11institutetext: I. Physikalisches Institut, Universität zu Köln, Zülpicher
Str. 77, 50937, Köln, Germany
11email<EMAIL_ADDRESS>22institutetext: Max-Planck-Institute for
Radio Astronomy, Auf dem Hügel 69, 53121 Bonn, Germany 33institutetext: Max
Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
44institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
55institutetext: Unidad Mixta Internacional Franco-Chilena de Astronomía
(CNRS, UMI 3386), Departamento de Astronomía, Universidad de Chile, Camino El
Observatorio 1515, Las Condes, Santiago, Chile 66institutetext: Dublin
Institute for Advanced Studies, 31 Fitzwilliam Place, D02 XF86 Dublin, Ireland
77institutetext: School of Physics, University College Dublin, Belfield,
Dublin 4,Ireland 88institutetext: Instituto de Astronomía, Universidad
Nacional Autónoma de México, Apdo. Postal 70264, Ciudad de México, 04510,
México 99institutetext: CENTRA, Centro de Astrofísica e Gravitação, Instituto
Superior Técnico, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
1010institutetext: Universidade do Porto, Faculdade de Engenharia, Rua Dr.
Roberto Frias, 4200-465 Porto, Portugal 1111institutetext: Universidade de
Lisboa - Faculdade de Ciências, Campo Grande, 1749-016 Lisboa, Portugal
1212institutetext: Max Planck Institute for Extraterrestrial Physics,
Giessenbachstrasse, 85741 Garching bei München, Germany 1313institutetext:
LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne
Universités, UPMC Univ. Paris 06, Univ. Paris Diderot,Sorbonne Paris Cité,
France 1414institutetext: Sterrewacht Leiden, Leiden University, Postbus 9513,
2300 RA Leiden, The Netherlands 1515institutetext: European Southern
Observatory, Karl-Schwarzschild-Str. 2, 85748, Garching bei München, Germany
1616institutetext: INAF – Osservatorio Astronomico di Capodimonte, via
Moiariello 16, 80131 Napoli, Italy

# The GRAVITY Young Stellar Object survey

VIII. Gas and dust faint inner rings in the hybrid disk of HD141569

GRAVITY Collaboration: V. Ganci GRAVITY is developed in a collaboration by the
Max Planck Institute for Extraterrestrial Physics, LESIA of Paris Observatory
and IPAG of Université Grenoble Alpes / CNRS, the Max Planck Institute for
Astronomy, the University of Cologne, the Centro de Astrofísica e Gravitação
and the European Southern Observatory.1122 L. Labadie 11 L. Klarmann 33 A. de
Valon 44 K. Perraut 44 M. Benisty 4455 W. Brandner 33 A. Caratti o Garatti
3366771616 C. Dougados 44 F. Eupen 11 R. Garcia Lopez 336677 R. Grellmann 11
J. Sanchez-Bermudez 3388 A. Wojtczak 11 P. Garcia 991010 A. Amorim 991111 M.
Bauböck 1212 J.-P. Berger 44 P. Caselli 1212 Y. Clénet 1313 V. Coudé du
Foresto 1313 P.T. de Zeeuw 12121414 A. Drescher 1212 G. Duvert 44 A. Eckart
1122 F. Eisenhauer 1212 M. Filho 991010 F. Gao 1212 E. Gendron 1313 R. Genzel
1212 S. Gillessen 1212 G. Heissel 1313 T. Henning 33 S. Hippler 33 M. Horrobin
11 Z. Hubert 44 A. Jiménez-Rosales 1212 L. Jocou 44 P. Kervella 1313 S. Lacour
1313 V. Lapeyrère 1313 J.-B. Le Bouquin 44 P. Léna 1313 T. Ott 1212 T. Paumard
1313 G. Perrin 1313 O. Pfuhl 1515 G. Heißel 1313 G. Rousset 1313 S.
Scheithauer 33 J. Shangguan 1212 T. Shimizu 1212 J. Stadler 1212 O. Straub
1212 C. Straubmeier 11 E. Sturm 1212 E. van Dishoeck 12121414 F. Vincent 1313
S.D. von Fellenberg 1212 F. Widmann 1212 J. Woillez 1515

(Received xx, xxxx; accepted xx, xxxx)

###### Abstract

Context. The formation and evolution of planetary systems impact the evolution
of the primordial accretion disk in its dust and gas content. HD 141569 is a
peculiar object in this context as it is the only known pre-main sequence star
characterized by a hybrid disk. Observations with 8 m class telescopes probed
the outer-disk structure showing a complex system of multiple rings and outer
spirals. Furthermore, interferometric observations attempted to characterize
its inner 5 au region, but derived limited constraints.

Aims.  The goal of this work was to explore with new high-resolution
interferometric observations the geometry, properties, and dynamics of the
dust and gas in the internal regions of HD 141569.

Methods.  We observed HD 141569 on milliarcsecond scales with GRAVITY/VLTI in
the near-infrared (IR) at low ($R\sim$20) and high ($R\sim$4000) spectral
resolution. We interpreted the interferometric visibilities and spectral
energy distribution with geometrical models and through radiative transfer
techniques using the code MCMax to constrain the dust emission. We analyzed
the high spectral resolution quantities (visibilities and differential phases)
to investigate the properties of the Brackett-$\gamma$ (Br$\gamma$) line
emitting region.

Results. Thanks to the combination of three different epochs, GRAVITY resolves
the inner dusty disk in the K band with squared visibilities down to $V^{2}$
$\sim$ 0.8. A differential phase signal is also detected in the region of the
Br$\gamma$ line along most of the six baselines. Data modeling shows that an
IR excess of about 6% is spatially resolved and that the origin of this
emission is confined in a ring of material located at a radius of $\sim$1 au
from the star with a width $\lesssim\,$0.3 au. The MCMax modeling suggests
that this emission could originate from a small amount ($1.4\times
10^{-8}\,$M⊕) of quantum-heated particles, while large silicate grain models
cannot reproduce at the same time the observational constraints on the
properties of near-IR and mid-IR fluxes. The high spectral resolution
differential phases in the Br$\gamma$ line clearly show an S-shape that can be
best reproduced with a gaseous disk in Keplerian rotation, confined within
0.09 au (or 12.9 R⋆). This is also hinted at by the double-peaked Br$\gamma$
emission line shape, known from previous observations and confirmed by
GRAVITY. The modeling of the continuum and gas emission shows that the
inclination and position angle of these two components are consistent with a
system showing relatively coplanar rings on all scales.

Conclusions. With a new and unique observational dataset on HD 141569, we show
that the complex disk of this source is composed of a multitude of rings on
all scales. This aspect makes HD 141569 a potentially unique source to
investigate planet formation and disk evolution in intermediate-mass pre-main
sequence stars.

###### Key Words.:

Proto-planetary disk – interferometry – near infra-red – stars: individual (HD
141569)

##  1 Introduction

The formation and evolution of protoplanetary disks are directly linked to
planet formation. The outer disk features of young stellar objects (YSOs) have
been thoroughly studied in the past through scattered light imaging (e.g.,
with SPHERE/VLT, Beuzit et al. 2019) and with ALMA in the (sub-)millimeter
range (ALMA Partnership et al., 2015). With both techniques, disk observations
have shown rings, gaps, and asymmetric structures up to a few hundred au
(e.g., Lodato et al., 2019; Benisty et al., 2017). The inner regions (at
$\sim$au scale) of such disks are also of prime interest since key processes
like gas accretion flows, winds, outflows, and dust sublimation take place.
All these processes affect the dynamics and evolution of the first few au
regions where terrestrial planets may form and/or migrate over few million
years. Constraints on these processes can be derived indirectly through
spectroscopic studies, but at typical distances of a few hundred parsecs only
observations with milliarcsecond (mas) resolution, which are required to probe
sub-au scales, can discriminate between competing models. Numerous
interferometric studies have been conducted in the past in the near- and the
mid-infrared (IR), for instance with IOTA (Millan-Gabet et al., 2001), the
Palomar Testbed Interferometer (PTI; Eisner et al., 2004), the Keck
Interferometer (Monnier et al., 2005; Eisner et al., 2014), and the VLTI (Menu
et al., 2015; Lazareff et al., 2017; Gravity Collaboration et al., 2019). To
date, the new generation of four-telescope instruments including GRAVITY,
operating in the K band (Gravity Collaboration et al., 2017), and MATISSE,
operating in L to N band (Lopez et al., 2014), are pushing further the
achievable spectral coverage, the sensitivity, and precision of
interferometric measurements. Even though statistical studies of large YSO
samples are of high relevance (Lazareff et al., 2017; Gravity Collaboration et
al., 2019), some of these objects require a more in-depth study enabled by the
improved data quality of recent interferometric observations. Some objects,
like HD 141569, are unusual in their evolutionary sequence and require
dedicated studies. For such systems the exact nature and properties of the
very inner regions are still a matter of debate and can be solved in part by
modeling the distribution of the warm dust traced in the K band. Furthermore,
it is still unclear how to characterize the star–disk mechanisms traced by
emission lines like the hydrogen Brackett-$\gamma$
(Br$\gamma\,\sim\,$8000-10000 K) and CO (T$\,\sim\,$2000-3000 K) bandheads,
and where this emission occurs. The high-quality spectroscopic capabilities of
the GRAVITY instrument allows us to study in detail the gas phase and its
spatial morphology thanks to the interferometric visibilities and differential
phase signals (Gravity Collaboration et al., 2020a, b, 2021).
HD 141569 is a Herbig star classified as a B9-A0 spectral type (Augereau &
Papaloizou, 2004), with an effective temperature of $9750\pm 250$ K, an
estimated age of $7.2\pm 0.02$ Myr, a luminosity between $16.60\pm 1.07\,$L⊙
(Vioque et al., 2018) and $27.0\pm 3.6\,$L⊙ (Di Folco et al., 2020), a mass of
$2.14\pm 0.01$ M⊙, and a GAIA distance of $110\pm 1$ pc (Arun et al.,
2019)111The recent distance estimate in the new EDR3 of Gaia suggests
111.6$\pm$0.4 pc. Considering the very close value to DR2, we decided to keep
the former distance throughout the paper.. It is a non-flaring disk system
with little mid-IR excess classified as a group-II source (Meeus et al.,
2001). It is the only known pre-main sequence star characterized by a hybrid
disk (Wyatt et al., 2015; Péricaud et al., 2017; Di Folco et al., 2020), an
evolutionary disk state between the protoplanetary and debris-disk regimes.
Near-IR imaging spatially resolved an optically thin disk consisting of two
rings located at about $\sim 280$ and $\sim 455$ au from the star Augereau et
al. (1999); Weinberger et al. (1999); Biller et al. (2015). A more complex
system is shown in the visible, consisting of multiple rings and outer spirals
that could be explained through perturbations by two nearby ($\sim 7.5$
arcsec) M dwarfs, or by planetary perturbations (Augereau & Papaloizou, 2004;
Wyatt, 2005; Reche et al., 2009). Fisher et al. (2000) found a warm disk
component up to 110 au at 10.8 and $18.2\,\mu$m, later confirmed by Marsh et
al. (2002). The short-wavelength counterpart of this component was detected by
Mawet et al. (2017) through L′ imaging and ranging between 20 and 85 au.
Emission at $8.6\,\mu$m was detected by Thi et al. (2014), and was interpreted
as emission from polycyclic aromatic hydrocarbons (PAHs). NOEMA and ALMA
observations in the millimeter range showed continuum emission equally shared
between a compact ($\lesssim 50$ au) and a smooth extended dust component
($\sim 350$ au), with large millimeter grains dominating the inner regions and
smaller grains in the outer ones (Di Folco et al., 2020). Finally, inner disk
features were detected by SPHERE in the Y, J, H, and K bands (Perrot et al.,
2016) and by Keck/NIRC2 in the L’ band (Currie et al., 2016) at physical
separations of 45, 61, and 88 au. These results point out the high
morphological complexity of the outer disk in the HD 141569 system.
Little is known about the central astronomical units of the system. The
spectral energy distribution (SED) of HD 141569 alone does not help us in this
sense since the IR excess is very small (see Fig. 9 of Thi et al. 2014). The
majority of the K-band measurements listed in Table 4 reflect a featureless
SED in the near-IR. Moreover, pure SED fits obtained by different authors (Li
& Lunine, 2003; Merín et al., 2004; Thi et al., 2014) may suggest at first
that the near-IR emission is exclusively photospheric in nature and that the
disk contributes only at longer wavelengths. The object was observed at
milliarcsecond resolution in the K band with the PTI and the Keck
interferometer, but it was spatially unresolved (Eisner et al., 2004, 2009).
Monnier et al. (2005) derived a 10 mas upper limit in radius for the spatial
extension of the K-band emission. Therefore, trustworthy information on the
first $5\,$au of the system are scarce, and the question arises of whether the
inner region of the disk could be already in a debris disk stage, where the
SED fits in the near-IR are not accurate enough to detect such a faint excess.
The circumstellar gas has been observed in both atomic and molecular form,
which suggests the system has not yet reached the gas-depleted stage
characteristic of a debris disk system. Mendigutía et al. (2017) set an upper
limit of $\sim 0.11$ au for the gas region responsible for the spatially
unresolved double-peaked H$\alpha$ emission. A comparable upper limit of $\sim
0.13$ au for the Br$\gamma$ line emitting region is suggested by Eisner et al.
(2009). Both lines are observed to be not variable over timescales of days and
years (Eisner et al., 2015; Mendigutía et al., 2011b). In addition to
hydrogen, CO ro-vibrational emission ($v\geq 1$, $\Delta\,v=1$) was observed
by many authors extending from 10 to 275 au (Dent et al., 2005; Goto et al.,
2006; Brittain et al., 2007; Flaherty et al., 2016; White et al., 2016; Miley
et al., 2018; Di Folco et al., 2020).
We present here the first GRAVITY interferometric observations of this disk,
with the goal of revealing the geometry and dynamics of the internal structure
of HD 141569, and of gaining insights about the dust and gas properties.
Section 2 describes the observations; Section 3 and Section 4 present the
observational data and the adopted methodology; Section 5 describes the
results of the possible scenarios along with the corresponding modeling; A
discussion is developed in Section 6.

Figure 1: HD 141569 FT data; squared visibilities (left panel), closure phases
(central panel), and U-V plane coverage (right panel), from all the
observation epochs. Colors refer to the different GRAVITY spectral channels.

##  2 Observations

HD 141569 was observed with VLTI/GRAVITY (Gravity Collaboration et al., 2017)
using the four 1.8 m Auxiliary Telescopes (ATs), on March 18 and July 12,
2019, in the intermediate D0-G2-J3-K0 configuration, and on May 23, 2019, in
the large A0-G1-J2-J3 configuration. The observations span a spatial frequency
range between about 15 M$\lambda$ and 65 M$\lambda$ (see Fig. 1, left panel)
with a maximum angular resolution of $\lambda/2B$, about 1.7 mas for the
longest baseline (B) of 130 m, which corresponds to about 0.19 au at a
distance of 110 pc. The data consist in high spectral resolution
(R$\,\sim\,$4000) observables recorded by the science channel (SC) detector
over the whole K band with individual integration times of 30 s and in low
spectral resolution (R$\,\sim\,$20) observables recorded by the fringe tracker
(FT) detector (five spectral channels over the K band at 1.908, 2.058, 2.153,
2.256, 2.336 $\mu$m) at frame rates of $\approx$ 300 and $\approx$ 900 Hz
(Lacour et al., 2019). Each observation block corresponds to 5 minutes on the
object. In total, three files were acquired in March 2019, one in May 2019,
and eight in July 2019. HD 141569 observations were preceded by the
observation of a point-source calibration star, close to our object on the
sky, and of similar spectral type and brightness in order to calibrate the
atmospheric and instrumental transfer function. We observed the calibrator HD
137006 in March and July 2019, and the calibrator HD 141977 in May 2019.
Further details on the executed runs and observing conditions are given in
Table 3.

##  3 Data

All the data were reduced and calibrated using the GRAVITY data reduction
software (Lapeyrere et al., 2014). For the low-resolution FT data we discarded
the first spectral channel, which can typically be affected by the metrology
laser operating at 1.908 $\mu$m. Figure 1 shows the U-V plane coverage and the
FT calibrated squared visibilities and closure phases (right, left, and center
panel, respectively). Following Gravity Collaboration et al. (2019), we
applied a floor value on the error bars of 2% for the squared visibilities and
1∘ on the closure phases as the error bars computed by the pipeline might be
underestimated or correlated. We observe that GRAVITY partially resolved the
near-IR emission in HD 141569 with squared visibilities between 0.8 and 1.0.
Therefore, the data can be used to estimate the characteristic size of the
dust environment (see Section 5.1). Moreover, with the inclusion of the May
2019 large configuration data, we observe that the visibility reaches a
plateau at almost all spatial frequencies, allowing us to constrain near-IR
flux contributions of the star and environment.

Figure 2:  Science channel data of July 2019. Top left: Wavelength-calibrated
continuum-normalized spectrum, corrected for telluric lines. Top right: Same
as top left, but corrected for the Br$\gamma$ photospheric absorption.
Differential phases (left column) and visibilities (right column) along the
six GRAVITY baselines. The red lines in the visibility plots show the pure-
line visibilities.

Since the closure phases are consistent with 0∘ at all baselines and for all
the epochs, we can confidently consider the emission to be centro-symmetric on
the spatial scale of our observations, and we therefore discard the hypothesis
of a close companion as the origin of the resolved emission, at least within
the 250 mas ($\sim$28 au) field of view of GRAVITY with the ATs.
For the high-resolution SC data, we concentrated on the July 2019 dataset only
since this is the epoch where we gathered the highest number of files. In
order to optimally exploit the SC dataset, the eight files from July 2019 have
been merged in order to increase the signal-to-noise ratio per spectral
channel. Considering that the maximum span in position angle between the two
extreme positions of the UV coverage is only $\sim$4 ∘, we do not expect any
visibility smearing of the data due to differences in hour angles. The error
bars were computed as the standard deviation between the eight files of the
corresponding differential quantity (visibility or differential phase) in each
spectral channel. For instance, we derived the absolute error on the
visibility to about 1.8%. Figure 2 shows the visibility amplitude (left
panels) and the differential phase (right panels) for the six baselines in the
region of interest of the Br$\gamma$ line, between 2.15 and 2.18 $\mu$m. The
top panels show the object spectrum normalized to the continuum, corrected for
telluric lines (left plot), and corrected for both telluric lines and
photospheric absorption (right plot). The visibilities appear to be spectrally
flat with no clear signature at 2.16612 $\mu$m. They are measured to vary
between 0.9 and 0.96 as a function of the baseline, which is indicative of a
compact region well inside the dusty disk. Interestingly, the differential
phase signal is more marked at the position of the Br$\gamma$ line. We observe
a clear S-shaped signal through baselines J3-D0 (100.1 m, 220∘) and J3-K0
(53.6 m, 152∘), and a weaker signal through baselines J3-G2 (62.1 m, 223∘) and
D0-G2 (38.2 m, 36∘). No clear signal is detected within the error bars through
the baselines D0-K0 (95.1 m, 72∘) and G2-K0 (68.1 m, 92∘).
In order to study the Br$\gamma$ line gas region using the SC high spectral
resolution data, we need precise measurements for the line-to-continuum flux
ratio. It is essential to perform a proper wavelength calibration of the
spectrum and to take out the contribution by the telluric lines. We describe
the whole procedure in Appendix C. Errors are from the original data, reduced
and calibrated through the GRAVITY data reduction software.
The high-resolution GRAVITY spectrum of HD 141569 (see top panels of Fig. 2)
shows a double-peaked Br$\gamma$ emission line. Since the error associated
with the GRAVITY wavelength calibration is $\sim 3\,$Å, the two peak positions
can be considered to be symmetric with respect to the Br$\gamma$ wavelength
rest position. Both the double-peaked emission line and the S-shaped feature
in the differential phase suggest a scenario where the gas emitting in the
Br$\gamma$ line could be in Keplerian rotation. We explore this hypothesis
further in the following sections.

##  4 Methodology

The properties of the spatially resolved continuum emission were first
investigated with the help of low spectral resolution FT data (see Sects. 4.1
and 5.1). The squared visibility curve was modeled through chromatic
geometrical models accounting for a point-like central star, and simple
geometrical shapes (Gaussian geometrically thin rings, Gaussian-convolved
infinitesimally thin rings) representing the circumstellar environment. Useful
information was obtained such as the star-to-dust flux ratio, the dust
spectral index, and the spatial distribution of the dust. To gain further
information on the dust emission properties, we investigated through radiative
transfer (RT) modeling the impact of such a component on the SED (see Section
5.2). We used the RT code MCMax (Min et al., 2009), which solves 2D RT (e.g.,
Bjorkman & Wood 2001) to calculate the dust density and temperature structure
of a given disk setup. In Sects. 4.2 and 5.3 we discuss the high-spectral
resolution SC data in the Br$\gamma$ region used to constrain the spatial
scale of the hot gas emitting component through modeling of the visibility
curves, and through the analysis of the Br$\gamma$ spectrum under the
assumption of a gas disk in Keplerian rotation. Further information on the gas
region size and its dynamical properties were derived through the analysis of
the differential phases and the resulting photocenters shifts. Finally, an
analytical axisymmetric Keplerian disk model is compared to our observations.

###  4.1 Dust continuum: low spectral resolution data

Following the work of Lazareff et al. (2017) and Gravity Collaboration et al.
(2019), we used geometric models that consist of a point-like central star,
assumed to be unresolved at all observed baselines, and a circumstellar
environment in order to fit the observed visibilities. The complex visibility
of the system at spatial frequencies $(u,\varv)$ and at wavelength $\lambda$
is therefore described by a linear combination of the two components as

$V(u,\varv,\lambda)=\frac{F_{\rm s}(\lambda)\,V_{\rm
s}(u,\varv,\lambda)+F_{\rm c}(\lambda)\,V_{\rm c}(u,\varv,\lambda)}{F_{\rm
s}(\lambda)+F_{\rm c}(\lambda)},$ (1)

where $V_{\rm c}$ is the visibility of the circumstellar environment, and
$F_{\rm s}$ and $F_{\rm c}$ the specific fractional flux contributions of the
star and of the circumstellar environment, respectively. The visibility of the
star $V_{\rm s}$ is equal to 1 since we assume it to be unresolved. Since our
GRAVITY FT data contains six visibility measurements and four closure phases
for each of the four spectral channels and for each file, we can derive the
spectral dependence of the circumstellar environment by modeling it as a power
law, defined by its spectral index $k_{\rm c}$, where
$k=d\,\mathrm{log}\,F_{\lambda}/d\,\mathrm{log}\,\lambda$, and by describing
the complex visibility of the system as

$V(u,\varv,\lambda)=\frac{F_{\rm s}\,(\lambda/\lambda_{0})^{k_{\rm s}}+F_{\rm
c}\,(\lambda/\lambda_{0})^{k_{\rm c}}\,V_{\rm c}(u,\varv)}{F_{\rm
s}\,(\lambda/\lambda_{0})^{k_{\rm s}}+F_{\rm c}\,(\lambda/\lambda_{0})^{k_{\rm
c}}},$ (2)

where $\lambda_{0}=2.15\,\mu$m is the wavelength of the central spectral
channel of the FT, and $k_{\rm s}$ the spectral index of the star derived
assuming that it radiates as a black body at the star effective temperature
$T_{\mathrm{eff}}$=9750 K, which translates into s spectral index $k_{\rm
s}=-3.62$ at $\lambda_{0}$ for the central star.
We chose to fit our visibility data with three different geometric models that
differ only by the $V_{\rm c}$ term in Eq. 2. Since the closure phases are
basically zero for every baseline, as shown in the central panel of Fig. 1, we
do not consider any azimuthal modulation in our models. Therefore, the
resulting brightness distributions are centro-symmetric.
The first model consists of a Gaussian disk whose visibility is described,
from Berger & Segransan (2007), as

$V_{\rm c}(u,\varv)=V_{\mathrm{gauss}}(u,\varv)={\rm
exp}\left(-\frac{(\pi\Theta r)^{2}}{4\,\mathrm{ln}2}\right),$ (3)

where $\Theta$ is the Gaussian full width at half maximum (FWHM) and
$r=\sqrt{u^{2}+\varv^{2}}=B/\lambda$, with $u$ and $v$ the spatial frequency
coordinates, and B the projected baseline.
The second model consists of a geometrically thin ring whose visibility is
described by subtracting an inner smaller uniform disk from a larger one:

$V_{\rm c}(u,\varv)=V_{\mathrm{ring}}(u,\varv)=\frac{F_{\rm outer}\,V_{\rm
outer}-F_{\rm inner}\,V_{\rm inner}}{F_{\rm outer}-F_{\rm inner}}.$ (4)

Here the subscript outer refers to the larger disk and the subscript inner
refers to the inner hole. After normalization and knowing that $F_{\rm
disk}=\pi\,D^{2}/4$, where $D$ is the diameter of the disk, we can express Eq.
4 as

$V_{\mathrm{ring}}(u,\varv)=\frac{D_{\mathrm{outer}}\,2\,\frac{J_{1}(\pi\,D_{\mathrm{outer}}\,r)}{\pi\,r}-D_{\mathrm{inner}}\,2\,\frac{J_{1}(\pi\,D_{\mathrm{inner}}\,r)}{\pi\,r}}{D_{\mathrm{outer}}^{2}-D_{\mathrm{inner}}^{2}},$
(5)

where $J_{1}$ is the first-order Bessel function. Finally, our last model
consists of an infinitesimally thin ring convolved by a Gaussian whose
visibility is described as the product between $V_{\mathrm{gauss}}(u,\varv)$
and

$V_{\mathrm{inf-ring}}(u,\varv)=J_{0}(2\pi ar),$ (6)

where $J_{0}$ is the zero-order Bessel function and $a$ is the radius of the
infinitesimally thin ring (Berger & Segransan, 2007). Since in the Fourier
space the convolution of two functions is simply their multiplication, we have

$V_{\rm c}(u,\varv)=V_{\mathrm{gauss-
ring}}(u,\varv)=V_{\mathrm{gauss}}(u,\varv)\cdot V_{\mathrm{inf-
ring}}(u,\varv).$ (7)

The inner hole radius and ring width are defined as $r_{i}=D_{\rm inner}/2$
and $w=(D_{\rm outer}-D_{\rm inner})/2$, respectively, for the geometrically
thin ring model while they are defined as $r_{i}=a-\Theta/2$ and $w=\Theta$,
respectively, for the Gaussian-convolved infinitesimally thin ring.
Inclination and position angle of the circumstellar environment are taken into
account through the parameter $r=\sqrt{u^{2}+\varv^{2}}$ following Berger &
Segransan (2007). The model fitting is based on a Markov chain Monte Carlo
(MCMC, Foreman-Mackey et al. 2013) numerical approach and was implemented on
the combined dataset of all three epochs in order to maximize the number of
experimental points against the number of free parameters. This assumes that
the near-IR emission and the disk structures are not variable over a five-
month period, which is strengthened by the fact that the star is not variable,
either spectroscopically in the optical (Mendigutía et al., 2011a) or
photometrically in the mid-IR (Kóspál et al., 2012). Once a global solution
was identified, we further checked how well the parameters are indeed
constrained by the data. For this purpose, we performed a series of squared
visibility fits to the model by fixing the tested parameter to different
values and left the other parameters free in the subsequent minimization. In
this way we obtain a $\chi^{2}$ curve as a function of the tested parameter
value.
Finally, the error on the $\chi^{2}$ was estimated by treating the quantity Ti
as a stochastic variable with

$\rm{T}_{i}=\rm{N}\,(\frac{\rm{y_{i}}-\rm{y_{model}}}{\rm{\sigma_{i}}})^{2},$
(8)

where $\rm{N}$ is the number of points in the dataset, $\rm{y_{i}}$ the
individual measurement, $\rm{y_{model}}$ the value of the model, and
$\rm{\sigma_{i}}$ the error associated to the measurement. The $\chi^{2}$
value is given by the mean of Ti,

$\chi^{2}=\frac{1}{\rm{N}}\sum_{\rm{i}=1}^{\rm{N}}\rm{T_{i}}=\sum_{\rm{i}=1}^{\rm{N}}(\frac{\rm{y_{i}}-\rm{y_{model}}}{\rm{\sigma_{i}}})^{2},$
(9)

and the $\chi^{2}$ error is given by the error on the mean of T${}_{\rm i},$

$\sigma_{\chi^{2}}=\sqrt{\frac{\sum_{\rm{i}=1}^{\rm{N}}\rm{T}_{i}^{2}}{\rm{N}(\rm{N}-1)}-\frac{(\chi^{2})^{2}}{\rm{N}-1}}.$
(10)

This applies to the reduced $\chi^{2}$ as well.

###  4.2 Gas: High spectral resolution data

To estimate the gas region size from the SC visibility we extrapolated the
pure-line contribution from the total visibility (line+continuum) displayed in
Fig. 2. To do this we modeled the total visibility with a three-component
model that accounts for the contributions from the star, the circumstellar
dust, and the line emitting gas. The total visibility is given therefore by

$\displaystyle V_{\rm tot}(u,\varv,\lambda)=$
$\displaystyle\frac{\alpha(\lambda)\,F_{\rm s}(\lambda)\,V_{\rm
s}(u,\varv,\lambda)+F_{\rm c}(\lambda)\,V_{\rm c}(u,\varv,\lambda)+F_{\rm
L}(\lambda)\,V_{\rm L}(u,\varv,\lambda)}{\alpha(\lambda)\,F_{\rm
s}(\lambda)+F_{\rm c}(\lambda)+F_{\rm L}(\lambda)},$ (11)

where $\alpha(\lambda)$ is the science star continuum-normalized photospheric
absorption (see Appendix C for more details), the subscript $c$ refers to the
dust component, and the subscript L refers to the Br$\gamma$ line gas. From
Eq. 4.2 it can be proven (see Appendix D) that the pure-line visibility is
given by

$\displaystyle V_{\rm L}(u,\varv,\lambda)=$ $\displaystyle\frac{V_{\rm
tot}(u,\varv,\lambda)\,\\{[1+\beta(\lambda)]\,F_{\rm
L/C}-1-\beta(\lambda)\\}-\alpha(\lambda)+1}{[1+\beta(\lambda)]\,F_{\rm
L/C}-\alpha(\lambda)-\beta(\lambda)},$ (12)

where $F_{\rm L/C}$ is the line-to-continuum flux ratio

$F_{\rm L/C}=\frac{\alpha(\lambda)\,F_{\rm s}(\lambda)+F_{\rm
c}(\lambda)+F_{\rm L}(\lambda)}{F_{\rm s}(\lambda)+F_{\rm c}(\lambda)}$ (13)

and $\beta(\lambda)$ is the disk-to-star flux ratio outside the line

$\beta(\lambda)=\frac{F_{\rm c}}{F_{\rm s}}(\lambda)=\frac{F_{\rm c}}{F_{\rm
s}}\,\left(\frac{\lambda}{\lambda_{0}}\right)^{k_{\rm c}-k_{\rm s}}.$ (14)

For clarity, we note that, outside the line emitting region, $\alpha(\lambda)$
and $F_{\rm L}(\lambda)$ tend toward 1 and 0, respectively. Equation 13
corresponds to the line-to-continuum ratio including the photospheric
absorption. Finally, we estimated the gas region size by modeling it with an
infinitesimal ring model given by Eq. 6.
In the same way, we needed to take out the continuum contribution from the
total differential phases (Fig. 2). Following Weigelt et al. (2011), the pure-
line differential phase is given by

$\mathrm{sin}\,\phi_{\rm L}=\mathrm{sin}\,\phi\,\frac{F_{\rm tot}\,V_{\rm
tot}}{F_{\rm L}\,V_{\rm L}}=\mathrm{sin}\,\phi\,\frac{V_{\rm tot}}{V_{\rm
L}}\,\frac{F_{\rm L/C}}{F_{\rm L/C}-(\frac{\alpha+\beta}{1+\beta})},$ (15)

where $\phi_{\rm L}$ is the pure-line differential phase, $\phi$ the total
differential phase, $F_{\rm tot}$ the total flux (star, dust, and gas), and we
write the ratio $F_{\rm tot}/F_{\rm L}$ through Eq. 22. Following Le Bouquin
et al. (2009), we can derive wavelength-dependent photocenter displacements
along each baseline from the pure-line differential phases by

$\mathbf{p}=\frac{-\phi_{\rm L}}{2\pi}\cdot\frac{\lambda}{\mathbf{B}},$ (16)

where $\mathbf{p}$ is the projection on the baseline $\mathbf{B}$ of the 2D
photocenter vector with origin on the central star. The error bars on the
pure-line differential quantities are computed through error propagation in
Eq. 4.2 and Eq. 15.

##  5 Results

###  5.1 Disk component inside 2 au

Table 1: FT squared visibility best-fit solution for the geometrically thin
ring model and the Gaussian-convolved ring model.

Geometrically thin ring |  |  | Gaussian-convolved ring |  |  |
---|---|---|---|---|---|---
Parameter | Unit | Fit solution | Parameter | Unit | Fit solution | Scan range
$F_{\rm s}$ | [%] | $93.81\pm 0.07$ | $F_{\rm s}$ | [%] | $93.81\pm 0.07$ | [0 ; 100%]
$F_{\rm c}$ | [%] | $6.19\pm 0.07$ | $F_{\rm c}$ | [%] | $6.19\pm 0.07$ |  1 - $F_{s}$
$k_{\rm c}$ |  | $-0.35\pm 0.21$ | $k_{\rm c}$ |  | $-0.35\pm 0.21$ | [-4 ; 4]
$r_{i}$ | [mas (au)] |  $7.40\pm 0.21$ (0.81) | $r_{i}$ | [mas (au)] |  $7.35\pm 0.21$ (0.81) | [0 ; 60]
$w$ | [mas (au)] |  $0.35\pm 0.35$ (0.04) | $w$ | [mas (au)] |  $0.24\pm 0.24$ (0.03) | [0 ; 100]
$i$ | [deg] | $58.47\pm 1.55$ | $i$ | [deg] | $58.45\pm 1.55$ | [30 ; 80]
$PA$ | [deg] | $-1.77\pm 1.11$ | $PA$ | [deg] | $-1.78\pm 1.11$ | [-40 ; 40]
$\chi_{r}^{2}$ |  | $4.67\pm 0.33$ | $\chi_{r}^{2}$ |  | $4.67\pm 0.33$ |

222$F_{\rm s}$ is the stellar flux contribution, $F_{\rm c}$ the dusty
circumstellar environment flux contribution, $k_{\rm c}$ the dust spectral
index at 2.15 $\mu$m, $r_{i}$ the ring inner hole radius, $w$ the ring width
($r_{i}$ and $w$ defined in different ways for the two models, see Section
4.1), $i$ the ring inclination from face-on, $PA$ the northeast position
angle, and $\chi_{r}^{2}$ the reduced chi-square. The uncertainties on the
fitted parameter correspond to the 1$\sigma$ error. Scan ranges refer to both
models.

Of the three models discussed in Sect. 4.1, the fit of the squared
visibilities to the Gaussian disk model (Eq. 3) did not converge to a solution
(i.e., the marginal posterior distributions for the Gaussian width,
inclination, and position angle are flat). Therefore, we discarded this model
in the rest of the work. Solutions were found for the geometrically thin ring
model (Eq. 5) and the Gaussian-convolved ring model (Eq. 7). The two models
converge basically toward the same solution, with a reduced $\chi_{r}^{2}$=
4.7 for both models.
We performed a wide scan range of the fitted parameters to find convergence
toward a global solution. The results of the minimization are presented in
Fig. 11 and Table 2 for the ring model along with the parameter scan range and
the 1$\sigma$ uncertainties. The resulting MCMC posterior distribution is
presented in Fig. 11 (top plots) and allows us to identify an optimal global
solution for the six parameters. The fitting process leads to a photospheric
near-IR flux contribution of $\sim 93.8\%$, and therefore to a dust ring flux
contribution of $\sim 6.2\%$ for both models. Interestingly, the degeneracy
typically found between the disk’s flux and the characteristic size in V2 is
broken here because the constant plateau as a function of spatial frequencies
unambiguously determines the level of the disk’s flux contribution. Both
models predict a spectral index $k_{\rm c}$ for the dust ring with a value of
$-0.35\pm 0.2$. To better illustrate the visibility plateau and the expected
modulation due to the modeled thin ring, we show in the bottom plots of Fig.
11 three visibility curves corresponding to the best model for three selected
baseline orientations and for a fixed wavelength value of 2.15 $\mu$m.
Regarding the geometrical shape of the circumstellar environment, both models
lead to a ring inclined from face-on by $58.5^{\circ}$, with a northeast
position angle $PA\sim 0^{\circ}$. The inner hole radius is estimated to be
$r_{i}\approx 7.4$ mas ($0.8\,$au) from both models, while the ring width is
estimated to be $\sim 0.24-0.35$ mas ($0.03-0.04\,$au, for the Gaussian-
convolved ring model and the geometrically thin ring model, respectively).
Importantly, Fig. 12 shows the $\chi_{r}^{2}$ curves of each parameter for the
geometrically thin ring model, which helps us to evaluate how well each
parameter is constrained by our data. The star near-IR flux contribution is
very well constrained as we expected from the plateau seen in the squared
visibility curve. The ring spectral index is more loosely constrained, since
the best value is consistent with values ranging between $-2$ and $2$. The
ring inclination is constrained between $\sim$ 45∘ and $\sim$75∘, while the
position angle is less well constrained with two possible minima at $\sim$ 0∘
and 120∘, the former considered to be the absolute minimum. The inner hole
radius is constrained to be inside the first $2\,$au of the system, with a
global minimum found around $0.8\,$au (7.4 mas) and a second (almost equally
possible) solution, at $\sim$1.7 au (15.4 mas). Taking into account the upper
limit of 10 mas for the radius of the K-band emission found by Monnier et al.
(2005), we decided to adopt $r_{i}$=0.8 au. According to Fig. 12, the ring
width $w$ tends toward small values, not larger than $\sim\,$0.3 au. This is
discussed further in Sect. 6.1.

###  5.2 Dust properties through radiative transfer modeling

To strengthen the obtained results and to assess the scenario of an inner ring
as close as $0.8\,$au, gaining further information on the emission properties,
we investigated through RT modeling the impact of such a component on the SED.
We note here that our aim was not to perform a detailed mineralogy study of
the system, but rather to understand how the detected inner dust is consistent
with both the near-IR flux and overall SED of the system.
We modeled the multiple and complex outer rings with only three rings based on
the results from Thi et al. (2014). In their model, the lower limit particle
size was set at 0.5 $\mu$m and the upper one at 0.5-1 cm for the two outermost
rings and the innermost one, respectively. The grain size follows a
distribution $\propto\,a^{-3.5}$, the surface density profile is a modified
version from Li & Lunine (2003), and the flaring index is $\gamma=1$. The
three rings peak at $\sim 15$, 185, and 300 au, with the first two rings
separated by a 75 au gap.
Our initial disk setup consists of a disk structure similar to Thi et al.
(2014), but with updated stellar parameters (see Sect. 1) and a grain
population based on DIANA standard dust grains (Woitke et al., 2016)
containing 75% amorphous silicates (e.g., Mg0.7Fe0.3SiO3), 25% porosity, and
no amorphous carbon. Our modified grain size distribution and surface density
is described in next paragraph. The computed SEDs account for both thermal
emission and scattered light contributions.
A silicate dust ring:  First, we attempted to reproduce the near-IR excess
detected by GRAVITY by including dust grains close to the star, but also
taking into account the fact that HD 141569 does not show a silicate emission
feature at 10 $\mu$m (Seok & Li, 2017), which is in part connected to the
grain size distribution. We tested several models of the inner ring with the
following properties: a varying lower-limit dust grain size of 0.6, 1.2, 2.5,
5.0, 10, 20, 40, 80, 158, and 316 $\mu$m; an equal upper-limit grain size of 1
cm, with a size distribution $\propto$ $a^{-3.5}$; a surface density $\propto$
$r^{-1}$; and an inner ring radius fixed at 0.8 au with a width of 0.04 au
according to our best-fit model.
All models with grains smaller than $\sim$ 20 $\mu$m in the inner $\sim$ 1 au
region can be tuned to reproduce the $\sim$ 6% near-IR excess, but at the same
time they still exhibit a clear 10 $\mu$m silicate feature, which is not in
adequacy with the observations.
Models accounting only for grains larger than 40 $\mu$m result in an almost
complete quenching of the 10 $\mu$m silicate feature. When testing the mass at
$\sim$ 1 au required not to exceed the mid-IR flux, we find that 10-10 M⊙ (or
3.3$\times$10-5 M⊕) would be compliant with this condition, but only a $\sim$
1% near-IR excess is generated. On the other hand, the dust mass required to
reach a $\sim$6% near-IR excess is well beyond 8.5$\times 10^{-10}$M⊙ (or
2.8$\times$10-4 M⊕), but this then produces a too large mid-IR emission
inconsistent with the known SED.
Finally, decreasing the percentage of silicates and increasing the carbon
percentage in the dust grains up to 25 % did not improve the SED fit.
Obtaining a more precise fit to the global SED would require further analysis
and tuning of the outer ring contribution in the mid-IR, which is beyond the
immediate goal of the paper. However, our modeling seems to point out that it
is difficult to reconcile the level of near- and mid-IR excess reported with a
model of solely silicate dust in the disk ring at $\sim$ 1 au. Three
representative cases of our modeling are presented in Fig. 14.

Table 2: List of parameters relevant to the RT modeling of the HD 141569 disk.
This model only describes the case of a QHP-dominated inner ring.

Star parameters |  |  |  |  |
---|---|---|---|---|---
Distance | [pc] | 110 |  |  |
Temperature | [K] | 9750 |  |  |
Luminosity | [L⊙] | 19 | (see Table notes) |  |
AV | [mag] | 0.095 |  |  |
Rings parameters
Parameter | Unit | 1st | 2nd | 3rd | 4th
Inner rim | [au] | 0.8 | 5 | 185 | 300
Outer rim | [au] | 0.84 | 110 | 500 | 500
Particle type |  | QHP | Silicate | Silicate | Silicate
Dust mass | [M⊕] | $1.4\times 10^{-8}$ | 0.020 | 1.665 | 0.233
Smallest particle size | [$\mu$m] | 0.006 | 1.26 | 0.63 | 20
Biggest particle size | [$\mu$m] | 0.006 | 10000 | 10000 | 10000
Inner surface density power law |  | -1 | -1 | -1 | -1
Size distribution power law |  | -3.5 | -3.5 | -3.5 | -3.5
Scale-height at 100 au | [au] | 1.4 | 1.4 | 7.5 | 7.5
Ring inclination | [degree] | 59 | 59 | 59 | 59
333The stellar luminosity has been revisited in this work as follows. We
determined a lower and upper limit of that value by matching in the K band the
photospheric flux plus the near-IR excess with the 2 MASS photometry within
its 5% uncertainty (see Table 4). This provided a stellar luminosity between
18 and 19 L⊙, in agreement with the revised value by Vioque et al. (2018).

A ring of quantum heated particles:  Another way to produce near-IR emission
consistent with the absence of the prominent 10 $\mu$m silicate feature and
with the presence of the mid-IR PAH bands is to consider quantum heated
particles (QHPs, Purcell 1976; Draine & Li 2001). In the context of
interferometric observations, this scenario was invoked for HD 100453 where
QHPs were detected in the disk gap (Klarmann et al., 2017), and for HD 179218
with the presence of hot QHPs inside the disk cavity (Kluska et al., 2018).
We tested different amounts of QHPs (from $10^{-13}$ to $10^{-15}\,$M⊙), and
two different QHP particles sizes ($10^{5}$ and $5\times 10^{5}$ carbon
atoms), based on the results obtained by Klarmann et al. (2017).

The highest masses ($\sim$ 10${}^{-13}\,$M⊙) produce an inner rim emission
which results in a near-IR excess that is too large and inconsistent with the
GRAVITY measurement. The smallest value ($\sim$ $10^{-15}\,$M⊙) corresponds
instead to an optically thin disk with negligible excess at 2 $\mu$m. For a
ring geometry in agreement with our best fit of the GRAVITY data, a sweet spot
is found for a mass of 4.3$\times$10${}^{-14}\,$M⊙ (or
1.4$\times$10${}^{-8}\,$M⊕) for which the resulting disk produces a near-IR
excess of $\sim$ 7 %. Smaller particle sizes (e.g., 102 carbon atoms) would
require a larger mass reservoir of QHPs to reach a near-IR excess of $\sim$
6%. As a consequence, a higher mass would result in stronger mid-IR PAH
features overestimating the HD 141569 SED observed in the Spitzer IRS spectrum
(Sloan et al., 2005). The order of magnitude of $\sim 10^{5}$ carbon atoms per
particle appears consistent with the SED profile estimated by GRAVITY and
IRS/Spitzer.
The resulting disk SED shows a spectral index (d $\mathrm{log}F_{\lambda}$/d
$\mathrm{log}\lambda$) of -1.4 at 2.15 $\mu$m which is consistent with the
minimization curve of the spectral index in Fig. 12. The details of the QHP
model parameters is shown in Table 3. In Fig. 15 we show the density and
temperature structure of the full-disk model. For the silicate dust in the
outer second ($\sim 15\,$au), third ($\sim 185\,$au), and fourth ring ($\sim
300\,$au), the equilibrium temperature varies between 125 K at $\sim$5 au to
20 K in the outermost disk regions. The QHPs used to model the innermost
optically thin ring are not in thermal equilibrium. Their temperature
distribution depends on their size (the smaller, the hotter) and on the
strength of the local ultraviolet (UV) radiation field as they absorb UV
photons to quickly re-emit in the near-IR, changing their temperature
drastically and very fast. The dark red color in Fig. 15 shows the location of
the QHPs, and according to our RT simulations their temperatures are in the
range between 1865 K and 95 K.

We find that the strongest constraint on the ring width is set by our
interferometric measurement. When considering widths from 0.04 to 0.3 au in
our RT modeling, the observed tendency remains the same: the population of
silicate dust grains does not satisfactorily reproduce the near- and mid-IR
excess, contrary to a population of a stochastically heated small grains.

Beside the ring’s width, the important result of this analysis suggests that a
tenuous, QHP-dominated, optically thin inner ring may provide a suitable
description of the close circumstellar environment of HD 141569 in agreement
with existing detailed modeling of the outer disks. On the contrary, a
silicate-dominated dusty inner ring, heavier by about four orders of magnitude
and composed of large grains, fails to provide a satisfying description of the
system, in particular in terms of flux contribution in the mid-IR spectral
range. Figure 3 presents the final result of our MCMax modeling with the
parameters of Table 3. We note that the strong PAH feature at 7.8 $\mu$m in HD
141569 is not seen in this model because the corresponding opacity has not
been added to our models of the outer rings, unlike the models of Thi et al.
(2014).

Figure 3: MCMax SED (black line) of the model described in Table 3. The yellow
circles are HD 141569 photometric data listed in Table 4. The blue line
represents the star SED modeled as a black body. The cyan line represents the
QHPs SED and the red line the silicates SED. The magenta line is the total
disk SED accounting for both QHP and silicate emission. The top right plot
focuses on the first $10\,\mu$m wavelength range of the SED.

###  5.3 Spatial scale of the Br$\gamma$-line emitting region

We exploited the high-spectral resolution data of GRAVITY in the Br$\gamma$
region to constrain the spatial scale of the hot gas emitting component,
following the formalism in Sect. 4.2. Conservatively, the resulting pure-line
visibilities plotted in red in Fig. 2 are very close to 1 for all the
baselines. Considering the error bars we can say that the gas region is at the
limit of spatially unresolved emission. We therefore propose to constrain the
size of the gas emitting region by considering the gas emitting at the
Br$\gamma$ line wavelength peaks (i.e., 2.1654 $\mu$m and 2.1672 $\mu$m) using
an infinitesimally thin ring model and estimating the upper-limit size that
would exceed the error bar of $\sim$2% on the pure-line visibilities. In this
way, we estimated a maximum radius of $\sim$ 0.35 mas (0.0385 au) for the gas
emitting region.
While the analysis of the visibility amplitudes only provides us an estimate
of the size scale of the gas emitting region, further information on the
spatial and kinematic properties of the gas component is found in the
differential phase signal. Typically, differential phases provide information
on photocenter displacements along the baselines on angular scales that can
surpass the nominal resolution of the interferometer. Figure 4 shows the
GRAVITY pure-line differential phases. After removal of the continuum
contribution (cf. Sect. 4.2), the S-shape becomes clearly visible for the
baselines J3-D0 (100 m, 220∘) and J3-K0 (54 m, 151∘) around the Br$\gamma$
line, while a weaker trend is seen for the baselines J3-G2 (62 m, 223∘) and
D0-G2 (38 m, 36∘). The strongest signature shows an amplitude in the
differential phase exceeding $\sim$20∘ for J3-K0. Since the differential phase
signals for the baselines D0-K0 (95 m, 72∘) and G2-K0 (68 m, 92∘) are
consistent with zero at all wavelengths, we fixed the pure-line differential
phases to 0∘. The typical uncertainties after correcting for the continuum
subtraction are $\sim 4^{\circ}$. The resulting deprojected photocenter shifts
per spectral channel, with reference frame fixed to the star location, are
shown in Fig. 5. We clearly observe that all points are aligned along the same
direction with an angle of -10${}^{\circ}\pm 7^{\circ}$. The redshifted points
are located along northwest, while the blueshifted points are located toward
the southeast. Based on the redshifted maximum extent of the photocenter
shifts, we estimated the radius of the gas region from the differential phases
to be $0.333\pm 0.039$ mas or $0.037\pm 0.004$ au. This appears consistent
with the less precise upper-limit size set through the analysis of the
visibility amplitudes. We recall however that the size estimate derived
through the photocenter shift does not correspond to the physical outer radius
of the gas region, but to the size where the gas emission is more intense for
a given wavelength.

Figure 4: HD 141569 spectrum (top), total (black circles and line), and pure-
line differential phases (colored signs) along the different baselines. D0-K0
and G2-K0 baseline pure-line differential phases are set to zero. The colors
refer to the different spectral channels. Dashed magenta lines represent the
pure-line differential phases of the analytical Keplerian disk model described
in Section 5.3. Figure 5: Deprojected photocenter shifts. The colors refer to
the different spectral channels and velocities, as shown in Fig. 4. The dashed
black line, derived through a linear fit of the photocenter shifts, represents
the gas region position angle.

The distribution of the 2D photocenter solution can be interpreted to the
first order as being caused by a gas disk in Keplerian rotation orbiting HD
141569. Under this assumption the analysis of the Br$\gamma$ emission line’s
shape provides further clues to the gas kinematics. We can indeed derive an
estimate of the gaseous disk’s radius from the separation between the two
peaks of the line and the rest position. From Beckwith & Sargent (1993) we use

$R_{g}=\frac{\rm{G\,M}_{\star}}{v_{obs}^{2}}\,\rm{sin^{2}}i,$ (17)

where $R_{g}$ is the radius, G is the gravitational constant, M⋆ the mass of
the star, $v_{obs}$ the projected velocity at the line peaks, and $i$ the disk
inclination. The $128\pm 42$ km/s average peaks shift of the Br$\gamma$ line
with respect to the line rest position leads to an outer limit for the gas
region of $0.766\pm 0.554$ mas in radius, or $0.084\pm 0.061$ au. The
resulting error accounts for the uncertainty on the stellar mass (0.01 M⊙), on
the distance (1 pc), on the disk inclination (15∘), and on the peak position
(3 Å), the last being the dominant one.
The three approaches presented above and based on the analysis of the
visibility amplitudes, differential phases, and the spectrum all seem
consistent with a gas component in Keplerian rotation confined within $\sim
0.8$ mas ($\sim 0.09$ au, $\sim 12.9$ R⋆) in radius. We compared our
differential phase signals, our strongest measurable quantity, to a simple
geometrical model of an axisymmetric disk in Keplerian rotation built with two
thin layers that account for the top and bottom sides of the disk,
parameterized by an inner radius $r_{\rm in}$ (varying from 0.008 to 0.03 au),
an outer radius $r_{\rm out}$ (varying from 0.033 to 0.8 au), and a power-law
exponent $\alpha$ (varying from 0 to 4.0) for the disk’s intensity radial
profile following $I(r)\propto\,r^{-\alpha}$. The inclination of the disk was
fixed to 58.5∘, based on the results of the FT data analysis and under the
assumption of coplanar dust-gas rings, and its position angle was fixed to
-10∘, based on the photocenter shift analysis. The hypothesis of an optically
thin disk is made, which implies that the disk intensity does not strongly
depend on the disk scale height, which is then fixed to ${\rm H/R}=0.1$
throughout the ring, but only on the surface area of the disk: $I\propto dS$
where $dS$ is an area element (for a complete description of the model, see de
Valon et al. in preparation). The model delivers a spectral profile that is
eventually normalized and fitted to our experimental double-peaked spectrum in
Fig. 6. A grid of 2700 models has been explored, and a best-fit model is found
for $r_{\rm in}$=0.011 au, $r_{\rm out}$=0.09 au, and $\alpha$=0.5. From this
best-fit model, we produced 2D velocity maps (Fig. 7) and retrieved the
theoretical pure-line differential phase signal (see Fig. 4). Our Keplerian
disk model reproduces qualitatively well the observed pure-line differential
phases both in the orientation of the sine wave relative to the wavelength and
in amplitude reinforcing the Keplerian gas disk scenario. We also observe that
the weakest signals indeed correspond to the baselines D0-K0 and G2-K0 for
which the peak amplitudes are $\sim$3–5∘. On a final note, for the best-fit
solution model we tested the optically thick hypothesis by accounting for
projection effects on the emissivity law,
$I\propto\mathbf{dS}\cdot\mathbf{e_{y}}=dS\cdot(\mathbf{n}\cdot\mathbf{e_{y}})$,
where $\mathbf{e_{y}}$ is the line-of-sight unit vector and $\mathbf{n}$ the
normal unit vector of the area element considered (see de Valon et al. in
preparation). The resulting model leads to a spectrum that is similar to that
derived from the optically thin model (a small difference is seen at the
lowest velocities), and to pure-line differential phases and photocenters
shifts consistent with those of the optically thin model.

Figure 6: HD 141569 GRAVITY spectrum (black line), and the spectra of the
Keplerian disk models described in Section 5.3. The blue line represents the
model in the optically thin scenario with residuals given by the red dashed
line, while the orange line represents the model in the optically thick
scenario with residuals given by the green dashed line. Figure 7:
Monochromatic images (from $2.16452$ to $2.16852\,\mu$m, i.e., from $221$ to
$332\,$km/s) of the Keplerian ring model described in Section 5.3. The colored
dashed lines refer to the different GRAVITY baselines. Circles represent the
2D photocenter shift for each baseline.

##  6 Discussion

###  6.1 A newly detected inner dust ring

Figure 8: Visualization of dust and gas distribution in HD 141569 (adapted
from Di Folco et al. 2020). Shown in light orange are the optical–IR dust
rings detected in scattered light by HTS (e.g., Augereau et al., 1999; Clampin
et al., 2003; Konishi et al., 2016). Overplotted in green are the three near-
IR dust ringlets detected by VLT/SPHERE (Perrot et al., 2016). The extended
millimeter continuum emission detected by ALMA (Miley et al., 2018) and NOEMA
(Di Folco et al., 2020) is shown in orange. The large brown ring represents
the mid-IR continuum emission detected by VLT-VISIR and modeled by Thi et al.
(2014). In dark red is shown the near-IR dust emission detected by GRAVITY and
studied in this work. In the same color is the Br$\gamma$ line emitting gas
region detected by GRAVITY, also analyzed in this work. In blue is depicted
the H$\alpha$ line region based on the upper-limit size estimated by
Mendigutía et al. (2017). Finally, in purple is shown the CO gas region whose
emissions were detected by ALMA (e.g., White et al., 2016; Miley et al., 2018)
and NOEMA (Di Folco et al., 2020).

Thanks to GRAVITY, an additional inner ring component at $\sim$ 1 au from the
star, narrower than $\sim\,$0.3 au, has been discovered, which adds to the
multi-ring picture identified for HD 141569. A schematic visualization of the
dust and gas distribution obtained from multi-epoch and multi-instrument
observations is depicted in Fig. 8. The low level of near-IR excess ($\sim$
6%) is comparable to the typical 5% accuracy of K-band photometric data (e.g.,
2 MASS), which led to considering the near-IR flux of HD 141569 as essentially
photospheric in earlier studies. With this result, the presence of dust as
close as $1\,$ au from the star, but well beyond the sublimation radius
$R_{\rm sub}$ $\sim$ 0.25 au is a more robust piece of evidence. The disk
inclination and position angle estimated through the GRAVITY FT data analysis
are consistent within the error bars with the values found for the outer rings
that make up the circumstellar environment of the system (see Table 5),
suggesting an almost coplanar system of circumstellar rings. Considering how
many constraints can effectively be set on the position angle (see Fig. 12),
it does not appear that any clear misalignment between the inner and outer
disks could be claimed.
The inclination and position angle could be the reason why PTI and the Keck
interferometer were not able to spatially resolve the near-IR emission of HD
141569 in earlier measurements, since the alignment of their single baseline
was around 42∘ from north to east, hence not far from the semi-minor axis of
the disk. However, the upper limit of 10 mas in radius for the location of the
dust proposed by Monnier et al. (2005) is in agreement with our results. They
also report a fractional excess of $\sim$5%, but compatible within the error
bars with pure photospheric flux in the K band.
Our results are substantially different from those obtained by Lazareff et al.
(2017) with PIONIER in the H band. The reported flux excess is $\sim$ 50%
larger than in our case and their half-flux radius for the dust emission is
only 0.03 au, well inside the sublimation radius of the system expected to be
at $\sim$0.21-0.26 au (for $L_{\star}=16.6-27.0\,$L⊙, Tsubl $\sim$ 1470 K and
a cooling efficiency $\epsilon$ $\sim$0.5). Lazareff et al. (2017) modeled the
H-band excess emission using an ellipsoid distribution and not a ring (see
their Table B.2 and B.3). The choice of the model and the low-quality PIONIER
data could potentially explain this unexpected result.
The values derived for the radius and width of this new innermost dusty
component allows us to compare the system with other YSOs. The analysis made
by Gravity Collaboration et al. (2019) for a sample of 27 Herbig Ae/Be stars
revealed dusty circumstellar environments with half-flux radii that range
between 0.1 and 6 au depending on the stellar luminosity with a median of 0.6
au, and a width-to-radius ratio $w$ ranging from 0.1 to 1 with a median of
0.83, which is interpreted as smooth and wide rings, even though with large
error bars.
Our radius estimate for HD 141569 ($\sim$0.8 au) is within the range found by
these authors, but its peculiarity is also reflected in the fact that its
position in the size-luminosity diagram does not coincide with the bulk of the
Herbig stars, reinforcing the idea that HD 141569 is a unique system in terms
of evolution. From Table 12, the width-to-size ratio is estimated to
0.05$\pm$0.05 for the best-fit model, but it is also noticeable from Fig. 12
that the ring’s width is difficult to clearly constrain in the region below
0.3 au. This implies that the width-to-size ratio could be seven times larger.
Therefore, our estimate of the width-to-size ratio goes on the lower end of
the range found in Gravity Collaboration et al. (2019), but remains comparable
to systems such as HD 114981 (0.10$\pm$0.03) or HD 190073 (0.14$\pm$0.03).
In addition to the disk’s well-known flux-size degeneracy, which in our case
is broken thanks to the constant plateau in the squared visibilities curve as
a function of spatial frequencies, Lazareff et al. (2017) found a negative
correlation between the ring width and its half-flux radius, meaning that it
gets more difficult to detect with the VLTI baseline small-radius ring-like
structures than small-radius ellipsoid structures for angular sizes of the
K-band emission of $\sim$1 mas or smaller. In our case the size is well
constrained being beyond the suggested limit, which is one argument favoring
the robustness of our modeling as opposed to the model of a Gaussian
brightness distribution that did not converge to a solution.
To further test our model findings, first we tried to add the contribution of
a fully resolved emission (also known as a halo) to our geometrically thin
ring model following Eq. 4 of Lazareff et al. (2017). Our best-fit solution
($\chi^{2}_{red}=4.69\pm 0.32$) led to results similar to our geometrically
thin ring model with a halo contribution that actually converges toward zero
($0.103^{+0.144}_{-0.075}\,\%$). Interestingly, this is in full agreement with
Lazareff et al. (2017) who found null halo flux contribution as well. Second,
we also tried a Lorentzian-convolved infinitesimally thin ring (see Table 5 of
Lazareff et al. 2017), which leads to practically the same solution
($\chi_{r}^{2}=4.68\pm 0.32$). We conclude from this detailed analysis that
the narrow ring-like shape with a width smaller than 0.3 au is a good
description of our observations.

###  6.2 Nature and origin of the detected ring

We advance in Section 5.2 the scenario of a population of stochastically
heated particles (e.g., PAH-like very small grains) as the cause for the near-
IR excess. A number of arguments can be discussed in this context.
Our best-fit chromatic model shows a spectral index $k_{c}$=-0.35$\pm$0.21 for
the circumstellar emission. Following Lazareff et al. (2017) and Gravity
Collaboration et al. (2019), we estimate from the parameter $k_{c}$ a
temperature of the radiating dust under the gray body hypothesis (i.e.,
wavelength-independent emissivity) and find $T_{\rm c}$=1460$\pm$70 K. In the
case of a silicate dust ring in thermal equilibrium at $\sim$0.8 au, we would
expect a cooler temperature444Eq. 14 in Lazareff et al. (2017) of $T_{\rm c}$
$\sim$ 650–850 K using the stellar parameter of Table 3 and a cooling
efficiency $\epsilon$ between 0.3 and 1. Therefore, we argue that the near-IR
emission is not dominated by emission of dust in thermal equilibrium, which
can be explained by the presence of these small particles that are quantum
heated by the stellar UV radiation. Even though the spectral index is found to
be not very well constrained (see Fig. 12), the spectral index corresponding
to dust in thermal equilibrium at 800 K would be around +3.4 at $\lambda_{0}$,
relatively far from the our best-fit model.
Maaskant et al. (2014) interpret the high intensity ratio between the PAH band
at 6.2 $\mu$m to the band at 11.3 $\mu$m as a tracer of predominantly ionized
PAH species located in a disk’s gap and exposed to the intense ionizing UV
radiation field of the central star. For instance, with a $I_{6.2}$/$I_{11.3}$
feature peak ratio of $\sim$3$-$4 (Seok & Li, 2017) the PAH sources IRS 48 and
HD 179218 present emission from such predominantly ionized PAHs located in
part inside the gap or disk cavity (Maaskant et al., 2014; Klarmann et al.,
2017; Kluska et al., 2018; Taha et al., 2018). Interestingly, the
$I_{6.2}$/$I_{11.3}$ peak ratio of HD 141569 derived from Seok & Li (2017) is
high as well, estimated to $\sim$ 5$-$6\. This may indicate the presence of
PAH species close to the star, with a predominantly ionized state due to the
direct irradiation by UV stellar flux, bringing further support to our QHP-
dominated inner ring model.
Comparing the IRS spectrum to our model, we find that the our model accounts
for 28%, 25%, and 4% of the observed PAH peak emission for the features at 6
$\mu$m, 8 $\mu$m, and 11 $\mu$m, respectively. The remaining emission would
come from PAHs located in the outer rings. Comparing the outer ring PAH mass
reservoir estimated by Thi et al. (2014) to that of our innermost ring model,
we find that our estimate ($4.3\times 10^{-14}$ M⊙) is smaller than their
$\sim 15\,$au and $\sim 300\,$au rings by three orders of magnitude
($2.0\times 10^{-11}$ M⊙ and $2.1\times 10^{-11}$ M⊙, respectively), and by
four orders of magnitude for their $\sim 185\,$au ring ($1.2\times 10^{-10}$
M⊙) and the entire outer disk environment ($1.6\times 10^{-10}$ M⊙). We recall
however that in their model Thi et al. (2014) do not account for any dust
located at $\sim$ 1 au and suggest a $5\,$au dust-free inner gap, which could
result in overestimated values on their side.
Regarding the origin of the ring structure, the case of HD 141569 is
particularly interesting under the aspect of our proposed QHP-dominated inner
component: one could question the presence of QHPs in an inner narrow ring
since this kind of particle is expected to be coupled to the gas component and
to date have been mostly invoked in more extended emission (e.g., Klarmann et
al. 2017; Kluska et al. 2018).
Several authors have detected CO emission beyond $\sim$10 au (see
Introduction), whereas little information is available on the presence of CO
within the first 10 au. It is likely however that this inner region is not gas
depleted. The [O$\,\rm{}_{I}$] $\lambda$6300 emission detected by Acke et al.
(2005), and suggested by these authors as a dissociation product of OH in the
circumstellar disk, would originate between $\sim$0.05 and 0.8 au under the
assumption of a gas disk in Keplerian motion (Brittain et al., 2007). Hydrogen
recombination and sodium lines were also detected (van der Plas et al., 2015).
Both Mendigutía et al. (2017) and our results (see Section 6.3) find excited
atomic hydrogen in the dust-free cavity, which implies the existence of a
replenishment mechanism from the outer regions. Moreover, Brittain et al.
(2007) set an upper limit on the column density of CO inside 6 au of
N(CO)$\,<10^{15}$ cm-2, which translates into a gas mass $<5.9\times
10^{-13}\,$M⊙. Considering our QHPs mass ($4.3\times 10^{-14}$ M⊙), a gas-to-
dust ratio of 100, and a [CO/H2] ratio of $10^{-4}$, this would translate in a
CO mass of $4.3\times 10^{-16}\,$M⊙, which is well below the upper limit and
therefore not detected yet. These arguments suggest that a gaseous component
may exist and coincide with the proposed QHP component to which it would be
coupled.
Furthermore, even though QHPs have been invoked in more extended emission in
past works, we cannot exclude the possibility of more compact or narrow
components. For example, Khalafinejad et al. (2016) modeled its inner
circumstellar region through an optically thin spherical halo extending from
0.1 to 1.7 au in order to explain the near-IR flux of HD 100453 and to fit
simultaneously its Q-band flux. The choice of the spherical halo was based on
the fact that their data poorly constrained the structure of the inner disk
(and so it is the halo extension estimate) and the optically thin hypothesis
was set in order to not affect the Q-band flux modeling. This component was
also suggested by Klarmann et al. (2017). Their QHP model for HD 100453
underestimates the observed flux in the 1-5 $\mu$m wavelength range by up to
30%, and slightly overestimates the long-baseline visibility data, indicating
that the missing flux is emitted on short spatial scales. Closer results to
those we obtained for HD 141569 were found by Maaskant et al. (2013). These
authors suggest a compact optically thin spherical halo for HD 169142 (0.1-0.2
au), HD 135344 B, and Oph IRS 48 (0.1-0.3 au) to reproduce the observed near-
IR flux. Several scenarios have been proposed to explain the different
structures that protoplanetary disks exhibit, such as gaps, spirals, or rings.
Fragmentation of wide rings into narrow ones by secular gravitational
instability (e.g., Tominaga et al. 2020), self-induced pileup of particles by
aerodynamical feedback (e.g., Gonzalez et al. 2017), and dust traps at local
maxima in the gas density due to a reversal of the pressure gradient by
dynamical clearing from a companion (e.g., Pinilla et al. 2012) could explain
a structured nature of the disk. Our observations leave this matter as an open
question since GRAVITY informs us solely on the spatial properties of the
detected K-band continuum emission.

###  6.3 HD 141569 Br$\gamma$-line emitting gas region

Our analysis of the kinematic and spatial distribution (via the differential
phase) of the hot hydrogen gas is in line with a scenario of a Keplerian disk
inside the dust-free cavity. The distribution of the photocenter shifts shown
in Fig. 5 agrees well with the behavior expected from a Keplerian disk
(Mendigutía et al., 2015). The position angle of the photocenter shifts
distribution (-10${}^{\circ}\pm 7^{\circ}$ north to east) is also found to be
in overall agreement with the position angle of the inner ring responsible for
the near-IR excess, and of the outer rings. Moreover, the photocenters of the
Br$\gamma$ line emitting gas region are located as the photocenters of the
outer CO regions, blueshifted ones along the southeast and redshifted ones
along the northwest (White et al., 2016). The profiles of pure-line
differential phase signals depart a bit from the perfect S-shaped signal
expected for a pure Keplerian disk. We believe that it is also limited
ultimately by our spectral calibration. In this sense, more accurate
measurement of the differential phases in HD 141569 using GRAVITY with the 8 m
Unit Telescopes could certainly improve the accuracy of this analysis. In
order to evaluate the quality of our spectral data, we chose to compare the
GRAVITY profile measured with the ATs to other high-quality spectra obtained
with the ISAAC spectrograph at the VLT (Garcia Lopez et al., 2006), the
NIRSPEC echelle spectrograph at the Keck Observatory (Brittain et al., 2007),
and with SINFONI/VLT from archival data. The comparison is shown in Fig. 9 and
we observe that the GRAVITY spectrum exhibits a mild asymmetry between the
blue and red peaks. This would suggest that our spectrum could be still
affected by some calibration effects, either telluric or instrumental. We then
further explored how far our resulting differential phases might be impacted
by the slight spectrum asymmetry and tested the derivation of the pure-line
differential phases using the SINFONI spectrum, which has a very similar
spectral resolution, instead of the GRAVITY spectrum. We found that the 2D
distribution of the photocenter shifts remains unchanged within the error bars
reported in Fig. 5.
The star is known to be a fast rotator ($222.0\pm 7.0\,$km/s, Folsom et al.
2012), which results in a small co-rotation radius around $2.38\pm 0.53\,$R⊙
($0.011\pm 0.002$ au), assuming $R_{\star}=1.5\pm 0.5\,$R⊙ (Fairlamb et al.,
2015). We cannot exclude that part of the Br$\gamma$ line emission comes from
magnetospheric accretion flows, but the small co-rotation radius compared to
the size of the Br$\gamma$ line emitting region estimated from the SC data
analysis ($\sim 0.09\,$au) would not favor this scenario, as opposed to what
has been recently found for TW Hya (Gravity Collaboration et al., 2020b).
Interestingly, the scenario of magnetospheric accretion was also tested by
Mendigutía et al. (2017) to explain the H$\alpha$ double-peaked emission line,
but they were not able to reproduce the observed profile with any set of input
parameters. Comparing the extent of the Br$\gamma$ emission to the continuum
emission (RBrγ/R${}_{\rm cont}\approx 0.1$), we find that the case of HD
141569 is in contrast with the findings of Kraus et al. (2008). These authors
found for a small sample (5 objects) of Herbig Ae/Be stars that those showing
a P Cygni H$\alpha$ line profile and a high mass-accretion rate ($>10^{-7}$
M${}_{\odot}\,$yr-1) seem to show compact Br$\gamma$-emitting regions
(RBrγ/R${}_{\rm cont}<0.2$), from which the emission stems from magnetospheric
accretion or recombination line from ionized hydrogen, while stars showing a
double-peaked or single-peaked H$\alpha$ line profile show a more extended
Br$\gamma$-emitting region ($0.6\leq$RBrγ/R${}_{\rm cont}\leq 1.4$), which
would trace a stellar or disk wind. Our system shows mixed features, a
Br$\gamma$ and H$\alpha$ double-peaked emission line that originates from
compact disks in Keplerian rotation where magnetospheric accretion is not the
most likely main emission mechanism. Therefore, recombination line emission
from ionized hydrogen in an inner gaseous accretion disk as hinted by GRAVITY
is a more supported scenario. If that is the case, considering the age of the
system and the reported accretion rates between $10^{-7}-10^{-11}\,$M⊙ yr-1
(Merín et al., 2004; Garcia Lopez et al., 2006; Mendigutía et al., 2011a; Thi
et al., 2014; Fairlamb et al., 2015), the inner gaseous disk requires some
sort of replenishment mechanism to explain its presence and for it to survive.
Replenishment flows, planet-boosted or not (Mendigutía et al., 2017),
connecting the inner and outer disk as already observed in other Herbig stars
like HD 142527 (Casassus et al., 2013) could be investigated in the future.

Figure 9: HD 141569 continuum-normalized spectrum taken at different epochs
and with different instruments. In blue is depicted the data taken on July 12,
2019, by GRAVITY; in red data taken in June 2019 by SINFONI; in green data
taken in 2002 by the KECK NIRSPEC (Brittain et al., 2007); and in cyan data
taken in 2004 by the VLT/ISAAC (Garcia Lopez et al., 2006).

###  6.4 Hybrid or debris disk?

As mentioned in the introduction, HD 141569 is the only known pre-main
sequence star characterized by a hybrid disk. The main characteristic of a
hybrid disk is the weak fractional excess of IR emission ($8.4\times 10^{-3}$
for HD 141569, Sylvester et al. 1996), that stems from the optically thin
second-generation grains dust component, coupled with the presence of a
significant gaseous component, believed to be primordial. Unfortunately the
other known best hybrid disk system candidates, which are 49 Cet, HD 21997
(Moór et al., 2011), HD 131835 (Moór et al., 2015), HD 121617, HD 131488 (Moór
et al., 2017), and HD 32297 (Moór et al., 2019), have not been studied in
their hot dust content. As HD 141569 does, they show a featureless SED in the
near-IR suggesting systems depleted of material inside the first 5 au.
However, as we saw in this work, interferometric observations could reveal as-
yet-undetected dust, so this scenario should not be excluded for the other
mentioned objects. The potential of long-baseline interferometry to detect
small levels of hot circumstellar dust emission in supposedly dust-free
systems has been indeed exploited for older main sequence stars (Ertel et al.,
2014).
In this context it is interesting to compare our results for HD 141569 with
Vega, the most iconic debris disk system with a similar spectral type (A0V),
but that is significantly older than our source (400 $-$ 700 Myr). Near-IR
excess from Vega was detected and constrained to $\sim$ 1.29$\pm$0.19% by
interferometric observations with CHARA/FLUOR in the K band (Absil et al.,
2006). These authors suggest by SED modeling that the excess comes from hot
small grains starting at $\sim$0.2-0.3 au with a total dust mass of
8$\times$10-8 M⊕. Knowing the age of Vega, it is clear that its circumstellar
dust is of second generation, a characteristic of debris disks. In the
scenario of a silicate-dominated inner ring for HD 141569, the total mass
required to induce a larger near-IR excess is, as expected, significantly
larger ($>$ 10-4 M⊕). Considering the younger age of HD 141569 ($\sim$7.2
Myr), a more massive inner disk is compatible with a system at an earlier
stage of disk evolution. Since the timescale for disk dissipation is known to
be about $5-10\,$Myr (Wyatt, 2008), it is plausible that part of the dust in
the inner region of HD 141569 is of first generation and a remnant of the
primordial circumstellar environment. This would confirm that HD 141569 is
closer to a system in the final stage of the protoplanetary disk phase than a
debris disk system.

The fact that gas, both H and CO, is detected in HD 141569 is an factor in
favor of a system in the (late) protoplanetary disk stage rather than in the
debris disk stage. The system, similarly to other hybrid disks, does not
follow the correlation between the CO flux density and the millimeter
continuum emission followed by T Tauri, Herbig Ae, and debris disks, but lies
systematically above the correlation line (Péricaud et al., 2017). The authors
suggest that the dust and gas evolution are decoupled, with the dust evolving
faster than the gas, leading to an unusual high gas-to-dust ratio (between 135
and 2370 for HD 141569, Di Folco et al. 2020). Other than the primordial
origin scenario, a secondary origin of CO was proposed (Kral et al., 2019) in
which the gas is self-shielded and shielded by accumulated neutral carbon
produced through photodissociation of molecular gas released by planetesimals.
This model is able to explain the estimated CO masses in all the hybrid disk
candidates (Kral et al., 2019; Moór et al., 2019) except for HD 141569, for
which it was not tested. Since CO molecule photodissociation occurs at UV
wavelengths, we note that C0 could shield these molecules, and also QHPs
(Woitke et al., 2016), which are known to absorb UV photons and cool down very
quickly by emitting photons in the near-IR. The new detected innermost ring,
which we propose in this work to be dominated by a small amount of QHPs, could
contribute to the shielding process.

##  7 Summary

We presented the first GRAVITY interferometric observations of HD 141569. Here
we summarize the main conclusions of our work:

  * •

The system was resolved by GRAVITY with squared visibilities down to
V${}^{2}\sim 0.8$. If before these observations the near-IR flux contribution
of a dust disk was considered absent because no feature was seen in the SED of
the object, now, thanks to interferometry, the presence of dust in the first
au of the system is a more robust piece of evidence and the flux excess is
clearly detected and constrained to be $\sim 6.2\,\%$ of the total flux.

  * •

Data modeling suggests that the dust is located in a thin ring ($\lesssim
0.3\,$au in width) at a radius of $\sim$1 au from the star. The ring shares,
within the errors of Fig. 12, the same inclination ($\sim$ $58^{\circ}$) and
position angle ($\sim$ $0^{\circ}$) as the outer rings observed in the past.

  * •

MCMax SED modeling suggests that this innermost ring could be made of a small
amount ($1.4\times 10^{-8}\,$M⊕) of QHPs. Large silicate grain models, with
and without carbon, can reproduce the 6% flux excess at 2.15 $\mu$m, but at
the same time they show a significant emission in the mid-IR not consistent
with the SED of the system.

  * •

The SC data analysis confirms the significant amount of Br$\gamma$ line
emitting gas already observed in the past. The gas region is spectrally
resolved, but spatially unresolved.

  * •

The pure-line differential phases constrain the gas to be in a Keplerian disk-
like structure, as hinted by the double-peaked line shape, confined within
$\sim 0.09\,$au ($\sim 12.9$ R⋆) and oriented in the same was as the outer
rings (${\rm PA_{NE}}\sim-10^{\circ}$).

These results confirm the complexity of the HD 141569 circumstellar
environment also at milliarcsecond scale, making the system a unique
astronomical laboratory to investigate the missing steps of disk evolution and
planet formation theories.

## Acknowledgements

We thank the anonymous referee for reviewing this work in detail and for
providing useful comments that have improved the quality of the paper. This
work is based on observations made with ESO Telescopes at the La Silla Paranal
Observatory under program IDs 0102.C-0408(D), 0103.C-0347(C), and
0103.C-0347(A). V.G was supported for this research through a stipend from the
International Max Planck Research School (IMPRS) for Astronomy and
Astrophysics at the Universities of Bonn and Cologne, and from the Bonn-
Cologne Graduate School of Physics and Astronomy (BCGS). A. C. G. received
fundings from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 743029).
L. K. acknowledges the helpful discussions that took place during the ISSI
meeting ”Zooming In On Rocky Planet Formation” (team 482). We thank Michiel
Min for allowing us to use his MCMax code and for his help on understanding
the code itself. We thank Anne-Marie Lagrange, Mickaël Bonnefoy and Florian
Peissker for providing the SINFONI spectrum of HD 141569. We acknowledge the
Gemini Observatory for the use of the IR spectrum model of the atmospheric
transmission above Cerro Pachon. This research has made use of the model
atmosphere grid NeMo, provided by the Department of Astronomy of the
University of Vienna, Austria (http://www.univie.ac.at/nemo/). NeMo was funded
by the Austrian Science Fonds.

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##  Appendix A GRAVITY observations

Table 3: Observation logs of the VLTI/GRAVITY HD141569 observations.

Date | UT | Configuration | N |  Calibrator (diam. [mas], SpT †) |  Seeing [′′] | Airmass |  $\tau_{0}$ [ms] | Frame rate [Hz]
---|---|---|---|---|---|---|---|---
2019-03-18 | 08:08 | D0-G2-J3-K0 | 3 |  HD 137006 (0.291$\pm 0.008$, A5 II/III) | 0.4 | 1.1 | 14 | 909
2019-05-24 | 04:15 | A0-G1-J2-J3 | 1 |  HD 141977 (0.251$\pm 0.006$, K3 III) | 1.4 | 1.1 | 3 | 303
2019-07-12 | 00:44 | D0-G2-J3-K0 | 8 |  HD 137006 (0.291$\pm 0.008$, A5 II/III) | 0.6 – 1.1 | 1.1 | 3 | 909

555The date format is year-month-day. N denotes the number of 5-minute files
that have been recorded on the target. Reference †: uniform disk diameter and
SpT derived from the software package SearchCal from the Jean-Marie Mariotti
Center (JMMC).

Figure 10: HD 141569 FT data, squared visibilities, closure phases, and U-V
plane coverage, from the three different observation epochs. The colors refer
to the different GRAVITY spectral channels.

##  Appendix B MCMC posterior distribution functions and visibility modeling

Figure 11: Results of the FT data fit. Top: MCMC marginal posterior
distributions of the fitted parameters. The blue lines identify the median of
the distributions for the geometrically thin ring model (left) and for the
Gaussian-convolved infinitesimally thin ring model (right). Center: Comparison
between the model squared visibilities (blue dots) and the observational data
(white–black dots). The red dots represent the absolute residuals. Bottom:
Vis2 best fit to the geometrically thin ring model (blue continuous line) at
$\lambda$=2.153 $\mu$m for three different position angles identified in the
insets. The visibility data are shown for all spectral channels by the red
circles. Figure 12: Reduced chi-squared $\chi_{r}^{2}$ from the geometrically
thin ring model squared visibility fit as a function of the different model
parameters. Each point represents a squared visibility fit with the respective
parameter fixed to that value while the other parameters are set free.

##  Appendix C Spectrum wavelength calibration, telluric line correction, and
photospheric absorption correction

The 2.15-$2.18\,\mu$m part of the HD 141569 spectrum is shown in Fig. 2 after
continuum-normalization, wavelength calibration, telluric line correction
(left panel), and photospheric absorption correction (right panel). The
continuum-normalization was done in both the science object spectrum and in
our inteferometric calibrator HD 137006 spectrum ($\alpha=230.932^{\circ}$,
$\delta=-01.022^{\circ}$, A5II/III SpT) by fitting the slope of each raw
spectrum and dividing the raw spectrum by the resulting fit. The first step of
the wavelength calibration was done by comparing the observed wavelength
positions of the telluric lines in the HD 141569 spectrum with respect to the
positions of the telluric lines present in the IR spectrum of the atmospheric
transmission above Cerro Pachon, generated using the ATRAN modeling software
(Lord 1992) accounting for a $4.3\,$mm water vapor column and a $1.5\,$mm
airmass column, available on the Gemini Observatory website. The atmospheric
transmission spectrum was convolved by a Gaussian with FHWM of $6\,$Å to have
the same resolution as GRAVITY. The same correction, which results in a $5\,$Å
blueshift, was also applied on the SC visibilities and differential phases,
and on the HD 137006 spectrum, our calibrator star. The calibrator was also
used as a telluric spectroscopic standard. It shows a photospheric absorption
feature at $2.16612\,\mu$m that was taken out before the telluric correction
on the science object through a NextGen spectrum model (Allard et al. 1997;
Hauschildt et al. 1999), available on the GAIA archive, with the following
parameters: $7600$ K effective temperature, 6.0 surface gravity logarithm, and
a $-2.0$ solar metallicity. Moreover, the spectrum shows an absorption feature
at around $2.16712\,\mu$m that is not observed in the spectrum of the other
calibrators observed that night and that significantly affects the shape and
the intensity of the HD 141569 Br$\gamma$ red peak. For this reason we chose
to calculate the atmospheric transmission function of the night by taking the
average of three calibrator spectra, HD 137006, HD 149789, and HD 157029,
shown in Fig. 131\. The HD 141569 spectrum was then corrected for its radial
velocity $(-6.4\,$km/s$)$ and its proper motion with respect to the local
standard of rest, resulting in a correction of $-8.7\,$km/s. Finally, the HD
141569 spectrum was corrected for atmospheric absorption through a spectrum
model from the Vienna New Model Grid of Stellar Atmospheres (Heiter et al.
2002)666available on the NeMo webpage (Ch. Stütz and E. Paunzen,
http://www.univie.ac.at/nemo/). The model accounts for a star effective
temperature of $9800$ K, a surface gravity logarithm of 4.4 (Fairlamb et al.
2015), solar metallicity, and microturbulence of 2.0 km/s (Folsom et al.
2012). Since the star is a fast rotator having a projected linear velocity of
$v\,sin\,i=222.0\pm 7.0$ km/s (Folsom et al. 2012), we included the rotation
broadening effect on the spectrum model using SPECTRUM (Gray & Corbally 1994).
We decreased the intensity of the absorption model by $3\,\%$ to better fit
our spectrum.

Figure 13: Atmospheric transmission functions derived from calibrator spectra:
HD 137006 (top), HD 149789 (center), and HD 157029 (bottom). The last plot is
the average transmission function of the three spectra.

##  Appendix D Pure-line visibility in presence of photospheric absorption

The total visibility three-component model used to fit the SC visibilities
accounts for the contributions from the star, the circumstellar dust, and the
line emitting gas. The total visibility as a function of the wavelength is
given by

$\displaystyle V_{\rm Tot}(u,\varv,\lambda)=$
$\displaystyle\frac{\alpha(\lambda)\,F_{\rm s}(\lambda)\,V_{\rm
s}(u,\varv,\lambda)+F_{\rm c}(\lambda)\,V_{\rm c}(u,\varv,\lambda)+F_{\rm
L}(\lambda)\,V_{\rm L}(u,\varv,\lambda)}{\alpha(\lambda)\,F_{\rm
s}(\lambda)+F_{\rm c}(\lambda)+F_{\rm L}(\lambda)}.$ (18)

The parameter $\alpha(\lambda)$ is the star continuum-normalized photospheric
absorption, which implies that $\alpha(\lambda)$ = 1 outside the Br$\gamma$
line and $\alpha(\lambda)$ ¡ 1 inside the line. This parameter is estimated
from the Vienna New Model Grid of Stellar Atmospheres fitting the stellar
parameters of HD 141569, as described in Section C. $F_{\rm s}(\lambda)$,
$F_{\rm c}(\lambda),$ and $F_{\rm L}(\lambda)$ are the wavelength-dependent
fluxes of, respectively, the stellar continuum (i.e., outside the line and
temperature-dependent), the dust ring continuum, and the Br$\gamma$ line
emitting gas. $F_{\rm L}(\lambda)$ varies across the line and vanishes to zero
outside the line. $V_{\rm s}(\lambda)$, $V_{\rm c}(\lambda),$ and $V_{\rm
L}(\lambda)$ are the intrinsic visibility functions of each of the three
components taken individually. From now on, the star is considered unresolved;
therefore, $V_{\rm s}$ is equal to 1, and we drop the explicit parameter
dependencies for convenience.
Outside the Br$\gamma$ line (i.e., in the continuum region) the total
visibility is given by

$V_{\rm Tot}^{\rm Cont}=\frac{F_{\rm s}+F_{\rm c}\,V_{\rm c}}{F_{\rm s}+F_{\rm
c}}.$ (19)

Using Eq. 19 to replace $V_{\rm c}$, we can rewrite Eq. D as

$V_{\rm Tot}=\frac{\alpha\,F_{\rm s}+V_{\rm Tot}^{\rm Cont}\,(F_{\rm s}+F_{\rm
c})-F_{\rm s}+F_{\rm L}\,V_{\rm L}}{\alpha\,F_{\rm s}+F_{\rm c}+F_{\rm L}},$
(20)

and using the definition of the line-to-continuum flux ratio (Eq. 13), Eq. 20
becomes

$V_{\rm Tot}=\frac{F_{\rm s}\,(\alpha-1)}{F_{\rm Tot}}+\frac{V_{\rm Tot}^{\rm
Cont}}{F_{\rm L/C}}+\frac{F_{\rm L}\,V_{\rm L}}{F_{\rm Tot}},$ (21)

where $F_{\rm Tot}=\alpha\,F_{\rm s}+F_{\rm c}+F_{\rm L}$. We note once again
that $F_{\rm L/C}$ is the raw line-to-continuum ratio including the
photospheric absorption. This quantity corresponds to the top left spectrum in
Fig. 2. We also note that outside the line (i.e., $\alpha$=1 and $F_{\rm
L}$=0), Eq. 21 simplifies to Eq. 19. Now, making use of the parameter $\beta$
of Eq. 14, we can write

$\frac{F_{\rm L}}{F_{\rm Tot}}=1-\frac{\alpha+\beta}{1+\beta}\,\frac{1}{F_{\rm
L/C}},$ (22)

and finally Eq. 21 becomes

$V_{\rm Tot}=\frac{\alpha-1}{1+\beta}\,\frac{1}{F_{\rm L/C}}+\frac{V_{\rm
Tot}^{\rm Cont}}{F_{\rm L/C}}+V_{\rm
L}\,(1-\frac{\alpha+\beta}{1+\beta}\,\frac{1}{F_{\rm L/C}}).$ (23)

Solving Eq. 23 for $V_{\rm L}$, and noting that from our data $V_{\rm
Tot}=V_{\rm Tot}^{\rm Cont}$ since the SC visibilities are spectrally flat for
all wavelengths and baselines, we obtain Eq. 4.2:

$V_{\rm L}=\frac{V_{\rm Tot}\,[(1+\beta)\,F_{\rm
L/C}-1-\beta]-\alpha+1}{(1+\beta)\,F_{\rm L/C}-\alpha-\beta}.$ (24)

This equation tells us that the pure-line visbility can be estimated from the
total visibility in the line (which in our case is comparable to the total
visibility in the continuum) if the photospheric absorption profile can be
estimated, the continuum disk-to-star flux ratio $\beta$ is known and the
continuum-normalized spectrum of the line is accessible. We note, in the case
where $V_{\rm Tot}=V_{\rm Tot}^{\rm Cont}$, that the absence of photospheric
absorption (i.e., $\alpha$=1) leads simply to $V_{\rm L}$ = $V_{\rm Tot}^{\rm
Cont}$=$V_{\rm Tot}$.
Finally, we see that the pure-line visibility $V_{\rm L}$ is lower than 1 only
when $F_{\rm L/C}$ is greater than 1, which is equivalent to detecting the
line above the continuum.

##  Appendix E MCMax silicate model

Here are shown three representative cases of our SED modeling through a
silicate dust ring, as described in Section 5.2. The first model accounts for
particles with a lower-limit size of $1.26\,\mu$m, a dust mass of
10${}^{-5}\,$M⊕, and $0\,$% carbon with a resulting temperature distribution
in the range $570-1070\,$K. The second model accounts for particles with a
lower-limit size of $40\,\mu$m, a mass of 1.17$\times 10^{-4}\,$M⊕, and $0\,$%
carbon with a resulting temperature distribution in the range $620-840\,$K.
The last model accounts for particles with a lower-limit size of $40\,\mu$m, a
mass of 3.33$\times 10^{-4}\,$M⊕, and $25\,$% carbon with a resulting
temperature distribution in the range $440-770\,$K. We note that in all three
cases the near-IR excess is smaller than that obtained through the GRAVITY
data analysis, while the mid-IR emission exceeds the photometry data.
Decreasing the mid-IR excess would at the same time decrease further the near-
IR value, suggesting that a pure silicate innermost ring model is not
compatible with the nature of the HD 141569 system.

Figure 14: MCMax models that account for the emission of the star and only the
first innermost silicate ring located at 0.8 au and 0.04 au in width. The
models differ for the particle size lower limit, dust mass, and carbon
percentage. The top plot model accounts for a dust mass of 10${}^{-5}\,$M⊕ and
no carbon; the center plot model accounts for a mass of 1.17$\times
10^{-4}\,$M⊕ and no carbon; the bottom plot model accounts for a mass of
3.33$\times 10^{-4}\,$M⊕ and 25 % carbon.

##  Appendix F MCMax model density and temperature structure for the QHP model

Figure 15: Density and temperature structure of the QHP model described in
Section 5.2 and Table 3. The x-axis shows the radius in au, the y-axis the
height divided by the radius. The colors of the top plots indicate the mass
density, the colors of the bottom plot the temperature. QHPs are located at
0.8 au in a very narrow (0.04 au) ring. The optical depth at the observation
wavelength is indicated by the radial (dotted line) $\tau=1$ surface. The
magenta dashed line represents the sublimation radius of the system. The dark
red color in the temperature plot refers only to the location of the QHPs and
not their temperature, since QHPs do not have an equilibrium temperature
value.

##  Appendix G Photometric data

Table 4: HD 141569 photometric data.

Band | $\lambda$ | Flux | Beam size and reference
---|---|---|---
|  [$\mu$m] | [Jy] |
IUE | 0.138 | 0.30 | archival data
IUE | 0.178 | 0.73 | archival data
IUE | 0.218 | 0.628 | archival data
IUE | 0.257 | 1.023 | archival data
IUE | 0.29 | 1.274 | archival data
U | 0.36 | 4.19 | Sylvester et al. (1996)
B | 0.436 | 8.37 | Sylvester et al. (1996)
V | 0.55 | 7.36 | Sylvester et al. (1996)
R | 0.708 | 5.92 | Sylvester et al. (1996)
I | 0.977 | 4.81 | Sylvester et al. (1996)
J | 1.22 | 3.1 | 2Mass
H | 1.65 | 1.8 | 2Mass
K | 2.16 | 1.29 | 2Mass
K | 2.16 |  1.26$\pm 0.03$ | 2Mass
K | 2.16 | 1.25 | 2Mass
K | 2.18 | 1.2 | 2Mass
K | 2.19 |  1.22$\pm 0.02$ | 2Mass
K | 2.19 | 1.23 | Penprase (1992)
K | 2.24 |  1.3$\pm 0.04$ | Malfait et al. (1998)
ISO | 2.45 | 1.07 | ESA archive
ISO | 3.23 | 0.76 | ESA archive
WISE | 3.4 | $0.79\pm 0.025$ | 6.1” NASA archive
UKIRT | 3.76 | 0.54 | Sylvester et al. (1996)
ISO | 4.26 | 0.44 | ESA archive
WISE | 4.6 | $0.49\pm 0.01$ | 6.4”
ISO | 5.89 | 0.40 | ESA archive
ISO | 6.76 | 0.43 | ESA archive
ISO | 7.76 | 0.82 | ESA archive
ISO | 8.70 | 0.62 | ESA archive
AKARI | 9.0 | $0.5178\pm 0.0104$ | NASA archive
ISO | 9.77 | 0.52 | ESA archive
ISO | 10.7 | 0.58 | ESA archive
OSCIR | 10.8 | $0.318\pm 0.016$ | Fisher et al. (2000)
Michelle | 11.2 | $0.338\pm 0.034$ | Moerchen et al. (2010)
ISO | 11.48 | 0.635 |  14’$\times$20’ ESA archive
IRAS | 12.0 | $0.55\pm 0.04$ |  1’$\times$5’ NASA archive
WISE | 12.0 | $0.38\pm 0.006$ | 6.5’ NASA archive
MIRLIN | 12.5 | $0.333\pm 0.022$ | Marsh et al. (2002)
MIRLIN | 17.9 | $0.936\pm 0.094$ | Marsh et al. (2002)
AKARI | 18 | $0.8655\pm 0.0168$ | NASA archive
Michelle | 18.1 | $0.883\pm 0.147$ | Moerchen et al. (2010)
OSCIR | 18.2 | $0.646\pm 0.035$ | Fisher et al. (2000)
MIRLIN | 20.8 | $1.19\pm 0.16$ | Marsh et al. (2002)
WISE | 22 | $1.44\pm 0.027$ | 12” NASA archive
MIPS | 24.0 | $1.47\pm 0.01$ | 6” Spitzer archive
IRAS | 25 | $1.87\pm 0.13$ |  1’$\times$5’ NASA archive
IRAS | 60 | $5.54\pm 0.49$ |  2’$\times$5’ NASA archive
PACS-Spec | 63.2 | $2.98\pm 0.01$ | Thi et al. (2014)
MIPS | 70 | $4.70\pm 0.02$ | 18” Spitzer archive
PACS-Spec | 72.8 | $3.91\pm 0.03$ | Thi et al. (2014)
PACS-Spec | 76.4 | $3.30\pm 0.03$ | Thi et al. (2014)
PACS-Spec | 90 | $2.80\pm 0.03$ | Thi et al. (2014)
IRAS | 100 | $3.48\pm 0.35$ |  4’$\times$5’ NASA archive
PACS-Spec | 145 | $1.1\pm 0.1$ | Thi et al. (2014)
PACS-Spec | 158 | $1.18\pm 0.02$ | Thi et al. (2014)
PACS-Spec | 180 | $0.83\pm 0.04$ | Thi et al. (2014)
SCUBA | 450 | $0.0649\pm 0.0133$ | Sandell et al. (2011)
SCUBA | 850 | $0.0140\pm 0.0020$ | Sandell et al. (2011)
LABOCA | 870 | $0.0126\pm 0.0046$ | Nilsson et al. (2010)
MAMBO | 1200 | $0.0047\pm 0.0005$ | Meeus et al. (2012)
SCUBA | 1350 | $0.0054\pm 0.0001$ | Sylvester et al. (2001)

Notes. Data without references are from Merín et al. (2004).

##  Appendix H Inclination and position angle of outer disks from the
literature

Table 5: Inclination and position angle of the HD 141569 outer rings. Reference | Radius | i |  PANE
---|---|---|---
| [au] | [deg] | [deg]
Dust components |  |  |
Augereau et al. (1999) | $361\pm 10$ | $52.5\pm 4.5$ | $-4.6\pm 1.0$
Weinberger et al. (1999) | $206$ | $51\pm 3$ | $-4.0\pm 5.0$
Biller et al. (2015) | $232\pm 3$ | $44.9\pm 0.5$ | $-8.9\pm 1.3$
| $385\pm 13$ | $47.3\pm 3.3$ | $-11.3\pm 6.1$
Perrot et al. (2016) | $185\pm 1$ | $56.9\pm 1.0$ | $-3.9\pm 0.4$
| $88\pm 2$ | $57.6\pm 1.3$ | $-4.0\pm 2.0$
| $61\pm 1$ | $56.0\pm 2.2$ | $-5.5\pm 1.0$
| $45\pm 1$ | $57.9\pm 1.3$ | $-6.3\pm 1.1$
Currie et al. (2016) | $37\pm 2$ | $56.0\pm 4.0$ | $-1.2\pm 2.4$
White et al. (2016) | $81$ | $55$ | $-8.8$
Mawet et al. (2017) | $37\pm 4$ | $53.0\pm 6.0$ | $-11.0\pm 8.0$
This work | $\sim 1$ | $58.5\pm 1.6$ | $-1.8\pm 1.1$
Gas components |  |  |
White et al. (2016) | $81-199$ | $53.4\pm 1.0$ | $-3.4\pm 0.6$
Di Folco et al. (2020) | $17-277$ | $56-58$ | $-4\pm 1$
| $35-232$ | $53\pm 2$ | $0\pm 2$
This work | $0.01-0.09$ | $58.5$ | $-10\pm 7$