Astronomy&
Astrophysics

A&A, 683, A242 (2024)
https://doi.org/10.1051/0004-6361/202347858© The Authors 2024

3D non-LTE modeling of the stellar center-to-limb variation fortransmission spectroscopy studies⋆
Na I D and K I resonance lines in the Sun
G. Canocchi1 , K. Lind1 , C. Lagae1 , A. G. M. Pietrow2 , A. M. Amarsi3 , D. Kiselman4 ,O. Andriienko4 , and H. J. Hoeijmakers5
1

Department of Astronomy, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden
e-mail: gloria.canocchi@astro.su.se
2
Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
3
Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden
4
Institute for Solar Physics, Dept. of Astronomy, Stockholm University, Albanova University Center, 106 91 Stockholm, Sweden
5
Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, 221 00 Lund, Sweden
Received 1 September 2023 / Accepted 8 December 2023
ABSTRACT
Context. Transmission spectroscopy is one of the most powerful techniques used to characterize transiting exoplanets, since it allows

for the abundance of the atomic and molecular species in the planetary atmosphere to be measured. However, stellar lines may bias thedetermination of such abundances if their center-to-limb variations (CLVs) are not properly accounted for.
Aims. This paper aims to show that three-dimensional (3D) radiation hydrodynamic models and the assumption of non-local thermodynamic equilibrium (non-LTE) line formation are required for an accurate modeling of the stellar CLV of the Na I D1 and K I
resonance lines on transmission spectra.
Methods. We modeled the CLV of the Na I D1 and K I resonance lines in the Sun with 3D non-LTE radiative transfer. The synthetic
spectra were compared to solar observations with high spatial and spectral resolution, including new data collected with the CRISP
instrument at the Swedish 1-m Solar Telescope between µ = 0.1 and µ = 1.0.
Results. Our 3D non-LTE modeling of the Na I D1 resonance line at 5896 Å and the K I 7699 Å resonance line in the Sun is in good
agreement with the observed CLV in the solar spectrum. Moreover, the simulated CLV curve for a Jupiter-Sun system inferred witha 3D non-LTE analysis shows significant differences from the one obtained from a 1D atmosphere. The latter does indeed tend tooverestimate the amplitude of the transmission curve by a factor that is on the same order of magnitude as a planetary absorption depth(i.e., up to 0.2%).
Conclusions. This work highlights the fact that to correctly characterize exoplanetary atmospheres, 3D non-LTE synthetic spectra
ought to be used to estimate the stellar CLV effect in transmission spectra of solar-like planet hosts. Moreover, since different spectral
lines show different CLV curves for the same geometry of the planet-star system, it is fundamental to model the CLV individually foreach line of interest. The work will be extended to other lines and FGK-type stars, allowing for synthetic high-resolution spectra to
mitigate the stellar contamination of low-resolution planetary spectra, for example, those drawn from JWST.
Key words. line: formation – line: profiles – techniques: spectroscopic – Sun: atmosphere – planets and satellites: atmospheres –planet–star interactions

1. Introduction
Transmission spectroscopy is a powerful tool used to measure
the abundances of atomic and molecular species in the atmosphere of transiting exoplanets, especially close-in gas giants or
hot Jupiters (Czesla et al. 2015; Casasayas-Barris et al. 2017).
When a planet transits across the stellar disk of its host star, therecorded stellar spectrum includes the planetary contribution.
The latter can be detected as an extra absorption at specific wavelengths corresponding to the species that are present in the upper
layers of the planetary atmosphere, through which the starlightis filtered during the transit. This technique works by subtracting
the light recorded out of transit from that recorded during thetransit, in order to detect the extra absorption due to the planet.
In practice, a high spectral resolution (i.e., R ≈ 100 000) is⋆
The SST datasets of the Na I 5896 Å and K I 7699 Å lines areavailable at the CDS via anonymous ftp to cdsarc.cds.unistra.fr(130.79.128.5) or via https://cdsarc.cds.unistra.fr/
viz-bin/cat/J/A+A/683/A242

required to probe such weak features, which are usually on theorder of only 0.01–0.1%. Therefore, high-resolution ultra-stable
spectrographs such as HARPS (Mayor et al. 2003), HARPS-N(Cosentino et al. 2012), CRIRES+ (Kaeufl et al. 2004; Dorn et al.
2014, 2023), PEPSI (Strassmeier et al. 2015), or ESPRESSO(Pepe et al. 2021) are needed for this kind of research.
To date, the most extensively studied atomic lines in transit
spectroscopy are the lines of some alkali metals: the Na I
resonance doublet at 5890 Å and 5896 Å, and the near-infrared
K I resonance line at 7699 Å. High-resolution studies haveallowed for the detection of sodium in the atmospheres ofseveral giant exoplanets, which are either hot Jupiters or
Neptunes, such as WASP-166b (e.g., Seidel et al. 2020),
WASP-121b (e.g., Hoeijmakers et al. 2020; Cabot et al. 2020;
Seidel et al. 2023), WASP-49b (e.g., Wyttenbach et al. 2017),
WASP-69b (e.g., Casasayas-Barris et al. 2017), WASP-76b (e.g.,
Seidel et al. 2019), KELT-9b (e.g., Hoeijmakers et al. 2019),
KELT-20b (e.g., Casasayas-Barris et al. 2019), HD 189733b(e.g., Wyttenbach et al. 2015; Yan et al. 2017), and HD 209458b
A242, page 1 of 23

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Canocchi, G., et al.: A&A, 683, A242 (2024)(e.g., Charbonneau et al. 2002; Snellen et al. 2008). The latteris the first exoplanet where an atomic species was detected forthe very first time by Charbonneau et al. (2002). These authors
detected neutral sodium through in-transit excess absorption
in the Na I D1 and D2 lines from observations with the Space
Telescope Imaging Spectrograph (STIS) on board the Hubble
Space Telescope (HST). Afterward, many other studies also
from ground-based telescopes were carried out and several otherspecies were detected including neutral potassium through the7699 Å resonance line (Casasayas-Barris et al. 2021). The firstdetections of K I occurred for XO-2b (Sing et al. 2011) and theeccentric hot Jupiter HD 80606b (Colón et al. 2012).
An important effect to consider when analyzing spectra withthis technique is the center-to-limb variation (CLV) of the stellarline profiles. Indeed, Casasayas-Barris et al. (2020) stressed
the importance of correctly accounting for this effect on theinterpretation of the spectra of HD 209458. The CLV describes
the change in both strength and shape of the lines forming in thestellar atmosphere as the observer’s line of sight moves from thecenter to the edge of the stellar disk.
By going toward the limb, the spectral lines are formed inhigher layers of the photosphere. Different spectral lines areaffected in different ways owing to their sensitivities on depthdependent quantities such as temperature, electron pressure, andvelocity, among others. This CLV effect is closely related to continuum limb darkening: by going toward the limb, the higher,cooler layers of the photosphere are exposed to the observer,decreasing the observed surface brightness. The CLV of stellarlines can affect the correct determination of elemental abundances in the planetary atmospheres if not properly corrected foror even mimic planetary signals, thus leading to a false detection; this is because, in some cases, it is on the same orderof magnitude as the planetary absorption feature (Czesla et al.
2015). During a transit, as the planet progresses, it obscures andblocks the light of parts of the stellar disk that contribute differently to the observed stellar line profile, which then varies among
the different transit phases. Simply subtracting the out-of-transit
light without considering the angle dependence will therefore
introduce an error.
Although the CLV effect is recognized as an important andcritical factor (e.g., Müller et al. 2013), it is often modeled byadopting the simple assumption of local thermodynamic equilibrium (LTE) and one-dimensional (1D) plane-parallel stellar
atmospheres (e.g., Casasayas-Barris et al. 2019; Chen et al.
2020), or sometimes with 1D non-LTE models (e.g., Yan et al.
2017; Casasayas-Barris et al. 2017; Morello et al. 2022; Sicilia
et al. 2022). However, several studies have shown that the commonly used 1D LTE models fail to reproduce spatially resolved
observations of the solar disk. Specifically, the CLV of several
atoms, such as Ca, C, O, Al, and Fe lines, has been investigatedin the Sun, finding that they are poorly modeled in 1D LTE,
with errors on the line strength that range from strong (100%) tomoderate (10%) at µ = 0.2 (e.g., Pereira et al. 2009; Lind et al.
2017; Nordlander & Lind 2017; Bjørgen et al. 2018; Amarsi et al.
2018a, 2019; Bergemann et al. 2021; Pietrow et al. 2023b,a).
The Na I doublet and the K I 7699 Å resonance lines arestrongly affected by departures from LTE; in particular overrecombination of the neutral species and photon losses in theresonance lines themselves (e.g., Lind et al. 2011; Reggiani et al.
2019). As such, 3D non-LTE modeling should be used for thehighest accuracy.
The Sun is an excellent test bench for observing andstudying the CLV of spectral lines (e.g., Pierce & Slaughter 1977; Neckel & Labs 1994). Several solar atlases were
A242, page 2 of 23

recently produced and made publicly available, for example, the
NSO/Kitt Peak FTS (Stenflo 2015) and the Gregory Coudé Telescope SS3 (Ramelli et al. 2017, 2019) atlases. There, we cansee that the intensity of many spectral lines as a function of
wavelength at different viewing angles (µ = cos(θ), with θ theheliocentric angle1 ) from the limb (i.e., µ = 0) to the solar disk
center (i.e., µ = 1.0) is recorded. Specifically, the FTS atlasextends in the range 4084–9950 Å, but it samples only two positions along the solar disk: one at the center and one at the edge(µ = 0.145). Conversely, in the SS3 atlas the intensity at ten different angles, spanning from µ = 0.1 to µ = 1.0 in steps of 0.1,
is recorded but the covered spectral range is narrower (4384–
6610 Å). Moreover, both these atlases show a noisy continuum(about 3%) and are not corrected for telluric lines. Systematicand significant uncertainties in the continuum of different atlasesof the solar disk-center intensity were also investigated in Doerret al. (2016). An updated high-resolution atlas has been compiledby the Institut für Astrophysik Göttingen (IAG), which includes
the CLV of several lines in the wavelength range of 4200–8000 Å(Ellwarth et al. 2023). Nevertheless, it does not reach the very
limb, with µ = 0.2 being the farthest angle sampled.
The CLV of stellar lines in stars other than the Sun has only
been studied very recently. Dravins et al. (2017, 2018) have tested
how to resolve the surface of a K-type star, namely HD 189733A,
using a transit of its exoplanet as a probe and performing differential spectroscopy of a strong Fe I line between different transit
phases. This technique requires high-resolution spectra as wellas very precise observations, but it remains a very promising
candidate for spatially resolving the surface of many other stars
in the near future. However, it should be noticed that the technique relies on the assumption that the planetary atmosphere
contribution for that specific Fe I line is null.
In this work, we study the 3D non-LTE CLV of absorption
lines of neutral sodium and potassium in the Sun, and explore theimplications for transmission spectroscopy studies. In Sect. 2, we
describe the observational data used in this work. Complementing data from the literature, we have carried out observations for
the Na I D1 line at 5896 Å and the K I line at 7699 Å between µ =0.1 and 1.0, obtained with the CRisp Imaging SpectroPolarimeter (CRISP; Scharmer et al. 2008) on the Swedish 1-m Solar
Telescope (SST; Scharmer et al. 2003). In Sect. 3, we describe
our 3D non-LTE calculations, as well as our 3D LTE, 1D nonLTE, and 1D LTE calculations. In Sect. 4, we use these models
to quantitatively analyze the solar observations, and demonstrate
the impact of 3D non-LTE modeling on a simulated CLV transmission light curve in a Sun-Jupiter system. Finally, in Sect. 5,
we summarize our conclusions and discuss future perspectives.

2. Observations and data processing
For this work, we employed data from the recently publishedsolar intensity atlas obtained with the Fourier Transform Spectrograph (FTS; Schäfer et al. 2020) at the IAG Vacuum Vertical
Telescope (VVT), hereafter referred to as IAG FTS CLV Atlas,
as well as observations from the SST/CRISP. The spectra fromthe IAG FTS CLV Atlas show a very high spectral resolutionof about R ≈ 700 000 at λ = 6000 Å, or ∆ν = 0.0024 cm−1
(Reiners et al. 2023) but they have not been corrected for
telluric lines from water (H2 O) and molecular oxygen (O2 ).
θ is defined as the angle between the outward normal of the stellar
atmosphere and the vector pointing from the center of the star toward
the observer.
1

Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. 1. Full-disk image of the Sun on 29 April 2023 in the AIA 171 Å
filter (Lemen et al. 2012). The black squares show all telescope pointings used during the observation campaigns, spanning from the solar
north pole to disk center.

Therefore, the regions affected by telluric lines were masked
out. The IAG Atlas spans from µ = 0.2 to 1.0, in step of0.1, with additional spectra at the following viewing angles:µ = 0.35, 0.95, 0.97, 0.98, 0.99, with a total of 14 µ-angles
covering the wavelength range 4200–8000 Å. These data aredescribed in detail in Ellwarth et al. (2023).
For comparison and completeness, we also took observations
with the CRISP instrument at the SST of the Na I D1 resonanceline at 5896 Å and the K I 7699 Å resonance line. The datasets forthese are described in Sect. 2.1, along with the reduction and calibration processes. A comparison between the two solar datasets,
the IAG Atlas and the SST data, is presented in Sect. 2.2.
2.1. SST dataset

We obtained the data for the Na I D1 and K I lines in two campaigns, hereafter referred to as Campaign I and Campaign II, butfollowing the same procedure. We pointed the telescope at several distinct positions on the disk, targeting quiet regions (i.e.,
without activity), from the solar north pole to the center, recording data for a time period of between 14 and 26 min at eachposition (i.e., µ-angle). Since the tracking was active for theentire observation time, this method allows us to perform temporal averages in order to correct for p-modes and oscillations. Thisis a well-established technique used in several previous studies(e.g., Pierce & Slaughter 1982; Ding & Fang 1991; Pereira et al.
2009; Lind et al. 2017). For the Na I D1 line, we pointed the telescope at µ = 0.2, 0.4, 0.6, 0.8 and 1.0, going along the meridianfrom the solar north pole to the disk center in order to avoid
Doppler shifts from rotation. For the K I line, we also pointed
at µ = 0.9, but the data from this pointing were not reduced in
the end. The pointing positions are highlighted in Fig. 1. Forµ = 0.2, 0.4 and 0.6, a wider µ range is covered by each pointing,due to the large Field of View (FoV) of the cameras. The FoV iscircular, with a diameter of about 87′′ on the solar disk.
In addition to Na I D1 and K I 7699 Å line, we also observed
the photospheric magnetically sensitive line of Fe I at 6173 Å inorder to check the level of solar activity during measurements.
This is the only line for which the polarimetric signal was measured. A summary of the data is reported in Table 1 and thespecific wavelength points sampled for each line are shown in
Table 2. These positions are given relative to their line center
and, especially for the Na I D1 line, they are not all symmetricallysampled2 .
The observations were carried out under variable seeing
conditions. This is acceptable because our first goal is to takelarge-scale averages along the FoV and use the fine-scale structure only for the alignment. Indeed, the only constraint was that
there had to be at least one frame with high enough seeing toalign it to co-temporal SDO/HMI observations, as explained
hereunder. Campaign I was performed at the end of August
2022, and specifically, the Na I D1 and Fe I 6173 Å data weretaken between 08:15 and 10:41 UT on 27 August 2022 with
SST/CRISP, as shown in Table 1. We obtained one scan at fiveµ-angles on the same day. Just before the start of Campaign I,
new detectors were installed at the SST, with a larger FoV compared to the previous set-up (1.5 ′ radius circular compared to
the 1′ × 1′ with the old detectors). The old detectors were from
Sarnoff and the new ones are called “Ximea CB262RG-GPX8G3”. The new detectors also allowed us to solve an issue
with the back-scattering light in the red part of the spectrum.
The data of Campaign I were the first to be taken with the newdetectors.
Campaign II occurred at the end of April 2023, between
April 27–30 and we were not able to get one scan in one day
due to a very variable seeing. Therefore, different pointings were
performed each day depending on the seeing conditions (seeTable 1), but the data with the best seeing conditions were takenin the morning of 29 and 30 April 2023 and so these are the K I
data analyzed in this work. During this campaign we observed
the K I 7699 Å and Fe I 6173 Å lines, obtaining in total two scansat six µ-angles (however, as previously mentioned, the data forµ = 0.9 are not used in this work).
We reduced the data of both Campaigns with the SSTRED
pipeline (de la Cruz Rodríguez et al. 2015; Löfdahl et al. 2021),
which has been specifically designed to process the data from the
SST. It corrects for dark and flat fields but also includes polarimetric calibrations and image restoration that removes opticalaberrations due to turbulence in the atmosphere. This step is performed by the Multi-Object Multi-Frame Blind Deconvolution(MOMFBD; Löfdahl 2002; Van Noort et al. 2005) algorithm,
which however can fail if the seeing conditions are poor (i.e.,Fried parameter, r0 ≤ 5 cm). However, this is not the case withrespect to our observations.
The SST pointing and tracking are good but in order to get avery precise estimate of the µ-angle corresponding to each pixelof the FoV, we performed an additional aligning step. Several
studies make use of cross-correlation algorithms to align theirdata (e.g., Cauzzi & Reardon 2012) but they only work wellwhen activity features like sunspots are clearly distinguishable in
the FoV. For quiet Sun observations, this is not usually the case,especially at the limb with a lower image contrast. Therefore weuse the same method as Pietrow et al. (2023b), which consists ofmanually aligning the images by looking for patterns among the
granules and bright points that match with SDO/HMI continuum
images (Lemen et al. 2012) which were enhanced in resolution
and contrast by means of a deep learning algorithm (Díaz Baso &Asensio Ramos 2018). Each HMI image was picked to be within
2

A value is symmetrically sampled if it has a ± symbol in front of it.
It is not if it has only a + or a −.
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Canocchi, G., et al.: A&A, 683, A242 (2024)
Table 1. Summary of the SST/CRISP data.

Observation date, time (UT)Campaign

Line

nP

nλ

Elevation angle (◦ )2022-08-27, 08:15–10:41
2022-08-27, 08:15–10:41
2023-04-27, 17:23–18:30
2023-04-27, 17:23–18:30
2023-04-28, 13:17–13:38
2023-04-28, 13:17–13:38
2023-04-29, 07:52–09:29
2023-04-29, 07:52–09:29
2023-04-30, 08:09–09:30
2023-04-30, 08:09–09:30

I
I
II
II
II
II
II
II
II
II

Na I D1 5896 Å
Fe I 6173 Å
K I 7699 Å
Fe I 6173 Å
K I 7699 Å
Fe I 6173 Å
K I 7699 Å
Fe I 6173 Å
K I 7699 Å
Fe I 6173 Å

5 (µ = 0.2, 0.4, 0.6, 0.8, 1.0)5 (µ = 0.2, 0.4, 0.6, 0.8, 1.0)
3 (µ = 0.2, 0.8, 0.9)3 (µ = 0.2, 0.8, 0.9)
1 (µ = 1.0)
1 (µ = 1.0)
4 (µ = 0.4, 0.6, 0.9, 1.0)
4 (µ = 0.4, 0.6, 0.9, 1.0)
4 (µ = 0.2, 0.4, 0.6, 0.8)4 (µ = 0.2, 0.4, 0.6, 0.8)

39
11
27
11
27
11
27
11
27
11

36.8
36.8
34.8
34.8
72.8
72.8
18.6
18.6
29.2
29.2

Notes. The columns show the date and time of observations, the campaign when they were taken, the observed line, the number of pointings in
the scan (nP), the number of wavelength points (nλ), and the mean value of the solar elevation angle during observations.

Table 2. FPI tuning positions for each observed line.

Na I D1 5896 Å
−2000, −1740, −1480, −1220, −960, −700, −550, −440,−374, ±300, ±248, ±196, −144, ±118, ±92, ±78, ±66, ±40,
±14, +140, +165, +381, +500, +578, +656, +812, +890,+968, +1070 and 0 mÅ.
K I 7699 Å±1380, ±1150, ±920, ±690, ±460, ±230, −178, ±138, ±115,±92, ±69, ±46, −23, +161 and 0 mÅ.
Fe I 6173 Å±450, ±175, ±105, ±70, ±35 and 0 mÅ.
Notes. The tuning positions are relative to a (daily) calibration on theline at disk center.

half a minute of the highest seeing-quality scan within the firstten scans of each pointing. In this way, we can straightforwardly
obtain absolute solar coordinates with respect to the disk center.
After that, we assigned a µ value (see Eq. (1)) at each pixel in theimages by simply using the Pythagorean theorem to calculate thedistance of each pixel from the disk center (r):
s
!2
r
µ= 1−
,
(1)
R⊙
where R⊙ is the angular radius of the Sun as seen from the Earthin arcseconds. In Campaign I the solar radius is R⊙ = 949.5′′ andit is 953′′ in Campaign II3 .
Then, as previously mentioned, for the pointing at µ = 0.2,
0.4, and 0.6, a wider µ range is covered due to the large FoVof the cameras. Consequently, we were able to extract averageline profiles for more µ-angles. Specifically, from the pointing atµ = 0.2, we obtained the line profiles at µ = 0.1, 0.15, 0.2, and0.25. Similarly, from the pointing at µ = 0.4, we got the profiles
at µ = 0.3, 0.35, 0.4 and 0.45. Lastly, from the pointing at µ = 0.6
we could also extract the profiles at µ = 0.55. Finally, we obtainaveraged line profiles for each spectral line, for the following 12µ-angles: 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.55, 0.6, 0.8,
3

From https://ssd.jpl.nasa.gov/horizons/app.html#/

A242, page 4 of 23

and 1.0. These are computed by averaging over time and overspace all the pixels in the frames that are within the selected µ
value ±0.02. The CRISP image scale is 0.044′′ pixels−1 .
After that, we performed another calibration of the disk center intensity (i.e., the µ = 1.0 line profile) using the ISPy4 library(Díaz Baso et al. 2021) by fitting several wavelength points of
the averaged disk center profile to the IAG Atlas convolved withCRISP instrumental profile5 . The calibration routine was modified because it usually employs the solar atlas by Neckel & Labs
(1984). We then applied the intensity and wavelength calibration
factors to all the other µ-angle line profiles. This is again thesame method employed in Pietrow et al. (2023b).
The resulting limb darkening curves and line profiles of the
Na I D1 line taken in Campaign I and of the K I line of Campaign
II are shown in Figs. A.2 and A.3, respectively.
2.2. Comparison of solar data

In this section, we compare our new SST/CRISP observations
to the IAG FTS Atlas through the measurement of equivalent
widths (Wλ ) at different µ-angles. First of all, we convolved theAtlas with CRISP instrumental profile in order to degrade it to
the same resolution as our SST data, which is about R ≈ 160 000
at 7699 Å. Such convolution is shown in Figs. 2 and 3 where theline profile of the Na I D1 and K I line, respectively, at disk center(i.e., µ = 1.0) is compared between the two datasets. From thesefigures, it is apparent that the agreement between the two datasetsat disk center is excellent, and specifically, it is within 2.5% for
the K I line and within 3% for the Na I D1 line.
Next, we computed the Wλ of all the Na line profiles at different µ-angles in the wavelength range 5895.21 – 5896.291 Å(shaded region in Fig. 2) in order to exclude the blending on theright wing due to an uncorrected telluric line that is clearly visible in Fig. 2. It is worth noting that we are not integrating theentire line profile of the Na I D1 line up to the continuum. The
Wλ of the Na I D1 line as a function of the µ-angle is shown in
Fig. 4.
From the latter, it is apparent that the Wλ of the SST datais slightly lower than the Wλ of the corresponding IAG profile. Even with this small difference linearly increasing toward4

https://github.com/ISP-SST/ISPy
The CRISP instrumental profile can be calculated with the “spec”
module of ISPy.
5

Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. 2. Intensity profile of the Na I D1 line at disk center. Top: comparison of the Na I D1 line profile at disk center between the SST/CRISP
data (black dots) and the IAG Atlas convolved with the instrumental
profile (red dashed line). The original profile from the IAG Atlas isalso plotted with a black solid line. The grey-shaded region highlights
the selected wavelength range for Wλ computation. A telluric spectrum(Hinkle et al. 2000) is overplotted as a grey line in the top panel, to showthat there is negligible contamination in the Wλ window. Bottom: difference between the convolved IAG Atlas and the SST data in percentage.

Fig. 3. Intensity profile of the K I resonance line at disk center. Top:
comparison of the K I 7699 line profile at disk center between the
SST/CRISP data (black dots) and the IAG Atlas convolved with the
instrumental profile (red dashed line). The original profile from the IAG
Atlas is also plotted with a black solid line. The grey-shaded region
highlights the selected wavelength range for Wλ computation. A telluric
spectrum (Hinkle et al. 2000) is overplotted as a grey line in the top
panel, to show that there is negligible contamination in the Wλ window.
Bottom: difference between the convolved IAG Atlas and the SST datain percentage.

the limb, the agreement between the two datasets is quite good,and specifically, it is within 0.3% at disk center (µ = 1.0) andabout 2.9% at the limb (µ = 0.2). It is worth noting that the
SST pointings are more precise than the IAG, especially at thelimb because of the method we used to obtain the µ-position ofeach pixel of the frames, that is the alignment with simultaneous
space-based data (i.e., SDO/HMI images) which translates into
sub-arcseconds precision at the limb. This procedure is accurately described in Sect. 2.1. Conversely, the limitations of theFig. 4. Comparison of the SST/CRISP data (black dots) with the IAG
FTS CLV Atlas (red squares) convolved with CRISP instrumental profile. Top: equivalent width of the Na I D1 line as a function of theµ-angle. Bottom: difference between the IAG and the SST data in percentage.

Fig. 5. Comparison of the SST/CRISP data (black dots) with the IAG
FTS CLV Atlas (red squares) convolved with CRISP instrumental profile. Top: equivalent width of the K I line at 7699 Å as a function ofthe µ-angle. Bottom: difference between the IAG and the SST data inpercentage.

IAG Atlas come from the use of fibers, which result in increasingly large uncertainties in the µ-positions closer to the limb, as
is apparent from the horizontal error bars in Fig. 4.
The same calculations were performed for the K I line, computing the Wλ in the wavelength range 7698.495–7699.415 Å, ashighlighted in Fig. 3. The resulting CLV of this line is shown in
Fig. 5, where Wλ at different µ-angles are compared between the
IAG and SST data. Also in this case the Wλ of the SST data issystematically lower than the IAG data but overall they agree
within 1–2% at all µ-angles. Differences in continuum fitting
may have contributed to this difference in line strength, but weemphasize that the atlases everywhere agree within the estimatedpointing errors.
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With regards to the uncertainties on the IAG datapoints,
the horizontal error bars on µ-angles were provided by
Ellwarth (priv. comm.) whereas the vertical error bars on equivalent widths were estimated from Cayrel’s formula (Cayrel 1988;
Cayrel et al. 2004). They are reported in Table A.2. The pointing of the SST is very precise; we estimated the horizontal error
bar on µ from the aligning procedure of SST frames with SDO
images described in Sect. 2.1. Indeed, the geocentric orbit of
SDO leads to a potential error of maximum 1.7′′ . In µ-space,this translates into a negligible value at disk center (about 10−6 )but becomes significant at the limb, reaching 0.02 at µ = 0.1.
The vertical error bars on Wλ of the SST data correspond to thestandard deviation of the individual pointings (σµ ) added to an
estimated error due to continuum placement (σcontinuum ), which
represents the main source of uncertainty for this dataset. Theseerrors were computed by vertically shifting the continuum within
1% and estimating how the Wλ change. Therefore the total error
becomes:
σ2tot = σ2µ + σ2continuum .

(2)

These values are shown in Table A.1.

3. Radiative transfer calculations
3.1. Synthetic spectra

Fig. 6. Ratio of the NLTE to LTE equivalent width of K I 7699 Å fora single Stagger snapshot. In the up-flowing granules, overionization
makes the line weaker in NLTE, whereas in the intergranular lanes the
NLTE Wλ is stronger.

The synthesis of the solar spectrum was performed with theradiative transfer (RT) MPI-parallelised code Balder (Amarsi
et al. 2018b), an extensively modified version of the Multi3d
code (Botnen & Carlsson 1999; Leenaarts & Carlsson 2009),
which can solve the restricted non-LTE problem for trace elements in 1D and 3D, for user-specified atomic elements. Balder
employs the BLUE (Amarsi et al. 2016a,b) opacity package, which
computes the line opacity on a grid of wavelength, density, andtemperature. A full 3D non-LTE analysis of several elements in
the Sun, including Na and K, was performed with the same codeby Asplund et al. (2021).
For the model atmosphere, a 3D radiative-hydrodynamics
simulation with the Stagger-code (Nordlund & Galsgaard
1995; Stein & Nordlund 1998; Collet et al. 2011) was used.
Stagger employs the “box-in-a-star” set-up, meaning that it
simulates a rectangular box, vertically open, with periodic horizontal boundaries, located in the upper part of the solar photosphere, without including any chromospheric layer. In this work,
we simulate the quiet Sun, without including magnetic fields6 .
The 3D model is set on a Cartesian grid of size 240×240×240,
corresponding to a physical size of 8×8×4 Mm. The verticaldimension is characterized by non-equidistant spacing in orderto obtain a higher resolution at the regions of interest and toresolve the strong temperature gradient of the photosphere. In
order to decrease the computational time for the spectral synthesis, all the atmospheric snapshots were resized to a resolution of
60×60×101, from their original value of 2403 . This approach has
been already tested in previous studies, such as Lind et al. (2017)and Lagae et al. (2023).
Ten snapshots were used, from the same simulations discussed already in several earlier analyses of the Sun (Amarsi
et al. 2018a, 2019, 2020, 2021; Asplund et al. 2021). We notethat this simulation is more recent than that in the Stagger-grid(Magic et al. 2013), with improvements to the opacity binningscheme (Collet et al. 2018) and extending for over one day ofsolar time. The mean effective temperature of these snapshots is
T eff = 5770 ± 5 K7 .
An example of the surface image of one of the snapshotsof the solar photosphere selected for this work is shown in
Fig. 6. It represents the 3D non-LTE to 3D LTE ratio of Wλof the K I 7699 Å line, showing that the non-LTE effects varywith the convection pattern: under-population in the up-flowing
granules and the opposite effect, overpopulation, in the intergranular lanes. Such a variation is expected and is due to themuch steeper temperature gradients of the granules compared tothose of the intergranular lanes. The granulation pattern of thesolar photosphere, which arises from the convective motion of
the hot plasma and gas, is clearly recognizable in this figure.
This is a natural 3D effect that leads to spectral line broadening and line asymmetries (e.g., Dravins et al. 1981). In the 3D
non-LTE models, the broadening effects of these temperature
inhomogeneities and convective velocity structures are naturally
included; therefore, no extra broadening is required. In 1D, however, the line broadening is instead roughly accounted for by thetunable parameters microturbulence (vmic ) and macroturbulence.
For the Sun, a microturbulence of 1.0 km s−1 is usually adopted
in the literature, and this was the one employed in this work forthe 1D simulations, independent of depth or viewing angle. For
the macroturbulence velocity (vmac ) a value of 3.5 km s−1 wasused. The 1D synthetic spectra have been computed with the
MARCS (Gustafsson et al. 2008) solar atmosphere.
The synthetic spectra were computed at three different abundances for each snapshot. Then the line profile was spatially andtemporally averaged, using all ten snapshots. After that, the diskcenter line profile was fitted to the observed data via a standard
χ(2) minimization routine. Linear interpolation between models
computed with several values of abundance was applied. In thisway, the elemental abundance was fit to the disk-center spectra
and then fixed to this value for the other µ-angles. Finally, we
computed the line profiles at the other µ-angles with the bestfit model and from those we calculated the Wλ of the synthetic

6

7

Stagger can compute M-RHD including magnetic fields, but we do
not use this feature here.
A242, page 6 of 23

The adopted value is close to the nominal T eff = 5772 K (Prša et al.
2016).

Canocchi, G., et al.: A&A, 683, A242 (2024)
Table 3. Atomic data of the sodium and potassium lines considered in the CLV analysis, as implemented in the model atoms of Lind et al. (2011)
and Reggiani et al. (2019).

λ (Å)

Transition

5896
5688
6154
6160
7699

3s2 S1/2 −3p2 P01/2
3p2 P03/2 −4d2 D5/2
3p2 P01/2 −5s2 S1/2
3p2 P03/2 −5s2 S1/2
4s2 S1/2 −4p2 P01/2(a)∆E (cm−1 )16 956.17
17 575.36
16 244.50
16 227.30
12 985.18(b)log (g f )–0.194–0.452–1.547–1.246–0.176(c)σ (a.u.)

(d)

α

407
1955
–
–
486

0.237
0.327
–
–
0.232

(e)

λint (Å)

5895.2100-5896.2910
5688.0546-5688.3546
6154.1053-6154.3453
6160.6270-6160.8670
7698.4953-7699.4153

Notes. (a) ∆E is the energy difference between the levels of the transition, from the NIST database (https://physics.nist.gov/PhysRefData/
ASD/lines_form.html; Ralchenko et al. 2005). (b) log (g f ) are from the VALD database (http://vald.astro.uu.se/; Ryabchikova et al.
2015), where f is the oscillator strength, or transition probability and g is the statistical weight. (c) σ is the broadening cross-section for elastic
collisions with hydrogen at a relative velocity of 104 m s−1 . (d) α is the exponent with which the cross-section σ varies with velocity, that is v−α(Anstee & O’Mara 1995). (e) λint shows the wavelength windows used in the integration for the equivalent width computation.

spectra by direct integration of the line within wavelength ranges
that were considered blend-free (see Sect. 4.1). For the study of
the Na I D1 and K I 7699 Å resonance lines, we also convolved
the synthetic spectrum with the CRISP instrumental profile inorder to compare Wλ with the observed SST data (see Sect. 4.1).
3.2. Model atoms

Several studies have shown that both Na I and K I atoms areaffected by strong non-LTE effects (i.e. over-recombination of
the ground state), meaning that non-LTE line formation isrequired to correctly reproduce observations. Lind et al. (2011)
has shown that Na stellar abundances can be overestimated by up
to 0.5 dex if calculations in 1D LTE are performed for saturatedlines of metal-poor dwarfs and giants ([Fe/H] = −2). In the caseof the Sun, they found a difference of about 0.02 dex between the
1D LTE and non-LTE inferred abundance. Reggiani et al. (2019)
showed that the abundance of K in the Sun is overestimated byabout 0.3 dex in 1D LTE compared to 1D non-LTE.
The non-LTE of any line is generally determined by thechange in formation depth, which is given by the departure coefficient of the lower level, and the change in line source function(S l ), whose ratio with respect to Bν is equal to the ratio between
the upper and lower level departure coefficients (neglecting stimulated emission). For the resonance lines (Na D and K I 7699 Å),
it has been seen that S l resembles that of a pure scattering line ina two-level atom at all depths, meaning that S l is equal to the
profile-averaged mean intensity ( J¯ϕ ) and strongly subthermal,that is J¯ν < Bν . Over-recombination and over-de-excitation (photon loss) overpopulate the ground levels of Na I and K I comparedwith LTE, pushing the mean formation depth outwards in theatmosphere. The combined effect is a stronger line in non-LTE
compared to LTE, and specifically a deeper line core. For moredetails about the non-LTE effects of Na I, we refer the reader toSect. 3.1 of Lind et al. (2011), and for K I to Sect. 4.3 of Reggiani
et al. (2019).
The model atom used in Balder for the Na atom is the one
developed and tested in Lind et al. (2011), characterized by astructure of 23 levels in total: 22 energy levels for Na I and theionized sodium (Na II) continuum. The lines of the Na doublet
originate from the transition between the levels 3s–3p1/2 and3s–3p3/2 . Both fine and hyper-fine structures are accounted forin the model atom. The latter describes both the radiative andcollisional transitions among the energy levels and, in particular,
detailed quantum mechanical calculations are used for the
inelastic collisions of Na respectively with electrons (Park 1971;

Allen & Rossouw 1993; Igenbergs et al. 2008) and hydrogenatoms (Belyaev et al. 2010; Barklem et al. 2010). Regarding
the latter, in this work, the impact of rate coefficients for thecollisions with neutral hydrogens from Kaulakys (1991) was
also explored. Indeed in the model atom developed in Lind et al.
(2011), these rates were only considered above the ionic limit,
but Amarsi et al. (2018a, 2019) argued that it could be necessary
to add these rate coefficients even below the ionic limit, at least
when asymptotic models such as those of Barklem (2016) orBelyaev (2013) are used. The results of Amarsi et al. (2018a,2019) for O and C, as well as Bergemann et al. (2019) for Mn,
suggest that doing so leads to better agreement of synthetic
spectra with observational data.
The model atom of the potassium atom was recently developed by Reggiani et al. (2019), who used the most up-to-date andaccurate atomic data, for both the radiative and collisional transitions (e.g., Belyaev & Yakovleva 2017; Yakovleva et al. 2018;
Barklem et al. 2017). The atomic model consists of 19 energy
levels, including the ionized potassium (K II) continuum. For thecollisions with hydrogen atoms, they adopted rate coefficientsfrom Yakovleva et al. (2018), based on the asymptotic models of
Barklem (2016). To these data, they also added those from thefree electron model of Kaulakys (1991), following Amarsi et al.
(2018a, 2019). From Reggiani et al. (2019), the K+H collisions
significantly affect the abundance determination. In particular,
they produce an abundance difference of about 0.03 dex in the
Sun if not taken into account. In the same work, they have shownthat the resonance line at 7699 Å has wings insensitive to departures from LTE, but a core that is strongly affected by non-LTE,
as it is much deeper than in LTE.
Both these model atoms were employed for full 3D non-LTE
calculations in a recent paper by Asplund et al. (2021) to study
and update the chemical composition of the Sun. However, thelines selected to compute abundances are not the same as thoseanalyzed in this work. The atomic data of the Na lines and K I
resonance line studied here are summarized in Table 3.

4. Analysis of the results
The CLV of solar lines represents a crucial test for line formation modeling since it is an excellent probe of the atmosphericstructure. Indeed, the light emitted at different µ-angles probe
different heights of the solar atmosphere: shallower layers close
to the limb and deeper layers at the disk center (µ = 1.0). The
3D radiative transfer shows that the profiles of the lines emitted
from different regions of the solar surface vary in both shape
A242, page 7 of 23

Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. 7. Center-to-limb variation of the solar Na I lines. Comparison of the equivalent widths of the IAG Atlas (filled squares) and the SST data
(open circles) with several different model atmospheres and atomic data, colored as in the figure’s legend.

and strength. This is important for transmission spectroscopy
studies when the stellar lines are distorted due to the planet transiting in front of its star and blocking the light from different
regions of the stellar surface. The deformation of the stellar lines
often complicates the interpretation of (chemical) spectral signatures caused by the planet’s atmosphere, even leading to thedetection of false positives for some chemical elements or to
the wrong derived abundance in some cases. Therefore highresolution ground-based spectra need to be corrected for the CLV
signature, using a model that is as realistic as possible.
For this purpose, in Sects. 4.1 and 4.2, we compare the spatially resolved solar observations of the IAG Atlas and of the SST
data with several theoretical models assuming different model
atmospheres (1D and 3D) and line formation (LTE and nonLTE). Furthermore, in Sect. 4.3 we simulate the impact on transit
phase curves for the Sun when the CLV is modeled with different
assumptions in a Sun-Jupiter system.
4.1. Center-to-limb variation of equivalent widths

In this section, we analyze the CLV of the equivalent widthsof four Na I lines and one K I line, presented in Table 3. Intransmission spectroscopy studies only the strong Na I D lines
are used for the detection of sodium in the planetary atmospheres since they are the only lines strong enough to show
a planetary signature. The Na I D2 line at 5890 Å in the IAG
A242, page 8 of 23

Atlas was contaminated by too many telluric lines that were cut
out from the dataset; hence, it was not possible to compare itwith our synthetic spectra. However, we analyzed the CLV ofthree other Na I optical lines, respectively at 5688 Å, 6154 Å and6160 Å, in order to further test our models. The Na I doublet
at 6154 Å and 6160 Å is also commonly used for abundance
analyses of solar-metallicity stars. The CLV of these two lines
was studied by computing the Wλ via direct integration of theline profile over a wavelength region extending ±0.12 Å from theline center, in order to exclude nearby lines from other elements.
For the Na I 5688 Å line a wavelength range of ±0.15 Å wasadopted instead. The wavelength range for integration is shownin the last column of Table 3. For the Na I D1 and K I resonancelines, the same wavelength range used in Sect. 2.2 was chosen
in order to compare the models with both SST and IAG data.
For this reason, the convolution of CRISP instrumental profile
was applied to the synthetic spectra of these two lines. In allcomparisons, the abundance is adjusted to fit the line profile
at disk center, as mentioned in Sect. 3.1. The resulting Wλ as afunction of µ-angles for the Na I lines and the K I line are shownin Figs. 7 and 8, respectively.
From both these figures, it is clear that 1D LTE modeling(dashed red line) heavily fails in describing the CLV of the solarlines, underestimating the line strengths at the limb as muchas 50.1% for the K I and 28.2% for Na I D1 at µ = 0.1, incomparison with the SST data. For the other Na I lines comparedCanocchi, G., et al.: A&A, 683, A242 (2024)

For other stars, we cannot resolve the stellar disk, and therefore
we cannot take observations at different µ-angles to fit the freeparameters of our 1D models with. For this reason, it is important to note how well we can reproduce the solar CLV without
any free parameter with our 3D non-LTE models.
4.2. Center-to-limb variation of line profiles

Fig. 8. Center-to-limb variation of the K I 7699 Å line. Comparison of
the IAG Atlas (filled squares) and the SST data (open circles) with several different model atmospheres and atomic data, colored as in the
figure legend.

with the IAG data at µ = 0.2, the underestimate is as large as 31%
for 5688 Å and about 29% for the 6154 and 6160 Å lines. Using a
3D atmosphere with LTE line formation (solid red line) slightly
improves the CLV but it still considerably under-performs at thelimb, with an equivalent width difference of about 20% for the
Na lines and 37% for the K line.
The observations are much better reproduced by the 1D nonLTE model (dashed blue line), which is in agreement with the
Na I D1 SST data at µ = 0.1 within about 8%. This demonstrates
that the non-LTE effect is a very important feature that must be
taken into account in order to accurately model the spectral lines.
The agreement with the data improves also for the other Na lines,
down to 7.2% for the 5688 Å and about 1% for the 6154 and6160 Å lines, at µ = 0.2. The equivalent width of the 1D nonLTE model is 17.6% lower than the SST data at µ = 0.1 for the
K I resonance line.
A large improvement is obtained by the 3D non-LTE
modeling (solid blue line) at the limb (µ = 0.2) for the K I (6%),
the 5688 Å (4%) line, and the Na I D1 line (2.5%). The CLVof the latter is also in excellent agreement with the SST data,with a difference of only 0.4% at µ = 0.1. Nevertheless, the 3D
non-LTE model slightly overestimates the strength of the other
Na I lines by about 8.5% (6154 Å) and 9.8% (6160 Å); that is,
performing slightly worse than 1D non-LTE. We stress that theslightly better agreement in 1D non-LTE is likely coincidental,
since it is dependent on the various parameters in the 1D models
that can be tuned to reproduce observations, such as vmic andvmac . The imperfect agreement of the 3D non-LTE model close
to the limb can be explained by several uncertain factors in
the modeling, such as the upper layers of the Stagger model
atmosphere, as well as the rates for collisions with H andelectrons in the model atom.
However, it is worth noting that in general the 3D non-LTE
prediction performs extremely well given that it does not rely on
any free parameter. Considering that for other stars we cannot
take spatially resolved observation of their disk, we would not
be able to constrain the parameters for the 1D modeling, like we
can do for the Sun.
Moreover, the CLV of the continuum and other observational diagnostics are not well reproduced by 1D models, suchas MARCS, not even in the Sun, as shown in Pereira et al. (2013).

While for some Na I lines, the 1D non-LTE modeling is predicting a very good CLV in terms of equivalent widths (Sect. 4.1),
the same is not true anymore for what regards the line profiles. The latter are shown in Figs. 9, and 10 for the Na I D1
and K I line, respectively, displaying the intensity of the µpointings in common between the IAG and the SST data, namelyµ = 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, and the µ = 0.1 profile in comparison with the SST profile. The line profiles of the other Na lines
compared with the IAG data only, are shown in Figs. A.4, A.5
and A.6. Moreover, Figs. A.7 and A.8 show the line profiles of
the Na I D1 and K I 7699 Å of the IAG data at original resolution compared with the non-convolved synthetic spectra. In all
the figures, from left to right, the panels display the data withthe best-fit 1D LTE, 1D non-LTE, 3D LTE and 3D non-LTE
synthetic spectra overplotted.
As is apparent from all these figures, the 3D non-LTE modeling is able to consistently reproduce the observed line profiles
from the disk center to the limb. On the other hand, the shape andasymmetries of the line profiles are not well modeled by the 1D
non-LTE spectra, getting worse at the limb even if the disk center intensity matches the data quite well. The LTE spectra instead
fail to correctly reproduce the shape of the lines even at disk center. The Wλ of the observations at different µ-angles are reported
in Tables A.1 and A.2 for the SST and IAG data, respectively. The
best-fitting abundances are reported in Table 4, from which we
notice that the average Na abundance from 3D non-LTE models
is in very good agreement with the value of A(Na) = 6.22 ± 0.03,
derived by Asplund et al. (2021) even if they used a different
line selection. The K I abundance instead, in this work is higher
by about 0.14 dex compared to the Asplund et al. (2021) value.
However, it is important to note that the Na I D and K I resonancelines are not ideal abundance diagnostics as they are saturatedlines. Hence, the values in Asplund et al. (2021) are preferred in
the context of the solar chemical composition.
The reason why the 3D non-LTE modeling overestimates
the line strengths Wλ at the limb in Na I lines (Fig. 7) is not fully
clear and we investigated the impact of collisions with neutralhydrogen (H). As discussed in Sect 3.2, earlier work has argued
that it could be necessary to add rate coefficients from asymptotic models such as those of Barklem (2016) or Belyaev (2013),
to those from the free electron model of Kaulakys (1991). Motivated by this, the Na model atom tested here includes additional
rate coefficients from Kaulakys (1991) for the Na+H collisions,even though the base data are not from an asymptotic model, but
come from detailed quantum mechanical calculations. We foundthat the additional rates have very little influence on the weak6154 and 6160 Å lines, whereas the effect is more noticeable
for the stronger 5688 Å and 5896 Å lines. Specifically, for thelatter the 3D non-LTE model matches the CLV of the equivalent
widths of the SST data much better, as can be seen from Fig. 7.
The comparison with the previous model atom developed in
Lind et al. (2011) is shown in Fig. A.1. With the previous model
atom, the Wλ of the 3D non-LTE model prefers the IAG data,
whereas in the new model atom (the one adopted in this work)
the 3D non-LTE model is in better agreement with the SST
A242, page 9 of 23

Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. 9. Normalized observed (black dots and blue squares) intensity and synthetic (red lines) center-to-limb profiles for the Na I D1 line, where
the numbers above each spectrum correspond to the µ-angle. In the left corner, the calibrated abundance is shown. Spectra for µ ≥ 0.2 have been
incrementally offset vertically by +0.3 for clarity.

Fig. 10. Normalized observed (black dots and blue squares) intensity and synthetic (red lines) center-to-limb profiles for the K I 7699 Å line, where
the numbers above each spectrum correspond to the µ-angle. In the left corner, the calibrated abundance is shown. Spectra for µ ≥ 0.2 have been
incrementally offset vertically by +0.3 for clarity.

data. Apart from uncertainties on the H collisions, other factorsthat might explain why the 3D non-LTE models do not perfectly
match observations of some Na lines and the K I line at the limb,
could be the uncertainties on the upper layers of the 3D model
atmosphere as well as the vertical resolution of the model.
In the next section, we demonstrate the effect of CLV in
narrowband transmission spectroscopy with the example of aA242, page 10 of 23

Jupiter-sized planet orbiting around a Sun-like star. For this purpose, the best-fitting synthetic spectra are used to evaluate the
CLV effect in a simulated Sun-Jupiter system in order to analyze
the impact of different models and different assumptions on themeasurement of planetary atmospheric species. Also, the CLV
impact on different geometries of the planet-star system and on
different lines is investigated.

Canocchi, G., et al.: A&A, 683, A242 (2024)
Table 4. Solar abundances determined from the best-fitting synthetic
spectrum at disk center for four Na I lines and one K I line.

λ (Å)

Table 5. Input parameters for the simulated Sun-Jupiter system.

Parameters
1D LTE

1D non-LTE

3D LTE

3D non-LTE

(a)

a/R⊙
e (a)
ω
Obliquityb = Ra⊙ · cos(i)Rp /R⊙ (a)Porb (days) (a)modevrot (m s−1 )

Na I
5896
6.23
6.12
6.36
6.24
5688
6.21
6.08
6.35
6.19
6154
6.26
6.20
6.31
6.26
6160
6.26
6.19
6.34
6.26
⟨A(Na)⟩ 6.24 ± 0.02 6.15 ± 0.05 6.34 ± 0.02 6.23 ± 0.03
KI
7699

5.31

5.00

5.49

5.21

4.3. Sun-Jupiter system

The simulation of a transit of a Jupiter-like planet on a circularorbit in front of a Sun-like star is performed by means of a modified version of the StarRotator10 python package. This code
simulates how the rotation-broadened stellar spectrum varies
during a planetary transit event. As the transit progresses, theplanet blocks the light from different parts of the stellar disk,each contributing differently to the line profile. Consequently,
the resulting stellar line profile varies in shape and strength during the transit phases in a non-homogeneous way, also dependingon the geometry of the star-planet system (i.e., the pattern ofthe planet across the stellar disk during the transit). We choseto simulate a circular orbit since most hot Jupiters are foundin low eccentric orbits that have been circularized due to perturbations from nearby stars or tidal forces (e.g., Fabrycky &Tremaine 2007). In an eccentric orbit, the transit duration would
vary, as well as the ingress/egress shape, while the amplitude of
the CLV light curve would not significantly change. In the simulation, the stellar disk is divided into a grid of (2×400)×(2×400)
cells (i.e., 800×800 square pixels), which provides sufficientspatial resolution for our simulations. Indeed, other works in literature have usually adopted a lower resolution of the grid of(2×100)×(2×100) cells (e.g., Wyttenbach et al. 2015; Yan et al.
2017). However, we performed a simulation with a lower resolution (i.e., (2×100)×(2×100)), finding a maximum differencein the resulting spectra of about 0.01%, which is non-negligible
for our CLV study. We also tested a higher resolution grid with
(2 × 800) × (2 × 800) elements and we found a maximum difference of only 0.001% among the spectra, which instead isinsignificant for our purposes. Therefore, we decided to run our
simulations with a (2×400)×(2×400) grid which shows a goodbalance of resolution and computational time.
Each of the elements of the stellar surface is characterized
by its own properties, in terms of µ-angle, corresponding radial
velocity (vµ ), and intensity (Iµ ). The latter comes from the best-fit
spectra synthesized with Balder in 3D non-LTE, linearly interpolated at 21 positions between the disk center and the edge, witha spacing of ∆µ = 0.05. Other studies employed the same number of spectra at different angles (e.g., Czesla et al. 2015; Sicilia
et al. 2022). In order to avoid numerical problems at the edgeof the stellar disk, the final angle was chosen to be µ = 0.001
instead of 0. For the cells at intermediate µ-angles, a linear interpolation between the spectra of the grid is performed. This is
the main difference with the original StarRotator package, inwhich the spectra describing the stellar atmospheres are modeledwith either the PHOENIX code (Husser et al. 2013), which does
10 https://github.com/Hoeijmakers/StarRotator

Value
1117.90.04870.0
0.0
0.0, 0.2, 0.4, 0.6, 0.8, 1.0
0.100394380
phases0, 2069.44(b) , 10000(c)Notes. (a) From the Jupiter Fact Sheet from NASA11 ; (b) from Meftah
et al. (2014), to simulate a slow-rotating Sun-like star; (c) to simulate afast-rotating star.

not take into account the CLV of the lines, or SPECTRUM (Szente
et al. 2019), which considers the CLV effect (i.e., the intensity isgiven at different µ-angles) but its RT computation is performed
in 1D LTE. In the following simulation instead, the stellar spectra
are computed in the more realistic 3D non-LTE assumption.
It is important to notice that the planet is only considered as
an opaque body without any atmosphere. In the simulation, thestellar flux is computed by summing up the spectra of the regionsof the discretized stellar disk that are not obscured by the planet(i.e., the “unobscured” stellar spectrum, Fs ), so that the observed
line profile varies at different orbital phases. This translates into
the following equation at i-th phase of transit:
Fs [i] = Fout − Fp [i],(3)where Fout is the out-of-transit flux and Fp is the integrated fluxthat is obscured by the planet during the transit at i-th orbital
phase. Then, by dividing Fs by its continuum, the normalized
spectrum for each given orbital phase is obtained.
Finally, the transmission curve or CLV curve is obtained as
the relative flux (Frel ), defined as the ratio of the normalized in(Fin ) to out-of-transit stellar flux (Fout ) integrated into a given
bandwidth centered on the line core, as a function of orbital
phase:
Frel =

Fin
.
Fout

(4)

The simulations were performed several times with the same
set-up, summarised in Table 5, but considering different impact
parameters (b), that is varying the planet trajectory from an edgeon transit (b = 0) to a grazing transit (b = 1) in steps of 0.2,
as outlined in the illustration in Fig. 11. This was done because
previous studies have shown that the CLV curve is affected byplanetary parameters, such as the planetary radius and impact
parameter. The effect from the former is straightforward: thelarger the radius, the stronger the CLV effect since the planet
will block more light during the transit. On the other hand, theeffect of different trajectories defined by the impact parameter ismore complicated to evaluate and we will investigate it further
in Sect. 4.3.1.
The final goal of this simulation is to investigate how the
CLV of the stellar lines influences the observed spectral line
profile during the transit of a Jupiter-sized planet in front of a11
https://nssdc.gsfc.nasa.gov/planetary/factsheet/
jupiterfact.html

A242, page 11 of 23

Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. 11. Transit schematic with different impact parameters.

Fig. 12. Transmission curves of the Na I 5896+,Å line for a Jupitersized planet transiting a Sun-like star in an edge-on transit (i.e., b = 0),evaluated in different bandwidths around the line center, colored as inlegend. 3D non-LTE synthetic spectra are used.

Fig. 13. Transmission curves of the Na I D1 (top), Na I D2 (middle) andK I line (bottom) for a Jupiter-sized planet transiting a Sun-like star indifferent trajectories, colored as in legend. The CLV curves are madeusing synthetic spectra in 3D non-LTE. The relative flux is computed ina bandwidth of 0.75 Å on the line center.

0.75 Å bandwidth, since the CLV effect is more prominent inthis band.
Sun-like star. This is done by evaluating how the relative flux
varies during the transit, that is at different orbital phases. Previous studies on the Na I D lines (e.g., Czesla et al. 2015; Yanet al. 2017) investigated passbands of 0.75, 1.5, and 3.0 Å, notingthat the CLV effect is more pronounced in the narrower bandwidth, which is also where the planetary absorption is more
prominent. We therefore selected the same three bandwidths toinvestigate how the line profile changes at different orbital phases
during the transit, and with different impact parameters. All theout-of-transit relative flux is normalized to unity.
In the simulations, we considered 66 time steps between
orbital phase −0.0005 and +0.0005, where the transit centeroccurs at phase zero. The CLV light curve, that is the relativeflux as a function of the orbital phase, using the best-fit 3D nonLTE model for the Na I D1 line and an edge-on transit is shownin Fig. 12. In this figure, the relative flux in three bandwidths
is highlighted in different colors. It can be noted that the CLV
curves show a similar shape but the effect is more pronounced forthe narrowest passband, that is 0.75 Å (green line in the figure),
which is consistent with other studies. However, minimizing the
CLV effect by choosing a larger passband might not be the best
approach to use, since the absorption from the planet’s atmosphere is usually concentrated in the core of the strong lines.
Therefore, in order to better understand how the CLV varies
among different trajectories, we analyzed only the flux in theA242, page 12 of 23

4.3.1. Different impact parameters

The CLV light curves for transits at different impact parameters
are shown in Fig. 13 for the Na I D1 , D2 and K I lines from top
to bottom, respectively. In Fig. 13, some interesting features incommon between the three analyzed lines can be noticed. Firstof all, as expected, the CLV effect is stronger for an edge-on andalmost edge-on transit (i.e., b = 0 or b = 0.2), represented by theblue and yellow solid lines. In particular, it is most pronounced
around mid-transit, that is at phase 0, when the planet blocks thelight from the very center of the stellar disk, where the intensityof the lines is stronger. Then, varying the impact parameter, thecurve varies in a fairly uniform way until b = 0.6 (red line), andthe weakest curve occurs for a grazing transit at b = 1.0 (brownline), as expected since the planet is obscuring only a small partof the stellar disk during the transit. However, in this specificcase, an interesting effect occurs for the Na I D1 and D2 lines,
namely that the CLV curve resembles a planetary absorption profile. This means that for a Jupiter-sized planet orbiting a Sun-likestar in this trajectory, the CLV effect can mimic a planetary atmospheric absorption, thus leading to a possible misattribution in
the planetary elemental abundance of Na if it is not properly corrected. This highlights the importance of properly modeling thestellar CLV in transmission spectroscopy studies.

Canocchi, G., et al.: A&A, 683, A242 (2024)

It is worth noting that even if the CLV curves of Na I D1 andD2 lines have similar shapes at different geometry, the values arestill slightly different. For example, in the case of b = 0 (blueline), the CLV curve of the Na I D1 line is about 0.1% lower thanthe Na I D2 at mid-transit. And the difference between the curveof the Na I D2 and the K I curve at the same point is even larger(about 0.27%). Indeed, the two curves exhibit a completely opposite behavior. For b = 0 to 0.6, the Na I doublet shows a peak withpositive values at mid-transit, whereas the peak for the K I line
occurs for negative values. This contrast arises from the distinct
CLV of the lines (Figs. 7 and 8), with the Na I D1 line weakening
as the µ-angles decrease, while the K I line strengthens toward
the limb. The different CLV can be explained since the two lines
have very different strengths (the Na I D lines are on the damping part of the curve-of-growth whereas the K I resonance line ison the flat, saturated part) and are forming at different layers inthe solar atmosphere, characterized by different values of atmospheric depth-dependent quantities, such as temperature, density,and velocity fields. Therefore, since different spectral lines show
different CLV light curves for the same geometry of the planetstar system, it is fundamental to model the CLV individually foreach line of interest.
4.3.2. Different stellar models

The emergent intensity spectra adopted to simulate the stellar
surface of the host star in StarRotator affect the resulting CLV
curve. As also noticed in a recent work by Reiners et al. (2023),
the amplitude and shape of the curve indeed change with different models, since the center-to-limb line profiles are different(see Sects. 4.1 and 4.2). In order to evaluate the impact of using
different stellar models, that is 1D instead of 3D or LTE insteadof non-LTE, we performed the same simulations of Sect. 4.3.1
again but using as input the best-fit stellar spectra from Baldercomputed in 1D LTE, 1D non-LTE and 3D LTE (see Sect. 4.1).
We also performed a simulation at b = 0 using the line profilesof the SST data. In Fig. 14 we show the CLV light curves of
the analyzed lines with different models and the SST data for anedge-on transit (b = 0), computed in a band of 0.75 Å, since itis the configuration with the strongest feature. The differences
between the models depend on each individual line but there areseveral similar patterns.
Yan et al. (2017) noticed that for the Na I D lines, the CLV
effect is less pronounced for a 1D non-LTE model compared tothe corresponding 1D LTE model. In this work we find a similar
result, that is clearly recognizable in the top panel of Fig. 14,
where the relative flux of the 1D LTE model (dashed red line)
is larger by about 0.15% than the corresponding 1D non-LTE
model (dashed blue line). This is valid also for the K I line in thebottom panel where the difference between the two 1D models
is about 0.1%.
For the models computed in non-LTE, in the Na I D1 , thedifference between a 3D (solid blue line) and a 1D (dashed blueline) atmosphere is that the former produces a less prominent
CLV feature of about 0.07%. On the other hand, the situation isreversed for the K I line, where the 3D non-LTE model not onlyshows a more pronounced feature compared to the 1D non-LTE
curve (of about 0.06%) but also an inversion in the relative flux
trend with respect to the 1D model, especially noticeable at midtransit. This is due to the different CLV trend of the synthetic
spectra, shown in Fig. 8 in Sect. 4.1, where the 3D non-LTE
model gets stronger at the limb, while all the other models exhibit
line weakening for decreasing µ-angles. Specifically, the weaker
the line gets toward the limb, the larger the amplitude of the CLV

Fig. 14. Transmission curves and residuals of the Na I D1 (top) andK I line (bottom) for a Jupiter-sized planet transiting a Sun-like star at
b = 0, using different models and the SST data to compute the stellar
spectrum, as in legend. The relative flux is computed in a bandwidth of
0.75 Å around the line center. The residuals are the difference between
the curve given by the SST data and the models in percentage.

curve will be at mid-transit. For both lines, the 1D LTE model isthe one that decreases the most in equivalent width (Figs. 7 and8), and therefore it is producing a higher peak in the CLV light
curve in Fig. 14.
In all cases, the 3D LTE model (solid red line) lies inbetween the 1D LTE and 1D non-LTE curves. The latter moreclosely resembles the 3D non-LTE model and the SST data. Consequently, from the analysis, it turns out that using a 1D non-LTE
model is more accurate than using a 3D stellar atmosphere withan LTE line formation.
The CLV light curve obtained using the SST line profiles isoverplotted in Fig. 14 (open bullets). In both panels, it closely
resembles the 3D non-LTE model, with a maximum differenceat mid-transit of only 0.003% for the Na I D1 . The 3D non-LTE
model of K I performs almost as well, with a maximum difference of about 0.008% at mid-transit, reflecting the slightly worse
fit to the CLV curve (see Sects. 4.1 and 4.2). In any case, theagreement between the CLV curves of the SST data and the 3D
non-LTE spectra is excellent, thus confirming that 3D non-LTE
spectra are necessary to model the CLV effect and to get the mostaccurate results in transmission spectroscopy.
This analysis shows that different assumptions on the computation of the stellar synthetic spectra (1D or 3D, LTE or
non-LTE) lead to different CLV curves and consequently todifferent estimates of the planetary elemental abundances. The
differences in the CLV curves between models, even if small, arestill important since they are on the same order of magnitude asa planetary absorption depth, which is very weak and typicallybetween 0.01–0.5% (e.g., Snellen et al. 2008; Yan et al. 2017;
A242, page 13 of 23

Canocchi, G., et al.: A&A, 683, A242 (2024)

Mounzer et al. 2022). Using a wrong CLV model could either
mimic the presence of a species, thus leading to a false detection, or under(over)-estimate the abundance of a species in theplanetary atmosphere. Therefore, a careful modeling as realistic and as precise as possible is required in order to correctly
characterize exoplanet atmospheres.
4.3.3. Rossiter McLaughlin effect

The Rossiter McLaughlin (RM) effect is an additional sourceof distortion of the stellar line profile that occurs during an
exoplanetary transit and that could be as critical as the CLV
effect, especially in fast-rotating stars like KELT-9 (Hoeijmakers
et al. 2019). When an exoplanet transits across the stellar disk,
it obscures different parts of the stellar surface, each having a
local rotational velocity vector that could be pointing toward oraway from the observer, depending on the transit phase. Consequently, red- or blue-shifted photons are removed from the totalstarlight, implying that the averaged wavelength is slightly red orblue-shifted as well. In order to investigate the combined effectof the CLV and the RM on the selected line profiles, we perform simulations of an edge-on transit (b = 0) with a non-zerorotational velocity (vrot ) using our 3D non-LTE synthetic spectra of the Na I D1 line. Specifically, one simulation assuming aslow-rotator like a Sun-like star, with a vrot equal to the rotational velocity of the Sun, that is 2069.44 m s−1 at the equator(Meftah et al. 2014), and another simulation for a fast-rotator(i.e., Prot ≤ 10 days), assuming a vrot = 10 km s−1 , which fora Sun-like star would correspond to a rotational period of about
five days. In these simulations we assume that the star rotates as asolid body, meaning that differential rotation is not implemented
in the code, and consequently, it is neglected in our analysis.
The results of these simulations are shown in Figs. 15 and 16,
in comparison with the same planet-star configuration with the
CLV effect only. The former figure shows the signal imprinted
by the planet when transiting the stellar disk, also often referred
to as “Doppler shadow”. It usually represents the combinationof the stellar CLV and RM, and this signal becomes stronger the
faster the star rotates. In Fig. 16, the transmission curve computed in a bandwidth of 0.75 Å is shown for the three models
with different stellar rotational velocities. As is apparent, thecurve becomes asymmetric due to the RM effect. The asymmetry is more relevant for higher vrot and the amplitude of thecurve becomes larger as well. In the Sun case, the CLV is thedominant source of distortion of the stellar lines, and this is also
valid for other slow-rotating Sun-like stars, such as HD 189733(Borsa & Zannoni 2018). On the other hand, in very fast-rotatingstars, such as KELT-9 where the lines are very broadened due tothe high rotational velocity (v sin(i) = 111.4 km s−1 , Gaudi et al.
2017), the RM becomes significantly more important than the
CLV.

5. Discussion and conclusions
In this work, we model the CLV of the Na I D1 5896 Å line andK I 7699 Å line using 3D hydrodynamic solar models and nonLTE line formation. We compare our synthetic spectra to solar
observations from the high-resolution IAG Atlas as well as to
SST/CRISP data taken in two different Campaigns in August2022 and April 2023. We include in the comparison also 1D
LTE, 1D non-LTE, and 3D LTE synthetic spectra. For the model
atmosphere, we use ten snapshots from the Stagger solar simulation in 3D and the MARCS solar model in 1D. We investigate theA242, page 14 of 23

Fig. 15. Models of the CLV and RM effect of the Na I D1 line for a Sunlike star orbited by a Jupiter-sized planet at b = 0. The upper panel is the
CLV effect-only, and the middle panel combines the CLV with the RM
effect for a slow-rotator like the Sun. Finally, the bottom panel shows
the two effects for a fast-rotator, that is a Sun-like star with a rotational
velocity of 10 km s−1 .
3D NLTE
CLV

0.15- CLV+RM (Sun)- CLV+RM (fast rotator)

0.10

.-1

0.05

I

�-:_ 0.00
-0.05

-0.10
-0.0004-0.00020.00000.00020.0004

Orbital Phases

Fig. 16. Transmission curves of the Na I D1 line for a Jupiter-sized planet
transiting a Sun-like star in an edge-on transit (b = 0), for different rotational velocities of the host star. 3D non-LTE synthetic spectra are used
and the relative flux is computed in a bandwidth of 0.75 Å.

line profile as well as the equivalent width (Wλ ) of the lines as afunction of the µ-angle. The 3D non-LTE model represents thebest match in both cases. It is able to reproduce the line shapefrom the center to the limb quite well and it is in reasonableagreement with the observed Wλ within 6% and 0.4% at µ = 0.1
for the K I and Na I D1 line, respectively. 1D models are symmetric by design therefore they are not able to reproduce theline asymmetries that are particularly strong in resonance lines.
A 3D stellar atmosphere and a proper treatment of convection
are required. The LTE models typically produce lines with cores
weaker than the observations but wings that are reasonably inagreement, at least for disk center line profiles. However, Wλ isseverely underestimated, with differences up to 37% for the K I
line at the limb.

Canocchi, G., et al.: A&A, 683, A242 (2024)

With the best-fit synthetic spectra, we also perform simulations of spectral lines of a Sun-like star transited by aJupiter-sized planet without an atmosphere in order to investigate
the impact of the CLV on the inference of planetary absorptions. In transmission spectroscopy the CLV effect can be on
the same order of magnitude as the planetary absorption feature,
therefore it is fundamental to adopt model spectra as accurate
as possible. We compute transmission curves for different trajectories, or impact parameters (b), and evaluate the flux in threedifferent bandwidths centered on the line cores. The LTE models, both with 1D and 3D atmospheres, produce a CLV curvewith an amplitude larger than in non-LTE, thus overestimating
the effect of CLV. In the presence of a planetary absorption,
this would eventually result in an underestimate of the abundance of the analyzed species in the planetary atmosphere. In
a recent work by Reiners et al. (2023), they also found that the
LTE curve amplitude is overpredicted when evaluating the CLV
effect in a simulated planet-star system. They analyzed lines ofseveral atomic species but they stress the fact that for Na I, nonLTE effects are definitely non-negligible in order to reproduce
observations. In this work, the differences between models, 1D
versus 3D and LTE versus non-LTE, are relatively small but on
the same order of magnitude as a planetary absorption depth.
The CLV curves also greatly vary in shape for different geometries of the star-planet system and for different lines. Even the
CLV curves of lines of the same species, that is Na I D1 and D2lines, are not identical. The CLV effect is indeed line-dependentand must be modeled line by line individually. We also perform atransit simulation with the SST line profiles and we find that the
3D non-LTE models are able to reproduce the CLV light curveof the observations extremely well.
Since the planetary parameters, such as the planet radius
and impact parameter, are also crucial to correctly modeling theshape and strength of the CLV curve (Pietrow & Pastor Yabar
2023), these values need to be known with a high level of accuracy. For instance, in our simulations, the CLV curve of the
Na doublet resulting from a grazing transit (b = 1.0) is very
similar to a planetary absorption signature and therefore could
be mistaken for a sodium detection if not properly corrected
for. Nowadays, having accurate planetary parameters is possible thanks to the many ground-based as well as space-based
telescopes performing high precision transit photometry, such as
TESS (Ricker et al. 2014), CHEOPS (Benz et al. 2021), and also
PLATO (Rauer et al. 2014) in the near future.
It is important to note that the simulation presented in thiswork represents a best-case scenario, given that only the quiet
solar photosphere is considered. Inhomogeneities on the stellar surface due to active regions, such as starspots or plages,
show a very different spectral line profile from the quiet regions(e.g., Oranje 1983; Dumusque et al. 2014; Pietrow et al. 2020,
2024; Rajhans et al. 2023; Howard et al. 2023; Cretignier et al.
2024). Observational data would allow for the effect of magnetic
fields on the line profiles to be investigated. Hence, separate CLVshould be measured in active regions in order to correctly incorporate them in the analysis of transmission spectra of active stars(Chakraborty et al. 2024). Future works should also include themodeling of synthetic spectra for active regions of different sizes
and at various locations on the stellar disk.
The differences in CLV transmission curves can be larger
for different kinds of stars, especially for low-temperature stars,
where the effect is stronger as shown in Yan et al. (2017). Consequently, a very good knowledge of the stellar parameters is as
important as the precision of the planetary parameters in order
to use a model stellar atmosphere as realistic as possible for themodeling of the CLV curves. A significant advance in this direction is given by the recent release of Gaia DR3 (Gaia Collaboration 2021; Katz et al. 2023) and in the very near future by large
spectroscopic surveys such as 4MOST (de Jong et al. 2019) and
WEAVE (Dalton et al. 2018; Jin et al. 2023), which will target
millions of FGK stars providing accurate stellar parameters.
With the continuous improvement of the instrumentation
used for transit spectroscopy, such as ESPRESSO on VLT or
ANDES on ELT, the CLV effect will soon become crucial not
only for the detection but especially for the characterizationof exoplanetary atmospheres. At present, some facilities havealready achieved the high signal-to-noise ratio necessary formeasuring the abundances of atmospheric species in hot Jupiters
and other gas giants. Several works have shown that using different models for the stellar spectrum can lead to very different
results. For example, the case of the benchmark exoplanet HD
209458b in which the presence of Na was debated in the literature. Initial works detected it in transmission (e.g., Charbonneau
et al. 2002; Jensen et al. 2011) but then later analysis argued
that it was just an artifact due to the modeled stellar spectrum(Casasayas-Barris et al. 2020, 2021). This highlights the importance of using correct stellar models to evaluate the subtle effects
such as the CLV, which could compromise the analysis of transit
spectroscopy observations.
The Sun is the only star we can spatially resolve, therefore, developing models able to reproduce solar observations isnecessary to improve the CLV model of other stars as well, especially considering that transmission observations are expected to
become more and more precise in the coming years. From ouranalysis, we can conclude that 1D models of Sun-like stars seem
to overestimate the CLV signature in transmission curves. We
can also say that, in accordance with previous works, the RM
effect has an impact on the transmission curve as well whichmust be taken into account (Reiners et al. 2023). Specifically,
it becomes stronger than the CLV effect for fast-rotating stars.
For Sun-like stars instead, the CLV remains the main source ofdistortion of the stellar lines during transit.
However, it is difficult to make predictions about other kindsof stars or even about specific exoplanets, since every planet-starsystem is different and we show that different configurations leadto very different transmission curves. Consequently, future work
includes the analysis of well-known and extensively studied hot
Jupiters, such as HD 189733b or HD 209458b, for which several
high-resolution observations are already available. The goal is
to apply a 3D non-LTE stellar model to evaluate the CLV effectof Na I and K I in these benchmark exoplanets and then compare
the results with previous works in the literature (e.g., Yan et al.
2017; Keles et al. 2019, 2020; Sicilia et al. 2022). The methodology presented in this work can be extended as well to other FGK
stars using 3D stellar atmospheres from the new full Stagger
grid (Rodríguez Díaz et al. 2023). It is worth noting that there are
only a few studies where Na I and K I have been detected simultaneously (e.g., Casasayas-Barris et al. 2021). Given the opposite
CLV curve of the resonance lines of the two ions, it would beinteresting to observe those lines for more planet-star systems.
The impact of a 3D non-LTE modeling of the CLV of thestellar lines needs to be studied in other species and other spectral lines as well. For transmission spectroscopy in particular,interesting species that have been detected in the atmosphereof several exoplanets include Ca I, Ca II (e.g., Yan et al. 2019),
Fe I (e.g., Hoeijmakers et al. 2018), Fe II (e.g., Bello-Arufe et al.
2022), and H I Balmer lines (e.g., Fossati et al. 2023).
Finally, with the recent launch of the JWST (Barstow et al.
2015), a new era of low-resolution transit spectroscopy has justA242, page 15 of 23

Canocchi, G., et al.: A&A, 683, A242 (2024)

begun. This space telescope will allow for the atmospheric characterization of a wide range of exoplanets, from hot Jupiters
down to rocky planets and super-Earths. Low-resolution spectra
are even more susceptible to the CLV effect in a wavelengthintegrated sense than the high-resolution ones. Several studies
have showm that high-resolution spectra, if adequately corrected
for the RM and CLV effects, can be used to isolate the different spectral components and mitigate the stellar contamination
in low-resolution spectra (e.g., Genest et al. 2022). Therefore,
exploring the effect of stellar CLV on low-resolution spectroscopy is an issue that has to be addressed in the near future
and this work is a step forward to achieving the best possible
results in exoplanet science with JWST.
Acknowledgements. We thank the anonymous referee for their comments, whichhave improved the manuscript. We thank A. Brandeker for the very useful comments on the first draft of the paper. We thank A. Reiners and M. Ellwarth forproviding us with the Na I D1 and K I line profiles before the publication of the
IAG Atlas, and for the errors on µ-angles. We also thank H. Reggiani for providing the potassium model atom that he developed. G.C. and K.L. acknowledge
funds from the Knut and Alice Wallenberg foundation. K.L. and C.L. acknowledge funds from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (Grant agreement
no. 852977). A.M.A. acknowledges support from the Swedish Research Council (VR 2020-03940). A.P. was supported at AIP by grants from the European
Commission’s Horizon 2020 Program under grant agreements 824064 (ESCAPE– European Science Cluster of Astronomy & Particle Physics ESFRI Research
Infrastructures) and 824135 (SOLARNET – Integrating High Resolution Solar
Physics). The computations were enabled by resources provided by the Swedish
National Infrastructure for Computing (SNIC) at UPPMAX, partially fundedby the Swedish Research Council through grant agreement no. 2018-05973. We
thank the PDC Center for High Performance Computing, KTH Royal Institute ofTechnology, Sweden, for providing access to computational resources and support. The Swedish 1-m Solar Telescope is operated on the island of La Palmaby the Institute for Solar Physics of Stockholm University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.
The Institute for Solar Physics was supported by a grant for research infrastructures of national importance from the Swedish Research Council (registration
number 2017-00625). The Swedish 1-m Solar Telescope, SST, is co-funded by
the Swedish Research Council as a national research infrastructure (registration
number 4.3-2021-00169). This research has made use of NASA’s Astrophysics
Data System (ADS) bibliographic services. This work has made use of the VALD
database, operated at Uppsala University, the Institute of Astronomy RAS in
Moscow, and the University of Vienna. We acknowledge the community efforts
devoted to the development of the following open-source packages that were usedin this work: numpy (numpy.org), matplotlib (matplotlib.org) and astropy(astropy.org). We also used the CRISPEX analysis tool (Vissers & Rouppe
van der Voort 2012) and the ISPy library (Díaz Baso et al. 2021).

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Appendix A: Additional tables and figures

Table A.2: Equivalent widths (in mÅ) measured for the observed
spectra of the IAG Atlas at different µ-angles via direct integration over the wavelength range specified by λint in Table 3.
line
µ
1.00
0.99
0.98
0.97
0.95
0.90
0.80
0.70
0.60
0.50
0.40
0.35
0.30
0.20

Fig. A.1: Comparison between 3D non-LTE models of the old
Na atom and the new one, with added H rate coefficients from
Kaulakys (1991).

Table A.1: Equivalent widths (in mÅ) measured for the observed
spectra of the SST data at different µ-angles via direct integrationof the line profiles.
µ
1.00
0.80
0.60
0.55
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10

W5896 (mÅ)
447.8 ± 4.7
446.3 ± 4.7
441.9 ± 4.7
440.7 ± 4.7
435.2 ± 4.8
428.7 ± 4.8
423.9 ± 4.8
414.7 ± 4.7
412.6 ± 4.7
400.7 ± 4.7
389.9 ± 4.8
378.2 ± 4.9

W7699 (mÅ)
157.5 ± 3.3
160.1 ± 3.3
166.2 ± 3.4
168.9 ± 3.4
171.3 ± 3.4
173.7 ± 3.4
175.6 ± 3.4
178.3 ± 3.5
181.2 ± 3.4
184.1 ± 3.4
188.2 ± 3.4
194.2 ± 3.4

Notes:The line profile of Na I 5896 Å is not integrated completely but it istruncated on the right wing to exclude the blending due to a telluricline.

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5896

5688

449.3 ± 2.1
448.6 ± 2.8
449.1 ± 2.8
450.6 ± 2.0
449.8 ± 2.3
449.3 ± 2.0
448.6 ± 2.4
447.3 ± 2.5445.8 ± 2.3
441.7 ± 2.3
436.2 ± 2.4
429.5 ± 2.1
424.5 ± 2.5
412.6 ± 2.9

122.1 ± 1.3
122.1 ± 1.7
122.5 ± 1.7
122.4 ± 1.2
122.2 ± 1.4
121.9 ± 1.2
121.2 ± 1.5
120.7 ± 1.6
120.3 ± 1.5
120.1 ± 1.5
119.7 ± 1.6
119.6 ± 1.4
119.3 ± 1.7
119.1 ± 2.06154
Wλ (mÅ)35.6 ± 1.1
35.8 ± 1.5
35.6 ± 1.5
35.7 ± 1.1
35.7 ± 1.2
35.9 ± 1.1
36.5 ± 1.3
37.1 ± 1.4
37.6 ± 1.4
38.5 ± 1.4
38.9 ± 1.5
39.2 ± 1.3
39.5 ± 1.6
40.1 ± 1.9

6160

7699

55.7 ± 1.2
55.8 ± 1.5
55.7 ± 1.5
55.7 ± 1.1
55.5 ± 1.3
55.6 ± 1.1
55.9 ± 1.3
56.4 ± 1.4
56.8 ± 1.4
57.4 ± 1.4
57.7 ± 1.5
57.8 ± 1.3
58.1 ± 1.6
58.5 ± 1.9

160.1 ± 1.4
160.8 ± 1.8
160.7 ± 1.8
160.4 ± 1.3
160.4 ± 1.5
161.1 ± 1.3
163.2 ± 1.6
165.7 ± 1.7
168.0 ± 1.6
172.3 ± 1.6
175.7 ± 1.8
178.0 ± 1.6
181.0 ± 1.9
186.3 ± 2.4

Notes:The line profile of Na I 5896 Å is not integrated completely but it istruncated on the right wing to exclude the blending due to a telluric
line.

Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. A.2: Na I D1 data taken in Campaign I. Left: Limb darkening data of each wavelength point in the Na I D1 line on an arbitrary
intensity scale. The curves are ordered by wavelength, with the top (red) curve representing the red continuum, the middle (green)
curve representing the line core, and the bottom (blue) curve representing the blue continuum. Right: Normalized intensity of theaveraged line profiles at different µ-angles. Spectra for µ ≥ 0.15 have been incrementally offset vertically by +0.3 for clarity.

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Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. A.3: K I data taken in Campaign II. Left: Limb darkening data of each wavelength point in the K I 7699 Å line on an arbitrary
intensity scale. The curves are ordered by wavelength, with the top (red) curve representing the red continuum, the middle (green)
curve representing the line core, and the bottom (blue) curve representing the blue continuum. Right: Normalized intensity of theaveraged line profiles at different µ-angles. Spectra for µ ≥ 0.15 have been incrementally offset vertically by +0.3 for clarity.

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Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. A.4: Normalized observed (black dots) intensity and synthetic (red lines) center-to-limb profiles for the Na I 5688 Å line, where
the numbers above each spectrum correspond to the µ-angle. In the left corner, the calibrated abundance is shown. Spectra forµ ≥ 0.3 have been incrementally offset vertically by +0.3 for clarity.

Fig. A.5: Normalized observed (black dots) intensity and synthetic (red lines) center-to-limb profiles for the Na I 6154 Å line, where
the numbers above each spectrum correspond to the µ-angle. In the left corner, the calibrated abundance is shown. Spectra forµ ≥ 0.3 have been incrementally offset vertically by +0.3 for clarity.

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Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. A.6: Normalized observed (black dots) intensity and synthetic (red lines) center-to-limb profiles for the Na I 6160 Å line, where
the numbers above each spectrum correspond to the µ-angle. In the left corner, the calibrated abundance is shown. Spectra forµ ≥ 0.3 have been incrementally offset vertically by +0.3 for clarity.

Fig. A.7: Normalized observed (black dots) intensity from the IAG Atlas and synthetic (red lines) center-to-limb profiles for the Na I
D1 at 5896 Å line, where the numbers above each spectrum correspond to the µ-angle. In the left corner, the calibrated abundance isshown. Spectra for µ ≥ 0.3 have been incrementally offset vertically by +0.3 for clarity.

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Canocchi, G., et al.: A&A, 683, A242 (2024)

Fig. A.8: Normalized observed (black dots) intensity from the IAG Atlas and synthetic (red lines) center-to-limb profiles for theK I 7699 Å line, where the numbers above each spectrum correspond to the µ-angle. In the left corner, the calibrated abundance isshown. Spectra for µ ≥ 0.3 have been incrementally offset vertically by +0.3 for clarity.

A242, page 23 of 23