11institutetext: HUN-REN Research Centre for Astronomy and Earth Sciences,
Konkoly Observatory, Konkoly-Thege Miklós út 15-17, H-1121 Budapest, Hungary
11email<EMAIL_ADDRESS>22institutetext: CSFK, MTA Centre of
Excellence, Konkoly-Thege Miklós út 15-17, H-1121 Budapest, Hungary
33institutetext: Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA
Leiden, The Netherlands 44institutetext: Institute for Mathematics,
Astrophysics and Particle Physics, Radboud University, P.O. Box 9010, MC 62
NL-6500 GL Nijmegen, the Netherlands 55institutetext: SRON Netherlands
Institute for Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands
66institutetext: Anton Pannekoek Institute for Astronomy, University of
Amsterdam, Science Park 904, 1090 GE Amsterdam, The Netherlands
77institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117
Heidelberg, Germany 88institutetext: Université Côte d’Azur, Observatoire de
la Côte d’Azur, CNRS, Laboratoire Lagrange, France 99institutetext: Univ.
Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 1010institutetext:
Institute of Theoretical Physics and Astrophysics, University of Kiel,
Leibnizstr. 15, 24118 Kiel, Germany 1111institutetext: ELTE Eötvös Loránd
University, Institute of Physics, Pázmány Péter sétány 1/A, 1117 Budapest,
Hungary 1212institutetext: Visiting astronomer, Laboratoire Lagrange,
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Boulevard de
l’Observatoire, CS 34229, 06304 Nice Cedex 4, France 1313institutetext: Max
Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85741
Garching bei München, Germany 1414institutetext: INAF-Osservatorio Astronomico
di Capodimonte, via Moiariello 16, 80131 Napoli, Italy 1515institutetext:
Institut de Recherche en Astrophysique et Planétologie, Université de
Toulouse, UT3-PS, OMP, CNRS, 9 av. du Colonel Roche, 31028 Toulouse Cedex 4,
France 1616institutetext: NASA Goddard Space Flight Center, Astrophysics
Division, Greenbelt, MD, 20771, USA 1717institutetext: Max-Planck-Institut für
Radioastronomie, Auf dem Hügel 69, 53121, Bonn, Germany 1818institutetext:
STAR Institute, University of Liège, Liège, Belgium 1919institutetext: I.
Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937, Köln,
Germany 2020institutetext: CENTRA, Centro de Astrofísica e Gravitação, IST,
Universidade de Lisboa, 1049-001 Lisboa, Portugal and Faculdade de Engenharia,
Universidade do Porto, Rua Dr. Roberto Frias, P-4200-465 Porto, Portugal
2121institutetext: AIM, CEA, CNRS, Université Paris-Saclay, Université Paris
Diderot, Sorbonne Paris Cité, 91191, Gif-sur-Yvette, France 2222institutetext:
European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching,
Germany 2323institutetext: Space Research Institute, Austrian Academy of
Sciences, Schmiedlstr. 6, 8042, Graz, Austria

# Mid-infrared evidence for iron-rich dust in the multi-ringed inner disk of
HD 144432 ††thanks: Based on observations collected at the European Southern
Observatory under ESO programmes 190.C-0963(D), 190.C-0963(E), 190.C-0963(F),
0100.C-0278(E), 0103.D-0153(C), 0103.C-0347(C), 0103.D-0153(G), 108.225V.003,
108.225V.011, and 108.225V.006.

J. Varga 112233 L. B. F. M. Waters 4455 M. Hogerheijde 3366 R. van Boekel 77
A. Matter 88 B. Lopez 88 K. Perraut 99 L. Chen 1122 D. Nadella 33 S. Wolf 1010
C. Dominik 66 Á. Kóspál 1122771111 P. Ábrahám 11221111 J.-C. Augereau 99 P.
Boley 1212 G. Bourdarot 1313 A. Caratti o Garatti 771414 F. Cruz-Sáenz de
Miera 11221515 W. C. Danchi 1616 V. Gámez Rosas 33 Th. Henning 77 K.-H.
Hofmann 1717 M. Houllé 88 J. W. Isbell 77 W. Jaffe 33 T. Juhász 11221111 V.
Kecskeméthy 33 J. Kobus 1010 E. Kokoulina 181888 L. Labadie 1919 F. Lykou 1122
F. Millour 88 A. Moór 1122 N. Morujão 2020 E. Pantin 2121 D. Schertl 1717 M.
Scheuck 77 L. van Haastere 33 G. Weigelt 1717 J. Woillez 2222 P. Woitke 2323
MATISSE & GRAVITY Collaborations

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. Rocky planets form by the concentration of solid particles in the
inner few au regions of planet-forming disks. Their chemical composition
reflects the materials in the disk available in the solid phase at the time
the planets were forming. Studying the dust before it gets incorporated in
planets provides a valuable diagnostic for the material composition.

Aims. We aim to constrain the structure and dust composition of the inner disk
of the young Herbig Ae star HD 144432, using an extensive set of infrared
interferometric data taken by the Very Large Telescope Interferometer (VLTI),
combining PIONIER, GRAVITY, and MATISSE observations.

Methods. We introduced a new physical disk model, TGMdust, to image the
interferometric data, and to fit the disk structure and dust composition. We
also performed equilibrium condensation calculations with GGchem to assess the
hidden diversity of minerals occurring in a planet-forming disk such as HD
144432.

Results. Our best-fit model has three disk zones with ring-like structures at
$0.15$, $1.3$, and $4.1$ au. Assuming that the dark regions in the disk at
$\sim$$0.9$ au and at $\sim$$3$ au are gaps opened by planets, we estimate the
masses of the putative gap-opening planets to be around a Jupiter mass. We
find evidence for an optically thin emission ($\tau<0.4$) from the inner two
disk zones ($r<4$ au) at $\lambda>3\ \mu$m. Our silicate compositional fits
confirm radial mineralogy gradients, as for the mass fraction of crystalline
silicates we get around $61\%$ in the innermost zone ($r<1.3$ au), mostly from
enstatite, while only $\sim$$20\%$ in the outer two zones. To identify the
dust component responsible for the infrared continuum emission, we explore two
cases for the dust composition, one with a silicate+iron mixture and the other
with a silicate+carbon one. We find that the iron-rich model provides a better
fit to the spectral energy distribution. Our GGchem calculations also support
an iron-rich and carbon-poor dust composition in the warm disk regions ($r<5$
au, $T>300$ K).

Conclusions. We propose that in the warm inner regions ($r<5$ au) of typical
planet-forming disks, most if not all carbon is in the gas phase, while iron
and iron sulfide grains are major constituents of the solid mixture along with
forsterite and enstatite. Our analysis demonstrates the need for detailed
studies of the dust in inner disks with new mid-infrared instruments such as
MATISSE and JWST/MIRI.

###### Key Words.:

protoplanetary disks – techniques: interferometric – stars: pre-main sequence
– stars: individual: HD 144432 – stars: variables: T Tauri, Herbig Ae/Be –
planets and satellites: formation

##  1 Introduction

Young low- and intermediate-mass pre-main-sequence stars are often surrounded
by a gas and dust disk. This disk forms early in the star formation process as
a result of angular momentum conservation, as matter from the collapsing
molecular cloud core accretes onto the forming protostar. Recent observations
with high-angular resolution facilities – for example, the Atacama Large
Millimeter/Submillimeter Array (ALMA) and the Spectro-Polarimetric High-
contrast Exoplanet REsearch at the Very Large Telescope (VLT/SPHERE) – have
revealed a rich variety of substructures in those disks (e.g., ALMA
Partnership et al., 2015; Andrews et al., 2018; Huang et al., 2018; Avenhaus
et al., 2018; Boccaletti et al., 2020). These observations show rings, gaps,
spiral arms, crescents, and clumps. This is accompanied with strong
observational evidence that planets form and evolve in those disks (Marois et
al., 2008; Keppler et al., 2018; Müller et al., 2018; Teague et al., 2018;
Pinte et al., 2019). Planet formation may start very soon after the formation
of the disk, as evidenced by structures detected in very young disks ($<2$
Myr) using high-resolution images at millimeter wavelengths (ALMA Partnership
et al., 2015; Sheehan & Eisner, 2018; Segura-Cox et al., 2020; Nakatani et
al., 2020; Sheehan et al., 2020, 2022).

These observations are put into context using thermochemical and
hydrodynamical disk models that describe the changes expected in the disk
structure and chemical composition of the gas and the solids as a result of
planet formation (e.g., Kley & Nelson, 2012; Flock et al., 2015; Woitke et
al., 2018). Key questions to address are how to link the observed diversity of
planetary systems to the properties of the disks in which they form; what are
the processes that determine the composition of exoplanets and their
atmospheres; and how can we constrain the chemical composition of the rocky
planets that form from the refractory dust in the inner regions of disks.

As the disk evolves, the small (sub-$\mu$m sized) and mostly amorphous
silicate dust grains that are accreted from the molecular cloud experience
grain growth through coagulation and begin to settle toward the midplane, and,
once reaching a critical size that depends on the local gas pressure, drift
inward toward the star (Testi et al., 2014). While in the outer disk grain
growth dominates, in the dense inner disk grain-grain collisions result in a
collisional equilibrium, which may include planetesimal-sized bodies. As the
temperature increases inward, the chemical composition and lattice structure
of the grains are modified, eventually leading to annealed, crystalline grains
(Bouwman et al., 2001; Henning, 2010). While irradiation from the central star
heats up the disk surface, viscous heating due to accretion acts in the
midplane. Both processes cause part of the dust to constantly evaporate and
reform (Min et al., 2011). Gas phase condensation of solid material in the
inner disk regions provides a source of freshly condensed crystalline material
that are mixed in with the grains that are accreted from the molecular cloud
(Sargent et al., 2009; Matter et al., 2020).

These processes tend to produce a radial distribution of silicate grains with
a gradient in size and crystallinity (van Boekel et al., 2004; Varga et al.,
2018). The opening of disk gaps and associated gas pressure bumps (Ruge et
al., 2016; Dullemond et al., 2018) can lead to size sorting and thus
substantially different spatial distributions of large and small grains in the
disk (Pinilla et al., 2012). In addition, parent body processing followed by a
collisional release of small grains may also modify the chemical composition
of the grains in the inner disk (Thébault & Augereau, 2007). Finally, episodic
eruptions (Ábrahám et al., 2009), photoevaporative winds (Owen et al., 2011),
and local heating processes (for instance, related to the formation of gas
giant planets, or electrical discharges in the disk) also affect the nature
and spatial distribution of the small grains (Pilipp et al., 1998; Desch &
Cuzzi, 2000; Harker & Desch, 2002).

Spectral analysis can uncover the mineral buildup of the dust in planet-
forming disks (van Boekel et al., 2005; Juhász et al., 2010; Olofsson et al.,
2010), thus providing clues about the composition of future planets that
formed from that material to be compared with the Solar System. Silicate dust
grains in the size range up to an $\sim$$5\ \mu$m radius are easily detected
in the surface layers of passively heated disks, because of the strong
vibrational resonances at infrared (IR) wavelengths that cause emission bands
and allow a measure of their chemical composition and size (e.g., Molster &
Waters, 2003). Observations have indeed shown the presence of $\mu$m-sized
grains in the disk surface that are substantially larger than those found in
the interstellar medium, and of crystalline silicates such as olivines and
pyroxenes with very low Fe/Mg ratios, consistent with forsterite (Mg2SiO4) and
enstatite (MgSiO3) (Henning, 2010). These minerals are also abundant in the
inner Solar System bodies, as an example, Mg, Si, and O constitute 59% of the
mass of Earth (Morgan & Anders, 1980). However, the single most abundant
element present in the inner Solar System planets is Fe (e.g., it makes up 32
% of the mass of Earth, Morgan & Anders, 1980), which is mostly concentrated
in the cores of the planets. This implies that during the formation of
planets, a large reservoir of iron-containing dust was available. Yet, the
detection of iron in the interstellar and circumstellar dust proves to be
elusive111There is some observational evidence for metallic iron and iron-
enriched silicate dust grains in the circumstellar envelopes of evolved stars
(e.g., Habing, 1996; Kemper et al., 2002; Marini et al., 2019).. Metallic iron
does not show any conspicuous spectral features at optical and IR wavelengths,
in contrast to silicates. The same applies for amorphous carbon dust, which is
also presumed to be present in the cosmic dust (e.g., Henning & Salama, 1998;
Tielens, 2022).

In this paper one of our main aims is to attempt to constrain the solid iron
content of a planet-forming disk, using IR photometric and spectro-
interferometric observations. For that we analyzed the spectral fingerprints
of the different dust species, aiming to constrain their mass ratios. The
novel aspect in this work is that we performed a consistent modeling of high
angular resolution IR interferometric data that enables us to uncover the disk
structure at sub-au resolution, and at the same time determine the composition
and radial distribution of the dust. Spatially resolving the warm inner parts
of the disk ($<10$ au) where the relevant dust emission features originate is
a key aspect, because it enables a more direct inference of the optical
properties of the solids, that can be compared to optical data of minerals
from laboratory experiments (Dorschner et al., 1995; Jäger et al., 1998a).

Here we study the planet-forming disk around HD 144432 A, a nearby Herbig Ae
star belonging to the Upper Sco subgroup of the Sco-Cen OB association (Galli
et al., 2018; Luhman & Esplin, 2020), at a distance of $154.1$ pc (Bailer-
Jones et al., 2021), with a spectral type of A9/F0V (Houk, 1982). Müller et
al. (2011) showed that HD 144432 is a hierarchical triple system, consisting
of the A-type primary, and a close binary system located
1$\aas@@fstack{\prime\prime}$47 ($227$ au) away. The close binary pair, HD
144432 B and C has a separation of 0$\aas@@fstack{\prime\prime}$1 ($15$ au).
Components B and C are T Tauri type stars of spectral type K7V and M1V,
respectively, and the triple system seems to be coeval with an age of $6\pm 3$
Myr (Müller et al., 2011). HD 144432 has a prominent silicate spectral
emission feature in the $N$ band ($8-13\ \mu$m), with one of the highest peak-
to-continuum ratios ($\sim$$4$) observed so far (van Boekel et al., 2005).
From a comparison of ISOPHOT-S and Spitzer InfraRed Spectrograph (IRS) spectra
of the object, Kóspál et al. (2012) pointed out that the $N$ band silicate
feature of HD 144432 is variable. van Boekel et al. (2004) presented $N$ band
($8-13\ \mu$m) spectro-interferometric observations of HD 144432 from the
VLTI/MIDI instrument. The authors found a radial mineralogy gradient in the
disk, where the region within $r\sim 2$ au is richer in crystalline silicate
grains, compared to the rest of the disk.

Table 1: Overview of VLTI observations of HD 144432 used in our work. The
$\tau_{0}$ is the atmospheric coherence time. LDD (limb darkened diameter) is
the estimated angular diameter of the calibrator. GRA4MAT is the new
instrument mode of MATISSE, which uses the GRAVITY fringe tracker to stabilize
the MATISSE fringes.

Instrument | Target | Calibrator | Band
---|---|---|---
| Date and time | Seeing | $\tau_{0}$ | Stations | Array | Name | LDD | Time |
| (UTC) | (″) | (ms) |  |  |  | (mas) | (UTC) |
PIONIER | 2013-06-07T02:52 | 0.7 | 8.0 | K0-A1-G1-J3 | large |  |  |  | $H$
PIONIER | 2013-06-17T02:46 | 0.6 | 5.2 | D0-G1-H0-I1 | medium |  |  |  | $H$
PIONIER | 2013-07-03T04:03 | 1.3 | 2.7 | D0-A1-C1-B2 | small |  |  |  | $H$
GRAVITY | 2018-03-05T08:29 | 0.5 | 11.0 | A0-G1-J2-J3 | large |  |  |  | $K$
MATISSE | 2019-05-06T07:33 | 0.5 | 6.2 | K0-G1-D0-J3 | large | 56 Hya | 1.2 | 07:17 | $L$
GRAVITY | 2019-05-24T04:38 | 0.9 | 2.8 | A0-G1-J2-J3 | large |  |  |  | $K$
MATISSE | 2019-09-02T01:19 | 0.6 | 6.2 | A0-B2-D0-C1 | small |  $H$ Sco | 4.7 | 01:57 |  $L$ (chopped)
GRA4MAT | 2022-03-09T09:10 | 0.4 | 16.8 | K0-G2-D0-J3 | medium | HD 138816 | 2.1 | 08:41 |  $L$ (chopped)(a)(a)footnotemark: $(a)$
GRA4MAT | 2022-03-09T09:10 | 0.4 | 16.8 | K0-G2-D0-J3 | medium | HD 138816 | 2.1 | 08:41 |  $M$ (chopped)(a)(a)footnotemark: $(a)$
GRA4MAT | 2022-03-22T09:11 | 0.9 | 3.8 | U1-U2-U3-U4 | UTs | HD 151051 | 2.3 | 08:39 | $L$
MATISSE | 2022-03-22T09:11 | 0.8 | 5.2 | U1-U2-U3-U4 | UTs | HD 151051 | 2.3 | 08:39 |  $M$ (chopped)
GRA4MAT | 2022-03-22T09:11 | 0.9 | 3.8 | U1-U2-U3-U4 | UTs | HD 151051 | 2.3 | 08:39 | $N$
GRA4MAT | 2022-04-04T08:23 | 0.8 | 3.8 | A0-G1-J2-J3 | large | 27 Sco | 2.6 | 08:51 |  $L$ (chopped)
GRA4MAT | 2022-04-04T08:23 | 0.8 | 3.8 | A0-G1-J2-J3 | large | 27 Sco | 2.6 | 08:51 |  $M$ (chopped)

222$a$$a$footnotetext: Only the BCD IN-IN data were used.

Chen et al. (2012) studied near-IR ($H$ and $K$ bands) VLTI/AMBER
interferometric observations of HD 144432. Using a geometric ring model they
found a nearly face-on disk (inclination $<28\degr$) with a $K$ band ring
radius of $0.17\pm 0.01$ au, which may be consistent with the location of the
dust sublimation radius. They also needed a spatially extended halo component
to fit their data. In their subsequent paper (Chen et al., 2016a), they
presented radiative transfer simulations with RADMC-3D using $H$ (IOTA,
AMBER), $K$ (Keck Interferometer, AMBER), and $N$ band (MIDI) data. Their
best-fit model features an optically thin inner disk component between $0.2$
and $0.3$ au, and an optically thick outer disk between $1.4$ and $10$ au.
They conclude that the disk of HD 144432 has a gap-like discontinuity at
around $1$ au radius, characteristic of pre-transitional disks. Such disks are
thought to represent an evolutionary step between an earlier stage where
optically thick material fills the entire warm region ($\lesssim 10$ au), and
the later transitional disk stage where the warm disk region is mostly cleared
of dust. Monnier et al. (2017) were not able to detect the disk of HD 144432
in $H$ band scattered light (using GPI), concluding that the nearby companion
may truncate the outer disk. HD 144432 was part of a VLT/SPHERE survey of
self-shadowed disks (Garufi et al., 2022). The authors suggest that the disk
is detected out to $0.25\arcsec$ ($\sim$$40$ au), and that the strong
asymmetry in the polarimetric image may be because of shadowing (see Fig. 2,
and page 5 therein).

In this paper we model new $L$, $M$, and $N$ band VLTI/MATISSE and $K$ band
VLTI/GRAVITY interferometric observations of HD 144432, complemented with
archival data from VLTI/PIONIER ($H$ band). To our knowledge, this is the most
complete IR interferometric data-set on a young stellar object analyzed so
far, in terms of wavelength coverage, ranging between $1.6-13\ \mu$m. We apply
a new flat disk temperature gradient model with simple radiative transfer to
fit the data, that is able to reveal disk substructures, and constrain the
dust composition. The paper is structured as follows: in Section 2 we describe
the observations and data processing, in Section 3 we present our modeling
approach, in Section 4 we show our results, followed by a discussion and
summary in Sections 5 and 6, respectively.

##  2 Observations and data processing

European Southern Observatory’s (ESO) Very Large Telescope Interferometer
(VLTI) is located at Paranal Observatory, Chile. Currently VLTI has three
instruments: PIONIER, working in $H$ band ($1.6\ \mu$m); GRAVITY, working in
$K$ band ($2.2\ \mu$m); and MATISSE, working in $L$ ($2.8-4\ \mu$m), $M$
($4.6-5\ \mu$m), and $N$ ($8-13\ \mu$m) bands. All of these instruments
combine the light of four telescopes, either of the $8.2$ m diameter Unit
Telescopes (UTs), or of the $1.8$ m diameter Auxiliary Telescopes (ATs). Using
interferometric beam combination, VLTI instruments sample the visibility
function, a complex quantity, which is the Fourier-transform of the object’s
brightness distribution on the sky. We collected a large set of VLTI data on
HD 144432 partly from our observations, partly from the Optical interferometry
DataBase (OiDB) at the Jean-Marie Mariotti Center. Table 1 shows the overview
of the VLTI observations used. Additionally, we collected archival IR
photometric and spectroscopic data using the VizieR catalog access tool.

VLTI instruments provide spectrally resolved measurements. As the probed IR
bands are quite broad, the angular resolution ($\vartheta$) varies
considerably over a band, modulating the spectral response. Additionally,
spectral modulations can arise due to well-resolved structures in the disk. In
the interferometric data, those spatial signatures are mixed with spectral
emission or absorption features of the dust grains. A key aspect in our
analysis is to disentangle the spatial signatures from true spectral
information.

The angular resolution of VLTI instruments depends on the wavelength and the
actual baseline lengths of the observations. The fixed UT array has a baseline
range of $46-130$ m, while, as of 2023, the movable ATs can be positioned to
three configurations: small ($11-34$ m), medium ($40-104$ m), and large
($58-132$ m). The approximate angular resolution333Following Matter et al.
(2020), the angular resolution is estimated as
$\vartheta={0.77}\lambda/\left(B_{\mathrm{p}}\right)$, where $\lambda$ is the
wavelength, and $B_{\mathrm{p}}$ is the projected baseline length. at long
baselines ($100-130$ m) is 2 mas for PIONIER, 3 mas for GRAVITY, 4.5 mas for
MATISSE $L$ band (at $3.5\ \mu$m), and 14 mas for MATISSE $N$ band (at $10\
\mu$m). These numbers translate to physical scales in the range of $0.3-2$ au
at the distance of our object.

###  2.1 MATISSE observations and data processing

We observed HD 144432 with MATISSE (Lopez et al., 2021) between 2019 and 2022
in the frame of our Guaranteed Time Observing (GTO) program for surveying
young stellar objects (YSOs). Data from 2019 were taken using the standard
instrument mode (MATISSE standalone), while the 2022 data were taken using
GRA4MAT. GRA4MAT (GRAVITY for MATISSE Woillez et al., in prep.) is a new
instrument mode, which uses the fringe tracker of the GRAVITY instrument
(Gravity Collaboration et al., 2017) to stabilize the MATISSE fringes. This
allows us to have longer integration time (on the order of seconds), in order
to reach higher signal-to-noise ratio (S/N). In contrast, the integration time
of MATISSE standalone is limited by the coherence time of the atmosphere,
typically $\sim$$0.1$ s in $L$ band. We reduced and calibrated the data, and
selected a set of data products to be used in this work. The selection was
necessary to ensure that only good quality data (taken in good atmospheric
conditions, and without technical issues) ended up in our final data set for
modeling. Table 1 lists those data sets. The selected data sets were observed
at seeing values ranging between $0.4\arcsec$ and $1.3\arcsec$, and
atmospheric coherence time ($\tau_{0}$) values between $2.8$ and $16.8$ ms.

We used versions 1.5.0 (for the 2019 data) and 1.5.8 (for the 2022 data) of
the standard MATISSE data reduction pipeline (DRS) (Millour et al., 2016) to
process the data444The pipeline versions 1.5.0 and 1.5.8 give essentially the
same results for $LM$ band data.. Additional processing steps, like Beam
Commuting Device (BCD) calibration and flux calibration was performed using
the MATISSE tools python software
package555https://gitlab.oca.eu/MATISSE/tools. Final processing (averaging and
merging different data types to a final OIFITS file) was done using our own
python tool. We applied the same data processing workflow as in Varga et al.
(2021). For more details, we refer to Sect. 3.1 in that work. Each calibrated
data set contains the following: spectrally resolved visibilities on six
baselines, closure phases on 4 baseline triplets, flux-calibrated correlated
spectra666The correlated flux is the fringes amplitude in flux units. If it is
normalized to the single-dish flux of the object, it is known as fringes
contrast or (absolute) visibility. By correlated spectrum we mean a series of
correlated flux values along the wavelength direction. For more details we
refer to Appendix C. on six baselines, and a flux-calibrated total spectrum
(i.e., single-dish spectrum). The final calibrated MATISSE data products are
shown in Appendix A, in Figs 10, 11, and 12777Our data products are publicly
available in OIFITS version 2 format (Duvert et al., 2017) at the Optical
interferometry Database (OiDB, http://oidb.jmmc.fr).. The spectral resolution
in $LM$ bands is 34, while it is 30 in the $N$ band. More details on the data
selection and processing can be found in Appendix B.

#### First look at the data.

The $LM$ band data show a moderately resolved structure, with the visibility
dropping to $0.3$ at the longest baseline (Fig. 10), with some contribution
from the unresolved central star which accounts for about $10\%$ of the total
$L$ band emission (for more details on how the stellar contribution is
calculated, we refer to Sect. 3). The $L$ band closure phases do not deviate
significantly from zero ($<6\degr$), indicating a brightness distribution
which is nearly centrally symmetric. The $L$ band spectra are featureless. In
the $N$ band, the visibilities on the UT baselines are between $0.03$ and
$0.35$ (Fig. 12), indicating a much more resolved brightness distribution,
compared to that in $L$ band. The contribution of the central star in $N$ band
is only $0.5\%-3\%$, practically negligible. The N-band closure phases reach
$\sim$$30\degr$, indicating some asymmetry of the N-band brightness
distribution. The $N$ band single-dish spectrum and correlated spectra are
dominated by the silicate spectral feature in emission (Fig. 1). The ratio of
the peak flux with respect to the continuum is $3.3$ in case of the MATISSE
single-dish spectrum. This remarkable feature strength has already been known
from earlier single-dish (van Boekel et al., 2005; Juhász et al., 2010) and
interferometric observations (van Boekel et al., 2004; Varga et al., 2018). As
a 0th order approximation, the correlated spectra may be interpreted as inner
disk spectra, corresponding to the radial region of the object which remains
unresolved at a given baseline. Thus, increasing the baseline length allows us
to hone in to the inner regions of the object (for more on the applicability
of this notion, we refer to Sect. 4.1). In the case of HD 144432, the shape of
the silicate feature shows a gradual change with baseline: at the largest
spatial scales (single-dish and short baseline spectra), we can see a
triangular shape peaking at around $10\leavevmode\nobreak\ \mu$m, while with
increasing baseline the shoulder at $11.3\leavevmode\nobreak\ \mu$m becomes
more and more prominent. At relatively long baselines ($90-100$ m) the
$11.3\leavevmode\nobreak\ \mu$m peak overtakes the $10\leavevmode\nobreak\
\mu$m peak. This behavior indicates a radial gradient in the silicate
mineralogy, in agreement with van Boekel et al. (2004).

###  2.2 GRAVITY and PIONIER data

Our target was observed in the $K$ band with the GRAVITY instrument (Gravity
Collaboration et al., 2017) in the large AT configuration (A0-G1-J2-J3) on
2018-03-05 and on 2019-05-24, as part of the GRAVITY GTO program for YSOs. We
used the single-field mode that feeds both the fringe tracker (FT) and the
science instrument (SC) with the same object. The FT operates at low spectral
resolution (5 spectral channels over the whole K-band) and high speed
($\sim$900 Hz; Lacour et al., 2019) to freeze the atmospheric effects and lock
the fringes for the SC instrument allowing long integrations of a few tens of
seconds and high spectral resolution observations to be performed. Each
observation file corresponds to five minutes on the object and contains the
interferometric observables for all spectral channels of both instruments (FT
and SC). We calibrated the instrument transfer function by observing
interferometric calibrators (HD 103125, HD 132763, and HD 163495)888All these
calibrators have been selected, with the help of the SearchCal tool of the
Jean-Marie Mariotti Center (JMMC) (https://www.jmmc.fr/search-cal/), to be
single (according to the Washington Visual Double Star Catalog, Mason et al.,
2001), non-variable, and bright stars ($K<5.6$ mag), close enough ($<60\degr$
angular separation) to the target. before and after our science target. We
reduced the data with the GRAVITY data reduction pipeline (Lapeyrere et al.,
2014) and following the approach of Gravity Collaboration et al. (2019);
GRAVITY Collaboration et al. (2021), we only used the FT data for our fits and
discarded the blueish spectral channel that is polluted by the fluorescence
induced by the metrology laser operating at 1.908 $\mu$m.

Calibrated PIONIER data were downloaded from the OiDB. We selected one data
set for each AT configuration, the best ones to avoid errors introduced by
lower-quality data. The 2013-06-07 and 2013-07-03 data sets were provided by
Jean-Philippe Berger, and the 2013-06-17 data by Bernard Lazareff. All of
these PIONIER data products are graded as L2, that is, calibrated but
unpublished data.

#### First look at the data.

GRAVITY and PIONIER data are shown in Appendix A, in Figs. 13 and 14. Both
instruments resolve the disk emission, with lowest visibilities of $0.25$
(GRAVITY) and $0.5$ (PIONIER) at the longest baselines. Considering the flux
contribution of the unresolved central star ($\sim$$45\%$ in $H$ band, and
$\sim$$30\%$ in $K$ band), we can see that the disk emission gets fully
resolved both in the GRAVITY and PIONIER data. The closure phases remain small
($<6\degr$), indicating that the near-IR disk emission is nearly centrally
symmetric.

###  2.3 Photometric data

We collected IR photometric and spectroscopic data in order to construct the
IR spectral energy distribution (SED) of HD 144432. The $JHK$ photometry comes
from the Two Micron All Sky Survey (2MASS) catalog (Cutri et al., 2003). The
$LMN$ band spectroscopy comes from our MATISSE observations. Finally we use a
low spectral resolution Spitzer IRS spectrum from 2004-08-08 (AOR: 3587072)
that covers the $5.3-36.9\ \mu$m wavelength range (Fig. 5 right panel). We use
the MATISSE $N$ band single-dish spectrum, which we average with the Spitzer
spectrum in the overlapping wavelength range. We deredden the photometric data
using the interstellar dust model of Jones et al. (2013), and assuming an
optical extinction of $A_{V}=0.4$ (Chen et al., 2016b).

##  3 Model-based imaging

Our aim is to describe the IR interferometric and photometric data of HD
144432 with a relatively simple flat-disk semi-physical model, in which the
physical quantities have radial dependence only. The model outputs images of
the object at the wavelengths of the data, as well as constrain several
physical parameters of the disk. For this purpose we have developed a new
temperature gradient model, which we call TGMdust, with the following main
characteristics:

  * •

The model disk starts at a specific inner radius, $R_{\mathrm{in}}$,
representing the inner edge of the dusty disk.

  * •

The disk can be divided into several radial zones, each having its own surface
density profile, allowing us to represent radial disk substructure. The
surface density profiles are power laws.

  * •

The disk temperature profile is described by a single power law over the whole
radial extent of the disk.

  * •

The emission of several dust components is taken into account. The dust
composition between radial zones can differ, accounting for mineralogy
gradients.

The following equation gives the radial temperature profile:

$T\left(r\right)=T_{\text{in}}\left(\frac{r}{R_{\text{in}}}\right)^{q},$ (1)

where $T_{\text{in}}$ is the temperature at the inner edge of the dusty disk,
and $q$ is the power law exponent of the temperature profile. In each radial
zone, the surface density profile is also a power law:

$\Sigma_{i}\left(r\right)=\Sigma_{0,i}\left(\frac{r}{R_{i}}\right)^{p_{i}}.$
(2)

Here $i$ denotes the i-th radial zone (between $R_{i}$ and $R_{i+1}$), where
the surface density at the zone inner edge ($R_{i}$) is $\Sigma_{0,i}$, and
the power law index is $p_{i}$. In the first, innermost zone $R_{1}\equiv
R_{\text{in}}$.

The various dust components are taken into account by their individual opacity
curves. We calculate the vertical optical depth of the disk in the following
way:

$\tau_{\nu,i}\left(r\right)=\sum_{j=1}^{N}{c_{i,j}\Sigma_{i}\left(r\right)\kappa_{\nu,j}},$
(3)

where $\kappa_{\nu,j}$ is the wavelength-dependent opacity and $c_{i,j}$ is
the mass fraction of the j-th dust component, in the i-th zone.

Assuming a geometrically thin and homogeneous emitting layer, we use the
following radiative transfer equation to calculate the surface brightness
profile of the disk:

$I_{\nu,i}\left(r\right)=\left(1-\text{e}^{\tau_{\nu,i}\left(r\right)}\right)B_{\nu}\left(T\left(r\right)\right),$
(4)

where the source function is the blackbody radiation ($B_{\nu}$), and
$\tau_{\nu,i}\left(r\right)$ is the radially and wavelength-dependent vertical
optical depth profile in the i-th zone.

To get observable quantities, we calculate the flux density ($F_{\nu}$) from
the surface brightness profiles of each zone, then add them together along
with the flux density of the central star ($F_{\nu,\text{star}}$):

$\displaystyle F_{\nu,i}=\int_{R_{i}}^{R_{i+1}}2\pi
rI_{\nu,i}\left(r\right)\text{d}r,$ (5) $\displaystyle
F_{\nu}=\sum_{i=1}^{M}F_{\nu,i}+F_{\nu,\text{star}}.$ (6)

We note that $F_{\nu}$ can be directly compared to single-dish photometric and
spectroscopic observations. For the outermost radius of the integration
($R_{\text{out}}$), we choose $30$ au which, at the distance of our object,
roughly corresponds to the field of view of MATISSE in the $N$ band on the
UTs. We expect that this radius is large enough that it encloses practically
all the flux emitted in the wavelength range of our data999We also tested a
larger outer radius, $60$ au, with which we got practically the same result as
with the smaller value.. For the calculation of the interferometric visibility
and correlated flux, we refer to Appendix C. Although our model has only
radial dependence, we can account for the inclination of the disk ($i$) by the
deprojection of the baselines (detailed in Appendix C). Several authors
analyzing near-IR VLTI data found that the disk is close to face-on (Chen et
al., 2012; Lazareff et al., 2017; Gravity Collaboration et al., 2019; Kluska
et al., 2020). We selected a subset of the MATISSE data with the highest
quality to fit the disk orientation with a simple geometric model. We applied
a Markov Chain Monte Carlo (MCMC) algorithm for the parameter search, and the
disk size and orientation were fit. We found a $\cos i$ of $0.79\pm 0.1$, and
the position angle of the major axis $\theta=108\degr\pm 15\degr$ (measured
from north to east). In our subsequent modeling runs we use these values,
keeping them fixed.

The flux contribution of the central star is also kept fixed in the modeling.
We estimate the stellar spectrum by fitting PHOENIX stellar atmosphere
models101010ftp://phoenix.astro.physik.uni-goettingen.de/HiResFITS (Husser et
al., 2013) to optical photometric data of HD 144432. The $B_{T}$ and $V_{T}$
magnitudes were obtained from the Tycho-2 catalog (Høg et al., 2000),
$G_{\text{BP}}$, $G$, and $G_{\text{RP}}$ magnitudes were obtained from the
Gaia Early Data Release 3 (EDR3, Gaia Collaboration et al., 2016, 2021), and
$U$ and $I$ magnitudes were obtained from Mendigutía et al. (2012). While Gaia
was able to spatially resolve the A and the BC components, as the EDR3 catalog
provides separate photometry for the objects, in the $B_{T}$, $V_{T}$, $U$,
and $I$ band observations the stars are likely measured together. According to
the Gaia EDR3, the BC binary in the $G$ band is $3.68$ mag fainter than the
primary. Using the stellar parameters by Müller et al. (2011), we estimate the
contamination by the BC components to be negligible ($<3\%$) in all our
photometric bands except the $I$ band where it is $\sim$$10\%$. The stellar
atmosphere fit was done with a modified version of SED Fitter (Robitaille et
al., 2007). We permitted a relatively small fit range around the typical
literature values of $T_{\text{eff}}$ and $\log g$. We got
$T_{\text{eff}}=8000$ K and $\log g=5.5$ as best fits. For comparison, the
typical range for the effective temperature of HD 144432(A) found in the
literature is $7100-7500$ K. A spectroscopic determination of $T_{\text{eff}}$
by Fairlamb et al. (2015) using XSHOOTER yielded $7500\pm 250$ K. Although our
fit is several hundreds of K higher, it is still reasonable.

###  3.1 Dust opacities

Table 2: Overview of the dust species used in this work.

Species | Chemical | Lattice | References
---|---|---|---
| formula | structure |
Mg-silicate of olivine stoichiometry |  Mg2SiO4 | amorphous | 1
Forsterite |  Mg2SiO4 | crystalline | 2
Mg-silicate of pyroxene stoichiometry |  MgSiO3 | amorphous | 3
Enstatite |  MgSiO3 | crystalline | 4
Iron | Fe | metallic | 5
Carbon | C | amorphous | 6

(1) Jäger et al. (2003); (2) Sogawa et al. (2006); (3) Dorschner et al.
(1995); (4) Jäger et al. (1998a); (5) Henning & Stognienko (1996); (6) Zubko
et al. (1996).

The $N$ band spectral window of MATISSE gives a handle on constraining the
dust composition of the disk. Silicate minerals have distinct spectral
emission features in this wavelength region. The shape of the feature mostly
depends on the following factors: (1) the chemical composition of the mineral,
(2) the crystallinity, and (3) the grain size and shape. Models aiming to
explain the SEDs of planet-forming disks have to include a continuum opacity
source to add more dust opacity at wavelengths where silicates are too
transparent. For that purpose, carbon is usually added to the dust mix (e.g.,
Woitke et al., 2016). However, metallic iron may also be considered 111111Dust
spectral features tend to be weaker with increasing grain size. Thus, very
large ($10-100\ \mu$m-sized) silicate grains may be also considered as the
source of the IR continuum emission. However, even at that grain sizes there
are distinct trends in the silicate opacity curves, such as a very broad dip
in the $\sim$$2-7\ \mu$m wavelength range, that could be distinguished from
the truly featureless opacity curves of, for instance, carbon or iron. Another
argument is that while $\mu$m-sized carbon grains have IR opacity values in
the range of $10^{4}-10^{5}$, the same range for $10-100\ \mu$m-sized silicate
grains is only $10^{1}-10^{2}$. This means that in order to produce the same
amount of radiation, we would need $\sim$$1000$ times as much big silicate
grains as $\mu$m-sized carbon grains, in terms of mass. . While the variety of
silicates found in the Solar System is enormous, in the interstellar space a
relatively small set of silicate minerals has been identified so far. In our
modeling we consider the following key dust components, thought to be main
ingredients of the cosmic dust, from earlier results of, for example, Bouwman
et al. (2001), van Boekel et al. (2005), Juhász et al. (2010), Jones et al.
(2017), and Tielens (2022): amorphous magnesium-silicate of olivine
stoichiometry (referred as olivine in the following), amorphous magnesium-
silicate with pyroxene stoichiometry (referred as pyroxene in the following),
crystalline forsterite, crystalline enstatite, metallic iron, and amorphous
carbon. In our modeling we assume that the dust species are isolated, that is,
they are not in thermal contact with each other. Carbonaceous material can
cover a wide range of properties, for instance, graphitic (conducting) or
diamond-like (not conducting) types have very different optical constants
(Jäger et al., 1998b). We use the optical data for amorphous carbon by Zubko
et al. (1996) who noted that their carbon samples were semiconductors rather
than metals. For each dust component we consider two grain radii, $0.1\ \mu$m
(”small”) and $2\ \mu$m (”large”), except for enstatite for which we have a
third size of $5\ \mu$m (”very large”) in addition. The motivation for the
latter choice came from our initial modeling tests where we noticed that
better fit can be achieved if we include $5\ \mu$m-sized enstatite grains in
the dust mix. In contrast, including $5\ \mu$m-sized grains of any other dust
species did not improve the fit noticeably.

Metallic iron might become oxidized. Iron oxide solids have several resonances
in their mid-IR spectra, mostly between $10$ and $50\ \mu$m. In Fig. 15 we
compare the observed Spitzer spectrum of HD 144432 in that spectral region
with the opacity curves of FeO (wüstite), Fe2O3 (hematite), and Fe3O4
(magnetite). As we see no signs of the iron oxide spectral features in the
Spitzer data, we conclude that iron oxides are not major constituents of the
dust in the inner disk of HD 144432.

In Table 2 we present an overview on the dust species used in our modeling,
with references to the refractive index data. The opacities were calculated
using the distribution of hollow spheres (DHS) scattering theory (Min et al.,
2005). The upper boundary for the hollow sphere distribution
($f_{\text{max}}$) is $0.7$ for the amorphous species and iron, and $1.0$ for
the crystalline silicates. We used
optool121212https://github.com/cdominik/optool (Dominik et al., 2021) to
calculate the opacity curves of carbon and iron grains. They are shown in Fig.
16 (N-band only) and in Fig. 17 (covering the full wavelength range of the
data). An important aim of this work is to check whether our data are
consistent with an iron-rich or carbon-rich dust composition. Therefore, we
run models where we include either iron or carbon, to represent the
extremities in the abundance ratios of those two species.

###  3.2 Optimization

We differentiate between model parameters describing the disk structure and
parameters of the dust composition. In case of a single zone, the model has 5
structural parameters ($T_{\text{in}}$, $q$, $R_{\text{in}}$, $\Sigma_{0,i}$,
$p_{i}$). Each further zone adds 3 parameters ($R_{i}$, $\Sigma_{0,i}$, and
$p_{i}$). The number of dust components in our initial model is 11, consisting
of four silicate species and one species for the featureless continuum
emission (either iron or carbon), in two or, in the case of enstatite, three
grain sizes. As we want to model radial variations in the mineralogy, each
radial zone may have a specific dust composition. We run fits with 1, 2, and 3
zones. In the 3-zone model, there are 11 structural parameters, and 33 dust
mass fractions (11 per each zone). Given the large number of free parameters,
we employ a two-step optimization procedure, in which we first constrain the
silicate composition using a spectral decomposition tool called specfit, then
in the second step, we fit the overall disk structure with TGMdust. specfit
employs a genetic fitting algorithm, while TGMdust performs MCMC sampling. For
more details, we refer to Appendix E.

An important note is that the dust mass fractions derived from our modeling
only make sense if the disk emission at the relevant wavelengths ($N$ band) is
optically thin. In the following section, we show whether this assumption
holds.

##  4 Results

Figure 1: Dust compositional fits to the N-band MATISSE data. Left panel: fit
to the single-dish spectrum. Right panel: fits to the six correlated spectra.
The solid yellow curve is the total model spectrum, while the green, red, and
purple lines show the contributions from the individual disk zones. The thin
blue line is the model stellar spectrum. Figure 2:  Correlated flux density as
function of the baseline length at a selected wavelength, $10\ \mu$m. The
contributions of the three disk zones are the green, red, and purple lines,
while the correlated flux curve of the whole disk is the orange solid line.
Filled circles indicate the MATISSE N-band data points. The zero baseline data
point is the single-dish flux. The thin blue line is the contribution of the
central star.

An important finding is that we need at least three zones to fit the data
well. In the following, we first present our results on the dust composition,
then on the disk structure.

###  4.1 Runs with specfit

Table 3: Results of the dust compositional fit. The abundances are given as fractions of the total dust mass in percent, excluding the hidden dust component responsible for the continuum emission. Species | Zone 1 | Zone 2 | Zone 3
---|---|---|---
|  $0.2-1.3$ au |  $1.3-4.1$ au |  $>4.1$ au
$\text{Olivine}\ 2.0\ \mu\text{m}$ | $25.2^{+3.6}_{-12.8}$ | $44.2^{+5.8}_{-15.0}$ | $81.2^{+0.1}_{-65.2}$
$\text{Pyroxene}\ 0.1\ \mu\text{m}$ | $5.1^{+3.0}_{-2.7}$ | $35.1^{+12.3}_{-7.0}$ | $0.0^{+53.0}_{-0.0}$
$\text{Pyroxene}\ 2.0\ \mu\text{m}$ | $8.3^{+9.3}_{-2.0}$ | $0.0^{+0.0}_{-0.0}$ | $0.0^{+0.0}_{-0.0}$
$\text{Forsterite}\ 0.1\ \mu\text{m}$ | $2.0^{+0.7}_{-0.2}$ | $0.8^{+0.7}_{-0.7}$ | $0.0^{+2.5}_{-0.0}$
$\text{Enstatite}\ 5.0\ \mu\text{m}$ | $59.3^{+3.7}_{-4.7}$ | $20.0^{+3.3}_{-10.5}$ | $18.8^{+11.6}_{-15.9}$
Crystallinity fraction | $61.3^{+3.8}_{-4.7}$ | $20.8^{+3.4}_{-10.5}$ | $18.8^{+11.9}_{-15.9}$

Figure 3: Best-fit model images at various data wavelengths with the TGMdust
run with iron grains. The brightness scaling is linear and homogeneous across
all wavelengths. For a better perception of the fainter structural features,
the images become saturated at $3.4\times 10^{-12}$ W m-2 Hz-1 sr-1. The gray
circles in the bottom left corner indicate the approximate beam size
(estimated as $0.77\lambda/(B_{\text{p,max}})$, where $B_{\text{p,max}}$ is
the maximum baseline length).  Figure 4: Radial surface brightness profiles of
the disk corresponding to the best-fit model with iron grains. The shaded area
around each curve is the $16-84$ percentile range from the last $5000$ samples
of the chain.

#### Fitting approach.

Initial runs with `specfit` showed that the fit prefers to have no or very
small amounts ($<0.05\%$) of small olivine, large forsterite, small enstatite,
and large enstatite. Thus, in order to decrease complexity of the model, we
removed those dust components from our further fits, leaving only six dust
components to fit (five silicates and the hidden dust component). A good fit
to the data is reached after a few dozen generations with a population of 100
models. In order to get a denser sampling of the parameter space, our final
modeling was run for 600 generations. The fits to the N-band MATISSE data are
shown in Fig. 1, and the best-fit silicate mass fractions are presented in
Table 3. The reduced $\chi^{2}$ corresponding to the best fit is $10.2$. In
contrast, with a two-zone model we got a much worse value of $40.6$. The two-
zone model could fit the single-dish spectrum, and the correlated spectra on
the three shortest baselines well, but the longer-baseline correlated spectra
are entirely missed. The three-zone model provides equally good fit on all
baselines. This demonstrates that the long-baseline ($>88$ m) MATISSE $N$ band
data contain crucial information on the inner-disk substructure of HD 144432.

Because the convergence of the genetic algorithm is different from that of an
MCMC sampler, we cannot simply use the sample distributions to estimate the
uncertainties. Thus, we filter the samples by imposing a very conservative
upper limit of 10 times the best-fit $\chi^{2}$, and the confidence intervals
are taken as the $16$th and $84$th percentiles of the filtered marginalized
parameter distributions.

#### Disk structure and radial dust composition.

The most important disk structural parameters which are constrained by the fit
are the zone radii: $R_{1}=0.22^{+0.01}_{-0.03}$ au,
$R_{2}=1.26^{+0.05}_{-0.08}$ au, and $R_{3}=4.13^{+0.03}_{-0.24}$ au. As shown
in Table 3, the most abundant dust components are large olivine, small
pyroxene, and very large enstatite. The dust composition of the individual
zones differs significantly. The fit results provide strong evidence for
radial mineralogy gradients. Specifically, the crystalline species
(forsterite, enstatite) are concentrated in the innermost zone ($<1.3$ au)
where around $61\%$ of the silicate dust mass is in crystalline form,
predominantly as enstatite. In contrast, the crystalline fraction in the outer
zones is only $\sim 20\%$ (Table 3). Most of the dust abundances are
relatively well constrained, except for large olivine and small pyroxene in
the third zone. We cannot exclude that the third zone contains significant
amounts of small pyroxene. We note that silicate compositional fits are
subject to some degeneracy, due to similarly shaped dust opacity curves. We
investigate this issue further in Appendix F.

Another notable finding is that the fraction of small grains ($0.1\ \mu$m
size) is rather small, only $\sim 7\%$ in the inner zone, and $\sim 36\%$ in
the second zone. The high crystallinity fraction and the high ratio of larger,
$\mu$m-sized grains can indicate that the dust in the inner au of the disk is
significantly processed. Alternatively, evaporation and recondensation may be
important in the innermost disk region, that can also produce crystalline
grains. The latter idea is further explored in Sect. 5.4. For the hidden dust
component, we got mass fractions of $\sim$$30\%$ in the first, $\sim$$50\%$ in
the second, and $\sim$$95\%$ in the third zone. As the opacity level of that
component was chosen arbitrarily, its mass fraction values have limited
physical meaning. However, it is worth to note that the mass fraction of the
hidden dust increases toward the outer zones.

Comparing the abundances of Mg2SiO4 (crystalline forsterite and amorphous
olivine together) vs. MgSiO3 (crystalline enstatite and amorphous pyroxene
together), we see that MgSiO3 is $2.7$ times more abundant in the innermost
zone, and $1.2$ times in the second zone. In the outer zone the same ratio is
$0.2$, so there Mg2SiO4 appears to be more abundant, although this is a highly
uncertain estimate. The elemental abundances of the disk zones are similar;
each zone contains roughly $30\%$ Mg, $24\%$ Si, and $46\%$ O by mass. These
numbers relate only to the silicates, as the elemental composition of the
hidden dust component is not considered at this stage. The abundance ratios of
Mg/Si and Si/O are $\sim 0.08$ dex and $\sim-0.3$ dex, respectively.

The left panel of Fig. 1 shows the model fit to the single-dish $N$ band
MATISSE spectrum. Green, red, and purple lines indicate the individual
contributions from each zone. It turns out that the silicate spectral feature
mostly originates from the second zone ($1.3$ au $-4.1$ au) where the
temperatures are in the range of $300-600$ K. There is a large difference in
the shape of the silicate spectral feature in the different zones. The inner
zone has a spectrum characteristic of crystalline silicates (especially
enstatite), the spectrum of the second zone is dominated by small and large
amorphous grains, while the third zone shows a very weak silicate feature with
an emission mainly from large amorphous grains.

Table 4: Final results of the fits.

| with Fe | with C
---|---|---
Best-fit parameter values with TGMdust
$R_{1}$ (au) | $0.15^{+0.01}_{-0.00}$ | $0.15^{+0.01}_{-0.00}$
$R_{2}$ (au)(a)(a)footnotemark: $(a)$ | $1.26^{+0.05}_{-0.08}$
$R_{3}$ (au)(a)(a)footnotemark: $(a)$ | $4.13^{+0.03}_{-0.24}$
$T_{\mathrm{in}}$ (K) | $1780^{+69}_{-193}$ | $1606^{+86}_{-111}$
$q$ | $-0.53^{+0.04}_{-0.05}$ | $-0.50^{+0.04}_{-0.06}$
$\Sigma_{0,1}$ ($10^{-3}\text{g\ cm}^{-2}$) | $1.21^{+0.35}_{-0.56}$ | $0.10^{+0.04}_{-0.07}$
$\Sigma_{0,2}$ ($10^{-3}\text{g\ cm}^{-2}$) | $0.37^{+0.06}_{-0.11}$ | $0.17^{+0.05}_{-0.06}$
$\Sigma_{0,3}$ ($10^{-3}\text{g\ cm}^{-2}$) | $10.5^{+632.5}_{-4.1}$ | $78.3^{+196.5}_{-76.2}$
$p_{1}$ | $-0.86^{+0.38}_{-0.29}$ | $-0.74^{+0.45}_{-0.28}$
$p_{2}$ | $-1.25^{+0.79}_{-0.17}$ | $-1.24^{+0.68}_{-0.21}$
$p_{3}$ | $-40.8^{+4.4}_{-49.1}$ | $-67.5^{+37.1}_{-23.1}$
$c_{1,\text{small}}$ ($\%$) | $0.04^{+1.20}_{-0.03}$ | $21.79^{+12.68}_{-7.05}$
$c_{1,\text{large}}$ ($\%$) | $91.98^{+2.63}_{-1.65}$ | $3.45^{+13.24}_{-1.75}$
$c_{2,\text{small}}$ ($\%$) | $45.48^{+7.30}_{-13.83}$ | $9.86^{+9.31}_{-7.89}$
$c_{2,\text{large}}$ ($\%$) | $10.56^{+12.85}_{-10.54}$ | $0.41^{+7.03}_{-0.39}$
$c_{3,\text{small}}$ ($\%$) | $2.92^{+44.54}_{-2.82}$ | $0.23^{+50.30}_{-0.14}$
$c_{3,\text{large}}$ ($\%$) | $0.01^{+14.46}_{-0.00}$ | $0.02^{+11.61}_{-0.00}$
Fixed parameters
$\theta$ ($\degr$) | $108.0$
$\cos\,i$ | $0.79$
$R_{\mathrm{out}}$ (au) | $30.00$
$d$ (pc) | $154.1$
Resulting half-flux radii (au)
$2.2\ \mu$m | 0.23 | 0.23
$3.3\ \mu$m | 0.31 | 0.30
$4.75\ \mu$m | 0.48 | 0.46
$11.3\ \mu$m | 2.25 | 2.31
$\chi^{2}_{V}/N_{V}$ | $4.9$ | $5.3$
$\chi^{2}_{\text{SED}}/N_{\text{SED}}$ | $5.5$ | $6.8$
$\chi^{2}_{\text{total}}$ | $20.2$ | $22.8$
131313$a$$a$footnotetext: These parameters were fit in the specfit run, but
were fixed in the TGMdust runs.

#### Spatial and spectral signatures in the correlated spectra.

In earlier IR interferometric studies (e.g., van Boekel et al., 2004 or Varga
et al., 2018) the correlated spectra were usually treated as they were proper
spectra. However, in the general sense that notion is not correct. If the
object has a significant substructure (e.g., rings, a stellar disk, or a
binary) which is well resolved by the interferometer, the spatial signature
emerges as a modulation in the correlated spectra that becomes mixed with the
real spectral signature of the object (e.g., Jamialahmadi et al., 2018). This
effect can be seen in the right panel of Fig. 1, showing the fits to the
correlated spectra, and in Fig. 2 where we plot the correlated flux as a
function of baseline length at a single wavelength of $10\ \mu$m. The
contributions of the various disk zones show the effects of resolution: while
the correlated flux of the inner zone barely decreases in the sampled baseline
range, the other two zones show a sharper decrease, and an oscillation pattern
associated with the resolved ring-like emission. The frequency of this pattern
is related to the location of the zones: the further out a ring is located,
the higher is the frequency141414The exact mathematical formula for the
correlated flux of an infinitesimally thin ring is
$F_{\nu,\text{corr}}=F_{\nu,\text{total}}J_{0}\left(2\pi
rB_{\text{p}}/\lambda\right)$ where $F_{\nu,\text{total}}$ is the single-dish
flux, $J_{0}$ is the Bessel function of the first kind of order zero, and $r$
is the ring radius.. The modulation effect gets more prominent with increasing
baseline.

The contributions of the disk zones are added up in the correlated flux of the
full disk (orange line). We note that the correlated fluxes of the different
disk zones may have negative values. At the longest baseline
($B_{\mathrm{p}}=129.1$ m, Fig. 1 right panel), we can even see the silicate
feature to completely disappear, due to the positive and negative correlated
flux contributions from the different rings canceling each other. This is
because in the general case the visibility function of a ring is the $J_{0}$
Bessel function which oscillates around zero. The correlated flux which we
measure with our interferometer is the absolute value of the sum of the
correlated fluxes of all zones, thus it is always positive151515There is an
inherent assumption in our analysis that the real value of the complex
visibility function of the whole object equals to its absolute value. This is
only true if the phase of the visibility function is zero. We are not able to
measure the visibility phase directly with VLTI, instead we measure the
closure phase which is the sum of the three phases along a closed triangle of
baselines (for more information, we refer to Monnier, 2003). As the closure
phases of our object are relatively small (typically $<10\degr$), we conclude
that our above assumption holds well..

The bottom line is that sharp edges in the object image cause strong
modulations in the visibility function, both in the spectral and spatial
directions. In the HD 144432 data this effect is prominent in the N-band
correlated spectra. As a general note, one should be aware of spatial
modulations when working with correlated spectra. Interpreting those data as
proper spectra should be generally avoided, instead, interferometric modeling
or image reconstruction should be applied. Still, the old notion that the
decreasing silicate feature amplitude with baseline indicates radial
mineralogy gradient or radiative transfer effects, can hold in systems where
the disk has no significant substructure at the sampled spatial scales.

###  4.2 TGMdust runs

Figure 5: Results of the TGMdust model run with iron grains. Left panel: fit
to the visibilities with respect to the effective baseline length. The points
are color coded for wavelength. The following wavelengths are plotted: $1.7\
\mu$m (PIONIER, gray), $2.2\ \mu$m (GRAVITY, purple), $3.3\ \mu$m (MATISSE
$L$, blue), $4.75\ \mu$m (MATISSE $M$, green), $8.3\ \mu$m (orange), $11.3\
\mu$m (red), and $12.3\ \mu$m (brown, last three are MATISSE $N$ data). Right
panel: fit to the SED. The first three data points are 2MASS $JHK$
measurements, the three points between $3$ and $5\ \mu$m are extracted from
our MATISSE $LM$-band data, and the points between $6$ and $18\ \mu$m are
taken from a Spitzer observation, except the $N$ band where the points are the
average of the Spitzer and our MATISSE single-dish spectra.

Figure 6: Same as Fig. 5, but for the TGMdust model run with carbon grains.

After the retrieval of the silicate composition with `specfit`, we run MCMC
fits with our `TGMdust` code, using the full data-set covering the $HKLMN$
bands. We fit the visibility data in the $1.6-12.3\ \mu$m wavelength range,
and the SED between $1.25$ and $18\ \mu$m. In these runs we fixed the relative
mass fractions between the different silicate components, and the inner radii
of the outer two zones. This is because those parameters are well constrained
by the N-band only modeling with `specfit`. We choose to fit the inner radius
of the disk, because that is better constrained with the shorter wavelength
VLTI data ($HKL$ bands). In addition to silicates, here we consider dust
components responsible for the continuum opacity of the disk, namely iron and
amorphous carbon. We attempt to determine their mass fractions, expecting that
the featureless opacity curves of those species (Fig. 17) might be better
constrained using the wide wavelength span of the full data set. We perform
two separate runs, one with silicates and iron, the other with silicates and
carbon grains. We choose a population of two grain sizes, $0.1\ \mu$m and $2\
\mu$m, like in the case of the silicate grains. We have $15$ fit parameters in
total: the inner radius of the disk ($R_{1}$), the temperature at the inner
radius ($T_{\text{in}}$), the power law exponent of the temperature profile
($q$), the three surface densities at the zone inner edges ($\Sigma_{0,i}$),
the three surface density gradients ($p_{i}$), and the mass fractions of
either iron or carbon per grain size per zone ($c_{i,\text{small/large}}$, six
mass fractions).

Each MCMC run consists of $50000$ steps with $32$ walkers. When estimating the
best-fit values, the first 20000 steps are discarded. For the $\chi^{2}$, we
first calculate separately the $\chi^{2}$ values for the SED
($\chi^{2}_{\text{SED}}$) and for the visibility data ($\chi^{2}_{V}$). In
case of the SED the $\chi^{2}$ is calculated using the logarithms of the
values, which helps to better fit the overall shape and flux levels of the
SED. To equalize the weight of the visibilities in the fitting, we multiply
$\chi^{2}_{V}$ by a factor of $w=3$. This ensures a similarly good fit for
both data types. With other values for $w$ we found that either the SED fit or
the visibility fit is much worse than the other. The total $\chi^{2}$, used in
the optimization, is then
$\chi^{2}_{\text{total}}=\chi^{2}_{\text{SED}}/N_{\text{SED}}+w\chi^{2}_{V}/N_{V}$,
where $N_{\text{SED}}$ and $N_{V}$ are the number of fit data points in the
SED and in the visibility data, respectively. The best-fit parameter values
are chosen to correspond to the sample with the lowest
$\chi^{2}_{\text{total}}$ in the chain, and the uncertainties of the
parameters are taken as the $16-84$ percentile ranges of the respective
posterior distributions.

In Table 4 we list the best-fit parameters from both modeling runs, along with
the fixed parameters and $\chi^{2}$ values. The results of the modeling with
iron grains are presented in the following figures: Fig. 3 shows the model
images at several wavelengths, Fig. 4 shows the radial surface brightness
profiles, Fig. 5 shows the fit to the data, and Fig. 7 shows the vertical
optical depth profiles. The corresponding figures for the model with carbon
grains are Fig. 21 (model images), Fig. 22 (radial surface brightness
profiles), Fig. 6 (fit to the data), and Fig. 23 (vertical optical depth). The
corner plots of the MCMC runs are presented in Fig. 20 (run with iron) and
Fig. 24 (run with carbon). The model images were produced by interpolating Eq.
4 on a rectangular grid, also applying the inclination effect.

Figure 7: Vertical optical depth of the best-fit disk model with iron grains.
Left panel: Radial optical depth profiles at selected wavelengths. Right
panel: Optical depth spectra at the inner edges of each zone.

The iron and carbon runs provide similar fits to the data, with slightly
better $\chi^{2}$ values for the run with iron (Table 4). Looking at the left
panels in Figs. 5 and 6, the fits to the visibilities are nearly identical.
The fit quality is generally good, especially in the $N$ band, while at
shorter wavelengths there are small to moderate systematic differences between
the model and the data. For instance, the $HKL$ data points at $<30$
M$\lambda$ spatial frequencies161616The spatial frequency is calculated by
dividing the baseline length with the wavelength of the data. are
systematically lower than the model points, indicating that at those
wavelengths the disk is actually more extended than the model prediction. In
contrast, in the $M$ band, the data points exceed the model points, indicating
that the disk emission in $M$ band is slightly more compact than the model.
These differences suggest that the disk has more substructure than what our
3-zone model can capture. Although the $\chi^{2}_{\text{SED}}$ values are
close to each other, the model with iron can reproduce the SED much better
than the model with carbon (right panels in Figs. 5 and 6). The difference is
most pronounced in the spectral region around the silicate feature, as the
model with iron correctly fits both the silicate peak and the continuum level,
while the model with carbon overestimates the continuum between $6$ and $15\
\mu$m.

###  4.3 Disk structure

Both models yield very similar model images (Figs. 3 and 21). There is a
general trend that the emission gets more extended with increasing wavelength.
In the $H$ and $K$ band, the flux comes almost exclusively from the innermost
zone. From around $3\ \mu$m the second zone starts to appear, and in the $N$
band emission all three zones contribute significantly. This is also reflected
in the half-flux radius of the disk emission, listed in Table 4 for several
wavelengths, that increases from $0.2$ au at $2.2\ \mu$m to $2.3$ au at $11.3\
\mu$m. The zone inner edges are sharply defined. The inner two zones are much
brighter at $11.3\ \mu$m than at $8.3$ or $12.5\ \mu$m; this is due to the
increased emission of the silicate grains at that wavelength. The same can
also be seen in the radial surface brightness profiles (Figs. 4 and 22). The
shaded areas in the surface brightness profiles, that reflect the uncertainty
of the model fit, suggest that the profiles are quite well constrained by the
data. We note that in the `TGMdust` run we do not fit the inner radii of the
second and third zones, so the uncertainty of those parameters are not
represented in the surface brightness profile plots. For the inner radius of
the disk ($R_{1}\equiv R_{\mathrm{in}}$), the `TGMdust` fit prefers a very
small value $<0.16$ au, which differs significantly the value found in the
`specfit` run ($0.22^{+0.01}_{-0.03}$ au). Moreover, $0.16$ au should be
regarded as an upper limit, as the fit range had a lower boundary of $0.15$
au, and the corner plots indicate that the fit would prefer even lower values.
However, permitting a smaller inner radius would be unphysical, as the dust
species in our model cannot exist at those regions because of the high
temperatures there ($>1500$ K). Consequently, the inner rim may have a
distinct radial brightness profile or a special dust composition, or both,
that our model cannot sufficiently reproduce.

Our models prefer a rather large brightness contrast at the inner edge of the
third zone ($\sim$$4$ au), and a very steep decrease of the surface brightness
outward. The edge at $4$ au is well constrained by the $N$ band data, however,
the physical parameters of the third zone, such as the surface density profile
and dust composition, are poorly constrained. This is because most of that
radial zone is likely too cold to emit significantly in the $N$ band. The
emission of that zone mostly contributes at wavelengths longer than $20\
\mu$m, for that spectral region the model underestimates the SED. To conclude,
our modeling can constrain the disk structure between $\sim$$0.15$ and
$\sim$$5$ au.

###  4.4 Optical depth, temperature, and density profiles

The surface brightness profile (Eq. 4) is constrained both by the SED (which
sets the flux density) and by the VLTI data (which constrain the area from
which we receive the flux). We assume that the source function is a blackbody,
which is generally accepted for the radiation of circumstellar dust at IR
wavelengths. The temperature profile is also well constrained by the
simultaneous modeling of the SED and VLTI data, thanks to the wide spectral
coverage of both data-sets. Thus, we argue that $\tau$, the only remaining
variable in Eq. 4, is robustly measured. Both Fig. 7 and Fig. 23 indicate that
the disk emission is not optically thick in the inner two zones ($r<4$ au), as
at $\lambda>3\ \mu$m $\tau$ remains below $0.4$. Consequently, our assumption
that the disk emission is optically thin, holds relatively well. The `TGMdust`
model runs suggest an optically thick inner edge of the third zone, even in
$N$ band, while in the silicate compositional fits we generally assumed
optically thin radiation. Thus, the `specfit` results for that zone are
probably biased. At the same time, we have already noticed in Sect. 4.1 that
the dust composition in the third zone is poorly constrained by `specfit`.

The temperature profile follows a relatively shallow decrease
($q\approx-0.5$). The temperature at the inner edge is considerably higher
than the generally assumed $1500$ K (dust sublimation temperature), as it is
$\approx$$1800$ K in the run with iron, and $\approx$$1600$ K in the run with
carbon. There are several minerals which can endure such high temperatures. We
discuss this subject further in Sect. 5.5.

The dust surface densities at the inner edges of the first two zones are in
the range of $10^{-4}-10^{-3}$ g cm-2. The iron-rich model gives a total dust
mass of $5.7\times 10^{-5}$ M${}_{\text{Earth}}$ in zone 1, and $2.6\times
10^{-4}$ M${}_{\text{Earth}}$ in zone 2, while with the carbon-rich model we
find $5.9\times 10^{-6}$ M${}_{\text{Earth}}$ in zone 1 and $1.2\times
10^{-4}$ M${}_{\text{Earth}}$ in zone 2. The model with carbon has a lower
surface density and dust mass, compared to the model with iron. This is
because the IR opacity of small carbon grains is roughly an order of magnitude
above that of iron, so in order to produce the same amount of radiation less
carbon is needed than iron, in terms of mass. Both `TGMdust` runs imply a very
steep falloff of the surface density after 4 au. However, as was mentioned
previously, our data cannot really constrain the disk parameters in the outer
zone.

We note that in our modeling we consider only three, relatively small grain
sizes ($0.1$, $2.0$, and $5.0\ \mu$m), while it is very likely that the disk
has considerable amount of larger ($>10\ \mu$m) grains which can hide a
significant amount of mass because of their very low IR opacities. Thus, our
derived masses may only reflect the mass present in small grains, and
underestimate the total dust mass. A more general approach for this problem
would be to apply a continuous grain size distribution (e.g., Mathis et al.,
1977), but that is out of scope of the current study.

The models indicate that we need significant amounts of either iron or carbon
in the dust mix in order to be able to reproduce the data. However, their
radial distribution and grain size ratios cannot be well ascertained. This is
primarily due to the featureless opacity curves of these dust types, that
makes it difficult to identify the specific dust components in our spectro-
interferometric data. Still, based on the fits to the SED (Sect. 4.2), we have
a preference for the model with iron grains. Looking at the iron dust mass
fractions ($c_{i,\text{small}}$, $c_{i,\text{large}}$) one by one, the large
grains in the inner zone ($92^{+3}_{-2}\%$), and the small grains in the
middle zone ($45^{+7}_{-14}\%$) are relatively well constrained, while for the
other components either we have upper limits, or no constraints within the fit
ranges at all. In case of the model run with carbon we find that only the
small grains in the inner zone ($22^{+13}_{-7}\%$) are constrained well.

##  5 Discussion

###  5.1 Disk structure

Our results indicate a structured inner disk of HD 144432, with not less that
three rings in the inner $5$ au. This is a robust finding, as our multiband
data set has a sufficient baseline sampling and resolution ($0.2-1.4$ au,
depending on the wavelength) to recover those structures. As the closure
phases of the VLTI data remain typically within $\pm 10\degr$, deviations from
central symmetry are small, hence the observed structural features indeed
should be concentric rings viewed slightly inclined. In our modeling the ring-
like features arise as a result of a sudden increase of the radial surface
brightness at the zone edges. In reality, the rings may have a rounded radial
profile, but it is out of the scope of this study to constrain that aspect.
Chen et al. (2016a) performed radiative transfer simulations to fit a set of
$H$, $K$, and $N$ band interferometric data on HD 144432. Their best model
(coined DA0) features an inner disk between $0.2$ and $0.3$ au, which is
optically thin at $\lambda=2\ \mu$m in the midplane in the radial direction;
and an optically thick outer disk between $1.4$ and $10$ au. Their inner disk
may be the same structure as our zone 1 ($0.15-1.3$ au), while their outer
disk coincides with our disk zones 2 and 3 ($r>1.3$ au). As they only had a
single-baseline $N$ band interferometric observation with MIDI, that was not
enough to detect the ring at $4$ au.

#### Nature of the dust inner rim.

We now investigate whether the inner radius of our model disk could be the
dust sublimation radius. The following equation from Dullemond & Monnier
(2010) describes the temperature – radius relation of the dust, assuming
thermal equilibrium with the stellar radiation:

$R_{\text{dust}}=\left(\frac{L_{*}}{16\pi\epsilon\sigma
T_{\text{dust}}^{4}}\right)^{1/2},$ (7)

where $T_{\text{dust}}$ is the dust temperature, $L_{*}$ is the stellar
luminosity, taken as $11.6\ \mathrm{L}_{\odot}$ from our stellar atmosphere
fitting (Sect. 3), $\epsilon$ is the cooling factor, and $\sigma$ is the
Stefan-Boltzmann constant. For regular grains we can assume sublimation at
$T_{\text{dust}}\approx 1500$ K, the usual value found in the literature
(e.g., Dullemond & Monnier, 2010). Following Eq. 10 in Dullemond & Monnier
(2010) we calculate the values of $\epsilon$ for the dust mixes in the inner
zone. For the dust mix with carbon, we get $\epsilon=0.23$, and for the mix
with iron $\epsilon=0.47$. With these values we estimate a sublimation radius
of $0.24$ au for the carbon-rich dust, and $0.17$ au for the iron-rich dust.
Thus, the preference for a relatively small inner radius ($<0.16$ au) by the
`TGMdust` fit can be still consistent with the dust sublimation radius.
However, the inner temperatures in our best fits are $100-200$ K above $1500$
K. We can use Eq. 7 to calculate the radius ($R_{\text{dust}}$) corresponding
to our fit inner temperatures ($T_{\text{in}}$). For the model with carbon-
rich dust we get $0.17$ au, and for the model with iron-rich dust we get
$0.15$ au. These values are remarkably consistent with our values for the
inner rim radius ($R_{\text{in}}$). Still, regular (sub-)$\mu$m-sized dust
grains are not expected to survive at temperatures significantly above $1500$
K. We note that the uniform sublimation temperature which we applied in our
modeling oversimplifies the complex nature of various grain materials with
significantly different evaporation temperatures (e.g., Gail, 2004). We
explore this issue further in Sect. 5.5. There is also a possibility that the
disk structure and dust composition at the inner rim is more complicated than
that our model can represent. For more accurate results on the physical
conditions near the dust inner rim, one has to run proper radiative transfer
models with scattering included, also accounting for the height and shape of
the inner rim.

To put our results into context, Chen et al. (2012) calculated the radius of
the $K$ band emission from interferometric model fitting. The $K$ band
radiation is expected to come almost exclusively from the dust inner rim, so
the $K$ band disk size is a good proxy for the dust sublimation radius. Their
value for the $K$ band ring radius ($0.17$ au) is in agreement with our
finding for the inner rim.

Figure 8: Sketch of the disk of HD 144432, as seen by VLTI. We show the three
ring-like structures, and the possible gaps between them. In the gaps we
indicate two putative Jupiter mass planets. The cold outer disk, shown as a
gray patch, cannot be constrained by our data.

#### Two planet-induced gaps?

One might ask whether the dark regions between the ring-like features are real
disk gaps, or just shadowed regions. As in our modeling we do not represent
the vertical extent of the disk, we cannot address this question. Chen et al.
(2016a) explored this aspect in their RADMC-3D simulations with a puffed-up
inner rim as the source of shadowing. They found that the self-shadowed model
does not fit the data well, as it provides too low flux in the near-IR, and
too much flux at $10-100\ \mu$m.

If the dark regions are indeed gaps, they might have been opened by massive
planets residing in the disk (Asensio-Torres et al., 2021). It has been shown
that the width and depth of disk gaps in the gas surface density profile are
closely related to the planetary mass (Kanagawa et al., 2015). We assume that
the $\mu$m-sized dust which we detect in our IR data is colocated with the
gas, as expected for such small grains (e.g., Fouchet et al., 2010; Pohl et
al., 2016). Here we use the Eq. 5 in Kanagawa et al. (2016) to estimate the
masses of the putative gap-opening planets residing in the inner disk of HD
144432. The equation requires the gap width which is not straightforward to
derive from our modeling, as we represent the surface brightness profile in
each disk zone by a power law, and the dark gap-like regions in our images
(Fig. 3) have just one well defined edge (the outer one). Here we assume that
the surface density at the gap inner edge in zone 1 is equal to the surface
density at the gap outer edge which coincides with the inner radius of zone 2.
Thus, we get a gap inner edge at $r\approx 0.6$ au, and an outer edge at
$1.26$ au. We place our planet in the middle position at $R_{\text{p}1}=0.93$
au. From the temperature profile in our best-fit model with iron grains we
calculate the aspect ratio ($H_{g}/R_{\text{p}1}=0.038$, where $H_{g}$ is the
gas pressure scale height) of the disk at the location of the planet assuming
hydrostatic equilibrium. For the stellar mass, we take $1.8\
\mathrm{M}_{\odot}$ (Guzmán-Díaz et al., 2021). The gap width is also
influenced by the $\alpha$ viscosity parameter of the disk which is the source
of considerable uncertainty. For a relatively low $\alpha$ value ($10^{-4}$)
we get a planet mass of $0.4$ M${}_{\text{Jup}}$, while for a higher value
($\alpha=10^{-3}$) we get $1.3$ M${}_{\text{Jup}}$.

For the gap-like region in zone 2, we cannot define the gap inner edge as we
did it in the first case, because the surface density at the outer edge of the
second gap (i.e., the inner edge of zone 3) is much higher than the surface
density anywhere in zone 2. Thus, for simplicity, we assume that the gap edges
coincide with the inner ($1.3$ au) and outer ($4.1$ au) limits of zone 2. We
place the planet in the middle of the zone at $R_{\text{p}2}=2.7$ au, where we
estimate the aspect ratio to be $H_{g}/R_{\text{p}2}=0.048$. We get a planet
mass of $1.3$ M${}_{\text{Jup}}$ and $4.1$ M${}_{\text{Jup}}$ for the low and
high viscosity case ($\alpha=10^{-4}$ and $10^{-3}$), respectively. Thus, we
argue that both gap-like regions (one at $\sim$$0.9$ au, the other at
$\sim$$3$ au) in the inner disk of HD 144432 might host gas giant planets,
assuming that the structures we detect are tracing the overall density
profile. In Fig. 8 we present a sketch of the disk of HD 144432, as we
conceive it from our VLTI data and modeling, with the locations of the
putative planets indicated.

###  5.2 An optically thin inner disk

In physical models of optically thick planet-forming disks the surface layer,
which is directly illuminated by the central star, gets superheated, while the
deeper regions remain cold (e.g., Chiang & Goldreich, 1997). In this case the
warm to hot surface layer dominates the near-IR continuum and also gives rise
to the dust spectral emission features, while the layer immediately below the
superheated disk photosphere is expected to be significantly colder and to
emit only at longer, mid-IR wavelengths. This requires a strong vertical
temperature gradient in the disk, otherwise the thermal IR emission becomes
partially optically thick, and the silicate spectral feature gets weaker. It
is not straightforward to distinguish between this scenario and the case for a
really optically thin disk. It might happen that the radiation we observe in
HD 144432 comes from a superheated layer, mimicking an optically thin
component. Still, the fact that we were able to reproduce both the level of
the IR continuum emission and the silicate feature, using only one temperature
gradient component per radial zone, is a strong indication for an optically
thin emission.

Furthermore, Chen et al. (2016a) have already demonstrated in their RADMC-3D
simulations that an optically thin inner component (between $0.2$ and $0.3$
au) is indeed compatible with the SED and IR interferometric data of HD
144432. For comparison, in our modeling we found evidence for optically thin
IR emission coming from the inner two disk zones ($<4$ au, Sect. 4.4). We note
that Chen et al. (2016a) reported the optical depth measured in the midplane
in the radial direction ($\tau_{1}\left(2\ \mu\text{m}\right)=0.14$), that is
not directly comparable to our $\tau$, which is measured in the vertical
direction. Thus, instead of the optical depths we opt to compare dust masses.
They reported a dust mass for their inner disk component of $1.8\times
10^{-11}\mathrm{M}_{\odot}$. Our dust mass estimations for the inner zone of
our model (between $0.15$ and $1.3$ au) are $1.7\times
10^{-10}\mathrm{M}_{\odot}$ with the iron-rich dust mix, and $1.8\times
10^{-11}\mathrm{M}_{\odot}$ with the carbon-rich dust. A word of caution here
is that the dust mixes in these models may have significantly different
opacities, so having similar dust masses does not guarantee that the optical
depths are also similar. Still, as Chen et al. (2016a) used carbon grains in
their modeling, it seems plausible that their opacities are similar to our
carbon-rich model opacities. In that case their model is in remarkably good
agreement with ours, in terms of total dust mass and optical depth. To further
investigate the optical depth aspect, we would need proper radiative transfer
modeling of our data set which is out of scope of this study.

As a final note, the reduced optical depth in the inner disk of HD 144432
might indicate that the disk is being depleted, emphasizing the transitory
nature of this object. Massive planets may halt the inward radial drift of
dust grains, trapping them in outer rings or vortices (Fedele et al., 2017).
Alternatively, photoevaporative and magnetohydrodynamic disk winds can be also
be responsible for the clearing of the inner disk (Pascucci & Sterzik, 2009;
Lesur et al., 2022).

###  5.3 Dust composition

#### Radial mineralogy gradients.

An important result from our modeling is the confirmation of radial mineralogy
gradients which were first observed by van Boekel et al. (2004). From an
analysis of VLTI/MIDI N-band spectro-interferometric data they measured a
crystallinity fraction of $55^{+30}_{-20}\%$ in the $1-2$ au disk region, and
$10^{+10}_{-5}\%$ in the $2-20$ au disk region. The radii of their disk
regions were estimated from the baseline lengths of the observations. This
method only gives a rough approximation, assuming that the interferometer
collects the light from an aperture corresponding to the spatial resolution.
In our modeling we were able to constrain the spatial extents of the disk
zones with different mineralogical compositions much more precisely. In the
`specfit` run, we found that the inner zone ($0.2-1.3$ au) has a crystallinity
fraction of $61^{+4}_{-5}\%$, while in the outer zones the same number is
$20\pm 10\%$. These numbers match well with the values found by van Boekel et
al. (2004) within the error bars, and also agree with our previous findings
presented in Varga et al. (2018).

Alternatively, optically thick emission from the hot inner rim can also cause
the inner disk spectra to show weaker silicate peaks, as shown by Meijer
(2007). However, in our `TGMdust` fits we found that the IR radiation from the
disk inner edge is optically thin ($\tau<0.4$). Thus, we do not see much
support for a bright, optically thick inner rim in the disk of HD 144432 from
our data.

#### Comparison with disk-integrated mid-IR spectroscopy.

van Boekel et al. (2005) and later Juhász et al. (2010) performed dust
compositional fits to N-band single-dish (TIMMI2 and Spitzer) spectra of HD
144432. van Boekel et al. (2005) found a fit with $53\%$ small ($0.1\ \mu$m)
amorphous olivine, $42\%$ large ($1.5\ \mu$m) amorphous pyroxene, and a small
fraction of crystalline dust, mainly small forsterite ($1.9\%$). Juhász et al.
(2010) in their fits to Spitzer spectra between wavelengths $5$ and $17\ \mu$m
found that the main amorphous constituents are large ($2\ \mu$m) amorphous
olivine ($42\%$), small ($0.1\ \mu$m) amorphous pyroxene ($19\%$), and large
amorphous pyroxene ($15\%$). Among the crystalline species, their fit
preferred more enstatite ($6.1\%$) than forsterite ($2.3\%$). The results of
these studies differ significantly, indicating that there is an inherent
ambiguity in dust compositional fits to mid-IR silicate spectra. There are
several contributing factors to this, like the selection of dust species to
fit, the choice on the sizes of the grains, how the opacity curves are
calculated (cf. Figs. 1 and 2 in Juhász et al., 2010), and the differences in
the underlying disk model. Our silicate fit is not directly comparable to the
aforementioned studies, because we dissected the disk into several zones, and
obtained a composition for each zone. Still, our values in the disk zone 2 are
comparable to the results by Juhász et al. (2010), as both feature a large
contribution from large olivine and small pyroxene grains, and a preference
for more enstatite than forsterite. Although our MATISSE $N$ band spectro-
interferometric data have lower S/N than the Spitzer spectrum, we argue that
our data have more constraining power in the compositional fits, because we
have seven spectra (one single-dish and six correlated spectra) probing the
same object.

###  5.4 Comparison with condensation sequences

Figure 9: Concentration of various condensates (normalized to hydrogen atom
number density) as a function of radius and temperature from our calculation
using the thermo-chemical equilibrium code GGchem (Woitke et al., 2018). Top:
plot showing the concentration using the photospheric elemental abundances of
HD 144432. Bottom: plot showing the concentration using Solar elemental
abundances. For clarity, we show a selection of the condensates: we plot only
those species which are among the top three most abundant condensates for at
least one temperature value. The $T\left(r\right)$ relation follows our best-
fit TGMdust model with iron grains. The gas pressure-temperature profile
follows a power law with $p=3.16\times 10^{-2}$ bar at the maximum temperature
$T=2000$ K, and $p=1.79\times 10^{-6}$ bar at the minimum temperature
$T=277.5$ K.

#### The expected rich mineralogy of disks.

While only a few dust species have been directly detected in the interstellar
and circumstellar medium so far (mainly forsterite, enstatite, silicon
carbide, and polycyclic aromatic hydrocarbons), there is evidence that the
diversity of minerals in the cosmic dust is far richer. The evidence mainly
comes from the analysis of Solar System interplanetary dust (IPD) grains,
thought to be of presolar origin (e.g., MacKinnon & Rietmeijer, 1987; Lodders
& Amari, 2005; Engrand et al., 2023). For example, carbon and iron are both
found in the IPD and in meteorites (Anders, 1964). Moreover, the planets in
the inner Solar System are mostly composed of refractory elements: the main
refractories Mg, Al, Si, Ca, Fe, Ni have a mass fraction of $67\%$ in Earth,
and $84\%$ in Mercury (Morgan & Anders, 1980). Most of these elements had been
locked in the dust grains of the presolar nebula, before they got incorporated
in planetary bodies. As many of the supposed materials in the cosmic dust have
only weak or nonexistent optical-IR spectral features, the rich mineralogy of
the interstellar and circumstellar dust mostly remains to be hidden from our
observations.

#### Modeling condensation sequences with GGchem.

Chemical modeling of dust condensation offers an alternative way to assess the
variety of minerals which could exist under the physical conditions found in
circumstellar disks. Here we investigate the expected diversity and
distribution of solid state materials in a disk like HD 144432, using the
thermo-chemical equilibrium code GGchem (Woitke et al., 2018). GGchem
calculates the chemical composition of astrophysical gases, with the option to
include the formation of condensates. We aim to address the following
questions with GGchem:

  * •

The radial distribution of forsterite (Mg2SiO4) and enstatite
(MgSiO3)171717GGchem cannot model the crystalline structure of the
condensates. Thus, in this section we use the terms forsterite and enstatite
to encompass both amorphous and crystalline forms of the materials..

  * •

The nature of dust species which can exist in the innermost disk regions where
$T>1500$ K.

  * •

The amount of carbon and iron which condensates to dust grains in the disk.

For the input gas composition, we use the stellar abundances of C, O, Fe, Mg,
Si reported by Kama et al. (2015) for HD 144432. For the rest of the elements
(H, He, N, Na, Al, Ca, Ti, S, Cl, K, Li, F, P, V, Cr, Mn, Ni, Zr, W), we use
Solar abundances by Asplund et al. (2009). GGchem calculates the concentration
of the end products as function of the temperature and gas pressure (or
hydrogen density). We define a pressure-temperature profile
($p_{\text{gas}}-T$) based on the minimum mass solar nebula description by
Hayashi (1981), but for the $T\left(r\right)$ and $\Sigma\left(r\right)$
radial profiles we use the power law exponents found for zone 1 in our
`TGMdust` run with iron grains. For the calculation of $p_{\text{gas}}$, we
used Eqs. B.5 to B.9 from Jorge et al. (2022), assuming that the disk is
vertically isothermal. The results are shown in the top panel of Fig. 9,
indicating that the most abundant condensates between $0.2$ and $1$ au are
iron and forsterite. We see no enstatite forming at all, which is also found
by Jorge et al. (2022) if the Mg/Si ratio is above the solar value by $0.17$
dex. In the case of HD 144432, the Mg/Si ratio is even higher, $0.26$ dex more
than in the Sun.

We also performed a chemical calculation with Solar abundances taken from
Asplund et al. (2009), shown in the bottom panel of Fig. 9. The plot indicates
that both forsterite and enstatite form in a solar-like gas mixture, that is
consistent with the results of our silicate compositional fits for HD 144432.
The figure also shows that the radial distribution of the two main silicate
minerals are different: while enstatite can be found between $0.3$ and $2$ au,
forsterite has a broader distribution from $0.2$ au to $5$ au and beyond. In
our results we do not find evidence for such radial distribution, as our model
fit requires significant amounts of enstatite in all three disk zones.
Finally, we note that our estimation ($0.08$ dex) for the Mg/Si ratio of the
disk (Sect. 4.1) is very close to the solar value ($0.09$ dex).

#### The problem of enstatite.

$N$ band spectroscopic and spectro-interferometric data clearly indicate that
there is significant amount of enstatite in the disk around HD 144432, in
stark contrast with the prediction of our GGchem calculation with the stellar
abundances of HD 144432. We consider two scenarios to resolve this
discrepancy: either the equilibrium condensation conditions do not hold in the
disk, or the chemical composition of the disk differs from that of the stellar
photosphere. The general notion is that the host star abundances are good
proxies for the chemical composition of the cloud from which both the star and
the disk are formed (Adibekyan et al., 2021). However, Kama et al. (2015)
found that stars hosting group I disks, known to have large gaps, are depleted
in Fe, Mg, and Si, compared to the Sun and to stars hosting group II (flat,
continuous) disks. They suggested that this depletion is caused by giant
planets which block the accretion of dust onto the star, while not affecting
gas accretion. The same planets can also open gaps in the disk, making it
exhibiting a group I type SED. Although HD 144432 has a group II disk, Kama et
al. (2015) detected a depletion of refractory elements similar to the stars
hosting group I disks. Thus, it seems plausible that the disk of HD 144432 has
a solar-like elemental chemical composition, and the depletion of the stellar
photosphere in Fe and Si is caused by a giant planet in the inner disk ($<5$
au) halting the radial drift of the dust. The planet, which might be
responsible for the inner disk substructure as well, could be located in one
of the gap-like regions between two rings (either at $\sim$$1$ or at $\sim$$3$
au). However, there are two problems with this idea: i) small
(sub-)$\mu$m-sized dust grains are relatively well coupled to the gas, and ii)
we detect enstatite even in the inner zone ($0.2-1$ au), where we do not
expect a planet blocking the accretion at the inner edge.

Alternatively, the physico-chemical conditions in disks might be more complex
than that we can represent with a simple disk model with equilibrium
chemistry. GGchem calculates the gas and dust composition at every distance
(temperature) by minimizing the Gibbs free energy. However, the dust will have
seen a complex history of growth, settling and mixing, which is not accounted
for in an equilibrium calculation. For instance, Gail (2004) concluded that by
taking into account radial mixing, the bulk of the grains at temperatures
lower than $\sim$$1000$ K should be enstatite. This is because all forsterite
grains, which move from the innermost regions outward, pass through a region
in which they are converted to enstatite, and at further out these reactions
could freeze out, so that conversion back to forsterite would not take place.
Furthermore, equilibrium theory cannot well describe the complex nucleation
processes of solids, and phase-equilibrium models have no history. As
illustrated in Fig. 11 of Herbort et al. (2020), condensation of $\mu$m-sized
grains is intrinsically fast ($<1$ s), but the timescale of annealing is much
longer (on the order of $1$ Myr at $T=900$ K). Therefore, if the disk has a
constant temperature structure, and the dust is at rest, only the innermost
part can provide conditions for phase-equilibrium. Pignatale et al. (2016)
demonstrated that equilibrium condensation can be a valid assumption for the
disk region where $T\gtrsim 600$ K ($r\lesssim 1$ au in our model).

###  5.5 Hot dust at the inner rim

In Sect. 4.4 we noted that our models prefer a temperature at the inner rim
that is $100-200$ K above the sublimation temperature ($\sim$$1500$ K) of
regular dust grains. From the GGchem calculation we can evaluate what kind of
minerals can endure such high temperatures. Calculations with solar and with
the HD 144432 abundances show similar results. The first two condensates
forming are tungsten at $\sim$$2000$ K, and zirconium dioxide (ZrO2) at
$\sim$$1900$ K, but they are not shown in our plots because of their very low
concentrations
($\log_{10}\left(n_{\text{solid}}/n_{\left<\text{H}\right>}\right)<-9.4$). At
slightly lower temperatures ($\sim$$1850$ K) corundum (Al2O3) forms, followed
by perovskite (CaTiO3, not shown because its low abundance) and gehlenite
(Ca2Al2SiO7) condensing at $\sim$$1700$ K. This condensation sequence is very
similar to what was found by Woitke et al. (2018) at $p_{\text{gas}}=1$ bar;
the only difference is that in our calculation the condensation occurs at
lower temperatures due to the lower pressures in our disk model.

The most abundant condensates (corundum and gehlenite) present at $T>1500$ K
have concentrations ($\sim$$-6$) about 30 times less than the major solid
state species iron and forsterite. Moreover, the near-IR opacity of corundum
($\sim$$100$ cm2g-1) is much lower than that of iron ($\sim$$2000$ cm2g-1).
Considering the difference in temperature and in the surface area, we estimate
that the $3\ \mu$m flux from corundum grains at $T=1800$ K located at $0.15$
au is only $0.3\%$ of the flux emitted by iron grains at $T=1500$ K at $0.2$
au radius. It remains to be determined, whether that apparently weak signal
from corundum and other refractory grains could be detected in observational
data.

###  5.6 Iron vs. carbon

#### A favored iron-rich composition.

Condensation of carbon does not occur in the warm disk regions ($r<5$ au,
$T>300$ K) if the gas has a solar C/O ratio ($\text{C}/\text{O}=0.55$) or
less. According to Kama et al. (2015), HD 144432 has a C/O ratio of $0.24$. If
we artificially increase the C/O ratio to $\sim$$0.9$, we start to see
graphite forming near a temperature of $T=1000$ K. If C/O reaches $1$,
graphite becomes a major constituent in the dust mix between $850$ K and
$1250$ K, corresponding to the radial region between $0.3$ and $0.6$ au. Such
high C/O ratio seems to be unlikely in the majority of stars in our Solar
neighborhood. In the Hypatia Catalog by Hinkel et al. (2014), which is a
compilation of stellar abundance data within $150$ pc of the Sun, only $2\%$
of the stars have $\text{C}/\text{O}>0.9$. Assuming a similar C/O distribution
for the nearby star forming regions and YSOs, we suggest that in the warm
inner disk regions ($r<5$ au) of planet-forming disks most, if not all, carbon
is in the gas phase, while iron and iron sulfide (FeS, troilite) grains are
major constituents of the solid mixture, along with forsterite and enstatite.
Thus, small (sub-)$\mu$m sized iron grains should be the main source of dust
continuum radiation at thermal IR wavelengths in disks. This is in line with
the conclusions of Matter et al. (2020) who suggested to replace carbon with
iron when modeling inner disk spectra of YSOs. For more discussion on the
effects of varying C/O in inner disks, we refer to Matter et al. (2020).

Carbon in solid form is expected to be abundant in the cold outer disk, locked
up in ices or as pre-stellar grains. Radial drift can cause that material to
reach the inner disk. As a result, the inner disk would be populated by a mix
of grains which had experienced chemical equilibrium conditions and grains
which came from the outer disk, which is a scenario that cannot be ruled out
for HD 144432. Indeed, the fact that we detect both crystalline and amorphous
silicate grains may already signify that such a mix exists. Still, the inner
Solar System planets are highly depleted in carbon, so there should be
processes resulting in carbon-depleted inner disk solid reservoirs (Öberg et
al., 2011). Klarmann et al. (2018) investigated the efficiency of oxidation
and photolysis in the removal of carbon grains from the solid phase, and they
found that radial transport of dust can easily prevent the depletion of
carbon. They conclude that more efficient carbon-depletion mechanisms along
with radial drift barriers are needed to prevent the replenishment of carbon
in the inner disk. In a carbon-enriched environment, we can expect the
presence of various hydrogenated carbon compounds which show spectral features
in the mid-IR (Jones et al., 2017). In HD 144432, those carbonaceous features
are much weaker than the silicate emission bands (Seok & Li, 2017). Further
observations by MATISSE using the medium spectral resolution mode ($R\approx
500$) in $L$ band may shed light on the spatial distribution of the
carbonaceous dust in the disk of HD 144432. As far as iron is concerned, iron-
rich grains should be present both in the population of grains which are
produced by recondensation in situ, as well as from the outer disk, drifting
inward.

Table 5: Comparison of elemental abundances between Mercury, Earth, CI chondrites, and our best-fit disk models for HD 144432. References for the abundances: Morgan & Anders (1980) (Mercury, Earth), and Vollstaedt et al. (2020) (CI chondrites). |  |  | HD 144432 |
---|---|---|---|---
|  |  | Disk zone 2 |
|  |  | Fe-rich | C-rich | CI
Element | Mercury | Earth | model | chondrite
Fe | $64.5$ | $32.1$ | $56$ | $0$ | $18.4$
O | $14.4$ | $30.1$ | $20$ | $41$ | $46.5$
Si | $7.1$ | $15.1$ | $11$ | $22$ | $10.7$
Mg | $6.5$ | $13.9$ | $13$ | $27$ | $9.6$
C | trace | $0.04$ | $0$ | $10$ | $3.2$
other | $7.5$ | $8.8$ | $0$ | $0$ | $11.6$

In our modeling we found evidence for a high abundance of enstatite in the
disk of HD 144432. The combination of enstatite with iron is reminiscent of
enstatite chondrites which are believed to be a major building block of the
Earth and other inner Solar System bodies (e.g., Javoy, 1995; Javoy et al.,
2010).

By deriving elemental abundances from the dust composition that we fit (Tables
3 and 4), we can investigate what kind of rocky planets could form in the
inner disk of HD 144432. In Table 5 we show our abundance values for HD
144432, compared with elemental abundances of Mercury, Earth, and CI
chondrites. We selected the second disk zone ($1.3<r<4.1$ au) for the
comparison, as that zone covers large part of the terrestrial planet forming
region. The table shows that our model with iron-rich dust represents an
elemental composition intermediate between that of Mercury and Earth. In
contrast, the elemental composition of our model with carbon-rich dust has
some similarities with the abundances of CI chondrites, although the
correspondence is not particularly good. We note that in our models either we
have iron but no carbon, or the other way around. With that we cannot account
well for the composition of CI chondrites, as they contain significant amounts
from both carbon and iron.

#### Carbon content and planet habitability.

To put the iron vs. carbon dichotomy in a wider context, Hakim et al. (2019)
has shown that exoplanets forming from carbon-enriched material will have,
after differentiation, a graphite outer layer. Such a shell has major effects
on the thermal evolution and geochemistry of the planet. Moreover, if the
graphite layer is thick enough, it might prevent the silicate-rich mantle
material to come to the surface. In this case the planet surface would be
devoid of essential life-bearing elements (except of course carbon), making
the planet potentially uninhabitable.

#### Chemical sorting by grain growth.

Johansen & Dorn (2022) explored the grain growth of condensates forming in a
gas of solar composition. They found that iron preferentially grows into
sparser, bigger pebbles, while enstatite, forming at the same location, has
smaller grain sizes. Consequently, the population of large iron grains becomes
chemically separated from the silicate grains having Stokes numbers $10$ times
lower. They further propose that under such conditions streaming instability
leads to the formation of iron-rich planets. They conclude that this formation
mechanism can explain why Mercury is so enriched in iron, without the need for
experiencing giant, mantle-stripping impacts. This study demonstrated both the
crucial role of iron in the formation of terrestrial planets, and the
importance of microphysics when dealing with dust evolution.

To conclude, our observations and modeling of HD 144432 show tentative
evidence for the presence of iron in its planet-forming disk. This claim is
partly based on the notion that our model with iron-rich dust provides a
better fit to the SED, compared to our other model with carbon-rich dust.
Furthermore, our chemical calculations in a disk model representing the
pressure-temperature conditions in the HD 144432 disk also give support for an
iron-rich chemistry, in agreement with numerous literature sources.

##  6 Summary

In this paper we studied the planet-forming disk around HD 144432 by modeling
an extensive set of photometric and spectro-interferometric data covering the
thermal IR wavelengths. Our data set includes new observations with
VLTI/GRAVITY ($K$ band) and VLTI/MATISSE ($L$, $M$, and $N$ bands), and
archival data from VLTI/PIONIER ($H$ band). We introduced a new disk model,
`TGMdust`, to image the interferometric data, and to constrain the disk
structure and dust composition. `TGMdust` can represent disk geometries
without (single zone) or with substructure. In the latter case the disk is
divided into multiple radial zones, each having its own surface density
profile and dust composition. Our main results are as follows:

  1. 1.

The disk within $<5$ au is structured. We need at least 3 zones to fit the
data. Our best-fit model features an inner zone between $0.15$ and $1.3$ au, a
second zone between $1.3$ and $4.1$ au, and an outer zone starting at $4.1$
au. Each zone has a bright inner edge, thus the disk appears to show a three-
ringed structure.

  2. 2.

From our silicate compositional fits we find that the dust composition of the
zones significantly differ from each other, indicating radial mineralogy
gradients. Specifically, the mass fraction181818These values are given as
fractions of the total silicate dust mass, excluding the dust components
responsible for the continuum emission. of crystalline silicates is about
$61\%$ in the innermost zone ($<1.3$ au), while only $\sim$$20\%$ in the outer
two zones.

  3. 3.

By inspecting the fits to the MATISSE $N$ band data, we find that the silicate
spectral feature mostly originates from the second disk zone ($1.3-4.1$ au).
We show that the silicate spectral feature in the correlated spectra is mixed
with strong spatial modulation signatures, an effect caused by the ring-like
substructures. We also find that the disappearance of the silicate feature in
the longest baseline ($B_{p}=129$ m) correlated spectrum is mostly due to
spatial modulation effects, not to mineralogy gradients or radiative transfer
effects, as was previously thought. As a general note, interpreting correlated
spectra as proper spectra should be generally avoided, instead,
interferometric modeling or image reconstruction should be applied.

  4. 4.

To identify the dust component responsible for the infrared continuum
emission, we explore two cases for the dust composition, one with a
silicate+iron, the other with a silicate+carbon mixture. We can fit the
interferometric data with both compositions equally well, however, the iron-
rich model provides a better fit to the SED.

  5. 5.

We find that the radiation from the inner two disk zones ($r<4$ au) is mostly
optically thin, having $\tau<0.4$ at wavelengths longer than $3\ \mu$m. This
finding is consistent with the radiative transfer modeling results of Chen et
al. (2016a).

  6. 6.

Assuming that the dark regions in the disk are gaps opened by planets, we
estimated the masses of the putative planetary bodies responsible for those
structures. We find that both gaps, the one in zone 1 at $\sim$$0.9$ au and
the other in zone 2 at $\sim$$3$ au, might be carved by approximately Jupiter
mass planets.

  7. 7.

In order to assess the hidden diversity of minerals in a planet-forming disk
like HD 144432, we performed chemical equilibrium calculations of dust
condensation with GGchem. Using solar elemental abundances in the GGchem
calculation results in a dust mixture with major constituents iron,
forsterite, and enstatite, in agreement with our modeling with the
iron+silicate dust mix.

  8. 8.

GGchem predicts the existence of some refractory minerals which can survive in
the $1500-2000$ K temperature range. The most abundant species under those
conditions are corundum (Al2O3) forming at $\sim$$1850$ K, and gehlenite
(Ca2Al2SiO7) condensing at $\sim$$1700$ K. We estimate that the $3\ \mu$m flux
emitted by corundum grains at $T=1800$ K located at $0.15$ au is only $0.3\%$
of the flux emitted by iron grains at $T=1500$ K located at $0.2$ au radius.
It is still unclear whether that weak emission could be detected with current
observational techniques.

  9. 9.

Our GGchem calculations show that condensation of carbon does not occur in the
warm disk regions ($r<5$ au, $T>300$ K) if the gas has a C/O ratio $\lesssim
0.9$. We propose that in the warm inner regions ($r<5$ au) of typical planet-
forming disks most if not all carbon is in the gas phase, while iron and iron
sulfide (FeS, troilite) grains are major constituents of the solid mixture,
along with forsterite and enstatite. Thus, small (sub-)$\mu$m sized iron
grains should be the main source of dust continuum radiation at thermal IR
wavelengths in disks. We repeat the recommendation of Matter et al. (2020) who
suggested to replace carbon with iron when modeling inner disk spectra of
YSOs.

Our analysis exemplifies the need for detailed studies of the dust in inner
disks with multiwavelength high angular resolution techniques. Currently,
MATISSE is the only instrument which can resolve disks on au-scales in the $N$
band, probing the warm, forsterite and enstatite rich dust, while at longer
wavelengths JWST/MIRI may give insights to the cold dust population, ices, and
gas chemistry (Kóspál et al., 2023; Grant et al., 2023; Perotti et al., 2023).
Additionally, ALMA could possibly detect larger scale substructures like the
$r\approx 4$ au ring of HD 144432 at its highest spatial resolution. As of
mid-2023, there is no high-resolution measurement of our target in the ALMA
archive, only a measurement with the 7m array, but that has a low spatial
resolution of only $4.5\arcsec$. In the coming years, more results on the
mineralogy of planetary building blocks will be expected from the MATISSE GTO
survey of YSOs.

###### Acknowledgements.

MATISSE was designed, funded and built in close collaboration with ESO, by a
consortium composed of institutes in France (J.-L. Lagrange Laboratory – INSU-
CNRS – Côte d’Azur Observatory – University of Côte d’Azur), Germany (MPIA,
MPIfR and University of Kiel), the Netherlands (NOVA and University of
Leiden), and Austria (University of Vienna). The Konkoly Observatory and
Cologne University have also provided some support in the manufacture of the
instrument. GRAVITY has been developed in a collaboration by the Max Planck
Institute for Extraterrestrial Physics, LESIA of Paris Observatory-
PSL/CNRS/Sorbonne Université/Université Paris-Cité and IPAG of Université
Grenoble Alpes/CNRS, the Max Planck Institute for Astronomy, the University of
Cologne, the Centro Multidisciplinar de Astrofisica Lisbon and Porto, and the
European Southern Observatory. Based on observations collected at the European
Southern Observatory under ESO programmes 190.C-0963(D), 190.C-0963(E),
190.C-0963(F), 0100.C-0278(E), 0103.D-0153(C), 0103.C-0347(C), 0103.D-0153(G),
108.225V.003, 108.225V.011, and 108.225V.006. P. Ábrahám, F. Lykou and J.
Varga are funded from the Hungarian NKFIH OTKA project no. K-132406. Á.
Kóspál, J. Varga, P. Ábrahám, F. Lykou, and F. Cruz-Sáenz de Miera have
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 716155 (SACCRED). The research of J. Varga and M. Hogerheijde is supported
by NOVA, the Netherlands Research School for Astronomy. J. Varga acknowledges
support from the Fizeau exchange visitors program. The research leading to
these results has received funding from the European Union’s Horizon 2020
research and innovation programme under Grant Agreement 101004719 (ORP). F.
Cruz-Sáenz de Miera received financial support from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (ERC Starting Grant “Chemtrip”, grant agreement No 949278). P.
Woitke acknowledges funding from the European Union H2020-MSCA-ITN-2019 under
Grant Agreement no. 860470 (CHAMELEON). N. Morujão acknowledges the financial
support provided by FCT/Portugal through grants PTDC/FIS-AST/7002/2020 and
UIDB/00099/2020. This research has made use of the services of the ESO Science
Archive Facility. This research has made use of the VizieR catalog access
tool, CDS, Strasbourg, France (DOI : 10.26093/cds/vizier). The original
description of the VizieR service was published in 2000, A&AS 143, 23. This
research has made use of the Jean-Marie Mariotti Center OiDB service available
at http://oidb.jmmc.fr . This work has made use of data from the European
Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This study has made use of
the GGchem code, available at https://github.com/pw31/GGchem.

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##  Appendix A VLTI data plots

Figure 10: Calibrated MATISSE $L$ band data sets. Left column: visibility as a
function of the spatial frequency. Middle column: Closure phase as a function
of the spatial frequency corresponding to the longest baselines of the
triangles. Right panel: Correlated flux as a function of the wavelength. We
also plot the single-dish spectrum, as the zero-baseline correlated spectrum.

Figure 11: Same as Fig. 10, but for the MATISSE $M$ band data.  Figure 12:
Same as Fig. 10, but for the MATISSE $N$ band data. We note that, as of DRS
version 1.5.8, the sign convention of the $N$ band phases are flipped with
respect to the $LM$ band phases (see also Gámez Rosas et al. 2022). This
causes a $180\degr$ rotation of the $N$ band images. This issue is no concern
for us, as we do not use the $N$ band closure phases in our modeling.

Figure 13: Calibrated GRAVITY data sets. Left column: visibility as a function
of the spatial frequency. Right column: Closure phase as a function of the
spatial frequency corresponding to the longest baselines of the triangles.

Figure 14: Same as Fig. 13, but for the PIONIER data.

##  Appendix B Additional notes on the MATISSE data processing

While the $L$ band sensitivity of the instrument is more than enough to record
good quality data on HD 144432 using the ATs, it is not the case in $N$ band
where only the UT data can be used. For obtaining correlated fluxes we apply
the so-called coherent processing provided by the DRS pipeline, while the
visibilities are processed using the incoherent method. The coherent algorithm
has a gain in S/N compared to the incoherent method. Due to this advantage,
coherently processed correlated fluxes are especially useful in the $N$ band.
For the `specfit` modeling, we directly used those data. However, the PIONIER
and GRAVITY data does not contain correlated flux, only visibilities, thus for
a homogeneous modeling of all VLTI data sets we need to use visibilities even
in the $N$ band. We opt not to use the incoherently processed $N$ band
visibilities, because of their lower quality. Instead we apply a special
treatment to the $N$ band correlated fluxes by dividing them with the average
single-dish flux, in this way producing coherently processed visibility data.
We use these specially calibrated $N$ band visibilities in the `TGMdust` runs.

The $L$ and $M$ band MATISSE data comes in two flavors: chopped and non-