11institutetext:  Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna,
Tenerife, Spain 22institutetext: Departamento de Astrofísica, Universidad de
La Laguna (ULL), 38206, La Laguna, Tenerife, Spain 33institutetext: Leiden
Observatory, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands
44institutetext: INAF- Palermo Astronomical Observatory, Piazza del
Parlamento, 1, 90134 Palermo, Italy 55institutetext: University of Palermo,
Department of Physics and Chemistry “Emilio Segrè”

# The strange case of Na i in the atmosphere of HD209458 b:

Reconciling low- and high-resolution spectroscopic observations

G. Morello${}^{{\ref{inst:iac},\ref{inst:ull},\ref{inst:inaf}}}$ N. Casasayas-
Barris${}^{{\ref{inst:lo}}}$ J. Orell-
Miquel${}^{{\ref{inst:iac},\ref{inst:ull}}}$ E.
Pallé${}^{{\ref{inst:iac},\ref{inst:ull}}}$ G.
Cracchiolo${}^{{\ref{inst:inaf},\ref{inst:unipa}}}$ G.
Micela${}^{{\ref{inst:inaf}}}$

###### Abstract

Aims. We aim to investigate the origin of the discrepant results reported in
the literature about the presence of Na i in the atmosphere of HD209458 b,
based on low- and high-resolution transmission spectroscopy.

Methods. We generated synthetic planetary atmosphere models and we compared
them with the transmission light curves and spectra observed in previous
studies. Our models account for the stellar limb-darkening and Rossiter-
McLaughlin (RM) effects, and contemplate various possible scenarios for the
planetary atmosphere.

Results. We reconciled the discrepant results by identifying a range of
planetary atmospheres that are consistent with previous low- and high-
resolution spectroscopic observations. Either both datasets are interpreted as
consistent with a total absence of Na i in the planetary atmosphere (with
Hubble Space Telescope data being affected by limb darkening), or the
terminator temperature of HD209458 b has to have an upper limit of about
1000K. In particular, we find that 1D transmission spectra with lower-than-
equilibrium temperatures can also explain the previously reported detection of
absorption signal at low resolution due to differential transit depth in
adjacent bands, while the cores of the Na i D lines may be masked by the
strong RM signal seen at high resolution. We also rule out high-altitude
clouds, which would otherwise mask the absorption signal at low resolution, as
the source of the discrepancies.

Conclusions. This work highlights the synergies between different observing
techniques, specifically low- and high-resolution spectroscopy, to fully
characterise transiting exoplanet systems.

###### Key Words.:

planetary systems – planets and satellites: individual: HD209458 b – planets
and satellites: atmospheres – techniques: spectroscopic – methods:
observational

##  1 Introduction

The characterisation of exoplanet atmospheres relies on precise measurements
of the spectroscopic signal that they imprint on the observed starlight. For a
planet transiting in front of its host star, the atmosphere acts as a
wavelength-dependent annulus that affects the fraction of occulted stellar
flux (Brown, 2001). If an atom or molecule is present in the planetary
atmosphere, it causes extra absorption at specific wavelengths corresponding
to electronic transition lines. The transit depth is defined as the apparent
planet-to-star area ratio, $p^{2}=(R_{p}/R_{*})^{2}$, $R_{p}$ and $R_{*}$
being the planet and star radii. Differences in transit depths obtained on
multiple passbands or wavelength bins have led to the detection of many
chemical species in the atmospheres of exoplanets (e.g. Charbonneau et al.,
2002; Vidal-Madjar et al., 2003; Fossati et al., 2010; Deming et al., 2013;
Damiano et al., 2017). High-resolution spectroscopy has enabled us to resolve
the atmospheric spectral features and to trace their Doppler shift as it
varies with the orbital phase (e.g. Snellen et al., 2010; Casasayas-Barris et
al., 2017, 2018, 2019; Palle et al., 2020; Stangret et al., 2020). Typically,
the signal from the planet atmosphere is of the order of 10-4 or less,
relatively to the host star flux. It is therefore necessary to disentangle
this small signal from other contaminating effects with similar amplitudes,
such as stellar limb darkening (Howarth, 2011; Csizmadia et al., 2013; Morello
et al., 2017; Yan et al., 2017), magnetic activity (Ballerini et al., 2012;
Oshagh et al., 2014; Cracchiolo et al., 2021a, b), and planet self- and phase-
blend effects (Kipping & Tinetti, 2010; Martin-Lagarde et al., 2020; Morello
et al., 2021).

HD209458 b is the first planet with a reported detection of a chemical species
in its atmosphere, namely Na i (Charbonneau et al., 2002). The observations
were performed with the Hubble Space Telescope (HST)/Space Telescope Imaging
Spectrograph (STIS) using the G750M filter. In particular, Charbonneau et al.
(2002) inferred Na i absorption from the larger transit depth on a narrow band
centred on the resonance doublet at 5893 $\AA$ relative to adjacent bands.
Sing et al. (2008) reanalysed the same data, confirming the previous Na i
detection and resolving the doublet lines. Other HST/STIS datasets led to the
possible detection of H i, O i, C ii (Vidal-Madjar et al., 2004), Mg i (Vidal-
Madjar et al., 2013), and Fe ii (Cubillos et al., 2020). The near-infrared
spectrum taken with the HST/Wide Field Camera 3 (WFC3) also revealed H2O
absorption (Deming et al., 2013; Tsiaras et al., 2016). Ground-based
observations provided independent confirmation of the Na i feature with higher
spectral resolution (Snellen et al., 2008) and also revealed other features
attributed to He i (Alonso-Floriano et al., 2019), Ca i, Sc ii, H i
(Astudillo-Defru & Rojo, 2013), CO (Snellen et al., 2010), H2O (Sánchez-López
et al., 2019), HCN, CH4, C2H2 and NH3 (Giacobbe et al., 2021). However, recent
studies cast doubts on the Na i detection as well as that of other atomic and
ionic species (Casasayas-Barris et al., 2020, 2021). In particular, the
transmission spectra observed with the High Accuracy Radial velocity Planet
Searcher for the Northern hemisphere (HARPS-N), mounted on the Telescopio
Nazionale Galileo (TNG) at the Observatory of Roque de los Muchachos (ORM) in
Spain, and the Echelle Spectrograph for Rocky Exoplanet and Stable
Spectroscopic Observations (ESPRESSO), mounted on the Very Large Telescope
(VLT) at the European Southern Observatory (ESO) of Cerro Paranal in Chile,
reveal several features due to the Rossiter-McLaughlin (RM) effect and no
evidence of absorbing species in the exoplanet atmosphere.

In this paper, we try to reconcile these apparently conflicting results that
have appeared using different instruments, observing techniques, and analysis
methods. Section 2 presents a reanalysis of the HST/STIS observations that led
to the first announcement of Na i in the atmosphere of HD209458 b with updated
data detrending techniques, transit modelling tools, and system parameters.
Section 3 describes our models of the planetary atmosphere. From them, we
extract time series and spectra following the same procedures adopted by
previously published studies of low and high spectral resolution observations
aimed at exoplanet atmospheric characterisation, along with other stellar and
planetary effects. Section 4 discusses the comparison between our simulations
and the observations, providing a range of planetary atmosphere scenarios that
could explain the apparently discrepant results. Section 5 summarises the
conclusions of our study.

##  2 Reanalysis of HST data

Table 1: Summary of HST observations for identification in the online
archives. More technical information is reported in Section 2.1.

Root name | Number of spectraaa$a$For each of the four orbits. | UT start date
---|---|---
o63302030 | 36, 36, 36, 35 | 2000-04-28
o63303030 | 36, 36, 36, 35 | 2000-05-05
o63304030 | 36, 36, 36, 35 | 2000-05-12
111

Table 2: Adopted HD209458 system parameters. Stellar parameters
---
T∗,eff [K] | 6065$\pm$50 | Torres et al. (2008)
$\log{g_{*}}$ [cgs] | 4.361$\pm$0.008 | ”
$[\mathrm{Fe/H}]_{*}$ [dex] | 0.00$\pm$0.05 | ”
$M_{*}$ [$M_{\odot}$] | 1.119$\pm$0.033 | ”
$R_{*}$ [$R_{\odot}$] | 1.155$\pm$0.016 | ”
Planetary parameters
$M_{\mathrm{p}}$ [$M_{\mathrm{Jup}}$] | 0.685$\pm$0.015 | ”
$R_{\mathrm{p}}$ [$R_{\mathrm{Jup}}$] | 1.359$\pm$0.019 | ”
$a$ [au] | 0.04707$\pm$0.00047 | ”
Transit parameters
$P$ [day] | 3.52474859$\pm$0.00000038 | Knutson et al. (2007)
$E.T.$ [HJD] | 2452826.628521$\pm$0.000087 | ”
$b$ [R∗] | 0.516$\pm$0.006 | Morello (2018)
$T_{0}$ [s] | 9500$\pm$6 | ”
RM parameters
$K_{\mathrm{p}}$ [km/s] | 145.0$\pm$1.6 | Casasayas-Barris et al. (2021)
$\lambda$ [deg] | 1.58$\pm$0.08 | ”
$v\sin{i_{*}}$ [km/s] | 4.228$\pm$0.007 | ”

###  2.1 Observations

We reanalysed three transits of HD209458 b observed with HST/STIS (GO-8789,
PI: Brown) using the G750M grism, taken on 28 April, 5 May and 5 December
2000. Table 1 provides their identifiers. The spectra cover the 5808-6380
$\AA$ wavelength range and have a resolving power of 5540 at the central
wavelength of 6094 $\AA$. Each visit contains 143 spectra distributed over
four HST orbits: one before, two during, and one after the transit event. The
four HST orbits were preceded by another one to enable instrumental settling,
but it was not included in the scientific data analysis. The integration time
is 60 s per frame, followed by a reset time of about 20 s. The interval
between consecutive HST orbits is about 50 min. The spectral trace forms an
angle of less than 0.5∘ with the longest side of the 64$\times$1024 pixel
detector, and it is stable within the same pixel rows during each visit.

###  2.2 Data analysis

####  2.2.1 Extraction

We downloaded the flat-fielded science images (extension: flt.fits) from the
Mikulski Archive for Space Telescopes (MAST)222https://archive.stsci.edu. We
considered a rectangular aperture of 17$\times$1024 pixels with the trace at
its centre and summed along the columns to extract the 1D spectra. As a
starting point, we adopted the wavelength solution from the corresponding
archive spectra (extension: x1d.fits) and then computed the cross-correlations
with a template spectrum to align the extracted spectra in the stellar rest
frame. The template spectrum was calculated as described in Section 3.1, and
purposely degraded to the same resolution as the observations.

Our analysis focused on a portion of the spectrum containing the Na i doublet.
We extracted the flux time series for three bands identical to the ‘narrow’
ones selected by Charbonneau et al. (2002) using two different sets of
apertures. The first set consisted of rectangular apertures, similar to the
ones used for the wavelength calibration, but limited in the dispersion axis
according to the bands definition. In this case, the flux is the simple sum
over the pixels contained in the rectangular aperture. The second set
consisted of tilted rectangular apertures with sides parallel and
perpendicular to the spectral trace, which was previously calculated by a
linear fit on the centroids resulting from Gaussian fits on the detector
columns. When calculating the flux from the fractions of pixels within the
tilted apertures we accounted for the non-uniform distribution inside the
pixel, that was approximated by a linear function in the cross-dispersion
direction. The resulting light curves have almost identical shapes, regardless
of the extraction method, but the tilted apertures led to higher fluxes by
$\sim$0.01$\%$, $\sim$0.04$\%$ and $\sim$0.10$\%$ for the left, central, and
right bands, respectively. This behaviour is expected as the small adjustment
for the orientation of the spectral trace reduces the flux losses from the
tails of the point spread function. For the rest of the analysis described in
this paper, both methods led to indistinguishable results.

####  2.2.2 Detrending

The HST time series exhibit well-known systematic effects, usually referred to
as short- and long-term ramps (Brown et al., 2001). The short-term ramp is a
flux variation that follows a highly repeatable pattern for each orbit of the
same visit, excluding the first orbit. The long-term ramp approximates a
linear trend in the transit timescale. We adopted the divide-oot method to
correct for the ramp effects (Berta et al., 2012). Following this method, the
two in-transit orbits are divided by the time-weighted mean of the two out-of-
transit orbits. Since the last orbit contains one less spectrum than the
others, we duplicated the last point to enable the operations described above.
We note that the added point falls into the plateau at the end of the ramp and
out of transit, so it is expected to be similar to the previous one.

Figure 1: Left: Best-fit transit depths to HST data around the Na I lines, as
reported in Table 3. Right: divide-oot detrended and phase-folded HST light
curves (see Section 2.2.2) for the 5818-5887 $\AA$ (blue crosses), 5887-5899
$\AA$ (red dots), and 5899-5968 $\AA$ (dodger blue pluses) passbands, along
with best-fit transit models (continuous coloured lines). The detrended out-
of-transit data are not represented, as they are identical to 1 by definition
of the divide-oot method (Berta et al., 2012). The bottom panel shows the
transit model differences with respect to that of the central band. The
slightly different shape between the two side bands is due to differential
limb darkening.

####  2.2.3 Fitting

We combined the detrended orbits from the three visits into phase-folded time
series to perform joint light-curve fits. The transit models were computed
with PYLIGHTCURVE333https://github.com/ucl-exoplanets/pylightcurve (Tsiaras et
al., 2016). Since we were primarily interested in the differences in transit
depth between the selected passbands, we kept the planet-to-star radii ratio
($p$) as the only free parameter. Table 2 reports the system parameters along
with the transit parameters that were fixed in the light-curve fits. The
stellar limb-darkening coefficients were fixed to the values obtained with
ExoTETHyS444https://github.com/ucl-exoplanets/exotethys (Morello et al.,
2020a, b)) using the four-coefficient law (Claret, 2000) and the STAGGER grid
of stellar spectra (Magic et al., 2015; Chiavassa et al., 2018). The fitting
procedure included a preliminary least-squares minimisation using
scipy.optimize.minimize with the Nelder-Mead method (Nelder & Mead, 1965).
Then we ran emcee555https://github.com/dfm/emcee (Foreman-Mackey et al., 2019)
with 80 walkers and 200 000 iterations. Each walker was initialised with a
random value close to the least-squares parameter estimate. The first 100 000
iterations were discarded as burn-in. The remaining samples were used to
determine the final best-fit parameters (median) and error bars (absolute
differences between the 16th and 84th percentiles and the median).

###  2.3 Results

Table 3 reports the measured transit depths for the three selected passbands
with the two extraction methods described in Section 2.2.1. Figure 1 (left
panel) shows the same results. The two sets of results are nearly identical
with 1-3$\%$ smaller error bars when using tilted apertures. They indicate a
greater transit depth in the central band containing the Na i doublet, and no
significant difference in transit depth between the two side bands. We
calculated the Na i absorption signal as the difference between the transit
depth in the central band and the average of the two side values, obtaining
232$\pm$62 ppm (tilted apertures) and 237$\pm$64 ppm (sum of pixel columns).
These results confirm that Na i is detected in the atmosphere of HD209458 b at
3.7$\sigma$.

We note that the differences between the best-fit light-curve models for the
side bands and for the central band are overall positive during transit, and
show a plateau at $>$200 ppm at orbital phases $|\Delta\phi|<$0.013 (see
bottom right panel of Figure 1). If stellar limb-darkening was the only
wavelength-dependent effect, the light-curve differences should have a typical
double-horned modulation with both positive and negative values (Rosenblatt,
1971; Tingley, 2004; Parviainen et al., 2019). The numerical closeness between
the plateau in the light-curve differences and the measured Na i absorption
feature also suggests that the results of the analysis are not strongly
influenced by the adopted stellar limb-darkening model. However, some authors
pointed out that stellar limb-darkening models may be biased, because of
unaccounted-for effects and lack of empirical validation (Csizmadia et al.,
2013; Espinoza & Jordán, 2015; Morello, 2018). We also attempted light-curve
fitting with free limb-darkening coefficients. The results point towards a
smaller (non-significant) Na i absorption feature and enhanced differential
limb darkening, but they are inconclusive because of the low signal-to-noise
ratio (S/N) of the data and strong parameter degeneracies. Better quality data
are needed to conclude as to the authenticity of the Na i detection at low
resolution.

Table 3: Best-fit transit depths to HST data and Na i absorption signal.

Passband | Transit Depth
---|---
($\mathrm{\AA}$) | tilted aper. ($\%$) | sums of col. ($\%$)
nl: 5818-5887 | 1.482$\pm$0.028 | 1.481$\pm$0.029
nc: 5887-5899 | 1.504$\pm$0.059 | 1.503$\pm$0.060
nr: 5899-5968 | 1.479$\pm$0.029 | 1.478$\pm$0.030
nc - (nl+nr)/2 | 232$\pm$62 ppm | 237$\pm$64 ppm
666Choice of passbands and apertures are detailed in Section 2.2.1.

##  3 Simulations

In the following sections, we describe our simulated time series of spectra
for the transit of HD209458 b, taking into account the stellar limb-darkening
and rotation, for a variety of planetary atmosphere models. Then, we processed
these synthetic data following typical procedures adopted in transmission
spectroscopy studies at low and high resolution, and in particular the average
spectra and light-curves presented as in Charbonneau et al. (2002) and
Casasayas-Barris et al. (2021). The main purpose of this exercise is to
investigate whether there are physical configurations that are compatible with
the observations both at low and high resolution. The methodology adopted to
simulate the spectra in this paper is a generalisation of that described by
Casasayas-Barris et al. (2019). The main upgrade is related to the use of a
wavelength-dependent planet radius to include the effect of its atmosphere.

###  3.1 Modelling the star

We modelled the sky-projected star disc as a grid of square cells, wherein
each cell has an associated spectrum. Each cell is identified by the Cartesian
coordinates of its centre $(x_{j},\,y_{j})$, assuming that the star disc is a
circle with unit radius centred on the origin. The cell spectrum depends on
its position because of the centre-to-limb variation (CLV, aka limb darkening)
and the local radial velocity due to the stellar rotation. We used the
Spectroscopy Made Easy (SME, Valenti & Piskunov, 1996; Piskunov & Valenti,
2017) package to generate intensity spectra for a star similar to HD209458
without rotational broadening. The SME calculation relies on a pre-computed
grid of MARCS stellar atmosphere models (Gustafsson et al., 2008). The
spectral resolving power is $\sim$8$\times$105. These intensity spectra were
calculated for 21 angles, equally spaced in $\mu$, except for $\mu=$0.005
instead of $\mu=$0 to avoid the singularity at the edge of the disc. Here,
$\mu=\cos{\theta}$, where $\theta$ is the angle between the line of sight and
the surface normal. The static cell spectrum is obtained by
$\mu$-interpolation from the pre-calculated intensity spectra, where for a
given cell

$\mu_{j}=\sqrt{1-x_{j}^{2}-y_{j}^{2}}.$ (1)

In the star rest frame, the radial velocity of a given cell is

$v_{j}=v\sin{i_{*}}\left(y_{j}\sin{\lambda}-x_{j}\cos{\lambda}\right),$ (2)

where $v\sin{i_{*}}$ is the radial component of the equatorial velocity, and
$\lambda$ is the sky-projected obliquity. We computed the Doppler shifted cell
spectra in the star rest frame over the same wavelengths of the SME models. In
this way, the disc-integrated stellar spectrum is the sum of the Doppler
shifted cell spectra multiplied by the cell area.

###  3.2 Modelling the planet

The planet is represented by a non-emitting opaque disc with a wavelength-
dependent radius to account for atmospheric absorption. We modelled the case
with constant radius equal to the value reported in Table 2, three cases with
an atmosphere including Na i absorption signature at different terminator
temperatures, and another case that also includes clouds. We used the
petitRADTRANS777https://gitlab.com/mauricemolli/petitRADTRANS/ (pRT, Mollière
et al., 2019) package to compute the apparent planetary radii at multiple
wavelengths including its atmosphere. The spectral resolving power was set to
106. Three of the simulated models assume clear atmospheres with solar
abundances and a range of isothermal temperatures ($T_{\mathrm{p,iso}}=$ 700,
1000, and 1449 K). The fourth model, with $T_{\mathrm{p,iso}}=$ 1449 K,
includes a grey cloud deck with top pressure of 1.38 mbar, as estimated by
recent retrieval studies (Tsiaras et al., 2018). The main effect of
temperature is that it changes the strength of the Na i lines, while the cloud
deck also dampens their tails (see Figure 2). We note that these are the radii
in the planet rest frame. Absorption from the planet atmosphere in the star
rest frame is Doppler shifted due to the relative radial velocity of the
planet,

$V_{\mathrm{p}}(\phi(t))=K_{\mathrm{p}}\sin{(2\pi\phi(t))},$ (3)

for a circular orbit. Here ,$K_{\mathrm{p}}$ is the orbital velocity of the
planet and $\phi(t)$ is the orbital phase,

$\phi(t)=\frac{t-E.T.}{P},$ (4)

with $E.T.$ being the reference epoch of mid-transit time, and $P$ being the
orbital period.

Figure 2: Planet-to-star radii ratios versus wavelength. Left: constant radius
(solid red line) and three pRT models with clear atmosphere and terminator
temperatures of 1449 K (dotted orange), 1000 K (dash-dotted cyan), and 700 K
(dashed blue). Right: pRT model assuming the equilibrium temperature at
terminator with a cloud deck (solid grey thick line), shifted and rescaled
model to have the continuum baseline and line peaks of the clear atmosphere
model (solid grey thin line), and cloud-free model with the same temperature
(dotted orange).

###  3.3 Modelling the transit

While transiting in front of its host star, the planet occults a time-varying
portion of the stellar disc. In our simulations, we compute the unocculted
spectrum at given instants with a cadence of 20 s. To do so, we first
calculate the sky-projected planet coordinates
$(X_{\mathrm{p}}(t),Y_{\mathrm{p}}(t))$ in units of the star radius using our
own modified version of the PYLIGHTCURVE routine. Our version takes into
account the light-travel delay (Kaplan, 2010; Bloemen et al., 2012), which has
a negligible impact in this study. Second, we compute the relative planetary
radii in the star rest frame,

$p(\lambda,t)=\frac{R_{\mathrm{p}}(\lambda,t)}{R_{*}},$ (5)

where $R_{\mathrm{p}}(\lambda,t)$ are either constant or Doppler-shifted pRT
models, interpolated over the same wavelengths of the cell spectra, and
$R_{*}$ is the star radius. Third, we apply the following mask to each cell
spectrum:

$m_{j}(\lambda,t)=\cases{0},&\text{if
}dist((X_{\mathrm{p}}(t),Y_{\mathrm{p}}(t)),\,(x_{j},y_{j}))<p(\lambda,t)\\\
1,\text{elsewhere}{},$ (6)

that is, the masked cell spectra are unchanged if the centre of the cell is
external to the planet discs at all wavelengths, partially or fully zeroed
otherwise. The stellar spectrum observed at a given instant is the sum of the
cell spectra multiplied by the mask and by the cell area.

Figure 3: Simulated transmission spectra of HD209458 b around the Na i D1
line, showing the RM+CLV effects along with atmospheric absorption. The
spectra are computed in the planet rest frame and averaged between transit
contact points as indicated in the top left or top right corner of each panel.
The different colours correspond to the same atmospheric models represented in
Figure 2. The ESPRESSO data are the same as those reported in Figure 5 of
Casasayas-Barris et al. (2021).  Figure 4: Simulated transmission light curve
of HD209458 b for passbands centred on any Na i D line with
$\Delta\lambda=\mathrm{0.4\,\AA}$ and $\Delta\lambda=\mathrm{0.7\,\AA}$,
showing the RM+CLV effects along with atmospheric absorption. The light-curves
are computed in the planet rest frame. The different colours correspond to the
same atmospheric models represented in Figure 2. The ESPRESSO data are the
same as reported in Figure 6 by Casasayas-Barris et al. (2021).  Figure 5:
Simulated transit light curves of HD209458 b for the same three passbands
adopted in the HST data analysis and five planetary atmosphere models. The
blue, red, and bright blue lines are obtained as described in Section 3, with
the continuous and dashed lines corresponding to clear and cloudy models,
respectively. The bottom panels show the transit model differences with
respect to that of the central band. The black and grey dotted lines are the
HST model differences from Figure 1. We note that the HST model differences
have a more trapezoidal shape than those obtained from the other set of
simulations, because of the different limb-darkening profiles of the
underlying STAGGER and MARCS stellar models.

###  3.4 Simulated transmission spectra and light curves

Starting from the time series of simulated spectra, we calculated time-
averaged spectra and band-integrated time series analogous to those that were
used for data analysis and/or visualisation in previously published papers
(Charbonneau et al., 2002; Casasayas-Barris et al., 2021). Figure 3 shows the
transmission spectra in the planet rest frame averaged between different
contact points, such as those reported in Figure 5 by Casasayas-Barris et al.
(2021). We zoomed in on the main Na i line to highlight the effects of
absorption in the planetary atmosphere, which are identical for both lines
based on our models. The Na i absorption imprints a dip near the peak of the
symmetric RM feature obtained in the T1-T4 and T2-T3 intervals, and an overall
offset due to its broad wings. The planetary Na i may also appear as a small
negative bump before or after the RM feature in the ingress or egress spectra.
We note that, depending on the S/N and resolving power of the spectroscopic
data, the double-peaked RM feature in T1-T4 or T2-T3, and the small negative
bumps in the T1-T2 and T3-T4 spectra may not be resolved. Also, the broadband
spectral offset may be altered by some data-processing steps, in particular
the continuum normalisation and the removal of instrumental systematic
effects. In these cases, the net effect of planet atmospheric absorption could
be that of reducing the amplitude of the RM feature in T1-T4 and T2-T3
spectra. However, the feature amplitudes are also sensitive to the underlying
stellar spectrum and limb-darkening profiles obtained with a different
synthesis code and/or input physics (Casasayas-Barris et al., 2021).
Consequently, the feature amplitude alone does not provide conclusive evidence
about the presence or absence of Na i absorption.

Figure 4 shows the transmission light curves for bandwidths of 0.4 and 0.7
$\mathrm{\AA}$ centred on the Na i D lines in the planet rest frame, as those
reported in Figure 6 by Casasayas-Barris et al. (2021). The Na i absorption
mostly affects the amplitude and vertical offset of the transmission time
series relative to the out-of-transit baseline, and introduces only modest
distortions during the ingress and egress phases. Even for these transmission
light curves, the effects of Na i in the planet atmosphere are degenerate with
details of the stellar spectrum template.

Moving to low resolution, Figure 5 shows the full transit light-curve models
for the three HST passbands analysed in this study (Section 2) and by
Charbonneau et al. (2002). Following the standard approach of low-resolution
transit spectroscopy studies, they consist of instantaneous band-integrated
fluxes relative to the stellar flux in the star rest frame. The decrease in
normalised flux during transit corresponds to the fraction of stellar flux
occulted by the whole planet disc, not just by its atmosphere. We subtracted
the light-curve model of the central band to highlight the wavelength-
dependent effects. If Na i is not present in the planet atmosphere, the
modulation obtained in the light-curve differences are solely due to the
stellar RM and CLV effects. In this case, we measured a peak-to-peak amplitude
of 330 ppm and mean value below 10 ppm for the modulation. If Na i is present
in the planet atmosphere, our models indicate positive light-curve differences
at all phases, with median values of $\sim$190, $\sim$300, and $\sim$520 ppm
assuming isothermal temperatures ($T_{\mathrm{p,iso}}$) of 700, 1000, and 1449
K, respectively. The cloudy model at 1449 K shows a median light-curve
difference of 300 ppm.

We note that the light-curves plotted in Figures 4 and 5 have been smoothed to
remove some fringing due to numerical errors, particularly in the models with
high-resolution signatures. We checked that fringing was significantly reduced
when decreasing the cell size adopted for the simulations, but the relevant
calculations soon became too heavy. The smoothed light curves are a good
compromise between precision of the models and computational costs.

##  4 Discussion

###  4.1 Comparison with published low-resolution spectroscopy

Charbonneau et al. (2002) announced the first detection of Na i in the
atmosphere of HD209458 b using the same HST data presented in Table 1. They
reported an absorption depth of 232$\pm$57 ppm based on the same passbands of
Table 3, but with a significantly different methodology. More specifically,
they computed the absorption light curve as a linear combination of the three
raw light curves, hence the depth is the difference between the out-of-transit
and in-transit mean values. Instrumental systematics and stellar limb-
darkening effects are ignored with their approach, although they were
mitigated by removing the first point of each HST orbit and those during the
transit ingress and egress. In this work, we instead measured the transit
depth for each passband by performing independent data detrending and light-
curve fitting, and assuming a STAGGER stellar limb-darkening model (see
Section 2). The absorption depth is a linear combination of the three transit
depths. Our best result of 232$\pm$62 ppm matches that reported by Charbonneau
et al. (2002). Turning to empirical limb-darkening coefficients in the light-
curve fits, however, we found that the HST data are consistent with the non-
detection of Na i in the atmosphere of HD209458 b.

By comparing the model light-curve differences shown in Figure 5 with the
best-fit results of Figure 1, the models that better reproduce the HST results
are those with cloud-free atmosphere and terminator temperatures of 700-1000
K, or that with partially cloudy atmosphere and terminator temperature equal
to the equilibrium temperature of 1449 K. We note that there is a degeneracy
between the terminator temperature and cloud top pressure for a given
amplitude of the Na i absorption signal at low resolution (Lecavelier Des
Etangs et al., 2008; Benneke & Seager, 2013; Heng & Kitzmann, 2017).

Sing et al. (2008) analysed the transmission spectrum of HD209458 b with
various resolving powers, obtaining similar results when considering the same
passbands as Charbonneau et al. (2002). Additionally, Sing et al. (2008)
reported a spectral signal in the opposite direction to that of Na i
absorption at the line cores, that they attributed to telluric contamination.
The removal of this effect would further increase the significance of the
measured Na i absorption. However, as pointed out by Casasayas-Barris et al.
(2020), a much more likely explanation for this signal is the combination of
RM+CLV effects instead of a telluric origin.

###  4.2 Comparison with published high-resolution spectroscopy

Snellen et al. (2008) and Albrecht et al. (2009) analysed ground-based spectra
taken with the High Dispersion Spectrograph (HDS) mounted on the Subaru
telescope at Mauna Kea in Hawaii, and with the Ultraviolet and Visual Echelle
Spectrograph (UVES) mounted on the Very Large Telescope (VLT) at Cerro Paranal
in Chile. Their results are in good agreement with those reported by Sing et
al. (2008) based on HST data, when considering similar passbands centred on
the Na i D lines.

Recently, Casasayas-Barris et al. (2020) analysed the high-resolution spectra
taken with TNG/HARPS-N, and with the Calar Alto high Resolution search for M
dwarfs with Exoearths with Near-infrared and optical Echelle Spectrographs
(CARMENES) located in Spain. Their approach differs from that of previous
studies as the transmission spectra are computed in the planet rest frame
instead of the stellar rest frame. They revealed the RM effect around the Na i
D, as well as other lines (H$\alpha$, Ca ii IRT, Mg i and K i D1), without
evidence of absorption in the exoplanet atmospheres. Casasayas-Barris et al.
(2021) confirmed these findings with higher confidence and also reported an
analogous behaviour for other lines (Fe i, Fe ii, Ca i, and V i), based on
data taken with VLT/ESPRESSO. The ESPRESSO datasets have higher S/N than any
other ground-based high-resolution dataset.

In Section 3.4, we discuss how Na i absorption may alter the RM+CLV signal in
transmission spectra. The error bars obtained by Casasayas-Barris et al.
(2021) with ESPRESSO range from 0.05$\%$ to 0.09$\%$ for bin widths of 0.03
$\AA$. Based on the models plotted in Figure 3, they should have resolved the
double peak caused by Na i absorption for both clear and cloudy atmospheric
models with equilibrium temperature of 1449 K, if present, in the ESPRESSO
data. We note that clouds tend to dampen the line tails, and hence they mostly
affect the low-resolution signal. Models with cooler temperatures pose a
greater challenge for the high-resolution signal, as the smaller dips on top
of the RM feature can be comparable with the spectral error bars. If
unresolved, the Na i dip would decrease the amplitude of the RM feature. The
ESPRESSO data produces a signal of greater amplitude than that predicted by
the models shown in Figure 3, contrary to expectations in the presence of
unresolved Na i absorption. However, the amplitude of the RM feature alone
cannot be used to validate Na i absorption in the atmosphere of HD209458 b,
given the uncertainties associated with stellar models and system parameters.
For example, Casasayas-Barris et al. (2021) obtained a larger RM feature by
assuming a MARCS model without local thermodynamic equilibrium (see their
Figure 10). We also note that the broadband spectral offset is $\sim$0.05$\%$
for the model with the strongest absorption and it can be attenuated by
ESPRESSO data processing, in particular due to continuum normalisation and
removal of observed fringe patterns (Casasayas-Barris et al., 2021).

Similar considerations apply to the transmission light curves as shown in
Figure 4. The possible Na i absorption affects the amplitude and offset of the
time signal, without significant distortions in most cases. The shape of the
transmission light curves is strongly dependent on the CLV effect, for which
the stellar models considered by Casasayas-Barris et al. (2021), which are the
same adopted in this work, do not provide a good match to the ESPRESSO data.

In conclusion, we find that atmospheric models with terminator temperature of
700-1000 K can be compatible with the non-detection of Na i at high
resolution. In Section 4.1, these models were also selected among the best
fits to the low-resolution data that led to the 3.7$\sigma$ detection of Na i.
The high-resolution observations taken with ESPRESSO added further
constraints; for example, discarding the hotter model with clouds.

###  4.3 Planet atmospheric temperature

Morello et al. (2021) calculated an equilibrium temperature of 1449 K for
HD209458 b in the case of zero reflectance and maximum circulation efficiency
(see their Table 3). Taking into account the Spitzer phase-curve measured by
Zellem et al. (2014), the terminator temperature should lie between the night-
side temperature of 970 K and the day-side temperature of 1500 K. This
temperature range can be further extended depending on the vertical profiles
(e.g. Venot et al., 2019; Drummond et al., 2020). In principle, the 3D
structure of the planetary atmosphere should be fully considered to simulate
accurate transmission spectra, but 1D models can also reproduce the same
spectra albeit with biases in physical and chemical parameters (Caldas et al.,
2019; Pluriel et al., 2020). In particular, MacDonald et al. (2020)
demonstrated that the 1D retrieval techniques can lead to significantly
underestimated temperatures for atmospheres with differing morning-evening
terminators. This effect may explain the trend that terminator temperatures
retrieved from transmission spectra are typically cooler than equilibrium
temperatures (Fisher & Heng, 2018; Tsiaras et al., 2018; Pinhas et al., 2019).
Our isothermal temperature estimate of 700-1000 K for the atmosphere of
HD209458 b aligns with this trend.

##  5 Conclusions

We investigated the puzzling question about the presence of Na i in the
atmosphere of HD209458 b, following conflicting reports from previous analyses
of transit observations with low- and high-resolution spectroscopy. By
comparing a set of models to the observations, we find that several
atmospheric scenarios are compatible with both low- and high-resolution data.
Overall, the HST and ESPRESSO datasets are consistent with a total absence of
Na i from the planetary atmosphere; otherwise, the terminator temperature of
HD209458 b has to have an upper limit of about 1000K. More precise knowledge
of the stellar intensity spectra, along with higher S/N data, is crucial to
selecting the best scenario. The lower-than-equilibrium temperature (1449 K)
on the terminator can be a consequence of heat redistribution processes or the
effect of 1D model bias. Clouds may be present, in agreement with previous
estimates, but they alone cannot help to reconcile the low- and high-
resolution observations. In some configurations, the presence of Na i is clear
from the broadband transmission spectrum, but, under certain physical
conditions, the RM effect can mask the line cores at high resolution. If the
overlapping signals at high resolution are not disentangled, the information
on the spectral continuum from low-resolution observations can be decisive in
detecting Na i absorption.

This study highlights the complementarity between low- and high-resolution
spectroscopic techniques for the characterisation of exoplanet systems. In the
literature, there are several contrasting results about exoplanet atmospheres
based on low- and high-resolution spectroscopy (Huitson et al., 2013;
Sedaghati et al., 2017; Espinoza et al., 2019; Chen et al., 2018; Allart et
al., 2020). We suggest that joint modelling efforts, such as the one proposed
in this paper, could lift the apparent discrepancies. Considering a broader
wavelength coverage should help reduce the degeneracy between stellar and
planetary signals by considering multiple lines, therefore leading to tighter
constraints on the possible exoplanet atmospheric models.

###### Acknowledgements.

The authors would like to thank the referee, Jens Hoeijmakers, for useful
comments that have improved the manuscript. G. Mo. has received funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No. 895525. N. C.-B. acknowledges
funding from the European Research Council under the European Union’s Horizon
2020 research and innovation program under grant agreement No. 694513. J.
O.-M. and E. P. were financed by the Spanish Ministry of Economics and
Competitiveness through grants PGC2018-098153-B-C31. G. Mi. e G. C.
acknowledge contribution from ASI-INAF agreement 2021-5-HH.0.

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