11institutetext: School of Physics, University College Dublin, Belfield,
Dublin 4, Ireland
11email<EMAIL_ADDRESS>22institutetext: Max Planck Institute for
Astronomy, Königstuhl 17, Heidelberg, Germany, D-69117 33institutetext: Dublin
Institute for Advanced Studies, 31 Fitzwilliam Place, D02 XF86 Dublin, Ireland
44institutetext: Instituto de Astrofísica de Andalucía, CSIC, Glorieta de la
Astronomía s/n, E-18008 Granada, Spain 55institutetext: Dept. of Space, Earth
& Environment, Chalmers University of Technology, SE-412 93 Gothenburg, Sweden
66institutetext: INAF-Osservatorio Astronomico di Capodimonte, via Moiariello
16, 80131 Napoli, Italy 77institutetext: I. Physikalisches Institut,
Universität zu Köln, Zülpicher Str. 77, 50937, Köln, Germany 88institutetext:
Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias,
P-4200-465 Porto, Portugal 99institutetext: CENTRA, Centro de Astrofísica e
Gravitação, Instituto Superior Técnico, Avenida Rovisco Pais 1, P-1049 Lisboa,
Portugal 1010institutetext: Max Planck Institute for Extraterrestrial Physics,
Giessenbachstrasse, 85741 Garching bei München, Germany 1111institutetext:
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
1212institutetext: Max-Planck-Institute for Radio Astronomy, Auf dem Hügel 69,
53121 Bonn, Germany 1313institutetext: Leiden Observatory, Leiden University,
PO Box 9513, 2300 RA Leiden, The Netherlands 1414institutetext: Univ. Grenoble
Alpes, CNRS, IPAG, F-38000 Grenoble, France 1515institutetext: Universidade de
Lisboa - Faculdade de Ciências, Campo Grande, P-1749-016 Lisboa, Portugal
1616institutetext: Instituto de Astronomía, Universidad Nacional Autónoma de
México, Apdo. Postal 70264, Ciudad de México, 04510, México 1717institutetext:
School of Physics & Astronomy, University of Leeds, Woodhouse Lane, LS2 9JT,
Leeds, UK

# The GRAVITY young stellar object survey

XII. The hot gas disk component in Herbig Ae/Be stars

GRAVITY Collaboration(): R. Garcia Lopez GRAVITY is developed in a
collaboration by the Max Planck Institute for Extraterrestrial Physics, LESIA
of Paris Observatory and IPAG of Université Grenoble Alpes / CNRS, the Max
Planck Institute for Astronomy, the University of Cologne, the Centro
Multidisciplinar de Astrofisica Lisbon and Porto, and the European Southern
Observatory.1122 A. Natta 33 R. Fedriani 4455 A. Caratti o Garatti 6622 J.
Sanchez-Bermudez 161622 K. Perraut 1414 C. Dougados 1414 Y.-I. Bouarour 11 J.
Bouvier 1414 W. Brandner 22 P. Garcia 99 8 8 M. Koutoulaki 1717 L. Labadie 77
H. Linz 22 E. Alécian 1414 M. Benisty 1414 J.-P. Berger 1414 G. Bourdarot 1010
P. Caselli 1010 Y. Clénet 1111 P. T. de Zeeuw 1313 R. Davies 1010 A. Eckart 77
12 12 F. Eisenhauer 1010 N. M. Förster-Schreiber 1010 E. Gendron 1111 S.
Gillessen 1010 S. Grant 1010 Th. Henning 22 P. Kervella 1111 S. Lacour 1111 V.
Lapeyrère 1111 J.-B. Le Bouquin 1414 D. Lutz 1010 F. Mang 1010 H. Nowacki 1414
T. Ott 1010 T. Paumard 1111 G. Perrin 1111 J. Shangguan 1010 T. Shimizu 1010
A. Soulain 1414 C. Straubmeier 77 E. Sturm 1010 L. Tacconi 1010 E. F. van
Dishoeck 1313 F. Vincent 1111 F. Widmann 1010

(Received ; accepted )

###### Abstract

Context. The region of protoplanetary disks closest to a star (within 1-2 au)
is shaped by a number of different processes, from accretion of the disk
material onto the central star to ejection in the form of winds and jets.
Optical and near-IR emission lines are potentially good tracers of inner disk
processes if very high spatial and/or spectral resolution are achieved.

Aims. In this paper, we exploit the capabilities of the VLTI-GRAVITY near-IR
interferometer to determine the location and kinematics of the hydrogen
emission line Br$\gamma$.

Methods. We present VLTI-GRAVITY observations of the Br$\gamma$ line for a
sample of 26 stars of intermediate mass (HAEBE), the largest sample so far
analysed with near-IR interferometry.

Results. The Br$\gamma$ line was detected in 17 objects. The emission is very
compact (in most cases only marginally resolved), with a size of 10-30 R∗(1-5
mas). About half of the total flux comes from even smaller regions, which are
unresolved in our data. For eight objects, it was possible to determine the
position angle (PA) of the line-emitting region, which is generally in
agreement with that of the inner-dusty disk emitting the K-band continuum.

The position-velocity pattern of the Br$\gamma$ line-emitting region of the
sampled objects is roughly consistent with Keplerian rotation. The exception
is HD 45677, which shows more extended emission and more complex kinematics.
The most likely scenario for the Br$\gamma$ origin is that the emission comes
from an MHD wind launched very close to the central star, in a region well
within the dust sublimation radius. An origin in the bound gas layer at the
disk surface cannot be ruled out, while accreting matter provides only a minor
fraction of the total flux.

Conclusions. These results show the potential of near-IR spectro-
interferometry to study line emission in young stellar objects.

###### Key Words.:

Stars: formation – Stars: circumstellar matter – Stars: variables: T Tauri,
Herbig Ae/Be – Techniques: interferometric

##  1 Introduction

Protoplanetary disks are ubiquitous around young stellar objects of very
different mass. In recent years, high spatial resolution observations at
infrared and millimetre wavelengths have shown that protoplanetary disks are
not smooth and homogeneous but show a variety of structures, from gaps to
rings to spirals and vortices (Garufi et al., 2018; Andrews, 2020). However,
very little is known of the inner disk, within 1–2 au from the central star,
due to the high angular resolution required to spatially resolve this region.

The inner disk of young stars is shaped by a number of different processes. As
matter in the disk accretes onto the central star, it is expected that a
significant fraction will be ejected in magneto-hydrodynamic (MHD)-driven
and/or photoevaporative winds, with specific mass-loss rates peaking in the
inner disk (e.g. Pascucci et al., 2022; Lesur et al., 2022). Observations of
evolved stars have demonstrated that planets can be located in the very inner
regions of disks. Disentangling and characterising these processes is
necessary if we want to understand the inner disk properties and their link to
planet formation and evolution.

Studying the inner disk requires milli-arcsecond (mas) spatial resolution,
which is only achievable with optical interferometry. The analysis of
available interferometric data has so far shown that the near-IR continuum
emission is dominated by the emission of dust near the sublimation radius,
with some contribution closer to the star, whose origin remains unidentified
(Millan-Gabet et al., 2007; Kraus et al., 2008; Benisty et al., 2010; Menu et
al., 2015; Lazareff et al., 2017; Gravity Collaboration et al., 2019; Davies
et al., 2020; Gravity Collaboration et al., 2021b; Marcos-Arenal et al., 2021;
Gravity Collaboration et al., 2021a).

Optical interferometry provides information not only on the continuum emission
but also on some emission lines. In particular, the Very Large Telescope
Interferometer (VLTI) K-band beam combiner GRAVITY and its precursor, VLTI-
AMBER, cover the hydrogen recombination line Br$\gamma$. The origin of the
hydrogen emission lines commonly detected in young stars is usually associated
with the presence of circumstellar material such as the surface of the
accretion disk itself and/or accreting and outflowing gas. In TW Hya, the
closest T Tauri star (d$\sim$60 pc), GRAVITY-VLTI Br$\gamma$ observations have
shown that the line traces the emission of the accretion columns and provided
a direct confirmation of the magnetic accretion models for T Tauri stars,
where the size of the accretion region is directly related to the strength of
the stellar magnetic field and the mass-accretion rate (Gravity Collaboration
et al., 2020). In more distant T Tauri star objects, we expect that the
current optical interferometers will not reach the spatial resolution required
to resolve the magnetosphere (GRAVITY Collaboration et al., 2017; Bouvier et
al., 2020; GRAVITY Collaboration et al., 2022a). Furthermore, existing studies
on more luminous Herbig AeBe stars have shown that most of the emission of the
Br$\gamma$ line originates in a wind, with a possible minor contribution from
unresolved emission from the accreting gas (Weigelt et al., 2011; Garcia Lopez
et al., 2015; Caratti o Garatti et al., 2015; Garcia Lopez et al., 2016; Hone
et al., 2017; Kreplin et al., 2018; Hone et al., 2019). So far, only very few
objects have been studied in detail (namely, MWC297, HD163296, HD98922, VVSer,
HD58647). However, these studies have proved that it is possible to use
interferometric data to constrain the wind’s physical properties (e.g. mass-
loss rate) and kinematics and directly test wind models.

This paper presents the results of a survey of 26 intermediate-mass stars
(HAEBE) observed in the Br$\gamma$ line with VLTI-GRAVITY. The objects cover a
mass range between $\sim$1.5 and $\sim$7 M⊙ and represent the largest sample
studied so far. The observations and data reduction procedure are described in
Sect. 2. Section 3 presents the properties of the stars in the sample. Section
4 describes the results obtained from the interferometric observations,
namely, the size of the Br$\gamma$ line-emitting region, its position angle on
the plane of sky, and the position-velocity diagrams. Section 5 is dedicated
to a discussion of the results and of the constraints they set on current
models of the line emission. Section 6 follows with a summary and conclusions.

Table 1: Properties of the sample.

| Name |  $d$aa$a$Distance, effective temperature, surface gravity, stellar luminosity and mass, and accretion luminosity from Vioque et al. (2018); Wichittanakom et al. (2020) |  $T_{\rm eff}$aa$a$Distance, effective temperature, surface gravity, stellar luminosity and mass, and accretion luminosity from Vioque et al. (2018); Wichittanakom et al. (2020) |  $\log g$aa$a$Distance, effective temperature, surface gravity, stellar luminosity and mass, and accretion luminosity from Vioque et al. (2018); Wichittanakom et al. (2020) |  $\log L_{\ast}$aa$a$Distance, effective temperature, surface gravity, stellar luminosity and mass, and accretion luminosity from Vioque et al. (2018); Wichittanakom et al. (2020) |  $M_{\ast}$aa$a$Distance, effective temperature, surface gravity, stellar luminosity and mass, and accretion luminosity from Vioque et al. (2018); Wichittanakom et al. (2020) |  $\log L_{acc}$aa$a$Distance, effective temperature, surface gravity, stellar luminosity and mass, and accretion luminosity from Vioque et al. (2018); Wichittanakom et al. (2020) |  Vbb$b$Visual magnitude from CDS-Simbad |  Avcc$c$Visual extinction from Vioque et al. (2018) |  Kdd$d$Inclination and PA of the continuum emission from Gravity Collaboration et al. (2019) |  v$sin$iee$e$see Sect. 4.5. |  vradee$e$see Sect. 4.5.
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | [pc] | [K] |  [cm/s2] |  [L⊙] |  [M⊙] |  [L⊙] | [mag] | [mag] | [mag] | [km/s] | [km/s]
[1] | HD37806 | 428 | 10500 | 4.0 |  2.17${}^{+0.19}_{-0.14}$ |  3.11${}^{+0.55}_{-0.33}$ |  1.46${}^{+0.14}_{-0.14}$ | 7.9 | 0.13 | 5.6 |  120${}^{+10}_{-3}$ | 47
[2] | HD38120 | 405 | 10700 | 4.0 |  1.72${}^{+0.31}_{-0.20}$ |  2.37${}^{+0.43}_{-0.24}$ |  1.24 ${}^{+0.17}_{-0.17}$ | 9.1 | 0.21 | 7.2 |  97${}^{+10}_{-3}$ | 28
[3] | HD45677 | 620 | 16500 | 4.0 |  2.88${}^{+0.32}_{-0.17}$ |  4.72${}^{+1.19}_{-0.34}$ |  2.07 ${}^{+0.16}_{-0.15}$ | 7.4 | 0.57 | 4.5 | 200 | 22
[4] | HD58647 | 319 | 10500 | 3.3 |  2.44${}^{+0.11}_{-0.09}$ |  3.87${}^{+0.33}_{-0.19}$ |  1.62 ${}^{+0.11}_{-0.11}$ | 6.8 | 0.37 | 5.4 |  118${\pm 4}$ |
[5] | HD85567 | 1023 | 13000 | 3.5 |  3.28${}^{+0.19}_{-0.20}$ |  6.79${}^{+1.16}_{-1.04}$ |  2.68 ${}^{+0.14}_{-0.13}$ | 8.5 | 0.89 | 5.8 |  31$\pm 3$ |
[6] | HD95881 | 1168 | 10000 | 3.2 |  3.13${}^{+0.12}_{-0.13}$ |  7.02${}^{+0.62}_{-0.67}$ |  2.37${}^{+0.16}_{-0.14}$ | 8.2 | 0.72 | 5.7 | 50 | 36
[7] | HD97048 | 185 | 10500 | 4.3 |  1.76${}^{+0.12}_{-0.12}$ |  2.40${}^{+0.15}_{-0.03}$ |  1.19${}^{+0.13}_{-0.13}$ | 8.4 | 0.90 | 5.9 |  140$\pm 20$ | 18
[8] | HD98922 | 689 | 10500 | 3.6 |  3.20${}^{+0.11}_{-0.11}$ |  7.17${}^{+0.67}_{-0.69}$ |  2.44${}^{+0.13}_{-0.13}$ | 6.7 | 0.09 | 4.4 |  50$\pm 3$ | 0
[9] | HD100546 | 110 | 9750 | 4.3 |  1.37${}^{+0.07}_{-0.05}$ |  2.06${}^{+0.10}_{-0.12}$ |  0.97${}^{+0.14}_{-0.14}$ | 6.7 | 0.00 | 5.7 |  60$\pm 5$ | 9
[10] | HD114981 | 705 | 16000 | 3.6 |  3.31${}^{+0.13}_{-0.12}$ |  6.46${}^{+0.68}_{-0.62}$ |  1.86${}^{+0.16}_{-0.16}$ | 7.2 | 0.15 | 7.4 |  239$\pm 13$ |  -50$\pm 11$
[11] |  HD135344B⋆ | 136 | 6375 | 4.0 |  0.94${}^{+0.13}_{-0.14}$ |  1.67${}^{+0.18}_{-0.16}$ |  0.09${}^{+0.21}_{-0.22}$ | 8.7 | 0.53 | 7.6 |  82$\pm$2 | 0
[12] |  HD139614⋆ | 135 | 7750 | 4.3 |  0.86${}^{+0.08}_{-0.09}$ |  1.61${}^{+0.02}_{-0.11}$ |  0.04${}^{+0.22}_{-0.23}$ | 8.4 | 0.21 | 6.7 |  24$\pm 3$ |  0$\pm 2$
[13] | HD141569 | 111 | 9500 | 4.2 |  2.34${}^{+0.06}_{-0.07}$ |  2.16${}^{+0.04}_{-0.15}$ |  0.35${}^{+0.21}_{-0.23}$ | 7.1 | 0.38 | 6.6 |  228$\pm 10$ |  -12$\pm 7$
[14] |  HD142527⋆ | 157 | 6500 | 3.9 |  1.39${}^{+0.09}_{-0.10}$ |  2.26${}^{+0.02}_{-0.13}$ |  0.65${}^{+0.18}_{-0.19}$ | 8.3 | 0.00 | 5.0 |  | -3.5
[15] |  HD142666⋆ | 148 | 7500 | 4.0 |  1.14${}^{+0.10}_{-0.11}$ |  1.69${}^{+0.13}_{-0.11}$ |  0.05${}^{+0.25}_{-0.28}$ | 8.8 | 0.95 | 6.1 |  65$\pm$3 |  -7$\pm$3
[16] |  HD144432⋆ | 155 | 7500 | 4.0 |  1.18${}^{+0.09}_{-0.09}$ |  1.74${}^{+0.13}_{-0.10}$ |  0.38${}^{+0.20}_{-0.21}$ | 8.2 | 0.48 | 5.9 |  79$\pm$4 |  -3$\pm$3
[17] |  HD144668⋆ | 161 | 8500 | 3.7 |  1.95${}^{+0.08}_{-0.08}$ |  2.97${}^{+0.17}_{-0.21}$ |  1.33${}^{+0.14}_{-0.15}$ | 7.2 | 0.87 | 4.4 |  199$\pm$11 |  -10$\pm$8
[18] |  HD145718⋆ | 153 | 8000 | 4.4 |  1.10${}^{+0.10}_{-0.10}$ |  1.67${}^{+0.06}_{-0.02}$ |  -0.04${}^{+0.28}_{-0.32}$ | 8.8 | 1.23 | 6.7 |  113$\pm 3$ |  -4$\pm$2
[19] | HD150193 | 151 | 9000 | 4.0 |  1.50${}^{+0.13}_{-0.14}$ |  2.05${}^{+0.19}_{-0.12}$ |  0.83${}^{+0.17}_{-0.18}$ | 8.8 | 1.80 | 5.3 |  108$\pm 5$ |  -5$\pm 4$
[20] | HD158643 | 123 | 9800 | 3.6 |  2.22${}^{+0.26}_{-0.07}$ |  3.35${}^{+0.79}_{-0.22}$ |  1.30${}^{+0.16}_{-0.15}$ | 4.8 | 0.00 | 4.3 | 256 | -11
[21] | HD163296 | 102 | 9250 | 4.3 |  1.36${}^{+0.08}_{-0.09}$ |  2.04${}^{+0.07}_{-0.10}$ |  0.62${}^{+01.8}_{-0.19}$ | 6.8 | 0.37 | 4.6 |  129$\pm 8$ |  -9$\pm 6$
[22] |  HD169142⋆ | 114 | 10700 | 4.0 |  1.31${}^{+0.12}_{-0.34}$ |  2.00${}^{+0.13}_{-0.13}$ |  0.59${}^{+0.15}_{-0.15}$ | 8.2 | 1.02 | 6.4 |  48$\pm$2 |  0$\pm$2
[23] | HD179218 | 266 | 9500 | 3.9 |  1.97${}^{+0.07}_{-0.07}$ |  2.86${}^{+0.16}_{-0.20}$ |  1.08${}^{+0.13}_{-0.13}$ | 7.8 | 0.33 | 5.8 |  69$\pm 3$ |  15$\pm 2$
[24] | HD190073 | 870 | 9750 | 3.4 |  2.84${}^{+0.16}_{-0.14}$ |  5.62${}^{+0.78}_{-0.65}$ |  2.31${}^{+0.18}_{-0.15}$ | 7.7 | 0.20 | 5.7 | 0-8 | 0
[25] | HD259431 | 721 | 14000 | 4.3 |  2.97${}^{+0.27}_{-0.40}$ |  5.2${}^{+1.8}_{-1.3}$ |  2.43 ${}^{+0.194}_{-0.13}$ | 8.7 | 1.11 | 5.7 |  83$\pm 11$ |  26$\pm 8$
[26] |  V1818Ori⋆ | 695 | 13000 |  |  2.96${}^{+0.24}_{-0.29}$ |  5.3${}^{+1.3}_{-1.1}$ |  | 11.0 | 3.72 | 6.0 |  46$\pm 10$ | 26

111. . . dd$d$Inclination and PA of the continuum emission from Gravity Collaboration et al. (2019)2MASS K-band magnitude. ee$e$see Sect. 4.5.v$sini$ and radial velocity from Miroshnichenko et al. (2001); Guimarães et al. (2006); Kharchenko et al. (2007); Montesinos et al. (2009); Alecian et al. (2013); Ilee et al. (2014).

Table 2: Results from the geometrical fitting and displacement analysis.

| Name |  FWHMaa$a$FWHM, inclination ($i$), PA, and relative flux contributions of the Br$\gamma$ line-emitting region as estimated from the analysis of the continuum-subtracted visibilities (see Sect. 4.3) |  FWHMaa$a$FWHM, inclination ($i$), PA, and relative flux contributions of the Br$\gamma$ line-emitting region as estimated from the analysis of the continuum-subtracted visibilities (see Sect. 4.3) |  $i$aa$a$FWHM, inclination ($i$), PA, and relative flux contributions of the Br$\gamma$ line-emitting region as estimated from the analysis of the continuum-subtracted visibilities (see Sect. 4.3) |  PAaa$a$FWHM, inclination ($i$), PA, and relative flux contributions of the Br$\gamma$ line-emitting region as estimated from the analysis of the continuum-subtracted visibilities (see Sect. 4.3) |  Fresaa$a$FWHM, inclination ($i$), PA, and relative flux contributions of the Br$\gamma$ line-emitting region as estimated from the analysis of the continuum-subtracted visibilities (see Sect. 4.3) |  Fcompaa$a$FWHM, inclination ($i$), PA, and relative flux contributions of the Br$\gamma$ line-emitting region as estimated from the analysis of the continuum-subtracted visibilities (see Sect. 4.3) |  Fhaloaa$a$FWHM, inclination ($i$), PA, and relative flux contributions of the Br$\gamma$ line-emitting region as estimated from the analysis of the continuum-subtracted visibilities (see Sect. 4.3) |  $\chi^{2}_{r}$bb$b$Reduced $\chi^{2}$ from our fitting procedure (see Sect. 4.3) |  PAdispcc$c$PA of the continuum-subtracted Br$\gamma$ line emission as derived from the line displacements (see Sect. 4.4) |  icontdd$d$Inclination and PA of the continuum emission from Gravity Collaboration et al. (2019) |  PAcontdd$d$Inclination and PA of the continuum emission from Gravity Collaboration et al. (2019)
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | [mas] | [au] | [°] | [°] |  |  |  |  | [°] | [°] | [°]
[1] | HD37806 |  1.4${}^{+1.1}_{-0.4}$ |  0.6${}^{+0.5}_{-0.2}$ |  40${}^{+24}_{-26}$ |  29${}^{+27}_{-79}$ |  0.36${}^{+0.26}_{-0.21}$ |  0.51${}^{+0.16}_{-0.25}$ |  0.12${}^{+0.05}_{-0.05}$ | 0.20 |  44${}^{+3}_{-2}$ |  60${}^{+3}_{-4}$ |  51${}^{+4}_{-1}$
[2] | HD38120 |  1.4${}^{+1.2}_{-0.8}$ |  0.6${}^{+0.5}_{-0.3}$ |  46${}^{+29}_{-31}$ |  167${}^{+56}_{-61}$ |  0.40${}^{+0.27}_{-0.27}$ |  0.54${}^{+0.17}_{-0.26}$ |  0.06${}^{+0.06}_{-0.04}$ | 0.75 | – |  49${}^{+2}_{-1}$ |  164${}^{+2}_{-2}$
[3] | HD45677 |  3.6${}^{+0.2}_{-0.2}$ |  2.2${}^{+0.1}_{-0.1}$ |  41${}^{+3}_{-3}$ |  55${}^{+6}_{-6}$ |  0.55${}^{+0.10}_{-0.17}$ |  0.19${}^{+0.04}_{-0.06}$ |  0.27${}^{+0.06}_{-0.08}$ |  48.30ee$e$see Sect. 4.5. | – |  60${}^{+1}_{-1}$ |  64${}^{+1}_{-1}$
[4] | HD58647 |  1.7${}^{+0.5}_{-0.5}$ |  0.5${}^{+0.2}_{-0.2}$ |  40${}^{+10}_{-17}$ | $--$ |  0.46${}^{+0.19}_{-0.20}$ |  0.47${}^{+0.18}_{-0.24}$ |  0.07${}^{+0.03}_{-0.02}$ | 1.00 |  22${}^{+3}_{-5}$ |  64${}^{+1}_{-1}$ |  15${}^{+1}_{-1}$
[5] | HD85567 |  2.8${}^{+1}_{-1}$ |  2.9${}^{+1.0}_{-1.0}$ |  32${}^{+7}_{-8}$ |  120${}^{+12}_{-13}$ |  0.22${}^{+0.26}_{-0.06}$ |  0.76${}^{+0.12}_{-0.20}$ | 0 | 0.91 |  119${}^{+1}_{-2}$ |  20${}^{+1}_{-2}$ |  96${}^{+7}_{-7}$
[6] | HD95881 |  1.7${}^{+1.0}_{-0.7}$ |  2.0${}^{+1.2}_{-0.8}$ |  46${}^{+28}_{-30}$ |  158${}^{+65}_{-56}$ |  0.44${}^{+0.24}_{-0.27}$ |  0.53${}^{+0.18}_{-0.27}$ |  0.03${}^{+0.04}_{-0.02}$ | 1.57 | – |  51${}^{+4}_{-1}$ |  167${}^{+3}_{-3}$
[7] | HD97048 |  1.5${}^{+1.8}_{-0.8}$ |  0.3${}^{+0.3}_{-0.2}$ |  50${}^{+26}_{-33}$ |  173${}^{+51}_{-59}$ |  0.35${}^{+0.29}_{-0.23}$ |  0.54${}^{+0.16}_{-0.25}$ |  0.11${}^{+0.09}_{-0.07}$ | 0.66 | – |  44${}^{+1}_{-2}$ |  9${}^{+3}_{-3}$
[8] | HD98922 |  2.8${}^{+0.3}_{-0.3}$ |  1.9${}^{+0.2}_{-0.2}$ |  16${}^{+12}_{-11}$ |  92${}^{+43}_{-43}$ |  0.37${}^{+0.08}_{-0.11}$ |  0.59${}^{+0.11}_{-0.18}$ |  0.04${}^{+0.02}_{-0.02}$ | 0.43 | (-23–9) |  49${}^{+1}_{-1}$ |  130${}^{+1}_{-1}$
[9] | HD100546 |  1.7${}^{+1.2}_{-0.5}$ |  0.2${}^{+0.1}_{-0.1}$ |  30${}^{+21}_{-20}$ |  108${}^{+48}_{-60}$ |  0.38${}^{+0.25}_{-0.22}$ |  0.54${}^{+0.16}_{-0.26}$ |  0.08${}^{+0.04}_{-0.03}$ | 0.22 |  141${}^{+1}_{-2}$ |  50${}^{+1}_{-1}$ |  147${}^{+1}_{-1}$
[10] | HD114981 |  $<$1.7 |  $<$1.2 | – | – | – | – | – | – | – | – |
[13] | HD141569 |  $<$1.7 |  $<$0.2 | – | – | – | – | – | – | – | – |
[19] | HD150193 |  4.6${}^{+0.3}_{-1.5}$ |  0.7${}^{+0.0}_{-0.2}$ |  42${}^{+21}_{-25}$ |  127${}^{+20}_{-32}$ |  0.33${}^{+0.21}_{-0.14}$ |  0.59${}^{+0.14}_{-0.21}$ |  0.08${}^{+0.03}_{-0.02}$ | 1.04 | – |  47${}^{+2}_{-2}$ |  6${}^{+3}_{-3}$
[21] | HD163296 |  3.0${}^{+0.4}_{-0.4}$ |  0.3${}^{+0.0}_{-0.0}$ |  30${}^{+13}_{-19}$ |  106${}^{+105}_{-20}$ |  0.32${}^{+0.11}_{-0.10}$ |  0.67${}^{+0.13}_{-0.20}$ | 0 | 1.11 |  137${}^{+1}_{-1}$ |  40${}^{+1}_{-1}$ |  136${}^{+2}_{-2}$
[22] |  HD169142⋆ | – | – | – | – | – | – | - |  |  35${}^{+5}_{-5}$ |  |
[23] | HD179218 |  1.4${}^{+1.6}_{-0.8}$ |  0.4${}^{+0.4}_{-0.2}$ |  54${}^{+25}_{-35}$ |  106${}^{+59}_{-84}$ |  0.38${}^{+0.23}_{-0.27}$ |  0.49${}^{+0.16}_{-0.25}$ |  0.12${}^{+0.07}_{-0.06}$ | 1.79 | - |  54${}^{+8}_{-10}$ |  49${}^{+9}_{-9}$
[24] | HD190073 |  1.2${}^{+0.9}_{-0.4}$ |  1.0${}^{+0.8}_{-0.4}$ |  44${}^{+30}_{-29}$ |  86${}^{+52}_{-45}$ |  0.41${}^{+0.28}_{-0.28}$ |  0.55${}^{+0.18}_{-0.28}$ |  0.03${}^{+0.04}_{-0.02}$ | 0.81 | - |  21${}^{+2}_{-1}$ |  64${}^{+9}_{-9}$
[25] | HD259431 |  2.0${}^{+0.4}_{-0.3}$ |  1.4${}^{+0.3}_{-0.2}$ |  17${}^{+14}_{-11}$ |  70${}^{+67}_{-37}$ |  0.45${}^{+0.18}_{-0.17}$ |  0.53${}^{+0.15}_{-0.21}$ | 0 | 0.61 |  33${}^{+3}_{-4}$ |  28${}^{+1}_{-1}$ |  50${}^{+3}_{-3}$

222. . . .

##  2 Observations and data reduction

The data in this paper were collected using the European Southern Observatory
(ESO) VLTI-GRAVITY (Gravity Collaboration et al., 2017) as part of the
Guaranteed Time Observations (GTO) program. Most of our targets were observed
using the four 1.8-m Auxiliary Telescopes (ATs), with the only exception being
HD 36917, for which unit telescope (UT) data are presented. The different
datasets shown in this paper were collected during a period of four years
(2017-2021). A full log of the observations can be found in Table 3.

The GRAVITY spectro-interferometric signals were recorded on six baselines
simultaneously using the fringe tracker (FT) detector at low spectral
resolution (six spectral channels) and the scientific (SC) detector at a
resolution of $R\sim$4000\. Typical integration times of 10 s–30 s were used
for the SC data, with each observation file corresponding to 300 s–360 s. The
FT uses much shorter integration times, with frame rates ranging from
$\sim$300 Hz to $\sim$900 Hz. To calibrate the atmospheric transfer function,
a minimum of one interferometric calibrator has been observed per target. The
calibrator was observed immediately after or before the science target at a
sky location very close to that of the science target (see Table 3 for a full
list of the calibrators used in this study).

The data have been reduced using the GRAVITY data reduction pipeline
(Lapeyrere et al., 2014). The GRAVITY observations provide the following
interferometric observables: the K-band spectrum, six spectrally dispersed
visibilities and differential phases, and four spectrally dispersed closure
phases. The wavelength calibration of the SC high-resolution (HR) observations
was refined by using the many telluric lines present in the K-band spectrum.
In addition, the calibrator spectra were used to correct the observed spectra
for the atmospheric telluric features, atmospheric transmission, and
instrumental response.

In this paper, we focus on the HR SC observations to constrain the properties
of the Br$\gamma$ line-emitting region in our sample. The FT observations used
to probe the inner dust rim in the K-band continuum have been presented in
Gravity Collaboration et al. (2019). The four interferometric variables in the
Br$\gamma$ region are shown for all the stars in the sample in Fig. B1–B8.

##  3 Sample

Our sample consists of 26 Herbig AeBe stars with effective temperatures
ranging from $\sim$6000 K to $\sim$16 000 K, and stellar luminosities from
$\sim$12 L⊙ to $\sim$1900 L⊙. The main properties of our sample are reported
in Table 1. Additional details can be found in Gravity Collaboration et al.
(2019).

In order to avoid spurious effects due to line and/or continuum variability,
datasets as close in time as possible have been selected, ideally including
data obtained with different configurations acquired during the same
observational campaign (a few weeks apart). In the case of data acquired in
different epochs, we excluded datasets showing evident line and/or continuum
variability. Variability studies will be performed in separate papers (see
e.g. Gravity Collaboration et al. 2021c).

Of the original sample of 26 stars, nine showed no or very faint Br$\gamma$
emission, even after correcting for the underlying photospheric absorption
line (see Sect. 4.1). These targets are marked with a star symbol in Table 1
and were omitted from further analysis.

##  4 Results

###  4.1 HI Br$\gamma$ line spectral signal

The spectrum of pre-main sequence intermediate-mass stars is usually
characterised by bright H i Br$\gamma$ line emission from circumstellar
material, namely, accreting and outflowing gas and/or the heated surface of
the accretion disk itself. Independent of the origin of this emission, the
interferometric observables across the line also have contributions from the
stellar photosphere and the continuum cicumstellar emission. The total
observed flux across the line can be expressed as

$F(\lambda)_{tot}=F_{disk}+F(\lambda)_{*}+F(\lambda)_{circ},$ (1)

where Fdisk is the continuum disk emission, F($\lambda$)∗ is the photospheric
component, and F($\lambda$)circ is the circumstellar Br$\gamma$ line emission.

Over our narrow wavelength range, the continuum disk circumstellar emission is
constant with wavelength. However, the contribution from the stellar
phostosphere is wavelength dependent, as Herbig AeBe stars have strong and
wide Br$\gamma$ line photospheric absorption features (Gray & Corbally, 1998).
To derive the values of the three components, we followed the procedure
described in Garcia Lopez et al. (2006). Firstly, synthetic spectra were
created for each star using the Castelli & Kurucz (2003) grid of
models333https://wwwuser.oats.inaf.it/castelli/ along with the appropriate
values of the effective temperature ($T_{eff}$), surface gravity ($g$), and
v$sin$i as reported in Table 1. For the case of HD 142527, where no v$sin$i
value is reported in the literature, a typical v$sin$i value of 100 km s-1was
assumed. The synthetic spectra were smoothed to the GRAVITY spectral
resolution of 4000. The source radial velocity (see Table 1) was also taken
into account to avoid shifts between the observed and synthetic spectra for
nearly all our sources, with the exceptions being HD 58647 and HD 85567, as no
radial velocities were available in the literature.

Under the further assumption that the stellar photosphere dominates the V-band
magnitude of the source, given the spectral type and extinction, the
photospheric component was computed for each star, corrected for the K-band
continuum, and removed from the observed spectrum. The amount of correction to
the observed Br$\gamma$ line depended on the infrared continuum excess of the
source. Sources with little continuum excess are more affected by the
correction, whereas sources with a high amount of excess did not show
significant changes in the Br$\gamma$ line flux.

The observed Br$\gamma$ lines are shown as black solid lines in the top panels
of Figs. 11–32, whereas the photospheric-corrected Br$\gamma$ line profiles
are shown as blue dashed lines. After photospheric correction, most sources
show a triangular Br$\gamma$ line shape profile, with the exception of four
sources (HD 58647, HD 114981, HD 141569, and HD 158643), which have double-
peaked line profiles, indicative of rotating emission regions.

###  4.2 Continuum-subtracted Br$\gamma$ line visibilities and differential
phases

As in the case of the Br$\gamma$ line flux, we computed the line visibility
and differential phase by removing the contribution of the other components.
The observed visibilities across the line (V($\lambda$)tot) have contributions
from the circumstellar disk continuum (Vdisk), the photospheric component
(V($\lambda$)∗), and the circumstellar Br$\gamma$ line emission
(V($\lambda$)circ). The relation between these quantities can be expressed in
a simple way in two extreme cases: when the differential phase is negligible
(diff. phase $\lesssim$5°–10°) or when the photospheric flux is negligible
with respect to the circumstellar continuum emission (i.e.
Ftot/F${}_{disk}\gtrsim$20%). In the first case, we have

$V(\lambda)_{tot}F(\lambda)_{tot}=V_{disk}F_{disk}+V(\lambda)_{*}F(\lambda)_{*}+V(\lambda)_{circ}F(\lambda)_{circ}$
(2)

and

$V(\lambda)_{circ}=V(\lambda)_{tot}\frac{F(\lambda)_{tot}}{F(\lambda)_{circ}}-V_{disk}\frac{F_{disk}}{F(\lambda)_{circ}}-V(\lambda)_{*}\frac{F(\lambda)_{*}}{F(\lambda)_{circ}}.$
(3)

We further assumed that the central star is not spatially resolved (i.e.
point-like; $V(\lambda)_{*}$=1) and measured $V_{disk}$ from the observed
visibility adjacent to the line $V_{tot}$:

$V_{disk}=V_{tot}\frac{F_{tot}}{F_{disk}}-\frac{F_{*}}{F_{disk}}.$ (4)

In the second case, if the photospheric flux is negligible at all wavelengths
but the differential phase ($\Phi$) is larger than 5°–10 °, Eq.(2) is replaced
by (Weigelt et al., 2007):

$\begin{split}V(\lambda)_{circ}F(\lambda)_{\mathrm{{circ}}}&=(\left|V(\lambda)_{tot}F(\lambda)_{tot}\right|^{2}+\left|V_{disk}F_{disk}\right|^{2}-\\\
&2V(\lambda)_{tot}F(\lambda)_{tot}V_{disk}F_{disk}\cos{\Phi})^{1/2}\end{split}.$
(5)

Figures 33–34 show the value of $V(\lambda)_{circ}$ calculated for all the
lines at zero velocity by averaging the signal over the three central points.
In the case of the continuum observables, the visibilities and flux values
were measured in the nearest line-free wavelength interval. The continuum-
subtracted visibilities were computed only for sources where the flux of the
peak of the circumstellar Br$\gamma$ line is larger than 10% of the continuum
flux (16 sources). The errors on the continuum-subtracted visibilities were
calculated by propagating Eq. 2 and Eq. 5. As a conservative value, the rms at
continuum wavelengths close to the line position was considered as the error
for the total and continuum visibilities and differential phases. The level of
the continuum visibilities is shown as a dashed line in the figures of
Appendix B.

By examining the dependence of $V(\lambda)_{circ}$ as a function of the
spatial frequency some information about the spatial extent of the Br$\gamma$
emission can be obtained without any assumption on the geometry of the
brightness distribution. In general, interferometry deals with two extreme
behaviours in the dependence of $V(\lambda)_{circ}$ versus the projected
baseline length (PBL). On the one hand, for a given PBL, an object is resolved
if its angular size (in units of radians) $\Theta\geq$ 1/2 $\lambda$/PBL. In
this case, visibilities will drop with an increasing baseline. The opposite
extreme case is that of an unresolved object (i.e. point source). In such a
case, the visibilities are equal to one and constant over all baselines. In
most cases, however, interferometry deals with marginally resolved objects.
This means that there is a drop in the visibility values with baseline but
without fully resolving the object. In this case, the size of the object is a
fraction of the PBL/$\lambda$ (see e.g. Monnier, 2007; Lachaume, 2003, for a
more formal treatment). This scenario could be even more complex when several
morphological components are traced by the visibilities (e.g. the overlap of
an over-resolved component, a resolved or marginally-resolved component, and a
point source).

In our case, Figs. 33 and 34 show that most of our objects are marginally
resolved as $V(\lambda)_{circ}$ drops with baseline length. Only two objects
(namely, HD114981 and HD141569) show a constant $V(\lambda)_{circ}$=1, within
the errors, indicating that they are unresolved. In order to derive further
information about the size and geometry of the marginally resolved Br$\gamma$
line-emitting region, geometric modelling is needed. A detailed analysis is
discussed in Sect. 4.3.

As for the line visibilities, the continuum contribution can also be
subtracted from the differential-phase signal across the line. Following
Weigelt et al. (2007), the continuum-subtracted differential phase
($\Delta\Phi$) can be written as

$\sin(\Delta\Phi(\lambda))=\sin(\Phi(\lambda))\cdot\dfrac{|F(\lambda)_{\mathrm{{tot}}}V(\lambda)_{\mathrm{{tot}}}|}{|F(\lambda)_{\mathrm{{circ}}}V(\lambda)_{\mathrm{{circ}}}|},$
(6)

where $\phi$ is the observed differential phase. The continuum-subtracted
differential phases were computed for all the sources where a differential-
phase signal was detected above the rms noise in the adjacent continuum (eight
sources).

###  4.3 The size of the Br$\gamma$ line-emitting region

The size of the Br$\gamma$ line-emitting region was derived using a Bayesian
approach by means of Markov chain Monte Carlo (MCMC) methods. The MCMC method
is preferred over ordinary least square methods when performing multiple
parameter fits, as the latter can be biased towards the initial conditions of
the fit and end up in local minima. We used the continuum-subtracted
Br$\gamma$ line visibilities at the peak of the line (i.e. at zero velocity)
for the 16 stars with detected Br$\gamma$ emission. In order to interpret the
Br$\gamma$ line visibilities, a brightness distribution of the emission needs
to be assumed. In our case, we describe the total Br$\gamma$ intensity
distribution as the sum of three components: a resolved (or marginally
resolved) component with a Gaussian brightness distribution of flux
$F_{\mathrm{res}}$ and visibility Vres; an unresolved component with flux
$F_{\mathrm{comp}}$ and visibility $V_{\mathrm{comp}}=1$; and an over-resolved
component with flux $F_{\mathrm{halo}}$ and visibility $V_{\mathrm{halo}}$=0.
This does not necessarily imply that the emission originates in distinct
physical regions (e.g. accretion rather than a wind), but it is simply a
result of the limited range of available baselines. The observed line flux at
the peak of the normalised line ($F_{line}$) is therefore
$F_{\mathrm{line}}=F_{\mathrm{res}}+F_{\mathrm{comp}}+F_{\mathrm{halo}}$=1,
and the corresponding observed line visibility is given by

$V_{\mathrm{line}}={{V_{\mathrm{res}}F_{\mathrm{res}}+F_{\mathrm{comp}}}\over{F_{\mathrm{res}}+F_{\mathrm{comp}}+F_{\mathrm{halo}}}}.$
(7)

We note that $F_{\mathrm{line}},V_{\mathrm{line}}$ are
$F(\lambda)_{\mathrm{circ}}$, $V(\lambda)_{\mathrm{circ}}$, respectively, as
defined in Sec.(4.2), measured at the peak of the line.

This simple model has six free parameters (five of them independent), namely,
the relative fluxes $F_{\mathrm{res}}$, $F_{\mathrm{comp}}$,
$F_{\mathrm{halo}}$; the size (i.e. full width at half maximum, FWHM), and the
geometrical parameters (i.e. inclination, $i$, and position angle, PA) of the
resolved (or marginally resolved) Gaussian component. In practice, we used the
Python module emcee to derive the free parameters in our model (see Foreman-
Mackey et al., 2013, for a detailed description of the module). We set the
prior distribution to be uniform (i.e. non-informative prior), the log-
likelihood to be simply a Gaussian including Eq. 7 in its definition, and the
posterior distribution was therefore given by the product between the prior
distribution function and the log-likelihood function. We initialised the MCMC
with 1000 walkers and ran them for 1000 steps. We set a burn-in phase of 10%
of the steps to account for the warm-up period of the chain (see GRAVITY
Collaboration et al., 2020, for details). Figures 33 and 34 show the best-
fitting visibility as a function of the projected baseline for the 16 objects.

The best-fitting value of the parameters and their uncertainties are given in
Table 2. Two objects (HD 114981 and HD 141569) with good quality
interferometric data are unresolved for our baselines. Considering our longest
baseline of 130 m, this translates into an angular size of less than 1.7 mas
(i.e. ¡ 1.2 au and ¡ 0.2 au for HD 114981 and HD 141569, respectively). We
note that though not fully discussed in this work, the fact that some objects
show evidence of asymmetries (i.e. non-zero differential phases and/or closure
phases) indicates that these sources are not point-like. Sources with detected
differential phases and/or closure phases are HD259431, HD190073, HD163296,
HD158643, HD150193, HD100546, HD98922, HD85567, HD58647, HD45677, and HD37806.

Due to the limited and sparse $uv$ coverage and the large errors associated
with the continuum-subtracted line visibilities, the errors on all parameters
are in general very large. We note that, as it is true in general, these
errors are just the errors provided by the fitting procedure, and they do not
take into account further sources of uncertainty (e.g. degeneracy between the
inclination, PA, and size in marginally resolved objects).

Nevertheless, there is clear evidence that the Br$\gamma$ line emission is
very compact. The FWHM of the Gaussian brightness distribution as described
above is plotted in Figure 1 as a function of the stellar distance reported in
Table 1. As shown in the figure, the FWHM of the Gaussian brightness
distribution component of the Br$\gamma$ line-emitting region has a typical
size of a few milli-arcsecond. Moreover, the unresolved component accounts for
about half of the total flux, while the halo contributes significantly to the
line flux in one source only (HD 45677). In addition, it should be noted that
none of our objects are fully resolved and that in most cases, the Br$\gamma$
line emission is only marginally resolved with our current spatial resolution.
Figure 2 shows the size of the resolved and marginally resolved components as
a function of the radius of each star. The values of $R_{res}$ are typically
10–30 R∗(see Fig. 2).

These results are usually in agreement with previous Br$\gamma$ line spectro-
interferometric studies done with VLTI-AMBER and the Keck Interferometer
(Kraus et al., 2008; Eisner et al., 2007; Garcia Lopez et al., 2015; Caratti o
Garatti et al., 2015; Kurosawa et al., 2016). Indeed, although limited in
number of baselines and sensitivity, those studies showed that the Br$\gamma$
line emission in early-type HAEBE stars is usually very compact, with an
extent of a few milli-arcseconds, which can usually be modelled with a disk
wind (see, e.g. Garcia Lopez et al., 2015; Caratti o Garatti et al., 2015;
Kurosawa et al., 2016; Hone et al., 2017) or a gaseous disk in Keplerian
rotation (Hone et al., 2019).

Figure 1: FWHM of the marginally resolved component as a function of the
distance. The numbers identify different stars (see first column in Table 2).
Figure 2: Radius of the marginally resolved Br$\gamma$ component as a function
of Rstar. When not visible, the errors are smaller than the size of the dots.
The number labels are as in Fig. 1.

###  4.4 Br$\gamma$ line displacements

The size of the Br$\gamma$ line-emitting region obtained from the analysis of
the continuum-subtracted line visibilities is limited by the maximum angular
resolution determined by our longest baselines. However, analysis of the
Br$\gamma$ line differential phases can give us information on smaller angular
scales through the study of the line astrometric displacements. This technique
is the spectro-interferometric equivalent to the analysis of spectro-
astrometric signals in standard spectroscopic observations (e.g. Lachaume,
2003; Whelan & Garcia, 2008; Le Bouquin et al., 2009).

At any wavelength, the displacement of the photo-centre of the continuum-
subtracted line emission with respect to the photo-centre of the continuum
($\delta$) is computed as

$\delta=-\Delta\Phi\frac{\lambda}{2\pi B},$ (8)

where $B$ is the length of the baseline (e.g. Lachaume, 2003). In this
expression $\delta$ represents the projection, in the baseline direction, of
the photo-centre in the plane of sky. A single spectro-astrometric solution
was found by performing a regression fit to all the baselines at different
velocity channels projected in the plane of sky. The results are shown in the
left panels of Figs. 3 and 4 for the eight stars with sufficiently good data
(half of the sample). Indeed, in order to detect differential-phase signals
across the line, a high S/N at the line position is required, along with
baselines long enough to be able to at least partially resolve the line
emission.

Figure 3: Two-dimensional displacements (left panel) and PVD (right panel) for
HD 37806, HD 45677, HD 58647, and HD85567 as computed from the differential
phase and described in Sec. 4.4. The data are colour-coded according to the
radial velocity (scale on the right). In the 2D-displacement figures, the
dashed lines show the best linear fit reported at the top of the panel. In the
PVD panels, the blue and red shaded regions show the Keplerian rotation
pattern (see text for more details).

Figure 4: Same as in Fig.3 but for HD 98922, HD 100546, HD 163296, and HD
259431.

####  4.4.1  The position angle of the Br$\gamma$ emitting region

From the astrometric line displacements, information about the projected PA of
the line emission in the plane of sky and on the centro-symmetry of the
brightness distribution can be obtained. The left panels in Figs. 3–4 show
that for seven out of eight stars, the displacements are roughly aligned along
one direction in the plane of sky (with the exceptions of HD 45677 and HD
98922; Gravity Collaboration et al. 2024b, a in prep.). Therefore, a linear
fit to the Br$\gamma$ line displacements provides the PA of the line-emitting
region.

The resulting PAs of the Br$\gamma$ line emission (PAdisp) are reported in
Table 2, Column 10. The linear fit is also shown as a dashed line overplotted
onto the line displacements in Figs.3–4. The PAdisp values have much smaller
uncertainties than those derived from the line visibility fits (Table 2,
Column 6; PA), but the agreement is quite good when both values are available.
The derived values are also in rough agreement with previous results found in
the literature for some of our sources (e.g. Kurosawa et al., 2006; Kluska et
al., 2020). In addition, the PA of the Br$\gamma$ emitting region is in good
agreement with the PA of the continuum emission in the K-band (PAcont in Table
2, Column 12), as computed in Gravity Collaboration et al. (2019). It should
be noted that in all cases, the errors on the PA reported in Table 2 are just
formal errors resulting from the fitting procedures. As for the visibility
fitting, these errors do not fully take into account other factors such as the
goodness of the $uv$ coverage or the presence of non-centrally peaked
emission.

Some information on the centro-symmetry of the brightness distribution can be
obtained by inspecting whether the Br$\gamma$ line displacements are centred
at ($\Delta\delta$(RA)=0, $\Delta\delta$(DEC)=0) (see left panels in Figs.
3–4). Centro-symmetric brightness distributions produce line displacements
centred at (0,0), whereas continuum and/or line brightness distributions, not
centro-symmetric ones, produce displacements shifted from (0,0).444It should
be noted that although the displacements were computed from the continuum-
subtracted line differential phases, the obtained displacements are still
differential, that is, with respect to the photo-centre of the brightness
distribution of the continuum. Examples of centro-symmetric objects are HD
163296 and HD 259431, whereas objects such as HD 98922 and HD 37806 show clear
deviations from centro-symmetry. Our simple analysis did not allow us to
discern which of the two components (i.e. the line and/or the continuum
emission) is not centro-symmetric (i.e. point-symmetric about the (0,0)
position). Further information can only be obtained from the analysis of
continuum and/or line reconstructed images (Sanchez-Bermudez, 2018; Sanchez-
Bermudez et al., 2018; Kluska et al., 2020). However, it should be noted that
two of our objects (namely, HD 45677 and HD 98922) show continuum closure-
phase signals of over 25°. Such huge closure-phase signals are difficult to
explain with simple inclination effects, and therefore they likely indicate an
asymmetric continuum structure (see Gravity Collaboration et al., 2019, for
further details).

####  4.4.2 The position-velocity diagrams

Additional information about the Br$\gamma$ line kinematics can be obtained
from position-velocity diagrams (PVDs). These diagrams are well-known
visualisation tools in spectroscopy. They generally display the absolute line
displacement with respect to the central source versus radial velocities
measured with respect to the source local standard of rest (LSR) and are
corrected for the source radial velocity when available from the literature
(see Table 1). The PVDs for our sample are shown in the right panel of Figs.
3–4. The 2D line displacements were converted to absolute displacements by
computing the module of the vector in RA and DEC.

The PVDs in Figs. 3–4 show evidence of motions dominated by a Keplerian
rotation pattern. If the emission is fully spatially and spectroscopically
resolved, the PVD of a Keplerian rotating disk is characterised by a typical
”butterfly” pattern, with the low-velocity emission located further from the
(0,0) position. In the other extreme case (namely, spatially unresolved
emission), the PVD is a straight line with the highest velocity further from
the (0,0) position. In the case of partially resolved emission, the PVD has an
S shape, with the low-velocity emission projected at decreasing displacements
(see, e.g., the PVDs of HD 163296 in Fig. 4). Of the eight objects for which
PVDs could be computed, only HD 98922 displays the butterfly pattern expected
when the emission is fully resolved, while others, such as HD 163296, HD
58647, HD 85567, and HD 259431, have the S-shape PVD of marginally resolved
sources. The sources HD 45677, HD 37806, and HD 100546 show evidence of
complex motions that are more difficult to interpret.

Detailed models are necessary to investigate the properties of individual
sources but are beyond the scope of this paper. However, a caveat must be
stated. Namely, the compactness of the Br$\gamma$ emission and the relatively
low spectral resolution of VLTI-GRAVITY limit the potential capability of this
tool.

###  4.5 HD 45677

The source HD 45677 (also known as FS CMa) has properties that differ from the
other HAEBEs in the sample. The evolutionary state of this object is debated
(Miroshnichenko, 2017). Very recent MATISSE interferometry at thermal infrared
wavelengths has already hinted at a rather complex geometry regarding the
circumstellar matter distribution (Hofmann et al., 2022). In our GRAVITY data,
HD 45677 has the largest extension of the Br$\gamma$ emitting region ($\sim$
140 R∗), has the lowest contribution from the unresolved component ($\sim$ 20%
of the total flux), and is the only object with clear evidence of an extended,
over-resolved component that contributes about 30% of the total flux. The
displacements and the PVD in Fig. 3 show non-orderly motions that are
difficult to interpret in terms of simple kinematics. A paper analysing, in
detail, the complex circumstellar environment of this object is currently in
preparation (Gravity Collaboration et al., 2024b), and we do not discuss the
topic further in this paper.

##  5 Discussion

Figure 5:  Comparison of the radius of the Br$\gamma$ line marginally resolved
component to that of the circumstellar continuum emission in the K-band
(Rcont). The number labels are as in Fig. 1.  Figure 6:  Radius of the
Br$\gamma$ line marginally resolved component versus the co-rotation radius
for all sources for which Rcorot is available (see Table 1). The number labels
are as in Fig. 1.  Figure 7:  Radius of the marginally resolved Br$\gamma$
component versus the gravitational radius of the star computed for a sound
speed of 10 km/s. The dashed line shows the locus of the points Rres=0.03
Rgrav (see text for details). The number labels are as in Fig. 1.

###  5.1 Relation to the dusty disk

The size of the Br$\gamma$ emitting region is often compared to the size of
the K-band continuum emission, which traces, in general, the silicate
sublimation radius ((Gravity Collaboration et al., 2019)). This, in itself, is
not immediately related to the line origin, which we discuss in Sect. 5.3;
however, it is an important factor to consider in modelling the properties of
the line-emitting region as grains control the thermal gas properties.

In general, the size of the Br$\gamma$ line extended component is smaller than
the $K$-band continuum emission (see Fig. 5). The exceptions are HD 259431
(#25) and HD 85567 (#5), for which Rres/R${}_{cont}\sim$2.0–2.4. These two
objects are among the hottest (T${}_{\mathrm{eff}}\sim$13 000 K–14 000 K), and
most luminous (log L${}_{*}\sim$3–3.3) of our sample.

Previous studies have shown that the size of the $K$-band continuum emission
in Herbig Ae and T Tauri stars tightly correlates with the stellar luminosity
as L${}_{*}^{1/2}$ and is close to the location of silicate sublimation
(Monnier & Millan-Gabet, 2002; Gravity Collaboration et al., 2019, 2021b).
This relationship breaks for some very luminous Herbig Be stars, for which the
size of the K-band continuum emission is smaller than predicted by the above
relationship. The sources HD 85567 and HD 259431 are the objects where this
effect is stronger (together with PDS 27; Gravity Collaboration et al. 2019).
A possible explanation for this is the presence of an inner gaseous disk
inside the dust sublimation radius, which produces a smaller continuum
emitting region than expected from simple dust sublimation physics. If this is
indeed the case, then the Br$\gamma$ line would also be emitted in a dust-free
region in HD 259431 and HD 85567. On the other hand, if the effect of a
gaseous inner disk is simply to screen the stellar radiation and move the dust
sublimation radius closer to the star, then grains could be present in the
line-emitting region.

As generally found for the K-band continuum emission, there is evidence in our
data that the size of the Br$\gamma$ line-emitting region is also related to
the stellar luminosity (see Fig. 9). The relation R${}_{res}\propto$ L∗0.5 is
expected if the physical conditions required to emit hydrogen recombination
lines are controlled by the strength of the geometrically diluted stellar
radiation field, as in the case of the dust sublimation temperature. We note
that the fact that the two processes (i.e. the hydrogen line emission and the
dust sublimation) are both controlled by the stellar radiation field does not
mean that they occupy the same spatial region, as they indeed do not. It
should be noted, however, that the correlation between the size of the
marginally resolved Br$\gamma$ line emission and the stellar luminosity is
most likely due to a strong distance dependence of these two quantities.

###  5.2 The kinematics of the innermost region

As shown in Sect. 4.3, about 50% of the Br$\gamma$ flux is not spatially
resolved by our observations. Therefore, we did not derive any of its
properties from the visibilities alone, and we can only give an approximate
upper limit to its size. However, further information on the kinematics of the
gas on very small scales can be extracted if the continuum-subtracted line
differential phases can be computed from the data. As discussed in Sects. 4.3,
and 4.4, this was possible for eight objects for which the PA of the major
axis of the emission (displacements) and the spectro-astrometric displacements
(PVDs) could be computed.

In most cases, the PVDs show the typical signature of gas in rotation, either
spectro-astrometrically resolved (HD 98922) or partially resolved (HD 163296,
HD 58647, HD 85567, and HD 259431). The PVD of HD 98922 is consistent with the
Keplerian rotation pattern of a star of HD 98922’s mass, as given in Table 1,
and the inclination derived from the visibility fits (solid curves in Fig. 4).
The predicted Keplerian rotation pattern is also shown for comparison for the
partially resolved sources.

A comparison of the line PA (PAdisp) with that of the K-band continuum disk
(PAcont; Table 2) showed that in several cases, both values coincide within
the errors. In partially resolved sources, this indicates that the Br$\gamma$
emitting region is aligned with the disk, that is, the line is emitted close
to the disk midplane, where Keplerian rotation dominates the gas motion. A
wind farther from the launching region, where poloidal motion dominates the
kinematics, would have a PA strongly inclined with respect to the disk, as
seen, for example, in jets (Pascucci et al., 2022; Lesur et al., 2022). Two
objects (HD 58647 and HD 85567) show evidence of partially resolved Keplerian
rotation. Objects that are partially resolved may have a complex behaviour
that needs detailed and more complex models in order to be understood (e.g.
the presence of variable rotating disk asymmetries Kobus et al., 2020; Varga
et al., 2021; Gravity Collaboration et al., 2021c, 2024a).

We also note that several objects have displacements not centred at (0,0)
coordinates. This is likely a consequence of the fact that the observed
continuum-subtracted differential phases measure displacements from the
adjacent continuum assumed to be symmetric around its photo-centre. In
practice, any asymmetric structure in the continuum emitting region, such as a
warp in the inner disk or a difference in disk flaring, could be at the origin
of the shift from the (0,0) position in the displacement and PVDs, even when
the line-emitting region itself is symmetric around the photocenter. For
instance, the presence of uneven disk-continuum illumination has been invoked
to account for the non S-shaped profile of the observed Br$\gamma$ line
differential phases in HD 98922 (Caratti o Garatti et al., 2015). Such a
scenario could also explain the Br$\gamma$ line displacements measured in this
source, where both the blue- and red-shifted velocity displacements are
oriented towards the south direction (see Fig. 4).

In contrast, Kluska et al. (2020) computed H-band continuum reconstructed
images of five of our sources (HD 144668, HD 98922, HD 100546, HD 163296, and
HD 150193). The authors classified these objects as symmetric or asymmetric
sources. Of those, HD 37806 was classified as a symmetric object, whereas HD
45677, HD 98922, HD 100546, and HD 163296 were classified as asymmetric
objects. These results are in agreement with our displacements with the
exception of HD 37806, for which a clear displacement offset was observed.
This might indicate that the compact Br$\gamma$ line emission is indeed
asymmetric or that the H-band continuum is asymmetric on spatial scales
smaller than those sampled in Kluska et al. (2020), i.e. $\lesssim$1.7 mas.

###  5.3 The origin of the H i Br$\gamma$ line emission

####  5.3.1 Magnetospheric accretion

The H i line emission, and in particular the Br$\gamma$ line emission, is
often considered a tracer of accretion. This is because of the well-
established empirical relationship between the H i line luminosity and the
accretion luminosity (see e.g. Muzerolle et al., 2004; Alcalá et al., 2014).
This correlation holds for a wide range of stellar masses, from brown dwarfs
up to HAEBE stars (see e.g. Natta et al., 2004; Gatti et al., 2008; Calvet et
al., 2004; Garcia Lopez et al., 2006).

This empirical relationship is expected in the current accretion paradigm, as
it is well established for T Tauri stars, where hydrogen lines are emitted by
matter accreting onto the star along stellar magnetic field lines (see e.g.
Hartmann et al., 2016, and references there in). The size of this region
depends on the strength of the magnetic field and on the mass-accretion rate
(see e.g. Bessolaz et al., 2008), and it is typically of $\sim$5 R∗ in T Tauri
stars. Herbig AeBe stars have much weaker magnetic fields than T Tauri sources
(e.g. Alecian et al., 2009; Hubrig et al., 2011; Alecian et al., 2013; Hubrig
et al., 2015) and comparatively high mass-accretion rates. Therefore, the
magnetospheric emitting region is much smaller, with an upper limit of
$\sim$1-2 R∗. Any emission from such a region will be unresolved in our
observations. Furthermore, accretion occurs if the magnetosphere is roughly
confined within the disk co-rotation radius, that is, within the distance from
the star where the Keplerian angular velocity equals the stellar angular
velocity (i.e. $R_{co}=(GM_{*})^{1/3}\Omega_{*}^{-2/3}$). This is not the case
for the HAEBE in our sample. Figure 6 clearly shows that the size of the
marginally resolved component is significantly larger than the value of the
co-rotation radius, with Rres/Rcorot$\sim$5–20. We then concluded that
magnetospheric accreting columns are not the source of the marginally resolved
Br$\gamma$ line emission.

On the other hand, the small size of the magnetosphere in HAEBE is, in
principle, consistent with the dimensions of the unresolved component but not
with the Keplerian rotation pattern that we detected in some objects, as most
line emission in magnetospheric columns occurs far from the disk surface,
where the gas motion is mostly radial (e.g. Muzerolle et al., 2004; Hartmann
et al., 2016). Recent studies have questioned the validity of the
magnetospheric accretion scenario in weakly magnetised Herbig AeBe stars and
point towards a change of the accretion paradigm within the Herbig AeBe mass
regime (Vink et al., 2002; Vink, 2015; Patel et al., 2017; Ababakr et al.,
2017; Marcos-Arenal et al., 2021) from magnetospheric to boundary layer
accretion (Lynden-Bell & Pringle, 1974; Takasao et al., 2018). A boundary
layer is expected to form very close to the stellar surface and to have a very
small radial extension. Any emission from this region will be by far too small
when compared to the size of the extended Br$\gamma$ line emission (e.g. Kley
& Lin, 1996). It may be possible that the unresolved Br$\gamma$ component
traces the boundary layer emission. Although, when the Keplerian pattern is
clear, it seems to indicate that the velocity increases at smaller radii,
rather than decreasing, as expected in boundary layer accretion. Detailed
models and better data are required if this hypothesis is to be explored
further.

####  5.3.2 Disk winds

A more convincing scenario that we can infer from our sample is that the
Br$\gamma$ line mostly forms in outflowing gas. In particular, two different
kinds of winds have been extensively studied, namely, photoevaporative winds
and the so-called magneto-centrifugally launched winds.

Photoevaporative winds are thermally driven winds launched where the radiative
pressure of the gas at the disk surface is larger than the gravitational
pressure. In this context, a fundamental parameter is the so-called
gravitational radius (Rgrav= G M$\ast$/$c_{s}^{2}$, with $c_{s}$ the sound
speed), where a gas parcel becomes unbound from the central star (Hollenbach
et al., 2000; Alexander et al., 2014; Picogna et al., 2021). The specific
mass-loss rate generally peaks at 0.1–0.5 Rgrav (see e.g. Alexander et al.,
2014). Depending on the adopted model and assuming a stellar mass of $\sim$2
M⊙ this translates into a peak of the mass-loss rate around $\sim$1.5–2 au for
the case of EUV-driven photoevaporation. This radius moves even further out
for X-ray-driven and FUV-driven winds (Alexander et al., 2014; Kunitomo et
al., 2021). Our measurements do not support these models, as the size of the
Br$\gamma$ line-emitting region is by far too small. This is clearly
demonstrated in Fig. 7, which shows that Rres is 30 to 100 times smaller than
the gravitational radius. As a consequence, photoevaporative winds cannot
account for the Br$\gamma$ line emission.

A more convincing interpretation is that the Br$\gamma$ line traces a magneto-
centrifugally launched wind. A fundamental ingredient of these models is a
magnetic field anchored to the disk, possibly a residual of the original
collapsing cloud magnetic field. In the simplest version, a wind is launched
if the disk magnetic field lines are inclined by more than 30° from the polar
axis (Blandford & Payne, 1982). In recent years, more sophisticated MHD winds
have received growing attention thanks to the improvement of simulations and
the suspected role of the winds in removing angular momentum and thus allowing
accretion to occur (see e.g. Bai, 2016; Zhu & Stone, 2018; Jacquemin-Ide et
al., 2021). Depending on the disk region from which these winds originate,
different launching mechanisms are involved (see e.g. Pascucci et al., 2022;
Lesur et al., 2022, for recent reviews).

In our case, to reproduce the extent of the Br$\gamma$ line-emitting region,
the wind emission needs to occur within a region of $\sim$10–30 R∗ from the
central source. Due to the high luminosity of our sources and their weak
magnetic fields, several wind launching mechanisms might be in place within
this region. A weakly magnetised disk rotating at Keplerian velocity and with
high enough levels of ionisation is known to be unstable against magneto-
rotational instability (MRI; Balbus & Hawley, 1991). Recent simulations have
shown that the levels of ionisation around typical Herbig AeBe stars up to
$\sim$1 au from the sources are sufficient to sustain MRI and thus launch an
MRI wind (Suzuki & Inutsuka, 2009; Bai & Stone, 2013; Flock et al., 2016,
2017; Jacquemin-Ide et al., 2019). These winds are launched above the disk
mid-plane, with their base rotating at the Keplerian velocity of the disk
footpoint at which the magnetic field lines are anchored. As matter blows up
following the magnetic field lines, the wind accelerates, and the poloidal
velocity becomes dominant over the tangential component.

Previous interferometric studies have successfully modelled the Br$\gamma$
emission properties using simplified models of disk winds launched
centrifugally from the disk surface (Weigelt et al., 2011; Garcia Lopez et
al., 2015; Caratti o Garatti et al., 2015; Garcia Lopez et al., 2016; Kurosawa
et al., 2016; Kreplin et al., 2018). The launching regions extend from 1–3 R∗
to $\sim$10–30 R∗, consistent with the location of the Br$\gamma$ emission
observations in this paper. The models assume that the line is emitted well
above the disk mid-plane (the so-called warm disk-wind models), where an
originally cold wind is heated by ambipolar diffusion as it moves above the
disk and reaches a roughly constant temperature of around 10 000 K (e.g.
Safier, 1993; Garcia et al., 2001). In these regions, the poloidal velocity
component may be large in comparison to the rotational one. Specifically, this
approach was used to reproduce the single-peaked Br$\gamma$ line profiles and
spectro-interferometric observations of several Herbig AeBe stars using VLTI-
AMBER observations at R$\sim$12 000 (Weigelt et al., 2007; Caratti o Garatti
et al., 2015; Garcia Lopez et al., 2015; Kurosawa et al., 2016). In fact, even
at GRAVITY’s modest spectral resolution, magneto-centrifugally launched winds
should produce double-peaked line profiles for disk inclinations larger than
$\gtrsim$40° if the line emission starts at the wind base (Tambovtseva et al.,
2014, 2016). Although all the objects with double-peaked Br$\gamma$ line
profiles in our sample (HD 58647, HD 114981, HD 141569, and HD 158643) have
continuum and/or Br$\gamma$ line inclinations $\gtrsim$40°, not all the
sources with inclinations $\gtrsim$40° have double-peaked line profiles (see
line profiles in Appendix B).

Wind models can then reproduce the basic features of the interferometric
observations presented here, in particular, the extent of the launching region
and the characteristics of the line profiles. Winds are ejected at all scales
between the inner and outer radius, with no gaps separating the resolved or
marginally resolved component from the unresolved component defined in our
visibility analysis. In fact, these separations are indeed arbitrary,
justified only by the length of the baselines available with VLTI-GRAVITY.
Support of the common origin of the two components is provided by the fact
that, as mentioned in Sect. 4.4, the PA derived from our line displacements is
generally in agreement with the PA obtained for the marginally resolved
Br$\gamma$ line-emitting region in Sect. 4.3.

The models can be revisited to include the results on the unresolved component
discussed above. The Keplerian pattern detected in the innermost regions and
the similarity between the line and continuum PA suggest that at small radii
the line emission originates close to the launching region, where the
Keplerian rotation still dominates the wind kinematics. On the other hand, the
Br$\gamma$resolved component has a relatively low rotational velocity, even
close to the launching point, as the Keplerian velocity at Rres is always
$<$150 km s-1 (see Fig. 8), and its component along the line of sight is $<$80
km s-1 in all objects. This is likely to be unresolved in our GRAVITY spectra,
given their low spectral resolution ($\sim$75 km s-1). This kind of data show
the potential of future high-resolution optical spectro-interferometers to
constrain MHD wind models and explore wind properties such as thermal
structure and dependence on the radius of the mass loading.

Figure 8: Keplerian velocity at the radius of the marginally resolved
Br$\gamma$ line emission Rres shown as a function of L∗. The number labels are
as in Fig. 1.

####  5.3.3 Disk surface

An additional possibility, which so far has been very little investigated, is
that some of the Br$\gamma$ emission comes from the disk surface, heated by
the stellar radiation. This region has similar physical properties to the base
of a disk wind, and thus separating gravitationally bound from unbound
material may be observationally very challenging. In particular, realistic
models of the gas temperature and density at the disk surface in regions
completely dust depleted are still lacking, and therefore, we lack accurate
line profile modelling accounting for the total flux expected from this
region. Nevertheless, as in wind models, one expects that the lines will
display a double-peaked profile, although the relative contribution to the
line emission at each radius will depend on the physical properties of the
disk. Therefore, it is also possible that the contribution of the bound gas is
relevant only at small scales, close to the source, where, indeed, we detect
the signature of Keplerian rotation in the line emission.

####  5.3.4 The Br$\gamma$ line luminosity versus accretion luminosity
correlation

After our previous discussion, it may be surprising that in young stars, the
luminosity of Br$\gamma$ (and, more in general, of other emission lines) shows
a very good correlation with the accretion luminosity. We find it worth
stressing that this is an ”empirical” correlation well established over a
large range of stellar and accretion properties (see, e.g. Garcia Lopez et
al., 2006; Alcalá et al., 2014, 2017; Fairlamb et al., 2017).

However, we have shown that a significant fraction of the Br$\gamma$ emitting
region in Herbig AeBe stars is extended and that around 50% or more of the
emission cannot be ascribed to magnetospheric or boundary layer accretion.
Moreover, most recent interferometric studies of the Br$\gamma$ line towards
classical T Tauri stars (CTTs) have shown that in only two cases so far (TW
Hya and DoAr44; see Gravity Collaboration et al., 2020; Bouvier et al., 2020,
respectively) the size of the Br$\gamma$ emitting region is consistent with
magnetospheric accretion alone. For a larger sample of seven CTTs, which also
includes those two sources, GRAVITY Collaboration et al. (2022a) find that,
while magnetospheric accretion is the primary driver of Br$\gamma$ emission in
those two weak accretors, the remaining CTTs have partial or strong
contributions from spatially extended Br$\gamma$ components, similar to the
HAEBE sample.

Figures 9 and 10 show the area of the Br$\gamma$ emitting region, defined as
$\pi\,R_{res}^{2}$, against both stellar and accretion luminosity,
respectively. The sample shown in these figures includes the HAEBE sample
presented in this paper and the CTTs sample presented in GRAVITY Collaboration
et al. (2022a) (see their Table 1 and 3 for stellar and Br$\gamma$
parameters). The Herbig AeBe stars are labelled as in Table 1, and CTTs labels
are provided in the caption of Figure 9. We chose to plot the area of the
emitting region rather than its radius, as both areas and luminosities have
the same dependence on the distance of the source ($\propto D^{2}$). This is
particularly important for the HAEBE sample, which is spread over a large
range of distances. Given the limited range of projected sizes available with
VLTI-GRAVITY (see Fig.1), this introduces a strong correlation between
$R_{res}$ and the distance of the star so that, to zero order,
$R_{res}\propto$ L∗0.5. Figure 10 suggests that there is a correlation between
the Br$\gamma$ emitting area and Lacc, as the best-fitting slope is different
than one (0.75$\pm$0.04). If the area of the emitting region is a good proxy
of the line luminosity, then the wind (or extended) component of the
Br$\gamma$ line is directly correlated with the accretion luminosity. We note,
however, that similar to the results obtained for emission lines in general,
the area of the marginally resolved Br$\gamma$ component is equally related to
L∗ (see Fig. 9) and has an identical slope (0.75$\pm$0.04). Indeed, similar
empirical relationships between the line luminosity of wind and/or jet tracers
(e.g. [O i]) and the stellar and accretion luminosity have been observed in
HAEBE and T Tauri samples (see, e.g. Fairlamb et al., 2017; Nisini et al.,
2018; Banzatti et al., 2019).

For now, we can only hypothesise that hydrogen recombination lines trace hot
gas located at the disk surface that is being accreted and/or ejected. All of
these processes are related and interdependent, and they correlate somehow
with the accretion luminosity. In order to further probe the origin of the
empirical correlation between Lacc and L∗and the Br$\gamma$ line luminosity,
additional interferometric surveys of complete samples located at similar
distances (i.e. same star-forming region) should be performed. Such surveys
would likely be possible with the advent of more sensitive intereferometers,
such as GRAVITY+, giving us the opportunity to study more homogeneous and
unbiased samples.

Figure 9: Area of the region emitting the marginally resolved Br$\gamma$
component ($\pi\,R_{rses}^{2}$) plotted as a function of L∗for both the Herbig
AeBe and CTTs samples. The number labels of the Herbigs are as in Fig. 1. CTTs
data are taken from GRAVITY Collaboration et al. (2022b). The number labels of
the CTTs are the following: T1: AS 353; T2: RU Lup (epoch 2018); T3: RU Lup
(epoch 2021); T4: TW Hya; T5: DG Tau; T6: DoAr 44; T7: S CrA; T8: VV CrA. The
dotted and solid lines show the best-fitting line of slope $0.75\pm 0.04$
(reduced $\chi^{2}$ of 0.14) and slope 1.0 (reduced $\chi^{2}$ of 0.25),
respectively.  Figure 10: Area of the region emitting the marginally resolved
Br$\gamma$ component as a function of $L_{acc}$ for both the Herbig AeBe and
CTTs samples. The number labels are as in Fig. 9. The dotted and solid lines
show the best-fitting line of slope $0.75\pm 0.04$ (reduced $\chi^{2}$ of
0.15) and slope 1 (reduced $\chi^{2}$ of 0.24), respectively.

##  6 Summary and conclusions

This paper presents the results of an interferometric study using VLTI-GRAVITY
of the region emitting the hydrogen Br$\gamma$ line in a sample of 26
intermediate mass stars. Of these, nine did not show any line emission at the
time of the observations, even after correcting for the underlying absorption
photospheric lines. For the remaining 17 targets, we derived continuum-
subtracted visibilities and differential phases when possible.

We derived the size of the line-emitting region by performing a 2D Gaussian
fit to the continuum-subtracted visibilities. The results show that 15 sources
are marginally resolved, with typical radii of 10–30 R∗, and two were not
resolved for our baselines. A significant fraction of the line flux, ranging
from $\sim$20% to 100%, comes from regions that are not resolved even with the
longest available baselines (B$\sim 130$ m). In one case only (HD 45677),
there is significant contribution ($\sim$30%) from a more extended component
that is over-resolved in our data. Further insight into the properties of the
innermost emitting region is provided by the continuum-subtracted line
differential phases that could be inferred for eight sources. These data were
used to compute the line displacement with respect to the continuum at
different velocities and the PA of the line-emitting region in the plane of
sky as well as to derive some kinematic information.

When compared to the dusty disk traced by the K-band continuum emission
(Gravity Collaboration et al., 2019), the Br$\gamma$ emission is generally
confined to a region inside the dust sublimation radius. The visibility fits
provided rough information on the inclination and PA of the marginally
resolved emitting region. However, for the eight objects for which continuum-
subtracted differential phases could be extracted, we measured the PA with a
much higher accuracy. These values are consistent with those found from the
visibility fits. When compared to the continuum disk, the continuum and
Br$\gamma$ emission PAs are roughly in agreement in all cases.

The PVDs were computed from the continuum-subtracted differential phases for
the same eight objects. They provide kinematic information on the innermost
disks on scales of a few R∗. Within the uncertainties, the diagrams are
typical of disks in Keplerian velocity, with the only clear exception being HD
45677, which shows evidence of very complex kinematics. The source HD 98922
shows the typical ”butterfly” pattern expected for disks in Keplerian rotation
when the emission is fully resolved, with values consistent with the stellar
mass and the inclination of the line-emitting region. The sources HD 163296,
HD 58647, HD 85567, and HD 259431 have the typical S shape of a disk in
Keplerian rotation shown in the PVD of marginally resolved sources.

A discussion on the possible origin of the Br$\gamma$ emission favours a disk-
wind origin. Photoevaporative winds are not a possible explanation, as they
are expected to originate at much larger scales than the 10–30 R∗ typical of
the resolved Br$\gamma$ component. The MHD disk winds are more likely, as they
can be ejected over a wider region, extending very close to the star. The
Keplerian rotation pattern, revealed by the PVDs in the innermost regions of
the few objects where this information is available, is expected if the
emission originates at the base of the wind, still close to the disk surface.
However, we cannot exclude that the Br$\gamma$ emission in the innermost disk
comes from gas on the surface of the disk itself, namely, gaseous layers that
are still ”bound” by the system’s gravity. It is worth stressing that the data
lack the required spectral and spatial resolution for more detailed
studies,which could reveal more complex kinematical patterns.

Finally, by combining the HAEBE sample presented here with the T Tauri sample
shown in GRAVITY Collaboration et al. (2022a), we noted a correlation between
the area of the Br$\gamma$ emitting region and both the accretion and stellar
luminosities. Although more homogeneous and unbiased samples are required to
prove whether such correlations are real, they might explain the existence of
the well-known empirical correlation between the Br$\gamma$ line luminosity
and the stellar accretion luminosity.

The results outlined in this paper show the potential of near-IR
interferometry to aid in understanding the phenomena that occur in the
innermost disk of young stars. The results also reveal the need to obtain
higher-quality data, both in terms of spectral and spatial resolution.
Moreover, the current effort to reconstruct images of both continuum and line
emission needs to be pushed farther. At the same time, models of the inner
disk’s physical properties, of the winds launched from the disk surface at
very small scales, and of the accretion region need to be improved so that
predictions of the emission line spectrum can be carried out and compared with
the data.

###### Acknowledgements.

This material is based upon works supported by Science Foundation Ireland
under Grant No. 18/SIRG/5597. R.F. acknowledges support from the grants Juan
de la Cierva FJC2021-046802-I, PID2020-114461GB-I00 and CEX2021-001131-S
funded by MCIN/AEI/ 10.13039/501100011033 and by “European Union
NextGenerationEU/PRTR”. R.F. also acknowledges funding from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 101032092. A.C.G. has been supported by
PRIN-INAF MAIN-STREAM 2017 “Protoplanetary disks seen through the eyes of new
generation instruments” and from PRIN-INAF 2019 “Spectroscopically tracing the
disk dispersal evolution (STRADE)”. T.P.R. has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 743029). A.N.
acknowledges the kind hospitality of DIAS. A.A., M.F. and P.J.V.G acknowledge
funding by Fundação para a Ciência e a Tecnologia, with grants reference
UID/FIS/00099/2013 and SFRH/BSAB/142940/2018. T.H. acknowledges support from
the European Research Council under the Horizon 2020 Framework Program via the
ERC Advanced Grant Origins 832428. We thank the technical, administrative, and
scientific staff of the participating institutes and the observatory for their
extraordinary support during the development, installation, and commissioning
of GRAVITY. This research has made use of the Jean-Marie Mariotti Center Aspro
and SearchCal services, 555Available at http://www.jmmc.fr/ and of CDS
Astronomical Databases SIMBAD and VIZIER 666Available at
http://cdsweb.u-strasbg.fr/.

## References

  * Ababakr et al. (2017) Ababakr, K. M., Oudmaijer, R. D., & Vink, J. S. 2017, MNRAS, 472, 854
  * Alcalá et al. (2017) Alcalá, J. M., Manara, C. F., Natta, A., et al. 2017, A&A, 600, A20
  * Alcalá et al. (2014) Alcalá, J. M., Natta, A., Manara, C. F., et al. 2014, A&A, 561, A2
  * Alecian et al. (2009) Alecian, E., Wade, G. A., Catala, C., et al. 2009, MNRAS, 400, 354
  * Alecian et al. (2013) Alecian, E., Wade, G. A., Catala, C., et al. 2013, MNRAS, 429, 1001
  * Alexander et al. (2014) Alexander, R., Pascucci, I., Andrews, S., Armitage, P., & Cieza, L. 2014, Protostars and Planets VI, 475
  * Andrews (2020) Andrews, S. M. 2020, ARA&A, 58, 483
  * Bai (2016) Bai, X.-N. 2016, ApJ, 821, 80
  * Bai & Stone (2013) Bai, X.-N. & Stone, J. M. 2013, ApJ, 767, 30
  * Balbus & Hawley (1991) Balbus, S. A. & Hawley, J. F. 1991, ApJ, 376, 214
  * Banzatti et al. (2019) Banzatti, A., Pascucci, I., Edwards, S., et al. 2019, ApJ, 870, 76
  * Benisty et al. (2010) Benisty, M., Natta, A., Isella, A., et al. 2010, A&A, 511, A74
  * Bessolaz et al. (2008) Bessolaz, N., Zanni, C., Ferreira, J., Keppens, R., & Bouvier, J. 2008, A&A, 478, 155
  * Blandford & Payne (1982) Blandford, R. D. & Payne, D. G. 1982, MNRAS, 199, 883
  * Bouvier et al. (2020) Bouvier, J., Perraut, K., Le Bouquin, J. B., et al. 2020, A&A, 636, A108
  * Calvet et al. (2004) Calvet, N., Muzerolle, J., Briceño, C., et al. 2004, AJ, 128, 1294
  * Caratti o Garatti et al. (2015) Caratti o Garatti, A., Tambovtseva, L. V., Garcia Lopez, R., et al. 2015, A&A, 582, A44
  * Castelli & Kurucz (2003) Castelli, F. & Kurucz, R. L. 2003, in Modelling of Stellar Atmospheres, ed. N. Piskunov, W. W. Weiss, & D. F. Gray, Vol. 210, A20
  * Davies et al. (2020) Davies, C. L., Kraus, S., Harries, T. J., et al. 2020, ApJ, 897, 31
  * Eisner et al. (2007) Eisner, J. A., Chiang, E. I., Lane, B. F., & Akeson, R. L. 2007, ApJ, 657, 347
  * Fairlamb et al. (2017) Fairlamb, J. R., Oudmaijer, R. D., Mendigutia, I., Ilee, J. D., & van den Ancker, M. E. 2017, MNRAS, 464, 4721
  * Flock et al. (2016) Flock, M., Fromang, S., Turner, N. J., & Benisty, M. 2016, ApJ, 827, 144
  * Flock et al. (2017) Flock, M., Fromang, S., Turner, N. J., & Benisty, M. 2017, ApJ, 835, 230
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Garcia et al. (2001) Garcia, P. J. V., Ferreira, J., Cabrit, S., & Binette, L. 2001, A&A, 377, 589
  * Garcia Lopez et al. (2016) Garcia Lopez, R., Kurosawa, R., Caratti o Garatti, A., et al. 2016, MNRAS, 456, 156
  * Garcia Lopez et al. (2006) Garcia Lopez, R., Natta, A., Testi, L., & Habart, E. 2006, A&A, 459, 837
  * Garcia Lopez et al. (2015) Garcia Lopez, R., Tambovtseva, L. V., Schertl, D., et al. 2015, A&A, 576, A84
  * Garufi et al. (2018) Garufi, A., Benisty, M., Pinilla, P., et al. 2018, A&A, 620, A94
  * Gatti et al. (2008) Gatti, T., Natta, A., Randich, S., Testi, L., & Sacco, G. 2008, A&A, 481, 423
  * Gravity Collaboration et al. (2017) Gravity Collaboration, Abuter, R., Accardo, M., et al. 2017, A&A, 602, A94
  * GRAVITY Collaboration et al. (2020) GRAVITY Collaboration, Caratti o Garatti, A., Fedriani, R., et al. 2020, A&A, 635, L12
  * Gravity Collaboration et al. (2024a) Gravity Collaboration, Ganci, V., & et al. 2024a, A&A
  * Gravity Collaboration et al. (2021a) Gravity Collaboration, Ganci, V., Labadie, L., et al. 2021a, A&A, 655, A112
  * Gravity Collaboration et al. (2020) Gravity Collaboration, Garcia Lopez, R., Natta, A., et al. 2020, Nature, 584, 547
  * GRAVITY Collaboration et al. (2017) GRAVITY Collaboration, Garcia Lopez, R., Perraut, K., et al. 2017, A&A, 608, A78
  * Gravity Collaboration et al. (2021b) Gravity Collaboration, Perraut, K., Labadie, L., et al. 2021b, A&A, 655, A73
  * Gravity Collaboration et al. (2019) Gravity Collaboration, Perraut, K., Labadie, L., et al. 2019, A&A, 632, A53
  * Gravity Collaboration et al. (2021c) Gravity Collaboration, Sanchez-Bermudez, J., Caratti O Garatti, A., et al. 2021c, A&A, 654, A97
  * Gravity Collaboration et al. (2024b) Gravity Collaboration, Sanchez-Bermudez, J., & et al. 2024b, A&A
  * GRAVITY Collaboration et al. (2022a) GRAVITY Collaboration, Wojtczak, J. A., Labadie, L., et al. 2022a, arXiv e-prints, arXiv:2210.13095
  * GRAVITY Collaboration et al. (2022b) GRAVITY Collaboration, Wojtczak, J. A., Labadie, L., et al. 2022b, arXiv e-prints, arXiv:2210.13095
  * Gray & Corbally (1998) Gray, R. O. & Corbally, C. J. 1998, AJ, 116, 2530
  * Guimarães et al. (2006) Guimarães, M. M., Alencar, S. H. P., Corradi, W. J. B., & Vieira, S. L. A. 2006, A&A, 457, 581
  * Hartmann et al. (2016) Hartmann, L., Herczeg, G., & Calvet, N. 2016, ARA&A, 54, 135
  * Hofmann et al. (2022) Hofmann, K. H., Bensberg, A., Schertl, D., et al. 2022, A&A, 658, A81
  * Hollenbach et al. (2000) Hollenbach, D. J., Yorke, H. W., & Johnstone, D. 2000, Protostars and Planets IV, 401
  * Hone et al. (2019) Hone, E., Kraus, S., Davies, C. L., et al. 2019, A&A, 623, A38
  * Hone et al. (2017) Hone, E., Kraus, S., Kreplin, A., et al. 2017, A&A, 607, A17
  * Hubrig et al. (2015) Hubrig, S., Carroll, T. A., Schöller, M., & Ilyin, I. 2015, MNRAS, 449, L118
  * Hubrig et al. (2011) Hubrig, S., Schöller, M., Ilyin, I., et al. 2011, A&A, 536, A45
  * Ilee et al. (2014) Ilee, J. D., Fairlamb, J., Oudmaijer, R. D., et al. 2014, MNRAS, 445, 3723
  * Jacquemin-Ide et al. (2019) Jacquemin-Ide, J., Ferreira, J., & Lesur, G. 2019, MNRAS, 490, 3112
  * Jacquemin-Ide et al. (2021) Jacquemin-Ide, J., Lesur, G., & Ferreira, J. 2021, A&A, 647, A192
  * Kharchenko et al. (2007) Kharchenko, N. V., Scholz, R. D., Piskunov, A. E., Roeser, S., & Schilbach, E. 2007, VizieR Online Data Catalog, III/254
  * Kley & Lin (1996) Kley, W. & Lin, D. N. C. 1996, ApJ, 461, 933
  * Kluska et al. (2020) Kluska, J., Berger, J. P., Malbet, F., et al. 2020, A&A, 636, A116
  * Kobus et al. (2020) Kobus, J., Wolf, S., Ratzka, T., & Brunngräber, R. 2020, A&A, 642, A104
  * Kraus et al. (2008) Kraus, S., Hofmann, K., Benisty, M., et al. 2008, A&A, 489, 1157
  * Kreplin et al. (2018) Kreplin, A., Tambovtseva, L., Grinin, V., et al. 2018, MNRAS, 476, 4520
  * Kunitomo et al. (2021) Kunitomo, M., Ida, S., Takeuchi, T., et al. 2021, ApJ, 909, 109
  * Kurosawa et al. (2006) Kurosawa, R., Harries, T. J., & Symington, N. H. 2006, MNRAS, 370, 580
  * Kurosawa et al. (2016) Kurosawa, R., Kreplin, A., Weigelt, G., et al. 2016, MNRAS, 457, 2236
  * Lachaume (2003) Lachaume, R. 2003, A&A, 400, 795
  * Lapeyrere et al. (2014) Lapeyrere, V., Kervella, P., Lacour, S., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9146, Optical and Infrared Interferometry IV, ed. J. K. Rajagopal, M. J. Creech-Eakman, & F. Malbet, 91462D
  * Lazareff et al. (2017) Lazareff, B., Berger, J.-P., Kluska, J., et al. 2017, A&A, 599, A85
  * Le Bouquin et al. (2009) Le Bouquin, J. B., Absil, O., Benisty, M., et al. 2009, A&A, 498, L41
  * Lesur et al. (2022) Lesur, G., Ercolano, B., Flock, M., et al. 2022, arXiv e-prints, arXiv:2203.09821
  * Lynden-Bell & Pringle (1974) Lynden-Bell, D. & Pringle, J. E. 1974, MNRAS, 168, 603
  * Marcos-Arenal et al. (2021) Marcos-Arenal, P., Mendigutía, I., Koumpia, E., et al. 2021, A&A, 652, A68
  * Menu et al. (2015) Menu, J., van Boekel, R., Henning, T., et al. 2015, A&A, 581, A107
  * Millan-Gabet et al. (2007) Millan-Gabet, R., Malbet, F., Akeson, R., et al. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 539
  * Miroshnichenko (2017) Miroshnichenko, A. S. 2017, in Astronomical Society of the Pacific Conference Series, Vol. 508, The B[e] Phenomenon: Forty Years of Studies, ed. A. Miroshnichenko, S. Zharikov, D. Korčáková, & M. Wolf, 285
  * Miroshnichenko et al. (2001) Miroshnichenko, A. S., Levato, H., Bjorkman, K. S., & Grosso, M. 2001, A&A, 371, 600
  * Monnier (2007) Monnier, J. D. 2007, New A Rev., 51, 604
  * Monnier & Millan-Gabet (2002) Monnier, J. D. & Millan-Gabet, R. 2002, ApJ, 579, 694
  * Montesinos et al. (2009) Montesinos, B., Eiroa, C., Mora, A., & Merín, B. 2009, A&A, 495, 901
  * Muzerolle et al. (2004) Muzerolle, J., D’Alessio, P., Calvet, N., & Hartmann, L. 2004, ApJ, 617, 406
  * Natta et al. (2004) Natta, A., Testi, L., Muzerolle, J., et al. 2004, A&A, 424, 603
  * Nisini et al. (2018) Nisini, B., Antoniucci, S., Alcalá, J. M., et al. 2018, A&A, 609, A87
  * Pascucci et al. (2022) Pascucci, I., Cabrit, S., Edwards, S., et al. 2022, arXiv e-prints, arXiv:2203.10068
  * Patel et al. (2017) Patel, P., Sigut, T. A. A., & Landstreet, J. D. 2017, ApJ, 836, 214
  * Picogna et al. (2021) Picogna, G., Ercolano, B., & Espaillat, C. C. 2021, MNRAS, 508, 3611
  * Safier (1993) Safier, P. N. 1993, ApJ, 408, 115
  * Sanchez-Bermudez (2018) Sanchez-Bermudez, J. 2018, in Imaging of Stellar Surfaces, 32
  * Sanchez-Bermudez et al. (2018) Sanchez-Bermudez, J., Millour, F., Baron, F., et al. 2018, Experimental Astronomy, 46, 457
  * Suzuki & Inutsuka (2009) Suzuki, T. K. & Inutsuka, S.-i. 2009, ApJ, 691, L49
  * Takasao et al. (2018) Takasao, S., Tomida, K., Iwasaki, K., & Suzuki, T. K. 2018, ApJ, 857, 4
  * Tambovtseva et al. (2014) Tambovtseva, L. V., Grinin, V. P., & Weigelt, G. 2014, A&A, 562, A104
  * Tambovtseva et al. (2016) Tambovtseva, L. V., Grinin, V. P., & Weigelt, G. 2016, A&A, 590, A97
  * Varga et al. (2021) Varga, J., Hogerheijde, M., van Boekel, R., et al. 2021, A&A, 647, A56
  * Vink (2015) Vink, J. S. 2015, Ap&SS, 357, 98
  * Vink et al. (2002) Vink, J. S., Drew, J. E., Harries, T. J., & Oudmaijer, R. D. 2002, MNRAS, 337, 356
  * Vioque et al. (2018) Vioque, M., Oudmaijer, R. D., Baines, D., Mendigutía, I., & Pérez-Martínez, R. 2018, A&A, 620, A128
  * Weigelt et al. (2011) Weigelt, G., Grinin, V. P., Groh, J. H., et al. 2011, A&A, 527, A103
  * Weigelt et al. (2007) Weigelt, G., Kraus, S., Driebe, T., et al. 2007, A&A, 464, 87
  * Whelan & Garcia (2008) Whelan, E. & Garcia, P. 2008, in Jets from Young Stars II, ed. F. Bacciotti, L. Testi, & E. Whelan, Vol. 742, 123
  * Wichittanakom et al. (2020) Wichittanakom, C., Oudmaijer, R. D., Fairlamb, J. R., et al. 2020, MNRAS, 493, 234
  * Zhu & Stone (2018) Zhu, Z. & Stone, J. M. 2018, ApJ, 857, 34

##  Appendix A Log of the observations

Table 3: Observation log of VLTI-GRAVITY observations.

HD | Date | Configuration |  $\lambda$/PBLmax | N | Calibrator
---|---|---|---|---|---
37806 | 2019-03-19 | D0-G2-J3-K0 | 4.7 | 5 | HD 64215
| 2020-01-22 | A0-G1-J2-K0 | 3.6 | 10 | HD 31623
38120 | 2019-03-18 | D0-G2-J3-K0 | 5.1 | 8 | HD 37356, HD 38225
45677 | 2020-01-27 | A0-G2-J2-J3 | 3.5 | 7 | HD 43847, HD 49794
| 2020-01-29 | K0-G2-D0-J3 | 4.5 | 6 | HD 44423
| 2020-02-04 | A0-B2-D0-C1 | 14.2 | 8 | HD 43847, HD 45420
58647 | 2020-01-27 | A0-G1-J2-K0 | 3.5 | 9 | HD 57939, HD 60325
| 2020-01-28 | A0-G2-J2-J3 | 4.4 | 7 | HD 60325
| 2020-02-04 | A0-B2-D0-C1 | 13.7 | 12 | HD 65810
85567 | 2021-02-09 | A0-G1-J2-J3 | 3.8 | 18 | HD 85313
| 2021-02-10 | A0-G1-J2-J3 | 3.5 | 9 | HD 85313
| 2021-02-20 | A0-B2-D0-C1 | 15.6 | 8 | HD 85313
95881 | 2018-03-02 | D0-G2-J3-K0 | 5.2 | 5 | HD 90452
97048 | 2017-03-20 | A0-G1-J2-K0 | 3.5 | 6 | HD 82554
98922 | 2019-05-24 | A0-G1-J2-J3 | 3.7 | 6 | HD 103125, HD 141977
| 2020-01-27 | A0-G2-J2-J3 | 3.5 | 5 | HD 103125, HD 99311
| 2020-02-04 | A0-B2-D0-C1 | 14.6 | 11 | HD 103125
100546 | 2020-01-28 | K0-G2-D0-J3 | 5.4 | 7 | HD 101531, HD 99556
114981 | 2019-03-19 | D0-G2-J3-K0 | 4.4 | 6 | HD 113776
135344B | 2018-03-05 | A0-G1-J2-J3 | 3.5 | 8 | HD 132763
139614 | 2019-03-19 | D0-G2-J3-K0 | 6.6 | 3 | HD 148974
141569 | 2019-07-12 | D0-G2-J3-K0 | 4.5 | 8 | HD 137006
142527 | 2017-03-18 | A0-G1-J2-K0 | 3.5 | 5 | HD 143118
142666 | 2018-06-16 | D0-G2-J3-K0 | 4.4 | 9 | HD 148605
144432 | 2018-03-05 | A0-G1-J2-J3 | 3.4 | 5 | HD 132763
144668 | 2017-03-19 | A0-G1-J2-K0 | 3.5 | 7 | HD 143118
145718 | 2019-03-18 | D0-G2-J3-K0 | 4.4 | 8 | HD 145809
150193 | 2018-06-15 | D0-G2-J3-K0 | 4.7 | 7 | HD 148605
| 2018-07-07 | D0-G2-J3-K0 | 5.0 | 3 | HD 181240
| 2019-06-05 | A0-B2-D0-C1 | 15.1 | 14 | HD 148974
158643 | 2017-05-29 | B2-D0-J3-K0 | 3.4 | 6 | HD 163955
| 2017-05-30 | B2-D0-J3-K0 | 3.4 | 14 | HD 163955
| 2017-05-31 | B2-D0-J3-K0 | 3.7 | 12 | HD 163955
| 2017-08-15 | A0-G1-J2-K0 | 4.8 | 7 | HD 163955
163296 | 2019-05-24 | A0-G1-J2-J3 | 3.5 | 4 | HD 163495
| 2019-06-05 | A0-B2-D0-C1 | 18.9 | 13 | HD 148974
| 2019-07-14 | D0-G2-J3-K0 | 5.0 | 7 | HD 172052
169142 | 2019-05-24 | A0-G1-J2-J3 | 3.4 | 6 | HD 169830
179218 | 2018-07-07 | D0-G2-J3-K0 | 5.3 | 9 | HD 181240
190073 | 2018-06-15 | D0-G2-J3-K0 | 4.4 | 8 | HD 183936
259431 | 2018-03-05 | A0-G1-J2-J3 | 3.5 | 12 | HD 43386, HD 50277
| 2019-01-13 | D0-G2-J3-K0 | 5.0 | 10 | HD 262137
V1818 Ori | 2019-03-19 | D0-G2-J3-K0 | 4.4 | 6 | HD 34045
777 ( Column 1: HD name of the target. Column 2: Date of the observations.
Column 3: Configuration of the observations. Column 4: $\lambda$/PBLmax in
units of mas; PBLmax is the maximum projected baseline available for that
configuration. Column 5: Number of five-minute long files that have been
recorded on the target. Column 6: Name of the calibrator.)

##  Appendix B GRAVITY data

The VLTI-GRAVITY observations of the full sample around the wavelength of the
Br$\gamma$ line are shown from Fig. 11 to Fig. 32. Each set of figures shows
the results for a different star and date, as labelled in the top-left panel.
There are three columns in each set of figures showing the visibility
amplitude (left), differential phase (middle), and closure phase (right). The
top panels show the observed (solid line) and the photospheric-corrected
spectrum (dashed blue line), which is the same in all cases. The length and PA
of the projected baselines (PBL) are given in the left panels. For reference,
the vertical dashed line indicates the location of the zero velocity channel
with respect to the local standard of rest.

Figure 11: VLTI-GRAVITY HR interferometric observables for the star HD 37806.
Each set of figures shows observations for a different date, indicated in the
top-left panel. The three columns in each figure show the visibility amplitude
(left), differential phase (middle), and closure phase (right). As reference,
the observed (black solid line) and photospheric corrected (dahsed blue line)
Br$\gamma$ line profiles are shown at the top of each panel. Vertical dashed
lines indicate the location of the zero velocity with respect to the local
standard of rest.

Figure 12: Same as Fig. 11 but for HD 38120 and HD 45677.

Figure 13: Same as Fig. 11 but for HD45677.

Figure 14: Same as Fig. 11 but for HD58647.

Figure 15: Same as Fig. 11 but for HD58647 and HD85567.

Figure 16: Same as Fig. 11 but for HD85567.

Figure 17: Same as Fig. 11 but for HD 95881 and HD 97048.

Figure 18: Same as Fig. 11 but for HD 98922.

Figure 19: Same as Fig. 11 but for HD 98922 and HD 100546.

Figure 20: Same as Fig. 11 but for HD 114981 and HD 135344B.

Figure 21: Same as Fig. 11 but for HD 139614 and HD 141569.

Figure 22: Same as Fig. 11 but for HD 142527 and HD 142666.

Figure 23: Same as Fig. 11 but for HD 144432 and HD 144668.

Figure 24: Same as Fig. 11 but for HD 145718 and HD 150193.

Figure 25: Same as Fig. 11 but for HD 150193.

Figure 26: Same as Fig. 11 but for HD 158643.

Figure 27: Same as Fig. 11 but for HD 158643.

Figure 28: Same as Fig. 11 but for HD 163296.

Figure 29: Same as Fig. 11 but for HD 163296 and HD 169142.

Figure 30: Same as Fig. 11 but for HD 179218 and HD 190073.

Figure 31: Same as Fig. 11 but for HD 259431.
Figure 32: Same as Fig. 11 but for V1818 Ori.

##  Appendix C Fits to continuum-subtracted visibilities

Figure 33: Continuum-subtracted Br$\gamma$ line visibilities for HD 37806, HD
38120, HD 45567, HD 58647, HD 85567, HD 95881, HD 97048, and HD 98922 as a
function of the projected baseline (blue filled points). The best fit to the
visibilities is overplotted as a red solid line (see Sect. 4.3 and Table 2 for
details).

Figure 34: Same as Fig. 33 but for HD 114981, HD 141569, HD 150193, HD 163296,
HD 179218, HD 190073, and HD 259431.