 | 1.52 | $962_{-14}^{+15}$ | $4.17_{-0.04}^{+0.04}$ | $1.2_{-0.1}^{+0.1}$ | $0.87_{-0.04}^{+0.03}$ | $1.02_{-0.03}^{+0.03}$ | $-5.09_{-0.04}^{+0.04}$ | $5.9_{-0.5}^{+0.5}$
| b.AB.11 | F/Z/f*FeMg_cd/ALL/ | -4 | 1.48 | $942_{-11}^{+11}$ | $4.15_{-0.03}^{+0.03}$ | $0.94_{-0.08}^{+0.08}$ | $0.77_{-0.04}^{+0.04}$ | $1.05_{-0.02}^{+0.02}$ | $-5.11_{-0.03}^{+0.03}$ | $5.9_{-0.3}^{+0.3}$
| b.A.12 | F/S3/eqMg_am/ALL/ | -4 | 1.51 | $965_{-13}^{+12}$ | $4.17_{-0.04}^{+0.04}$ | $1.1_{-0.1}^{+0.1}$ | $0.85_{-0.04}^{+0.03}$ | $1.02_{-0.02}^{+0.02}$ | $-5.09_{-0.03}^{+0.04}$ | $6_{-0.4}^{+0.4}$
| b.AB.13 | D/M/eq*FeMg_am/ALL/ | -6 | 1.54 | $944_{-10}^{+11}$ | $4.06_{-0.03}^{+0.03}$ | $1.11_{-0.08}^{+0.08}$ | $0.72_{-0.02}^{+0.02}$ | $1.17_{-0.03}^{+0.03}$ | $-5.03_{-0.04}^{+0.04}$ | $6_{-0.3}^{+0.3}$
| b.A.14 | F/S4/CLR/ALL/ | -6 | 1.49 | $966_{-12}^{+13}$ | $4.17_{-0.04}^{+0.04}$ | $1.15_{-0.1}^{+0.1}$ | $0.86_{-0.04}^{+0.03}$ | $1.02_{-0.02}^{+0.02}$ | $-5.09_{-0.04}^{+0.04}$ | $6_{-0.5}^{+0.4}$
| b.A.15 | F/S4/eqMg_am/ALL/ | -6 | 1.52 | $968_{-12}^{+13}$ | $4.17_{-0.04}^{+0.04}$ | $1.15_{-0.09}^{+0.1}$ | $0.86_{-0.03}^{+0.03}$ | $1.02_{-0.02}^{+0.02}$ | $-5.08_{-0.04}^{+0.04}$ | $6_{-0.4}^{+0.4}$
| b.AB.16 | D/M/eq*FeMg_cd/ALL/ | -6 | 1.55 | $911_{-13}^{+14}$ | $4.05_{-0.04}^{+0.04}$ | $1.16_{-0.08}^{+0.08}$ | $0.73_{-0.02}^{+0.02}$ | $1.18_{-0.04}^{+0.04}$ | $-5.09_{-0.04}^{+0.04}$ | $6_{-0.3}^{+0.3}$
| b.AB.17 | D/M/eq*FeMg_am_P/ALL/ | -7 | 1.55 | $941_{-12}^{+12}$ | $4.07_{-0.03}^{+0.03}$ | $1.08_{-0.09}^{+0.08}$ | $0.73_{-0.02}^{+0.02}$ | $1.16_{-0.03}^{+0.03}$ | $-5.05_{-0.03}^{+0.04}$ | $6_{-0.3}^{+0.3}$
| b.AB.18 | D/Z/eq*FeMg_am/ALL/ | -7 | 1.57 | $934_{-10}^{+10}$ | $4.06_{-0.03}^{+0.03}$ | $1.06_{-0.07}^{+0.06}$ | $0.73_{-0.02}^{+0.02}$ | $1.17_{-0.03}^{+0.03}$ | $-5.05_{-0.03}^{+0.03}$ | $6_{-0.3}^{+0.3}$
| b.A.19 | F/S5/CLR/ALL/ | -7 | 1.50 | $955_{-14}^{+13}$ | $4.16_{-0.04}^{+0.04}$ | $1.1_{-0.1}^{+0.1}$ | $0.84_{-0.05}^{+0.04}$ | $1.03_{-0.02}^{+0.02}$ | $-5.09_{-0.04}^{+0.04}$ | $6_{-0.5}^{+0.4}$
| b.A.20 | F/S5/eqMg_am/ALL/ | -7 | 1.51 | $958_{-14}^{+13}$ | $4.17_{-0.04}^{+0.03}$ | $1.09_{-0.1}^{+0.1}$ | $0.84_{-0.04}^{+0.03}$ | $1.03_{-0.02}^{+0.02}$ | $-5.09_{-0.03}^{+0.04}$ | $6_{-0.4}^{+0.4}$
| b.AB.21 | D/M/eqKCl/ALL/ | -8 | 1.56 | $906_{-13}^{+12}$ | $4.08_{-0.03}^{+0.03}$ | $1.22_{-0.08}^{+0.08}$ | $0.7_{-0.02}^{+0.02}$ | $1.13_{-0.03}^{+0.03}$ | $-5.13_{-0.04}^{+0.04}$ | $6_{-0.3}^{+0.2}$
| b.A.22 | F/S6/CLR/ALL/ | -9 | 1.50 | $960_{-13}^{+13}$ | $4.17_{-0.04}^{+0.04}$ | $1_{-0.1}^{+0.1}$ | $0.83_{-0.05}^{+0.04}$ | $1.03_{-0.02}^{+0.02}$ | $-5.09_{-0.04}^{+0.04}$ | $6_{-0.4}^{+0.4}$
| b.AB.23 | D/Z/f*FeMg_am/ALL/ | -9 | 1.53 | $954_{-59}^{+45}$ | $4.17_{-0.03}^{+0.03}$ | $1.22_{-0.06}^{+0.06}$ | $0.76_{-0.02}^{+0.01}$ | $1.02_{-0.02}^{+0.02}$ | $-5.1_{-0.1}^{+0.08}$ | $5.9_{-0.3}^{+0.3}$
| b.A.24 | F/S6/eqMg_am/ALL/ | -9 | 1.49 | $928_{-15}^{+17}$ | $4.01_{-0.04}^{+0.04}$ | $0.82_{-0.07}^{+0.07}$ | $0.63_{-0.05}^{+0.05}$ | $1.21_{-0.04}^{+0.04}$ | $-5.03_{-0.04}^{+0.04}$ | $5.8_{-0.4}^{+0.4}$