11institutetext: Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, France,
email<EMAIL_ADDRESS>22institutetext: European Southern Observatory,
Karl-Schwarzschild-Str. 2, 85748 Garching, Germany 33institutetext: Max-
Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
44institutetext: Leiden Observatory, Leiden University, Niels Bohrweg 2,
NL-2333 CA Leiden, the Netherlands 55institutetext: LESIA, Observatoire de
Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place
Jules Janssen, 92195 Meudon, France 66institutetext: Nicolaus Copernicus
Astronomical Centre, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa,
Poland 77institutetext: Unidad Mixta Internacional Franco-Chilena de
Astronomía (CNRS UMI 3386), Departamento de Astronomía, Universidad de Chile,
Camino El Observatorio 1515, Las Condes, Santiago, Chile 88institutetext:
Departamento de Astronomía, Universidad de Concepción, Casilla 160-C,
Concepción, Chile 99institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000,
Grenoble, France 1010institutetext: NASA Goddard Space Flight Center,
Astrophysics Division, Greenbelt, MD 20771, USA 1111institutetext: Anton
Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904,
1090 GE Amsterdam, The Netherlands 1212institutetext: Department of
Astrophysics, University of Vienna, Türkenschanzstrasse 17 1313institutetext:
AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne
Paris Cité, F-91191 Gif-sur-Yvette, France 1414institutetext: Institute for
Mathematics, Astrophysics and Particle Physics, Radboud University, P.O. Box
9010, MC 62 NL-6500 GL Nijmegen, the Netherlands 1515institutetext: SRON
Netherlands Institute for Space Research, Sorbonnelaan 2, NL-3584 CA Utrecht,
the Netherlands 1616institutetext: Institut für Theoretische Physik und
Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstraße 15, 24118,
Kiel, Germany 1717institutetext: Konkoly Observatory, Research Centre for
Astronomy and Earth Sciences, Eötvös Loránd Research Network (ELKH), Konkoly-
Thege Miklós út 15-17, H-1121 Budapest, Hungary 1818institutetext: CSFK
Lendület Near-Field Cosmology Research Group, Budapest, Hungary
1919institutetext: Max Planck Institute for Astronomy, Königstuhl 17, D-69117
Heidelberg, Germany 2020institutetext: Institute for Astrophysics, University
of Vienna, 1180 Vienna,Türkenschanzstrasse 17, Austria 2121institutetext: I.
Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937, Köln,
Germany 2222institutetext: Zselic Park of Stars, 064/2 hrsz., 7477
Zselickisfalud, Hungary 2323institutetext: Sydney Institute for Astronomy
(SIfA), School of Physics, The University of Sydney, NSW 2006, Australia
2424institutetext: NOVA Optical IR Instrumentation Group at ASTRON, Oude
Hoogeveensedijk 4 7991 PD Dwingeloo, the Netherlands 2525institutetext: ASTRON
(Netherlands), Oude Hoogeveensedijk 4 7991 PD Dwingeloo, the Netherlands

# Mid-infrared circumstellar emission of the long-period Cepheid $\ell$
Carinae resolved with VLTI/MATISSE††thanks: Based on observations made with
ESO telescopes at Paranal observatory under program ID 0104.D-0554(A)

V. Hocdé 11 N. Nardetto 11 A. Matter 11 E. Lagadec 11 A. Mérand 22 P.
Cruzalèbes 11 A. Meilland 11 F. Millour 11 B. Lopez 11 P. Berio 11 G. Weigelt
33 R. Petrov 11 J. W. Isbell 1919 W. Jaffe 44 P. Kervella 55 A. Glindemann 22
M. Schöller 22 F. Allouche 11 A. Gallenne 11667788 A. Domiciano de Souza 11 G.
Niccolini 11 E. Kokoulina 11 J. Varga 441717 S. Lagarde 11 J.-C. Augereau 99
R. van Boekel 44 P. Bristow 22 Th. Henning 1919 K.-H. Hofmann 33 G. Zins 22
W.-C. Danchi 111010 M. Delbo 11 C. Dominik 1111 V. Gámez Rosas 44 L. Klarmann
1919 J. Hron 1212 M.R. Hogerheijde 441111 K. Meisenheimer 1919 E. Pantin 1313
C. Paladini 22 S.Robbe-Dubois 11 D. Schertl 33 P. Stee 11 R. Waters 14141515
M. Lehmitz 1919 F. Bettonvil 44 M. Heininger 33 P. Bristow 22 J. Woillez 22 S.
Wolf 1616 G. Yoffe 44 L. Szabados 17171818 A. Chiavassa 11 S. Borgniet 55 L.
Breuval 55 B. Javanmardi 55 P. Ábrahám 1717 S. Abadie 22 R. Abuter 22 M.
Accardo 22 T. Adler 1919 T. Agócs 2121 J. Alonso 22 P. Antonelli 11 A. Böhm
1919 C. Bailet 11 G. Bazin 22 U. Beckmann 33 J. Beltran 22 W. Boland 55 P.
Bourget 22 R. Brast 22 Y. Bresson 11 L. Burtscher 44 R. Buter 22 R. Castillo
22 A. Chelli 11 C. Cid 22 J.-M. Clausse 11 C. Connot 33 R.D. Conzelmann 22 M.
De Haan 2424 M. Ebert 1919 E. Elswijk 2424 Y. Fantei 11 R. Frahm 22 V. Gámez
Rosas 55 A. Gabasch 22 E. Garces 22 P. Girard 11 A. Glazenborg 2525 F.Y.J.
Gonté 22 J.C. González Herrera 22 U. Graser 1919 P. Guajardo 22 F. Guitton 11
H. Hanenburg 2424 X. Haubois 22 N. Hubin 22 R. Huerta 22 J. Idserda 2525 D.
Ives 22 G. Jakob 22 A. Jaskó 1616 L. Jochum 22 R. Klein 1919 J. Kragt 2121 G.
Kroes 14142020 S. Kuindersma 2525 L. Labadie 1717 W. Laun 1919 R. Le Poole 55
C. Leinert 1919 J.-L. Lizon 22 M. Lopez 22 A. Marcotto 11 N. Mauclert 11 T.
Maurer 1919 L.H. Mehrgan 22 J. Meisner 55 K. Meixner 1919 M. Mellein 1919 L.
Mohr 1919 S. Morel 11 L. Mosoni 2222 R. Navarro 2424 U. Neumann 1919 E.
Nußbaum 33 L. Pallanca 22 L. Pasquini 22 I. Percheron 22 T. Phan Duc 22 J.-U.
Pott 1919 E. Pozna 22 A. Ridinger 1919 F. Rigal 2424 M. Riquelme 22 Th.
Rivinius 22 R. Roelfsema 2424 R.-R. Rohloff 1919 S. Rousseau 11 N. Schuhler 22
M. Schuil 2424 K. Shabun 22 A. Soulain 1919 C. Stephan 22 R. ter Horst 2424 N.
Tromp 2424 F. Vakili 11 A. van Duin 2525 L. B. Venema 2525 J. Vinther 22 M.
Wittkowski 22 F. Wrhel 1919

(Received … ; accepted …)

###### Abstract

Context. The nature of circumstellar envelopes (CSE) around Cepheids is still
a matter of debate. The physical origin of their infrared (IR) excess could be
either a shell of ionized gas, or a dust envelope, or both.

Aims. This study aims at constraining the geometry and the IR excess of the
environment of the bright long-period Cepheid $\ell$ Car (P=35.5 days) at mid-
IR wavelengths to understand its physical nature.

Methods.  We first use photometric observations in various bands (from the
visible domain to the infrared) and Spitzer Space Telescope spectroscopy to
constrain the IR excess of $\ell$ Car. Then, we analyze the VLTI/MATISSE
measurements at a specific phase of observation, in order to determine the
flux contribution, the size and shape of the environment of the star in the
$L$ band. We finally test the hypothesis of a shell of ionized gas in order to
model the IR excess.

Results. We report the first detection in the $L$ band of a centro-symmetric
extended emission around $\ell$ Car, of about $1.7\,R_{\star}$ in full width
at half maximum, producing an excess of about $7.0\%$ in this band. This
latter value is used to calibrate the IR excess found when comparing the
photometric observations in various bands and quasi-static atmosphere models.
In the $N$ band, there is no clear evidence for dust emission from
VLTI/MATISSE correlated flux and Spitzer data. On the other side, the modeled
shell of ionized gas implies a more compact CSE ($1.13\pm 0.02\,R_{\star}$)
and fainter (IR excess of 1% in the $L$ band).

Conclusions. We provide new evidences for a compact CSE of $\ell$ Car and we
demonstrate the capabilities of VLTI/MATISSE for determining common properties
of CSEs. While the compact CSE of $\ell$ Car is probably of gaseous nature,
the tested model of a shell of ionized gas is not able to simultaneously
reproduce the IR excess and the interferometric observations. Further Galactic
Cepheids observations with VLTI/MATISSE are necessary for determining the
properties of CSEs, which may also depend on both the pulsation period and the
evolutionary state of the stars.

###### Key Words.:

Techniques : Interferometry – Infrared : CSE – stars: variables: Cepheids –
stars: atmospheres

##  1 Introduction

Circumstellar envelopes (CSEs) around Cepheids have been spatially resolved by
long-baseline interferometry in the $K$ band with the Very Large Telescope
Interferometer (VLTI) and the Center for High Angular Resolution Astronomy
(CHARA) (Kervella et al. 2006; Mérand et al. 2006). They were detected around
four Cepheids including $\ell$ Car, in the $N$ band with VLTI/VISIR and
VLTI/MIDI (Kervella et al. 2009; Gallenne et al. 2013). In the $K$ band, the
diameter of the envelope appears to be at least 2 stellar radii and the flux
contribution is up to 5% of the continuum (Mérand et al. 2007). Both the size
and flux contribution are expected to be larger in the $N$ band, as
demonstrated by Kervella et al. (2009) for $\ell$ Car, Gallenne et al. (2013)
for X Sgr and T Mon, and by Gallenne et al. (2021, submitted) for other
Cepheids. Moreover, a CSE has also been discovered in the visible domain with
the CHARA/VEGA instrument around $\delta$ Cep (Nardetto et al. 2016). The
systematic presence of a CSE is still a matter of debate. While some studies
have found no significant observational evidence for circumstellar dust
envelopes in a large number of Cepheids (Schmidt 2015; Groenewegen 2020),
Gallenne et al. (2021, submitted) found a significant IR excess attributed to
a CSE for 10 out of 45 Cepheids.
Recently, Hocdé et al. (2020b) (hereafter Paper I) used an analytical model of
free-free and bound-free emission from a thin shell of ionized gas to explain
the near and mid-IR excess of Cepheids. They found a typical radius for this
shell of ionized gas of about $R_{\mathrm{shell}}$=1.15$\,R_{\star}$. This
shell of ionized gas could be due to periodic shocks occurring in both the
atmosphere and the chromosphere, which heat up and ionize the gas. Using
VLT/UVES data, (Hocdé et al. 2020a) found a radius for the chromosphere of at
least $R_{\mathrm{chromo}}$=1.5$\,R_{\star}$ in long-period Cepheids. In
addition, they found a motionless H$\alpha$ feature in UVES high-resolution
spectra obtained for several long-period Cepheids, including $\ell$ Car. This
absorption feature was attributed to a static CSE surrounding the chromosphere
above at least 1.5$\,R_{\star}$, and reported by various authors around $\ell$
Car (Rodgers & Bell 1968; Bohm-Vitense & Love 1994; Nardetto et al. 2008).
Determining the IR excess and the size of the CSE is a key to understand the
physical processes at play. In this paper, we study the long-period Cepheid
$\ell$ Car. Our aim is to determine its IR excess from photometric
measurements in various bands and Spitzer spectroscopy, while inferring the
size of the CSE and its flux contribution thanks to the unique capabilities
provided by the Multi AperTure mid-Infrared SpectroScopic Experiment
(VLTI/MATISSE; Lopez et al. (2014); Allouche et al. (2016); Robbe-Dubois et
al. (2018)) in the $L$ (2.8-4.0$\,\mu$m), $M$ (4.5 - 5$\,\mu$m) and $N$ bands
(8 - 13$\,\mu$m).
We first reconstruct the IR excess in Sect 2. Then, we present the data
reduction and calibration process of VLTI/MATISSE data in Sect. 3. We use a
simple model of the CSE to reproduce the visibility measurement of
VLTI/MATISSE in Sect. 4. In Sect. 5 we discuss the physical origin of this
envelope, and present a model of a shell of ionized gas. We summarize our
conclusions in Sect. 6.

##  2 Deriving the infrared excess of $\ell$ Car

###  2.1 Near-IR excess modeling with SPIPS

Due to their intrisic variability, the photospheres of the Cepheids are
difficult to model along the pulsation cycle. However, it is an essential
prerequisite for deriving both the IR excess in a given photometric band, and
the expected angular stellar diameter at a specific phase of interferometric
observations (required in Sect. 3). In order to model the photosphere we use
SpectroPhoto-Interferometric modeling of Pulsating Stars (SPIPS), which is a
model-based parallax-of-pulsation code which includes photometric,
interferometric, effective temperature and radial velocity measurements in a
robust model fitting process (Mérand et al. 2015).

SPIPS uses a grid of ATLAS9 atmospheric
models111http://wwwuser.oats.inaf.it/castelli/grids.html (Castelli & Kurucz
2003) with solar metallicity and a standard turbulent velocity of 2 km/s.
SPIPS was already extensively described and used in several studies (Mérand et
al. 2015; Breitfelder et al. 2016; Kervella et al. 2017; Gallenne et al.
2017), Javanmardi et al. (2021, accepted), Gallenne et al. (2021, submitted),
and Paper I. SPIPS also takes into account the possible presence of an IR
excess by fitting an ad-hoc analytic power law $\mathrm{IR}_{\mathrm{ex}}$
(see green curve in Fig. 1(a)). We note that Gallenne et al. (2021, submitted)
have shown that $\ell$ Car possibly has an IR excess using SPIPS, but its low
detection level (2.9 ± 3.2% in $K$ band) is not significant compared to a
single-star model (i.e. without IR excess model). Breitfelder et al. (2016)
also provided a SPIPS fitting of $\ell$ Car which did not present any
significant IR excess. However, this work was done by considering only the
$V$, $J$, $H$ and $K$ photometric bands, i.e. not the mid-infrared bands,
which are critical to detect such an IR excess. Finally, since a CSE around
$\ell$ Car has been detected by interferometry in the infrared (Kervella et
al. 2006, 2009), we enable SPIPS to fit an IR excess in the following
analysis. SPIPS provides a fit to the photometry along the pulsation cycle,
which is in agreement with the observational data (description in Appendix 7).
From Fig. 7, we emphasize that the SPIPS fitted model is well constrained by
the numerous observations, and thus the physical parameters of the photosphere
are accurately derived. However, as noted in the SPIPS original paper, the
uncertainties on the derived parameters are purely statistical and do not take
into account systematics. While the fitting is satisfactory from the visible
domain to the near infrared, SPIPS fits an IR excess for wavelengths above
about 1.2$\,\mu$m. Indeed the observed brightness ($m_{\mathrm{obs}}$)
significantly exceeds the predictions ($m_{\mathrm{kurucz}}$). The averaged IR
excess over a pulsation cycle is presented in Fig. 1(a). In the next section
we combine this average IR excess with Spitzer mid-infrared spectroscopy.

(a) SPIPS cycle-averaged IR excess

(b) Reconstructed IR excess with SPIPS and Spitzer

Figure 1: (a) Averaged IR excess of $\ell$ Car as derived with the SPIPS
algorithm (red dots) and presented with the fitted ad-hoc analytic law
$\mathrm{IR}_{\mathrm{ex}}=-0.053(\lambda-1.2)^{0.501}\mathrm{mag}$. For each
photometric band (described in Appendix A), the red dots are the mean excess
value averaged over the cycle of pulsation of $\ell$ Car while the blue
uncertainties correspond to the respective standard deviation. (b) The IR
excess of $\ell$ Car is reconstructed at the specific phase of Spitzer
observation. It includes the averaged IR excess from SPIPS (red dots) and the
Spitzer observations (blue line) corrected for the silicate absorption of the
ISM along the line of sight (orange curve). The silicate extinction has been
derived using silicate refractive index from Draine & Lee (1984). Green and
gray vertical stripes represent the $LM$ and $N$ bands of MATISSE,
respectively.

###  2.2 Mid-IR excess from Spitzer observations

Space-based observations such as those of obtained with the Spitzer space
telescope are useful to avoid perturbation caused by the Earth atmosphere,
which is essential for the study of dust spectral features. We use
spectroscopic observations made with the InfraRed Spectrograph IRS (Houck et
al. 2004) on board the Spitzer telescope (Werner et al. 2004). High spectral
resolution (R=600) from 10$\,\mu$m to 38$\,\mu$m was used with Short-High (SH)
and Long-High (LH) modules placed in the focal plane instrument. The full
spectra were retrieved from the CASSIS atlas (Lebouteiller et al. 2011) and
the best flux calibrated spectrum was obtained from the optimal extraction
using differential method, which eliminates low-level rogue pixels
(Lebouteiller et al. 2015). Table 1 provides an overview of the Spitzer
observation. The reference epoch at maximum light (MJD0=50583.742), pulsation
period ($P$=35.557 days) and the rate of period change ($\dot{P}$=33.4$\pm
1.0\,$s/yr) used to derive the pulsation phase corresponding to the Spitzer
observation are those computed by the SPIPS fitting of $\ell$ Car. We note
that the most up-to-date and independent value of rate of period change,
derived by Neilson et al. (2016) ($\dot{P}=20.23\pm 1.38\,$s/y) is slightly
lower than the value estimated by SPIPS. However adopting this latter value
would not change the result of this paper.

Table 1:  Spitzer data set of $\ell$ Car (program number 40968). The Astronomical Observation Request (AOR), the date of observation, the corresponding Modified Julian Date (MJD), and the pulsation phase ($\phi$) are indicated. The physical parameters of $\ell$ Car derived by SPIPS at the Spitzer observation phase are indicated: the effective temperature ($T_{\mathrm{eff}}$), the surface gravity (log $g$) and the limb-darkened angular diameter ($\theta_{\mathrm{LD}}$). Statistical uncertainties on $T_{\mathrm{eff}}$ and $\theta$ are provided by SPIPS, whereas uncertainty on log $g$ is arbitrarily set to 10%. Object | AOR | Date | MJD | $\phi_{\textit{Spitzer}}$ | $T_{\mathrm{eff}}(\phi_{\textit{Spitzer}})$ | log $g(\phi_{\textit{Spitzer}})$ | $\theta_{\mathrm{LD}}(\phi_{\textit{Spitzer}})$
---|---|---|---|---|---|---|---
|  | (yyyy.mm.dd) | (days) |  | (K) | (cgs) | (mas)
$\ell$ Car | 23403520 | 2008.05.03 | 54589.502 | 0.648 | $4630^{+15}_{-15}$ | $0.98^{+0.10}_{-0.10}$ | $2.897^{+0.002}_{-0.002}$

In order to derive the IR excess of Spitzer observation, and correct for the
absorption of the interstellar matter (ISM), we followed the method developed
in Paper I. First, we derived the IR excess of $\ell$ Car at the specific
Spitzer epoch using

$\Delta\mathrm{mag}=m_{\textit{Spitzer}}-m_{\mathrm{kurucz}}[\phi_{\textit{Spitzer}}]$
(1)

where $m_{\textit{Spitzer}}$ is the magnitude of the Spitzer observation and
$m_{\mathrm{kurucz}}[\phi_{\textit{Spitzer}}]$ is the magnitude of the ATLAS9
atmospheric model interpolated at the phase of Spitzer observations
($\phi_{\textit{Spitzer}}$) using the parameters derived by SPIPS and given in
Table 1. We discarded wavelengths of Spitzer observations longer than
30$\,\mu$m due to extremely large uncertainties. Secondly, we corrected the
spectrum for ISM extinction by subtracting a synthetic ISM composed of
silicates from Draine & Lee (1984) (see orange curve in Fig. 1(b)). This
calculation assumes a relation between the extinction E($B-V$) derived by
SPIPS (i.e. E($B-V$)=0.148 mag) and the silicate absorption from diffuse
interstellar medium (see Eq. 4 in Paper I). Finally, we combine this result
with SPIPS averaged IR excess in order to reconstruct the IR excess from the
visible to the mid-infrared domain. The determined IR excess is presented in
Fig. 1(b) (red dots and blue line).
Similarly to the five Cepheids presented in Paper I, we observe a continuum IR
excess increasing up to $-$0.1 mag at 10$\,\mu$m. In particular within the
specific bands of MATISSE (see vertical strips in Fig. 1(b)), the IR excess is
between $-$0.05 and $-$0.1 mag at 3.5$\,\mu$m and 4.5$\,\mu$m ($L$ and $M$
bands) which represents an excess of 5 to 10% above the stellar continuum. In
addition, we find no silicate emission feature in the Spitzer spectrum around
10$\,\mu$m ($N$ band) within the uncertainties, which points toward an absence
of significant amount of circumstellar dust with amorphous silicate
components.

##  3 VLTI/MATISSE interferometric observations

MATISSE is the four-telescope beam combiner in the $L$, $M$ and $N$ bands of
the Very Large Telescope Interferometer (VLTI). The VLTI array consists of
four 1.8-meter auxiliary telescopes (ATs) and four 8-meter unit telescopes
(UTs), and provides baseline lengths from 11 meters to 150 meters. As a
spectro-interferometer, MATISSE provides dispersed fringes and photometries.
The standard observing mode of MATISSE (so-called hybrid) uses the two
following photometric measurement modes: SIPHOT for $L$ and $M$ bands, in
which the photometry is measured simultaneously with the dispersed
interference fringes, and HIGH SENS for $N$ band, in which the photometric
flux is measured separately after the interferometric observations. The
observations were carried out during the nights of 27 and 28 February 2020
with the so-called large configuration of the ATs quadruplet (A0-G1-J2-J3,
with ground baseline lengths from 58 to 132 m) in low spectral resolution
(R=$\lambda/\Delta\lambda\approx$30). The log of the MATISSE observations is
given in Table 2. The raw data were processed using the version (1.5.5) of the
MATISSE data reduction software 222The MATISSE reduction pipeline is publicly
available at http://www.eso.org/sci/software/pipelines/matisse/.. The steps of
the data reduction process are described in Millour et al. (2016).

The MATISSE absolute visibilities are estimated by dividing the measured
correlated flux by the photometric flux. Thermal background effects affect the
measurement in the $M$ and $N$ bands significantly more than in the $L$ band.
Thus, we use only the $L$ band visibilities in our modeling and analysis,
while the $L$ and $M$ photometries are used for deriving the observed total
flux (see Sect. 3.2). In the $N$ band, the total flux of $\ell$ Car
($\approx$17 Jy) is at the lower limit of the ATs sensitivity with MATISSE for
accurate visibility measurements ($\sim$ 20 Jy, as stated on the ESO webpage
of the MATISSE instrument). Thus, we rather use the correlated flux
measurements (not normalized by the photometric flux) as an estimate of the
$N$ band photometry of $\ell$ Car. In the following of the paper, we discarded
the 4.1 to 4.5$\,\mu$m spectral region, where the atmosphere is not
transmissive, and also the noisy edges of the atmosphere spectral bands. As a
result, we analyze the following spectral region in $L$ (3.1-3.75$\mu$m) for
absolute visibility and flux, $M$ (4.75-4.9$\mu$m) for flux, and in $N$
(8.2-12$\mu$m) for correlated flux.

###  3.1 Calibration of the squared visibility in the $L$ band

Calibrators were selected using the SearchCal tool of the JMMC333SearchCal is
publicly available at https://www.jmmc.fr/english/tools/proposal-
preparation/search-cal/, with a high confidence level on the derived angular
diameter ($\chi^{2}\leq 5$, see Appendix A.2 in Chelli et al. (2016)).
Moreover, calibrator fluxes in the $L$ and $N$ bands have to be of the same
order than for the science target (i.e. $\approx$10 to 100 Jy in $L$ and $N$)
to ensure a reliable calibration (Cruzalèbes et al. 2019). These flux
requirements induce the use of partially resolved calibrators at the 132-m
longest baseline. The $L$ band uniform-disk (UD) angular diameters for the
standard stars as well as the corresponding $L$ and $N$ band fluxes and the
spectral type are given in Table 3.
We analyze these calibration stars with a special care. The first calibrator
of this night, q Car, was observed in the frame of a different program and was
not used in the interferometric visibility calibration process. Indeed, it is
a supergiant star of spectral type KII, which makes its angular diameter of
5.2 mas too large (compared to $\ell$ Car), and rather uncertain in the $L$
band. However, q Car appears to be a suitable total flux calibrator for $\ell$
Car in both $L$ and $M$ bands (see Sect. 3.2). Indeed, although q Car is an
irregular long-period variable, its photometric variation in the visible is
$\Delta$mag$\approx$0.1
444http://www.sai.msu.su/groups/cluster/gcvs/gcvs/iii/iii.dat (GCVS, Samus’ et
al. 2017), thus its flux variation in the Rayleigh-Jeans domain, i.e. in the
$L$ and $M$ bands, only represents about 1%. Moreover q Car was observed at an
air mass very close to the one of $\ell$ Car (see Table 2). In the analysis,
we discarded the calibrator B Cen since it has inconsistent visibilities as
explained in Appendix B. Thus, we used $\epsilon$ Ant to calibrate observation
$\\#$1\. For observation $\\#$2, we also used $\epsilon$ Ant. For observation
$\\#3$, we bracketed the science target with the standard CAL-SCI-CAL strategy
calibration using $\beta$ Vol. The calibrated squared visibilities ($V^{2}$)
in $L$ band are presented in Fig. 2. The visibility curve associated with the
limb-darkened angular diameter of the star (without CSE), which was derived
from the SPIPS analysis at the specific phase of VLTI/MATISSE ($\phi$=0.07),
is shown for comparison.

Figure 2: The calibrated squared visibilities of $\ell$ Car as a function of
the spatial frequencies in the $L$ band for observations $\\#1$, 2 and 3. The
theoretical visibility curve corresponding to a uniform disk of
$\theta_{\mathrm{UD}}=2.887\pm 0.003\,$mas in the $L$ band, as derived from
the SPIPS analysis (i.e. without CSE) is indicated for comparison.  Table 2:
Log of the observations for nights of 27 February (observations $\\#$1 and 2)
and 28 February 2020 (observation $\\#$3) with the Modified Julian Date (MJD),
the pulsation phase $\phi$, the seeing (in arcsecond), the air mass (AM),
integrated water vapour (IWV) at 30 deg elevation (in millimeter) and the
coherence time $\tau_{0}$ (in millisecond) are indicated. Stars used in the
visibility calibration strategy are marked with an asterisk (see Sect. 3.1). q
Car and $\beta$ Vol are used to calibrate the total flux (see Sect. 3.2),
while e Cen is used to check the consistency of the other calibrators (see
Appendix B). The calibrator B Cen is not used, see Appendix B

.

# | Target | Date | $\phi$ | seeing | AM | $\tau_{0}$
---|---|---|---|---|---|---
|  | (MJD) |  | (′′) |  | (ms)
| q Car | 58907.11 |  | 0.78 | 1.40 | 6.0
1 |  $\ell$ Car* | 58907.12 | 0.05 | 0.66 | 1.32 | 5.3
| B Cen | 58907.13 |  | 0.85 | 1.39 | 6.1
2 |  $\ell$ Car* | 58907.14 | 0.05 | 1.15 | 1.27 | 6.0
|  $\epsilon$ Ant* | 58907.15 |  | 0.69 | 1.02 | 6.9
| e Cen | 58907.28 |  | 1.26 | 1.10 | 4.8
| bet Vol* | 58907.99 |  | 0.79 | 1.51 | 6.3
3 |  $\ell$ Car* | 58908.00 | 0.07 | 0.81 | 1.66 | 6.3
| bet Vol* | 58908.01 |  | 0.91 | 1.42 | 15.7

Table 3:  Calibrator properties. $\theta_{\mathrm{UD}}$ is the uniform disk (UD) angular diameter in $L$ band from the JSDC V2 catalogue (Bourgés et al. 2014), $F_{L}$ and $F_{N}$ are the flux in the $L$ and $N$ bands from the Mid-infrared stellar Diameters and Fluxes compilation Catalogue (MDFC) (Cruzalèbes et al. 2019). Calibrator |  $\theta_{\mathrm{UD}}$(mas) |  $F_{L}$ (Jy) |  $F_{N}$(Jy) | Sp. Type
---|---|---|---|---
q Car | 5.19$\pm$0.60 | 299.5$\pm$50.6 | 45.8$\pm 8.2$ | K2.5 II
B Cen | 2.54$\pm$0.28 | 72.3$\pm 11.8$ | 10.6$\pm 1.0$ | K3 III
$\epsilon$ Ant | 2.86$\pm$0.30 | 88.1$\pm 1.2$ | 12.2$\pm 1.8$ | K3 III
e Cen | 2.97$\pm$0.29 | 88.9$\pm 10.9$ | 13.2$\pm 1.6$ | K3.5 III
bet Vol | 2.90$\pm$0.29 | 95.4$\pm 23.5$ | 14.6$\pm 3.4$ | K2 III

####

MATISSE also provides closure phase measurements, which contain information
about the spatial centro-symmetry of the brightness distribution of the
source. For all the closure phase measurements, we find an average closure
phase of about 0∘ in the $L$, $M$ and the $N$ band from 8.2 to 9.25$\,\mu$m
(see Fig. 3). In the $N$ band we indicated the typical peak-to-valley
dispersion of the MATISSE closure phase measurements with an error bar
($\approx$10 degrees) and we discarded the spectral region beyond 10$\,\mu$m
due to extremely large uncertainties (see Fig. 3). Down to a sub-degree level
(resp. 10 degrees level), our closure phase measurements are consistent with
the absence of significant brightness spatial asymmetries in the environment
around $\ell\leavevmode\nobreak\ $Car in $L$ and $M$ bands (resp. $N$ band).
That justifies our use of centro-symmetric models in the following of the
paper.

(a) #1 : $LM$ bands

(b) #1 : $N$ band

(c) #2 : $LM$ bands

(d) #2 : $N$ band

(e) #3 : $LM$ bands

(f) #3 : $N$ band

Figure 3: Closure phase for observations #1, 2 and 3 for $LM$ and the $N$
bands for each ATs triplet. The double arrows in the $N$ band panels represent
the typical peak-to-valley variation. Figure 4: Averaged calibrated total flux
for observations $\\#1$, 2 and 3 in $LM$ bands plus the correlated flux in $N$
band for observation $\\#3$. Panels (a) and (b) refer to $LM$ and $N$ bands
respectively. SPIPS photosphere model is interpolated at the phase
corresponding to the MATISSE the observation with parameters:
$T_{\mathrm{eff}}(\phi_{\mathrm{obs}})$=$5014\pm 15\,$K; log
$g(\phi_{\mathrm{obs}})$=$0.96\pm 0.10$; and
$\theta_{\mathrm{UD}}(\phi_{\mathrm{obs}})$=$2.887\pm 0.003\,$mas. The Spitzer
spectrum used in Sect. 2.2 is plotted for comparison.

###  3.2 Flux calibration

In $L$ and $M$ bands we calibrate the total flux $F_{\mathrm{tot,sci}}$ of
$\ell$ Car using

$F_{\mathrm{tot,sci}}=\frac{I_{\mathrm{tot,sci}}}{I_{\mathrm{tot,cal}}}\times
F_{\mathrm{tot,cal}}$ (2)

where $I_{\mathrm{tot,sci}}$ and $I_{\mathrm{tot,cal}}$ are the observed total
raw flux of the science target and the calibrator respectively.
$F_{\mathrm{tot,cal}}$ is the known flux of the standard star. We calibrate
$\ell$ Car using standard stars with the closest air mass which are q Car and
$\beta$ Vol for the observations of the first and second night respectively.
$F_{\mathrm{tot,cal}}$ is given by their atmospheric templates from Cohen et
al. (1999). Since the air mass of both target and calibrator are comparable we
do not correct for the air mass. Moreover, while a chromatic correction exists
for the $N$ band (Schütz & Sterzik 2005) it is not calibrated (to our
knowledge) for the $L$ and $M$ bands. Then we average the three observations,
adding uncertainties and systematics between each measurement in quadrature.

Since $\ell$ Car is too faint in $N$ band for accurate photometry (and thus
absolute visibility) measurements, we use the correlated flux for the $N$
band, which is the flux contribution from the spatially unresolved structures
of the source. We calibrated the correlated flux of the science target
$F_{\mathrm{corr,sci}}$ following :

$F_{\mathrm{corr,sci}}=\frac{I_{\mathrm{corr,sci}}}{I_{\mathrm{corr,cal}}}\times
F_{\mathrm{tot,cal}}\times V_{\mathrm{cal}}$ (3)

where $I_{\mathrm{corr,sci}}$ and $I_{\mathrm{corr,cal}}$ are the observed raw
correlated flux of the science target and the calibrator respectively, and
$V_{\mathrm{cal}}$ the calibrator visibility. For such an absolute calibration
of the correlated flux, we need a robust interferometric calibrator with an
atmospheric template given by Cohen et al. (1999). Only $\beta$ Vol, observed
during the second night, meets these two requirements. Thus we only calibrate
the correlated flux measurements from observation $\\#3$ using the known flux
of $\beta$ Vol. The calibrated correlated fluxes from the different baselines
do not present any resolved features, allowing us to average the correlated
flux from the different baselines. Note that we have discarded the spectral
region between 9.3 and 10$\,\mu$m, due to the presence of a telluric ozone
absorption feature around 9.6$\,\mu$m that strongly impacts the data quality.
The calibrated total fluxes in $L$ and $M$ bands and the averaged calibrated
$N$ band correlated flux are presented in Fig. 4.

####

We draw here several intermediate conclusions. First, the agreement between
the measured flux with MATISSE, Spitzer and the SPIPS atmosphere model of
$\ell$ Car is qualitatively satisfactory within the uncertainties. However,
the MATISSE flux measurements have rather large uncertainties ($\geq$ 10%),
making it difficult to determine the IR excess of $\ell$ Car with a precision
at the few percent level. Secondly, we can see that the $N$ band correlated
flux follows a pure Rayleigh-Jeans slope. This indicates the absence of
silicate emission, which is consistent with the Spitzer data (Sect. 2.2). This
is also in agreement with previous MIDI spectrum given by Kervella et al.
(2009). Third, a resolved environment around $\ell$ Car is seen in the
visibility measurements in the $L$ band. Indeed, the calibrated visibilities
are significantly lower than the visibilities corresponding to a model of the
star without CSE, as derived from the SPIPS algorithm at the specific phase of
VLTI/MATISSE observations (see Fig. 2). Since the interferometric closure
phase is zero the resolved structure around the star is centro-symmetric. In
the next section we apply a centro-symmetric model of envelope on the observed
$V^{2}$ measurements in $L$ band in order to investigate the diameter and flux
contribution of the envelope.

##  4 Gaussian envelope model

In this section we fit a geometrical model on the measured $L$ band
visibilities. $\ell$ Car and its CSE are modeled with an uniform disk (UD) and
a surimposed Gaussian distribution, as in the previous studies on $\ell$ Car
CSE (Kervella et al. 2006, 2009). The UD diameter corresponds to the one at
the specific phase of the observation derived by SPIPS that is
$\theta_{\mathrm{UD}}=2.887\pm 0.003\,$mas in the $L$ band. To model the CSE,
we superimposed a Gaussian intensity distribution centered on the stellar
disk. This CSE Gaussian model has two parameters which are the CSE diameter
$\theta_{\mathrm{CSE}}$ taken as the Full Width at Half Maximum (FWHM), and
the CSE flux contribution normalized to the total flux. We then performed a
reduced $\chi_{r}^{2}$ fitting to adjust the total squared visibility of the
model $V^{2}_{\mathrm{tot}}$ over the MATISSE observation in $L$ band. We
computed the total squared visibility of the star plus CSE model using

$V_{\mathrm{tot}}^{2}(f)=\Big{(}F_{\star}\,\left\lvert
V_{\mathrm{UD}}(f)\right\rvert+F_{\mathrm{CSE}}\,\left\lvert
V_{\mathrm{CSE}}(f)\right\rvert\Big{)}^{2}$ (4)

where $f$=$B_{p}$/$\lambda$ is the spatial frequency ($B_{p}$ the length of
the projected baseline), $F_{\star}$ and $F_{\mathrm{CSE}}$ are the normalized
stellar and CSE flux contribution to the total flux respectively, normalized
to unity $F_{\star}$ \+ $F_{\mathrm{CSE}}$ = 1. $V_{\mathrm{UD}}(f)$ is the
visibility derived from the UD diameter of the star given by

$V_{\mathrm{UD}}(f)=2\frac{J_{1}(\pi\theta_{\mathrm{UD}}f)}{\pi\theta_{\mathrm{UD}}f}$
(5)

where $J_{1}$ is the Bessel function of the first order. $V_{\mathrm{CSE}}(f)$
is the visibility of a Gaussian intensity distribution

$V_{\mathrm{CSE}}(f)=\exp\Big{[}-\frac{(\pi\theta_{\mathrm{CSE}}f)^{2}}{4\mathrm{ln}2}\Big{]}.$
(6)

We perform the fit on observations $\\#1$, 2 and 3 simultaneously, since
$\ell$ Car is not expected to vary significantly between $\phi$=0.05 and 0.07.
The derived visibility model is shown in Fig. 5 and we present the best-fit
parameters in Table 4. The visibility model turns out to be consistent, within
the error bars, with the three observations. However, the reduced
$\chi_{r}^{2}$ is low (i.e. 0.2), thus the data are overfitted by the model
and the uncertainties on the best-fit parameters are underestimated. In that
case, to obtain more reliable uncertainties on the best-fit parameters we fit
independently the three observations and we take the uncertainty as the
standard deviation as of the resulted parameters. We resolve a CSE with a
radius of 1.76$\pm 0.28\,R_{\star}$ accounting for about 7.0$\pm 1.4\%$ of the
flux contribution in the $L$ band, that gives an IR excess of $-$0.07 mag.
Both the size and the flux contribution are in agreement with previous studies
which have found a compact environment of size 1.9$\pm$1.4$\,R_{\star}$ in $K$
band with VINCI and 3.0$\pm$1.1$\,R_{\star}$ in $N$ band with MIDI (Kervella
et al. 2006), with few percents in flux contribution in both cases. As noted
by Kervella et al. (2006), the large uncertainties in the $K$ band diameter is
due to a lack of interferometric data for baselines between 15 to 75 m. We
also note that cycle-to-cycle amplitude variations discovered in $\ell$ Car
(Anderson 2014; Anderson et al. 2016) could slightly affect the diameter in
the $K$ band. On the other hand Kervella et al. (2009) found extended emission
between 100 and 1000 AU ($\approx$100-1000$\,R_{\star}$) using MIDI and VISIR
observations in the $N$ band. Thus it is possible that the CSE have both a
compact and an extended component which are observable at different
wavelengths. The CSE flux contribution of 7% is also in agreement with the
MATISSE total flux within the uncertainties (see Fig. 4), as well as the
determined IR excess of about $-$0.05 to $-$0.1 mag we derived from Fig. 1(b)
in Sect. 2.2.

Table 4: Fitted parameters of a Gaussian CSE with flux contribution $F_{\mathrm{CSE}}(\%)$ and FWHM $\theta_{\mathrm{CSE}}$(mas) which is also expressed in stellar radius $R_{\mathrm{CSE}}$($R_{\star}$). Parameters $F_{\mathrm{CSE}}$ and $\theta_{\mathrm{CSE}}$ are weakly correlated with a correlation coefficient of 0.09. The uncertainties on the parameters are obtained using the standard deviation of the best-fit parameters for the three observations fitted independently. These results are compared with those obtained with VINCI in $K$ band and MIDI in $N$ band from Kervella et al. (2006). | This work | VINCI | MIDI
---|---|---|---
| $L$ | $K$ | $N$
$f_{\mathrm{CSE}}(\%)$ | 7.0$\pm$1.4 | 4.2$\pm$0.2 | -
$\theta_{\mathrm{CSE}}$(mas) | 5.06$\pm$0.81 | 5.8$\pm$4.5 | 8.0$\pm$3.0
$R_{\mathrm{CSE}}$($R_{\star}$) | 1.76$\pm$0.28 | 1.9$\pm$1.4 | 3.0$\pm$1.1
$\chi_{r}^{2}$ | 0.2 | 0.65 | -
Figure 5: Fitted Gaussian CSE around $\ell$ Car for the combined observations
$\\#1$, 2 and 3 in the $L$ band. The error on the visibility model is obtained
using the covariance matrix of the fitting result.

##  5 Physical origin of the IR excess

###  5.1 Dust envelope

The absence of emission features in the $N$ band in both MATISSE and Spitzer
spectra rules out the presence of dust from typical oxygen rich star
mineralogy with amorphous silicates or aluminum oxide, which present a
characteristic spectral shape in $N$ band. A pure iron envelope, which
presents a continuum emission, is also unlikely to be created. We already
considered then rejected these possibilities in Paper I (see the Figures 8 and
9 in it). We note that the presence of large grains of silicate, with a size
of about 1$\,\mu$m and more, would lead to a broader emission which could
completely disappear (Henning 2010). However the reason why large grains would
preferentially be created in the envelope of Cepheids remains to be explained.
In addition, considering the best-fit of the Gaussian CSE model, at a distance
of 1.76$\,R_{\star}$ from the stellar center, i.e. 0.76$\,R_{\star}$ from the
photosphere, the temperature would exceed 2000 K, therefore dust cannot
survive since it would sublimate (Gail & Sedlmayr 1999). These physical
difficulties to reproduce the IR excess with dust envelopes were also recently
pointed out by Groenewegen (2020) who fitted the SEDs of 477 Cepheids
(including $\ell$ Car) with a dust radiative transfer code.

###  5.2 A thin shell of ionized gas

In Paper I we suggested the presence of a thin shell of ionized gas, with a
radius of about 1.15$\,R_{\star}$, to explain the reconstructed IR excess of
five Cepheids. We use the same model for $\ell$ Car with a shell of ionized
gas, to test its consistency with IR excess and interferometric observations.
In this part we perform a reduced $\chi_{r}^{2}$ fitting of the shell
parameters on the IR excess reconstructed in Sect. 2.2. This simple model has
three physical parameters which are: the shell radius; the temperature; and
the mass of ionized gas. An additional parameter is applied as an offset to
the whole reconstructed IR excess from Fig. 1(b), allowing the model to
present either a deficit or an excess in the visible domain if necessary.
Indeed, the key point is when considering a shell of ionized gas, this shell
is expected to absorb the light coming from the star in the visible domain,
which is currently not considered in the SPIPS algorithm when reconstructing
the IR excess (the $V$ magnitude is forced to be zero, see Fig. 1(a)).
However, thanks to the VLTI/MATISSE measurements of flux contribution of the
CSE in the $L$ band (7%), we can fix this offset parameter consistently to
0.03 magnitude, to force the IR excess in $L$ band from SPIPS (that is
$\approx$10%) to be also 7%, within the uncertainties. We present the result
of the modeling ($\chi_{r}^{2}$=0.55) constrained by the spectrum of $\ell$
Car in Fig. 6. We find a thin shell radius of $R_{\mathrm{shell}}$=1.13$\pm
0.02\,R_{\star}$ which is lower than the Gaussian CSE of 1.76$\pm
0.28\,R_{\star}$ constrained by MATISSE observations. On the other hand this
model also reproduces the Spitzer spectrum better than the SPIPS predicted
photometries. In particular its value in the $L$ band is about 1% (see Fig. 6)
which is not consistent with the value derived using $L$ band visibilities
($\sim 7$%). This model has also derived weak near-infrared excess for several
stars in Paper I (see its Figure 11). This behaviour is physically explained
by the free-bound absorption of hydrogen before the free-free emission
dominates at longer wavelength. We have explored the spatial parameters to
test if a larger ionized envelope has the potential to match MATISSE
observations with the IR excess. However, our model is not able to accurately
reproduce all the observations. Also, even if our model is complex considering
the physics involved (free-free and bound-free opacities description), its
geometrical description is rather simple, with a constant density and
temperature distribution. Thus, it might not be adapted to the modeling of a
larger envelope.

###  5.3 Limit of the model

The derived parameters of the shell of ionized gas depend on the initial IR
excess derived by SPIPS. Any systematics in this calculation could in turn
affect the IR excess and the parameters of the shell of ionized gas. As noted
by Hocdé et al. (2020b), the derived distance of SPIPS would be systematically
affected if the shell of ionized gas is not taken into account, because of its
absorption in the visible. This systematic should be only few percents
($10^{\Delta\mathrm{m}/2.5}$) all parameters being unchanged. The direct
implementation of the gas model into the SPIPS fitting has to be tested in
forthcoming studies to correct this uncertainty. Moreover, Gallenne et al.
2021 (submitted) studied the ad-hoc IR excess derived by SPIPS for a larger
sample of stars and found that angular diameter and the colour excess are the
most impacted parameters when no ad-hoc IR excess model is included. However
in the case of $\ell$ Car these parameters appear to be well constrained by
angular diameter measurements, the reddening and the distance are also in
agreement with values found in the literature (see Appendix A).

Figure 6: $\ell$ Car IR excess fitting result of ionized gas shell following
the method described in Paper I. Yellow region is the error on the magnitude
obtained using the covariance matrix of the fitting result. We find a thin
shell of ionized gas with radius
$R_{\mathrm{shell}}$=1.13$\pm$0.02$\,R_{\star}$; temperature
$T_{\mathrm{shell}}$=3791$\pm$85$\,K$; mass of ionized gas $M$=9.10$\times
10^{-8}\pm$8.5$\times 10^{-9}\,$$\mathrm{M}_{\odot}\,$, using an ad-hoc offset
of +0.03 mag on the data in order to match the IR excess found by VLTI/MATISSE
and the one of SPIPS (see the text); and $\chi_{r}^{2}$=0.55. The subplot
shows the comparison between the flux contribution derived from the Gaussian
CSE (dashed line), the IR excess from SPIPS (red point) and IR excess from a
shell of ionized gas in the $LM$ band (red line).

###  5.4 Perspectives

An interesting physical alternative is to consider free-free emission produced
by negative hydrogen ion H-, with free electrons provided by metals. Indeed
for an envelope of 2$\,R_{\star}$ the average temperature is about 3000 K and
the hydrogen is neutral. Thus, in that case, most of the free electrons would
be provided by silicon, iron and magnesium which have a mean first ionization
potential of 7.89 eV. As neutral hydrogen is able to form negative hydrogen
ion, H- could generate an infrared excess with free-free emission. This
phenomenon is known to produce a significant IR excess compared to dust
emission in the extended chromosphere of cool supergiant stars such as
Betelgeuse (Gilman 1974; Humphreys 1974; Lambert & Snell 1975; Altenhoff et
al. 1979; Skinner & Whitmore 1987). Moreover, the photo-detachment potential
occurs at 1.6$\,\mu$m for the H- mechanism, thus free-free emission should
dominate the IR excess above this wavelength, as it is suggested by the SPIPS
cycle-averaged IR excess. This single photo-detachment has also the advantage
to avoid the important bound-free absorption in the near-infrared produced in
the preceding model. We thus suggest that the compact structure we resolved
around $\ell$ Car has the potential to produce IR excess through H- free-free
mechanism by analogy with extended chromosphere of cool supergiant stars.
Further investigations are necessary to confirm this hypothesis.

##  6 Conclusions

Determining the nature and the occurrence of the CSEs of Cepheids is of high
interest to quantify their impact on the Period-Luminosity relation, and also
for understanding the mass-loss mechanisms at play. In this paper, we
constrain both the IR excess and the geometry of the CSE of $\ell$ Car using
both Spitzer low-resolution spectroscopy and MATISSE interferometric
observations in the mid-infrared. Assuming an IR excess ad-hoc model, we used
SPIPS to derive the photospheric parameters of the star in order to compare
with Spitzer and MATISSE at their respective phase of observation. This
analysis allows to derive the IR excess from Spitzer and also to derive the
CSE properties in the $L$ band spatially resolved with MATISSE. These
observations lead to the following conclusions on the physical nature of
$\ell$ Car’s CSE:

  1. 1.

We resolve a centro-symmetric and compact structure in the $L$ band with
VLTI/MATISSE that has a radius of about 1.76$\,R_{\star}$. The flux
contribution is about 7%.

  2. 2.

We find no clear evidence for dust emission features around 10$\,\mu$m ($N$
band) in MATISSE and Spitzer spectra which suggests an absence of
circumstellar dust.

  3. 3.

Our dedicated model of shell of ionized gas better reproduces the mid-infrared
Spitzer spectrum and implies a size for the CSE lower than the one derived
from VLTI/MATISSE observations (1.13$\pm 0.02\,R_{\star}$ versus
$1.76\pm$0.28$\,R_{\star}$), as well as a lower flux in the $L$ band (1%
versus 7%).

  4. 4.

We suggest that improving our model of shell of ionized gas by including the
free-free emission from negative hydrogen ion would probably help reproducing
the observations, i.e. the size of the CSE and the IR excess, in particular in
the $L$ band.

While the compact CSE of $\ell$ Car is likely gaseous, the exact physical
origin of the IR excess remains uncertain. Further observations of Cepheids
depending on both their pulsation period and their location in the HR diagram
are necessary to understand the CSE’s IR excess. This could be a key to unbias
Period-Luminosity relation from Cepheids IR excess.

###### Acknowledgements.

The authors acknowledge the support of the French Agence Nationale de la
Recherche (ANR), under grant ANR-15-CE31-0012- 01 (project UnlockCepheids).
The research leading to these results has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program under grant agreement No 695099 (project CepBin). This
research was supported by the LP2018-7/2019 grant of the Hungarian Academy of
Sciences. This research has been supported by the Hungarian NKFIH grant
K132406. This research made use of the SIMBAD and VIZIER555Available at
http://cdsweb.u-strasbg.fr/ databases at CDS, Strasbourg (France) and the
electronic bibliography maintained by the NASA/ADS system. This research also
made use of Astropy, a community-developed corePython package for Astronomy
(Astropy Collaboration et al. 2018). Based on observations made with ESO
telescopes at Paranal observatory under program IDs: 0104.D-0554(A). This
research has benefited from the help of SUV, the VLTIuser support service of
the Jean-Marie Mariotti Center (http://www.jmmc.fr/suv.htm). This research has
also made use of the Jean-Marie Mariotti Center Aspro service 666Available at
http://www.jmmc.fr/aspro.

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##  Appendix A SPIPS data set and fitted pulsational model of the star sample.

Figure 7 is organized as follows: pulsational velocity, effective temperature
and angular diameter curves according to the pulsation phase are shown on the
left panels, while the right panels display photometric data in various bands.
Above the figure, the projection factor set to $p=1.270$ is indicated, along
with the fitted distance $d$ using parallax-of-pulsation method, the fitted
color excess E($B-V$), and the ad-hoc IR excess law. In this work we
arbitrarily set the p-factor to 1.27 following the general agreement around
this value for several stars in the literature (Nardetto et al. 2004; Mérand
et al. 2005; Nardetto et al. 2009). However, there is no agreement for the
optimum value to use, and it is also the case for $\ell$ Car. In the case of
this star (Nardetto et al. 2007, 2009) found a p-factor of 1.28$\pm$0.02 and
1.22$\pm$0.04 using different observational techniques which both are in
agreement with 1.27 within about 1 sigma. Thus, a possible systematic cannot
be excluded although the distance derived by SPIPS ($d=519.2\pm 4.5\,$pc) is
in agreement for example with the distance obtained using Gaia parallaxes
(505$\pm$28 pc) (Gaia Collaboration et al. 2018) or the one derived using a
recent period-luminosity calibrated for Cepheids in the Milky Way (544$\pm$32
pc) (Breuval et al. 2020).

Gallenne et al. (2021, submitted) have shown that the absence of IR excess
model in the SPIPS fit can affect the derivation of the angular diameter and
the colour excess. In our fit using IR excess model, we emphasize the good
agreement of the angular diameter model with the data (panel b), and also the
consistency of the derived colour excess with the one given by Stilism 3D
extinction map (Capitanio et al. 2017), that is $0.148\pm 0.006$ versus
$0.116\pm 0.26$. In the photometric panels, the gray dashed line corresponds
to the magnitude of the SPIPS model without CSE. It actually corresponds to
the magnitude of a Kurucz atmosphere model, $m_{\mathrm{kurucz}}$, obtained
with the ATLAS9 simulation code from Castelli & Kurucz (2003), with solar
metallicity and a standard turbulent velocity of 2 km/s. These models are
indeed suitable for deriving precise synthetic photometry (Castelli 1999), in
comparison with MARCS models giving rough flux
estimates777https://marcs.astro.uu.se/. The gray line corresponds to the best
SPIPS model, which is composed of the latter model without CSE plus an IR
excess model. In the angular diameter panels, the gray curve corresponds to
limb-darkened (LD) angular diameters. For stars with solar metallicity, when
effective temperature is low enough, CO molecules can form in the photosphere
and absorb light in the CO band-head at 4.6 $\mu m$ (Scowcroft et al. 2016).
This effect is observed in the Spitzer I2 IRAC dataset. In this case, these
data were ignored during the fitting of SPIPS. Panels presenting a horizontal
blue bar contain only one data with undetermined phase of observation, thus
these photometric mid-infrared bands were not used in the IR excess
reconstruction in Sect. 2.1 and 2.2

Figure 7: The SPIPS results of $\ell$ Car. Velocity: Anderson (2014).
Effective temperature: No data. Angular diameter: Kervella et al. (2006).
Photometry: Madore (1975), Pel (1976), Neugebauer et al. (1984), Laney &
Stobie (1992), ESA (1997), Berdnikov & Turner (2002), Smith et al. (2004),
Ammons et al. (2006), Wright et al. (2010), Ishihara et al. (2010), Monson et
al. (2012), Gaia Collaboration et al. (2018).

##  Appendix B Verification of the diameters of the calibrators

In order to verify the consistency of the standard stars of the first night
(27 February), we consider all calibrators available during the night (B Cen,
e Cen, $\epsilon$ Ant) and we inter-calibrate them. Thus, each calibrator is
calibrated using the two others. We performed six different calibrations in
total, following Table 5 and Fig. 8. The results are also compared with the
known values from the literature.

From these results, the diameters of $\epsilon$ Ant and e Cen are in excellent
agreement with those given by the JSDC when they are calibrated from one to
another, and the visibilities are well fitted by an uniform disk (UD) model
(see Figures 8(a) and 8(b)). On the contrary, we find important
inconsistencies when deriving the diameter of B Cen or using it as a
calibrator (see Figures 8(c), 8(d), 8(e) and 8(f)). In particular this
standard star is suspected to be 20$\%$ larger tan the diameter given by the
JSDC ($\approx\,$3 mas instead of 2.5 mas). Moreover, a simple UD model seems
to be unsuitable for fitting the observed visibilities (see black curve in
Figures 8(e) and 8(f)). On the other hand the diameter of B Cen is also well
established by interferometry in the $K$ band (Richichi et al. 2009) and was
used many times as an interferometric calibrator (Kervella et al. 2006;
Kervella 2007). We emphasize that the transfer function was very stable during
this night, thus this result cannot be attributed to the atmospheric
conditions (see coherence time in Table 2). Since the origin of this
discrepancy remains unknown, this analysis prevented us for using B Cen to
calibrate the Cepheid $\ell$ Car.

Table 5:  Diameters in milliarcseconds obtained by the inter-calibration between B Cen, $\epsilon$ Ant and e Cen. The targets of the calibration are indicated in the upper part of the table together with their angular diameter in $L$ band from the JSDC catalogue in parentheses. The first column indicates the calibrator used to obtain the derived UD angular diameters in each cell. The red cells represent discrepant values compared to the JSDC catalogue at more than 1$\,\sigma$. This deviation from the derived $\theta_{\mathrm{UD}}$ to the JSDC value is derived by $|\theta_{\mathrm{UD}}-\leavevmode\nobreak\ \theta_{\mathrm{JSDC}}|/\sigma_{\mathrm{JSDC}}$. | e Cen |  $\epsilon$ Ant | B Cen
---|---|---|---
| (2.97$\pm$0.29) | (2.86$\pm$0.30) | (2.54$\pm$0.28)
e Cen | - | 2.91$\pm$0.01 |  red!25 3.09$\pm$0.02
$\epsilon$ Ant | 2.96$\pm$0.01 | - |  red!25 3.10$\pm$0.02
B Cen |  red!25 2.41$\pm$0.02 |  red!25 2.39$\pm$0.02 | -

(a) $\epsilon$ Ant calibrated with e Cen

(b) e Cen calibrated with $\epsilon$ Ant

(c) $\epsilon$ Ant calibrated with B Cen

(d) e Cen calibrated with B Cen

(e) B Cen calibrated with $\epsilon$ Ant

(f) B Cen calibrated with e Cen

Figure 8: Inter-calibration of the different calibrators and comparison with
JSDC diameters. (a) and (b) : $\epsilon$ Ant and e Cen diameters are in
agreement with the JSDC when they are calibrated from each other. (c) and (d)
: On the contrary the diameter of $\epsilon$ Ant and e Cen are inconsistent
with the JSDC catalogue when they are calibrated with B Cen. (e) and (f) : B
Cen diameter is inconsistent with the JSDC catalogue.