11institutetext:  Instituto de Física y Astronomía, Facultad de Ciencias,
Universidad de Valparaíso, Av. Gran Bretaña 1111, Playa Ancha, Valparaíso,
Chile
11email<EMAIL_ADDRESS>22institutetext: Núcleo Milenio Formación
Planetaria - NPF, Universidad de Valparaíso, Av. Gran Bretaña 1111,
Valparaíso, Chile 33institutetext: Departamento de Ciencias Básicas,
Universidad Viña del Mar, Viña del Mar, Chile 44institutetext: Max Planck
Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany
55institutetext: Leiden Observatory, Leiden University, PO Box 9513, 2300 RA,
Leiden, The Netherlands 66institutetext: European Southern Observatory, Alonso
de Córdova 3107, Casilla, 19001, Vitacura, Santiago, Chile 77institutetext:
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ.
Paris Diderot,
Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
88institutetext: Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France
99institutetext: Geneva Observatory, University of Geneva, Chemin des
Mailettes 51, 1290 Versoix, Switzerland 1010institutetext: Konkoly
Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege
Miklós út 15-17, H-1121 Budapest, Hungary 1111institutetext: Institute of
Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
1212institutetext: Anton Pannekoek Institute for Astronomy, Science Park 904,
NL-1098 XH Amsterdam, The Netherlands 1313institutetext: DOTA, ONERA,
Université Paris Saclay, F-91123, Palaiseau France 1414institutetext: Núcleo
de Astronomía, Facultad de Ingeniería, Universidad Diego Portales, Avenida
Ejercito 441, Santiago, Chile 1515institutetext: Aix Marseille Université,
CNRS, LAM - Laboratoire d’Astrophysique de Marseille, UMR 7326, 13388,
Marseille, France 1616institutetext: CRAL, UMR 5574, CNRS, Université de Lyon,
Ecole Normale Supérieure de Lyon, 46 Allée d’Italie, F-69364 Lyon Cedex 07,
France 1717institutetext: Escuela de Ingeniería Industrial, Facultad de
Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejercito 441, Santiago,
Chile

# Characterizing the morphology of the debris disk around the low-mass star
GSC 07396-00759††thanks: Based on SPHERE observations made with the Very Large
Telescope of the European Southern Observatory. Program ID: 1100.C-0481(R).

C. Adam Characterizing the morphology of the debris disk around the low-mass
star GSC 07396-00759††thanks: Based on SPHERE observations made with the Very
Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R). J. Olofsson Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R). R. G. van Holstein Characterizing the morphology
of the debris disk around the low-mass star GSC 07396-00759††thanks: Based on
SPHERE observations made with the Very Large Telescope of the European
Southern Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology
of the debris disk around the low-mass star GSC 07396-00759††thanks: Based on
SPHERE observations made with the Very Large Telescope of the European
Southern Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology
of the debris disk around the low-mass star GSC 07396-00759††thanks: Based on
SPHERE observations made with the Very Large Telescope of the European
Southern Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology
of the debris disk around the low-mass star GSC 07396-00759††thanks: Based on
SPHERE observations made with the Very Large Telescope of the European
Southern Observatory. Program ID: 1100.C-0481(R). A. Bayo Characterizing the
morphology of the debris disk around the low-mass star GSC
07396-00759††thanks: Based on SPHERE observations made with the Very Large
Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R). J. Milli Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R). A. Boccaletti Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R). Q. Kral Characterizing the morphology
of the debris disk around the low-mass star GSC 07396-00759††thanks: Based on
SPHERE observations made with the Very Large Telescope of the European
Southern Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology
of the debris disk around the low-mass star GSC 07396-00759††thanks: Based on
SPHERE observations made with the Very Large Telescope of the European
Southern Observatory. Program ID: 1100.C-0481(R). C. Ginski Characterizing the
morphology of the debris disk around the low-mass star GSC
07396-00759††thanks: Based on SPHERE observations made with the Very Large
Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R). Th. Henning Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R). M. Montesinos Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R). N. Pawellek Characterizing the
morphology of the debris disk around the low-mass star GSC
07396-00759††thanks: Based on SPHERE observations made with the Very Large
Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R). A. Zurlo Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R). M. Langlois Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R). A. Delboulbé Characterizing the
morphology of the debris disk around the low-mass star GSC
07396-00759††thanks: Based on SPHERE observations made with the Very Large
Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R). A. Pavlov Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R). J. Ramos Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R). L. Weber Characterizing the
morphology of the debris disk around the low-mass star GSC
07396-00759††thanks: Based on SPHERE observations made with the Very Large
Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R). F. Wildi Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R).Characterizing the morphology of the debris disk
around the low-mass star GSC 07396-00759††thanks: Based on SPHERE observations
made with the Very Large Telescope of the European Southern Observatory.
Program ID: 1100.C-0481(R). F. Rigal Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R).Characterizing the morphology of the
debris disk around the low-mass star GSC 07396-00759††thanks: Based on SPHERE
observations made with the Very Large Telescope of the European Southern
Observatory. Program ID: 1100.C-0481(R). J.-F. Sauvage Characterizing the
morphology of the debris disk around the low-mass star GSC
07396-00759††thanks: Based on SPHERE observations made with the Very Large
Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).Characterizing the morphology of the debris disk around the
low-mass star GSC 07396-00759††thanks: Based on SPHERE observations made with
the Very Large Telescope of the European Southern Observatory. Program ID:
1100.C-0481(R).

(Received Month DD, YYYY; accepted July 05, 2021)

###### Abstract

Context.  Debris disks have commonly been studied around intermediate-mass
stars. Their intense radiation fields are believed to efficiently remove the
small dust grains that are constantly replenished by collisions. For lower-
mass central objects, in particular M-stars, the dust removal mechanism needs
to be further investigated given the much weaker radiation field produced by
these objects.

Aims.  We present new observations of the nearly edge-on disk around the pre-
main sequence M-type star GSC~07396-00759, taken with VLT/SPHERE IRDIS in
Dual-beam Polarimetric Imaging (DPI) mode, with the aim to better understand
the morphology of the disk, its dust properties, and the star-disk interaction
via the stellar mass-loss rate.

Methods.  We model the polarimetric observations to characterize the location
and properties of the dust grains using the Henyey-Greenstein approximation of
the polarized phase function. We use the estimated phase function to evaluate
the strength of the stellar winds.

Results.  We find that the polarized light observations are best described by
an extended and highly inclined disk ($i\approx 84.3\,\mbox{${}^{\circ}$}\pm
0.3$) with a dust distribution centered at a radius $r_{0}\approx 107\pm 2$
au. Our modeling suggests an anisotropic scattering factor $g\approx 0.6$ to
best reproduce the polarized phase function $S_{12}$. We also find that the
phase function is reasonably reproduced by small micron-sized dust grains with
sizes $s>0.3$ $\mathrm{\SIUnitSymbolMicro m}$. We discuss some of the caveats
of the approach, mainly that our model probably does not fully recover the
semi-major axis of the disk and that we cannot readily determine all dust
properties due to a degeneracy between the grain size and the porosity.

Conclusions.  Even though the radius of the disk may be over-estimated, our
best fit model not only reproduces well the observations but is also
consistent with previous published data obtained in total intensity. Similarly
to previous studies of debris disks, we suggest that using a given scattering
theory might not be sufficient to fully explain key aspects such as the shape
of the phase function, or the dust grain size. Taking into consideration the
aforementioned caveats, we find that the average mass-loss rate of
GSC~07396-00759 can be up to 500 times stronger than that of the Sun,
supporting the idea that stellar winds from low-mass stars can evacuate small
dust grains in an efficient way.

###### Key Words.:

Stars: individual: GSC~07396-00759 – stars: winds, outflows – stars:
circumstellar matter – Techniques: high angular resolution – techniques:
polarimetric

##  1 Introduction

Debris disks are the natural by-product of the planet formation process. They
are second generation, dusty circumstellar disks created by collisions of
planetesimals that have formed in previously existing planet-forming disks
(Kral et al. 2018; Hughes et al. 2018). This process releases a large amount
of micron-sized dust grains, whose presence either can be inferred through the
infrared excess over the photospheric level of their host star, or from
spatially resolved scattered (linearly polarized) light observations in the
optical or near-infrared. The average lifetime of those small dust grains,
however is much shorter than the typical age of the star, and thus they have
to be replenished continuously from the larger bodies. The most efficient
process to remove these micron-sized dust grains from the system is usually
the pressure exerted by the radiation field of the central star (e.g., Krivov
2010). For low-mass stars, however, the irradiation field generally is much
weaker, while stellar winds can be strong (see e.g., Reidemeister et al.
2011). Thus stellar winds are most likely the dominant, yet poorly
constrained, mechanism to drive the rapid removal of particles. The question
remains poorly addressed because debris disks around low mass stars are rarely
detected.

Debris disks are found around about 20% of A-type stars (Su et al. 2006; Eiroa
et al. 2013; Chen et al. 2014; Matthews et al. 2014), mostly detected through
the excess emission in the far-infrared. Similar to the searches targeted at
solar-type and more massive stars, several large surveys have been conducted
to search for cold debris disks around M dwarfs in the mid- and far-infrared
(Gautier et al. 2007; Avenhaus et al. 2012; Kennedy et al. 2018) as well as
the sub-millimeter domain (Lestrade et al. 2006, 2009). These and other
surveys yielded numerous debris disk candidates around young M-stars (for
details see e.g. Luppe et al. 2020, and references therein). However, up to
now, only 5 disks around young M-stars have been confirmed based on more than
one independent observation, such as spatially resolved imaging (either in
scattered light and/or the sub-mm to mm domain): AU Mic (Kalas et al. 2004;
Boccaletti et al. 2015), TWA 7 (Choquet et al. 2016; Olofsson et al. 2018;
Bayo et al. 2019; Esposito et al. 2020; Ren et al. 2021), TWA 25 (Choquet et
al. 2016), GJ 581 (Lestrade et al. 2012),Fomalhaut C (LP 876-10, Cronin-
Coltsmann et al. 2021), and, GSC 07396–00759 (Sissa et al. 2018). Given the
low number of spatially resolved detections, we still know very little about
debris disks around low-mass stars, and yet several of these disks show very
interesting and peculiar features, such as possible spiral-like structures and
even gas around TWA 7 (Olofsson et al. 2018; Matrà et al. 2019), or fast-
moving arc-like structures around AU Mic (Boccaletti et al. 2018). Debris
disks around low-mass stars are also very interesting targets, especially with
respect to the possible connection between their occurrence and the possible
presence of planets (Raymond et al. 2011). Although only about 2% of the M
stars have been found to host giant planets (Johnson et al. 2007), rocky
planets appear to be more frequently detected around low-mass stars, as
pointed out by Dressing & Charbonneau (2015), who estimated a cumulative
planet occurrence rate of $2.5\pm 0.2$ planets per M dwarf with radii
$1-4\,R_{\oplus}$ and periods shorter than 200 days.

However, the reason for the paucity of debris disk detections around low-mass
M-type stars remains unclear (see e.g. Luppe et al. 2020), since stars of all
spectral types appear to have a similar detection frequency as protoplanetary
disks in the earlier stages of their evolution (see e.g. Andrews & Williams
2005). In fact, studies of the Lamba Orionis star-forming region carried out
on a wide set of spectroscopically confirmed members of the central star
cluster Collinder 69 (${\sim}$5–12 Myr) by Bayo et al. (2012), or the ALMA 887
$\mathrm{\SIUnitSymbolMicro m}$ survey of the disk population in the nearby 2
Myr-old Chamaeleon I star-forming region, conducted by Pascucci et al. (2016),
indicate that disks around young, low-mass stars with
$\mbox{$M_{\star}$}\lesssim 0.6$ $M_{\odot}$ are in fact more frequent than
those around higher-mass hosts. Therefore it is more likely that current
observations may simply not be sensitive enough because dust experiences
significantly less heating around low-luminosity M dwarfs than around more
massive stars. Consequently, these colder disks are significantly more
difficult to detect at the typical wavelengths (e.g. 24 – 160
$\mathrm{\SIUnitSymbolMicro m}$) that were used to build statistics of debris
disks, making the excess emission hard to detect in the spectral energy
distribution (SED) of these stars (Morey & Lestrade 2014). This might increase
the chances to miss potential targets when compiling a sample for an
observation/survey, and better suited alternatives are needed.

One alternative, as mentioned, are high angular resolution imaging
observations in total intensity or polarized light, at optical or near-
infrared wavelengths. Pioneered by Hubble Space Telescope observations (e.g.,
Graham et al. 2007; Stark et al. 2014), the availability and the recent
advances in high contrast imaging instruments, such as the Gemini Planet
Imager (GPI, Macintosh et al. 2006) or the Very Large Telescope/Spectro-
Polarimetric High-contrast Exoplanet REsearch (VLT/SPHERE; Beuzit et al. 2019)
provide direct access to the absorption and scattering properties of the
grains and have opened new avenues to detect, resolve, and investigate debris
disks at high angular resolution. Studying how stellar light is scattered off
of the dust grains, either using the color of the disk between different bands
(e.g., Debes et al. 2008; Rodigas et al. 2015), or through studying the phase
function over a wide range of scattering angles, i.e. the angle between the
star, the dust grain and the observer (e.g., Olofsson et al. 2016; Milli et
al. 2017, 2019; Ren et al. 2019) allows better constraints to be put on the
properties of the dust such as their typical grain sizes as well as their
porosity, shape, and composition.

The target of our study is GSC~07396-00759, a young, nearby (71.43$\pm$0.26
pc, Gaia Collaboration et al. 2018) M1-type star. The star, classified as
weak-line T-Tauri star (Kastner et al. 2011), is probably a member of the
$\beta$ Pictoris Moving Group ($\beta$ Pic MG, $\approx 18\pm 2$ Myr, Miret-
Roig et al. 2020), which is known to harbor numerous debris disk host stars
like AU Mic, or CP-72 2713 (Moór et al. 2020). Total intensity observations of
GSC~07396-00759 obtained by Sissa et al. (2018) have revealed an extended
($r_{0}=70$ au) and nearly edge-on disk ($i=83$∘), probably containing sub-
micron-sized grains, a possible indicator of a strong interaction of the
stellar radiation field with the disk. By comparing their best-fitting model
with the observed SED of GSC~07396-00759 up to 22 $\mathrm{\SIUnitSymbolMicro
m}$ (WISE/W4, ¡4.49 mJy Cutri & et al. 2014), the longest wavelength available
at this point, Sissa et al. (2018) estimated an upper limit for the dust mass
in the disk around GSC~07396-00759 of
$M_{\mathrm{dust}}{\sim}$0.33\text{\,}\mathrm{M}_{\oplus}$$, corresponding to
an upper limit for the fractional luminosity
$L_{\mathrm{disk}}/L_{\star}\leqslant 4\times 10^{-3}$ due to the lack of far-
IR photometry data points. The disk furthermore displays a small brightness
asymmetry, swept-back wings (warps) as well as ripples in the spine of the
disk on both sides of the disk. The available proper motion and radial
velocity data suggest that GSC~07396-00759 is also likely associated with
V4046 Sgr AB, forming a loosely bound hierarchical multiple system
($a_{\mathrm{proj.}}\approx 0.06$ pc, Torres et al. 2006; Kastner et al. 2011;
Sissa et al. 2018). V4046 Sgr AB itself a close binary with accretion
signatures (Stempels & Gahm 2004) and a gas-rich circumbinary disk (Rosenfeld
et al. 2013; Rapson et al. 2015).

In this paper, we aim to compare newly obtained near-infrared polarimetric
observations at high angular resolution of GSC~07396-00759 with radiative
transfer modeling to study the morphology of the disk, and to probe dust
properties under the influence of radiation pressure and stellar winds, very
rare constraints for low mass stars, and determined to date only for a few
debris disks, such as AU Mic (Augereau & Beust 2006), or $\varepsilon$ Eri
(Reidemeister et al. 2011). In Sect. 2, we describe the observations obtained
with the SPHERE/ZIMPOL instrument at near-infrared wavelengths. The model used
to analyze the observations is presented in Sect. 3, followed by the results
reported in Sect. 4 and their discussion in Sec. 5. We conclude with a short
summary and our conclusions in Sect. 6.

##  2 Near-infrared polarimetric imaging

###  2.1 Observations and data reduction

Table 1: Summary of the SPHERE-IRDIS observations for GSC~07396-00759.

Date | DIT | $N_{\mathrm{tot}}$ | $N_{\mathrm{pol}}$ | $t_{\mathrm{exp}}$ | Filter | $\tau_{0}$ | Seeing | Am | Strehl
---|---|---|---|---|---|---|---|---|---
(UTC) | (s) |  |  | (s) |  | (ms) | (″) |  | ($H$)
2018 Jun 22 | 64.0 | 48 | 12 | 3072 | $H$ | 5.3 | 0.84 | 1.31 | 0.67
111The average DIMM-seeing was measured at $\lambda=500$ nm. The $H$-band
Strehl ratio was measured from the observed IRDIS flux image for
GSC~07396-00759.

The observation of GSC~07396-00759 took place on 2018 June 22 (UTC, Programme
ID: 1100.C-0481(R), P.I.: J.L. Beuzit) as part of the SPHERE Guaranteed Time
Observations. The data were obtained with the SPHERE InfraRed Dual-band Imager
and Spectrograph (IRDIS, pixel scale of 12.25 mas, ${\sim}11\arcsec$ $\times$
11″ field of view; FoV, Dohlen et al. 2008) using the field-stabilized, dual-
beam polarimetric imaging mode (DPI, Langlois et al. 2014; de Boer et al.
2020; van Holstein et al. 2020), and employing the $H$-band ($BB\\_H$) filter
with a central wavelength $\lambda_{\mathrm{c}}$=1625 nm and a width
$\Delta\lambda$=290 nm. To further increase the contrast the Apodized Lyot
Coronagraph (mask diameter: 185 mas, Carbillet et al. 2011; Guerri et al.
2011) was used to mask the central star (Hmag=8.76, Rmag=12.01, Cutri et al.
2003; Zacharias et al. 2017). This allows for an efficient suppression of the
stellar light which is assumed to be only marginally linearly polarized, while
keeping the scattered, i.e. polarized, light from the circumstellar disk
relatively unaffected.

Each polarimetric observation consists of a set of four linear-polarization
components, called Stokes $Q^{+}$, $Q^{-}$, $U^{+}$, and $U^{-}$, that are
obtained by subtracting the two beams with orthogonal polarization states
recorded simultaneously on the detector and tuning their polarization
direction with a half-wave plate (HWP) with positions of 0°, 45°, 22.5°, and
67.5°, respectively. We took one exposure for each of the Stokes $Q^{+}$,
$Q^{-}$, $U^{+}$, and $U^{-}$ components, each with a detector integration
time (DIT) of 64 s. The polarization cycle of $Q^{+}$, $Q^{-}$, $U^{+}$, and
$U^{-}$ was then repeated twelve ($N_{\mathrm{pol}}$) times, adding up to a
total of 48 exposures with a total integration time ($t_{\mathrm{exp}}$) of
51.2 minutes. An overview of our observations is presented in Table 1. We also
list the observing conditions at the time of observation, such as the average
coherence time ($\tau_{0}$), the average seeing, estimated from the
Differential Image Motion Monitor (DIMM), the airmass (Am) and the Strehl
ratio in the $H$-band, measured from the flux image.

Since the star is obscured by the coronagraphic mask in the science images, a
flux and center calibration frame were taken in addition to the science
observations. The flux calibration frames were taken with the central star
moved away from the coronagraphic mask. The center calibration frames were
taken after the star was aligned behind the coronagraphic mask and with the
deformable mirror system (SAXO, Fusco et al. 2006) used to introduce a waffle
pattern to create equidistant calibration spots outside of the coronagraphic
mask. This allows us to accurately determine the stellar position behind the
coronagraph from the center image while we compute the Strehl ratio in the
$H$-band for GSC~07396-00759 directly from the flux image with the star off-
centered from the coronagraph. The $H$-band Strehl ratios estimated by the
adaptive optics system during the observation are recorded in separate FITS
files222“Classified as OBJECT, AO” in the ESO data archive (called GEN-SPARTA
data). From these data we find Strehl ratios for GSC~07396-00759 ranging from
0.6 to 0.77, and in good agreement with our measured Strehl ratio of 0.67 from
the flux image.

The data were reduced using the IRDAP333https://irdap.readthedocs.io (IRDIS
Data reduction for Accurate Polarimetry, van Holstein et al. 2020; de Boer et
al. 2020) pipeline. IRDAP is a dedicated pipeline for the reduction of
polarimetric data obtained with IRDIS, capable of differentiating and
correcting instrumental and stellar polarization. For a detailed description
of the reduction procedure and a discussion of the applied corrections we
refer the reader to van Holstein et al. (2020) and de Boer et al. (2020). The
reduction can be summarized as follows.

After applying the standard calibration routines, including sky-frame
subtraction, flat-fielding and bad-pixel correction, the images are split into
two individual frames representing the left and right sides of the IRDIS
detector, corresponding to the parallel and perpendicular polarized beams,
respectively. Then, the precise position of the central star is measured using
the star center calibration frames on both image sides separately, and the
right side of the image is shifted to a common center and subtracted from the
left side. The pipeline then applies the double-difference method (see, e.g.
Tinbergen 1996) to obtain the linear Stokes parameters $Q$ and $U$ corrected
for instrumental polarization created downstream of the HWP and the
corresponding total-intensity images computed with the double sum,
respectively. However, the double difference does not remove instrumental
polarization caused by the telescope and instrument mirrors upstream from the
HWP which is assumed to be proportional to the total intensity image, as shown
in Canovas et al. (2011), nor does it remove the most important cross talk
contributions (de Boer et al. 2020). IRDAP uses a Mueller matrix model to
determine the polarimetric response function for the polarimetric imager (de
Boer et al. 2020) and to correct for these instrumental polarization effects.
This model describes the complete optical path of SPHERE/IRDIS, i.e.,
telescope and instrument, and has been fully validated with measurements using
SPHERE’s internal source and observations of unpolarized standard stars (van
Holstein et al. 2020). The images of Stokes $Q$ and $U$ incident on the
telescope are computed by setting up a system of equations describing every
measurement of $Q$ and $U$ and solving it – for every pixel individually–
using linear least-squares. The $Q$ and $U$ images thus created, however, may
still contain some stellar polarization that is constrained by measuring the
flux in the $Q$ and $U$ images around regions that should be virtually devoid
of polarized signal from the disk. The final product of the reduction pipeline
are the images of the azimuthal Stokes parameter $Q_{\phi}$ and $U_{\phi}$
(for definition see de Boer et al. 2020), where $Q_{\phi}>0$ is equivalent to
a azimuthal polarization component (with respect to the position of the star),
$Q_{\phi}<0$ to a radial component, and $\pm U_{\phi}$ signal is equivalent to
polarization angles oriented at $\pm 45$∘with respect to the azimuthal
component, respectively.

###  2.2 Observational results

Figure 1: SPHERE/IRDIS polarimetric imaging observations of GSC~07396-00759.
Shown from left to right are the total intensity $I_{\mathrm{tot}}$ (log
stretch) as well as the $Q_{\phi}$ and $U_{\phi}$ images (both in linear
stretch). The $Q_{\phi}$ and $U_{\phi}$ images were convolved with a Gaussian
PSF ($\sigma=2\,$pixel) to increase the visibility in this plot. The
coronagraphic mask is indicated by the green circular region in each panel.
North is to the top and east to the left in each panel. Please note that the
disk is not visible in the total intensity image (left panel). The structure
appearing to extend from the NW to the SE is a PSF artifact. Figure 2: Signal-
to-noise map estimated from the $Q_{\phi}$ and noise images (without
convolution) of our IRDIS DPI observations. The plot is shown with a linear
stretch between [-3$\sigma$, 3$\sigma$]. The inner and outer model boundaries
are indicated by the circle and the ellipse in the image. North is to the top
and east to the left.

Figure 1 shows the final collapsed total-intensity, $Q_{\phi}$ and $U_{\phi}$
images of our IRDIS DPI $H$-band observations of GSC~07396-00759. As can be
seen, the total-intensity image (left panel) shows numerous point sources, and
in the FoV (cropped to ${\sim}\,2.2\arcsec\times 2.2\arcsec$) of our IRDIS
observation alone, we identified more than 10 different sources. In fact,
Sissa et al. (2018) detected a total of 109 point sources in their dual-band
imaging observations taken with IRDIS as part of the SHINE survey (Vigan et
al. 2020), in the night of June 15, 2017. However, by using observations of
GSC~07396-00759 taken in the International Deep Planet Survey (IDPS; Vigan et
al. 2012; Galicher et al. 2016) with the NIRC2 instrument at Keck, and with a
time difference of about 10 years, they were able to identify 70 objects as
background objects based on common proper motion. Another 32 sources could be
rejected as companions based on their location in the color-magnitude diagram
(CMD). For the 7 remaining objects the results could not be determined
unambiguously, however all but one have separations above 5″
(a${}_{\mathrm{proj.}}>400$ AU), and the colors of those objects are similar
to that of the identified background objects (Sissa et al. 2018). It is thus
highly likely that these are unrelated background objects too.

The middle-panel of Fig. 1 shows the reduced $Q_{\phi}$ image. Noticeable is
the almost edge-on disk extending about 1.3″ to the South-East (SE) and North-
West (NW). Similar to the findings by Sissa et al. (2018), we detect an
asymmetric brightness distribution, with the SE side of the disk appearing
slightly brighter than the NW side. The disk also appears to be warped on the
NW side, indicating a more complicated structure of the disk. The image
however shows one of the major advantages of polarimetric observations. That
is, since stellar light is usually not polarized, the image is not
contaminated by background stars which makes it easier to detect circumstellar
matter. In fact, except for one source/artifact located about 0.5″ North of
the central star, the disk appears to be free of contamination from other
sources visible in the total intensity image. We masked this source separately
during our subsequent analysis to minimize its contribution.

We also note that on average the $U_{\phi}$ image (Fig. 1, right panel) is
mostly devoid of signal. This is consistent with what we expect in case of
single scattering in a centrally illuminated, optically thin disk. We, thus,
assume that the $U_{\phi}$ image to the first order contains only noise and
measure these uncertainties from the standard deviation in concentric annuli
with a width of 1 pixel in the $U_{\phi}$ image to create a noise map. This
allows us to estimate the goodness of fit for our models as well as to create
the signal-to-noise ratio (S/N) map shown in Fig. 2, computed as the ratio of
$Q_{\phi}$ image and the estimated noise map. To minimize the influence of the
higher noise close to the central star on the modeling, we introduce a
circular numerical mask with $r=0.3$″ as inner boundary as well as an ellipse
as outer limit for the modeling, which are also indicated in Fig. 2. Values
inside the circular mask and outside the ellipse are not considered in the
modeling.

###  2.3 Stellar properties

Table 2: Summary of stellar parameters of GSC~07396-00759.

Parameter | Value | Reference
---|---|---
SpT | M1Ve | a
$T_{\mathrm{eff}}$ [K] | 3800$\pm$100 | -
$D$ [pc] | 71.43$\pm$0.26 | b
$L_{\star}$ [$L_{\odot}$] |  ${\sim}$0.13 | -
$R_{\star}$ [$R_{\odot}$] |  ${\sim}$0.71 | -
RV [km s-1] | -5.7$\pm$0.8 | c
$v\sin i$ [km s-1] | 3.0$\pm$1.5 | c
$P_{\mathrm{rot}}$ [d] | 11.63$\pm$0.02 | -
Age [Myr] |  $\lesssim$20 | -
$M_{\star}$ [$M_{\sun}$] | 0.62${}^{+0.04}_{-0.02}$ | -
444SpT: Spectral type. $D$: Distance to the Sun. $T_{\mathrm{eff}}$: Effective
temperature. $L_{\mathrm{bol}}$: Bolometric luminosity. $R_{\star}$: Stellar
radius. $M_{\star}$: Stellar mass. RV: Radial velocity. $P_{\mathrm{rot}}$:
Rotation period. (a): Pecaut & Mamajek (2013); (b): Gaia Collaboration et al.
(Gaia DR2, 2018); (c): Malo et al. (2014a); (-): This work.

Along with the observations of GSC~07396-00759, and a thorough analysis for
possible companions, Sissa et al. (2018) collected and estimated various
stellar parameters, and we refer the reader to their paper for a detailed
description of these parameters. Nevertheless, we revised the most important
parameter, i.e. the distance to GSC~07396-00759, using new astrometric data
from the Gaia satellite mission data release 2 (Gaia DR2, Gaia Collaboration
et al. 2018). In Table 2, we summarize the resulting stellar parameters which
we estimated as follows.

Employing the VO SED Analyzer tool555Version 7 (soon to be released),
http://svo2.cab.inta-csic.es/theory/vosa/ (VOSA, Bayo et al. 2008), we first
collected photometric data ranging from the ultraviolet (UV) at ${\sim}0.297$
$\mathrm{\SIUnitSymbolMicro m}$ (XMM-OM/UVW1, 0.38 mJy, Page et al. 2012) to
the mid-infrared (MIR) wavelengths, with the longest wavelength being 22
$\mathrm{\SIUnitSymbolMicro m}$ at this point. We then applied a BT-Settl-
CIFIST model grid of theoretical spectra (Baraffe et al. 2015) to derive the
stellar parameters from the constructed SED. The data indicate a small UV
excess which we account for by reducing the weight of the UV contribution to
the fit. Furthermore allowing the interstellar extinction ($A_{\mathrm{v}}$)
to vary between 0 and 0.4, we find an effective temperature of
$\mbox{$T_{\mathrm{eff}}$}=3800\pm 100$ K and a stellar luminosity of
$\mbox{$L_{\star}$}=0.132$ $L_{\odot}$ for GSC~07396-00759. From the position
in the Hertzsprung-Russell diagram (HRD) combined with theoretical isochrones
and mass tracks (Baraffe et al. 2015) we estimate an upper limit for the age
of $\lesssim$ 20 Myr and a stellar mass of ${\sim}0.62\,{\mbox{$M_{\odot}$}}$
(but with a flat distribution between 0.6 and 0.68 $M_{\odot}$). From the
dilution factor of the SED fit, combined with the distance of 71.43 pc, we
estimate a stellar radius of $\mbox{$R_{\star}$}\sim
0.71\,\mbox{$R_{\odot}$}$.

When compared with the previous estimates by Sissa et al. (2018), we find a
slightly higher effective temperature, but a significantly smaller stellar
radius for GSC~07396-00759. This, however, is not entirely surprising if we
take into account that i) the luminosity, and thus the stellar radius, depends
on the distance to the star, and ii) in particular young stars that are
members of young moving groups, such as the $\beta$ Pic MG, appear over-
luminous, and hence inflated, when compared to older (field) stars (Malo et
al. 2014b). This, combined with the choice of the model grid could explain the
discrepancy in the estimated stellar properties between our results and those
obtained by Sissa et al. (2018).

Additionally, we analyze available light-curves of GSC~07396-00759 observed
with the Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015), and
during the All-Sky Automated Survey for Supernovae (ASAS-SN, Shappee et al.
2014; Kochanek et al. 2017), respectively, to measure the stellar rotation
period. GSC~07396-00759 was observed with TESS (600–1000 nm), covering an
observation period of about 27 days, between 2019-06-19 and 2019-07-17,
whereas the ASAS-SN light curve was obtained in the $V$-band over a time
period of 2.5 years between 2016-03-10 and 2018-09-22. Using Lightkurve, a
Python package for Kepler and TESS data analysis (Lightkurve Collaboration et
al. 2018), we estimate a stellar rotation period $P_{\mathrm{rot}}=11.63\pm
0.02$ d from the Lomb–Scargle periodogram (Lomb 1976; Scargle 1982) of the
TESS light curve (see Fig. 12), and $P_{\mathrm{rot}}=12.06\pm 0.02$ d from
the ASAS-SN light curve (see Fig. 13), respectively. The uncertainty
associated with each period was estimated by re-sampling the light curve using
the bootstrap method (with replacement), and corresponds to the 95% confidence
interval of our sample estimates. We note, however, that these are only
statistical uncertainties and they do not represent the totality of the error
budget of the light curves. For example, the errors do not reflect properly
that the TESS light curve only covers about two periods, or that the period
estimates are probably affected by two additional sources
($\Delta\mathrm{mag}=2$) that fall on the same TESS pixel as GSC~07396-00759,
given the pixel scale of 21$\arcsec$ per pixel. Nevertheless, both period
estimates are in good agreement with the period of $P_{\mathrm{rot}}=12.05\pm
0.5$ d reported by Messina et al. (2017). From the period estimated from the
TESS light curve, together with the evaluated radius
($\mbox{$R_{\star}$}{\sim}0.71\,\mbox{$R_{\odot}$}$) and a $v\sin i=3.0\pm
1.5$ km s-1(Malo et al. 2014a) we estimate a stellar inclination of
$i{\sim}80\mbox{${}^{\circ}$}\pm 20$∘ (see e.g. Eq. 1; Justesen & Albrecht
2020), consistent with a stellar rotation that is likely co-planar with that
of the disk, as proposed by Sissa et al. (2018). Very few disks have a
measured stellar inclination to which the disk inclination can be compared
(see e.g. Greaves et al. 2014). Our results therefore might be helpful in
future studies of the relation between the stellar inclination and the
inclination of the disk.

As mentioned above, the model fit of the SED of GSC~07396-00759 suggests a
small UV excess. We also found that GSC~07396-00759 has various detections in
the X-ray band, reported in the fourth generation of serendipitous source
catalogs (4XMM, Traulsen et al. 2020). The detected release of energy in the
X-ray and UV portion of the stellar spectrum suggests that GSC~07396-00759 is
subject to strong coronal emission, and possibly intense and frequent flares,
common for young and low-mass stars such as GSC~07396-00759. These stellar
flares are caused by the re-connection of magnetic field loops on the surface
of the star (Mondrik et al. 2019). How frequently these stellar flares occur
also depends on how magnetically active the star is. For example, AU Mic which
is similar in luminosity and age to GSC~07396-00759 (both are members of the
$\beta$ Pic MG) is known to frequently present X-ray and EUV flares at a rate
of about 0.9 flares per hour, during which the EUV and X-ray luminosities
increase by a typical factor 10 (see, e.g., Augereau & Beust 2006, and
references therein). In fact, recent studies of the flare-activity of young
($\leq 100$ Myr) K- and M-stars in the Upper Sco region using Kepler K2 data
found that early and late-type M-stars might have 10000 to 80000 times as many
high-energy flares, so-called super-flares with $E\geq 5\times 10^{34}$ erg,
than solar like stars (Guenther et al. 2019). Such an enormous activity will
certainly affect the direct environment around the stars, especially potential
disks and/or planets that orbit the star.

Although we do not detect flares in the available light curves of
GSC~07396-00759, such an increased flare-rate might also be the case for
GSC~07396-00759, because not only is the star presumably young ($\approx 20$
Myr), but it is likely also magnetically active, as indicated by the detection
of the H$\alpha$ and H$\beta$ lines in emission (Sissa et al. 2018), and the
moderate rotation period of $P_{\mathrm{rot}}\approx 12$ d. Thus, even if
GSC~07396-00759 is currently in a quiescent state, the present coronal
activity of the star and its effect on the disk may not be negligible, given
the observed variations in the X-ray flux and the apparent UV excess detected
in GSC~07396-00759.

##  3 Analysis

Besides gravity, the orbital parameters of small $\mathrm{\SIUnitSymbolMicro
m}$-sized particles, observed in the visible and near-infrared using scattered
and linear-polarized light, are mainly affected by radiation pressure and
stellar winds. In particular the latter can dominate the effects on
circumstellar grains in debris disks around late-type, low-mass stars as first
pointed out by Plavchan et al. (2005). Hence, in this study we try to
constrain the dust properties, as well as the distribution of dust grains in
the debris disk around GSC~07396-00759, taking into account not only radiation
pressure but also the effect from the stellar wind of the central star.

###  3.1 Code description

The code used in this study is based on the same code as was described in
Olofsson et al. (2019). However, since we have made some changes to the code,
in the following, we give a short summary of the code description and the
modifications we made to also account for the effects of stellar wind pressure
on the dust grains.

The observed dust is released from a belt of planetesimal-sized parent bodies
(their numbers, sizes and masses do not need to be specified) which is defined
by 6 parameters: a reference radius $r_{0}$ which is the semi-major axis of
the orbit, eccentricity $e$, inclination $i$, argument of periapsis $\omega$,
the position angle on the sky $\varphi$, measured positive from north to east,
and the width of the ring $\delta_{r}$. All the dust grains originate from
those parent bodies that a priori, all share the same $e$ and $\omega$, and
have a radial distribution that follows a normal distribution centered at a
radius $r_{0}$ with a standard deviation of $\delta_{r}$. The dust grain-size
distribution, used in the code, follows a differential power-law

$\displaystyle dn(s)\propto s^{-p}ds\,,\;$ (1)

where $s$ is the grain size, and $p=3.5$ is the slope index of a particle
distribution derived from an idealized collisional cascade, following Dohnanyi
(1969). This distribution is divided in $n_{\mathrm{g}}$ intervals
(equidistant in logarithmic space between $s_{\mathrm{min}}$ and
$s_{\mathrm{max}}$, respectively), and the number density of grains in each
bin is computed using Eq. (2) of Dullemond & Dominik (2008).

For each grain size the code then computes the dimensionless parameter $\beta$
which, in previous versions of the code, was defined as the ratio between the
radiation pressure and the gravitational forces (Burns et al. 1979). If this
ratio exceeds 0.5 for an initially circular orbit, the dust particle gets
pushed into a hyperbolic orbit and leaves the system (Krivov et al. 2006).
Hence, for a given grain composition and porosity, this $\beta$ parameter cut-
off can be used as a very close approximation of the effective radius below
which dust particles will leave the system (Arnold et al. 2019). Following the
investigation of AU Mic (Augereau & Beust 2006; Sezestre et al. 2017) we
modified the code to also include effect of the stellar wind pressure force on
the dust grains. The net pressure force acting on a grain is then defined by
$\beta=\beta_{\mathrm{RP}}+\beta_{\mathrm{SW}}$ (Sezestre et al. 2017), where
$\beta_{\mathrm{RP}}$ is the ratio between the radiation pressure forces and
the gravitational forces, and $\beta_{\mathrm{SW}}$ is the ratio between the
wind pressure and gravitational forces. The individual contributions of the
two pressure forces to $\beta$ can be estimated via

$\displaystyle\beta_{\mathrm{SW}}$ $\displaystyle=$
$\displaystyle\frac{3}{32\pi}\frac{\mbox{$\dot{M}_{\star}$}V_{\mathrm{SW}}C_{\mathrm{D}}}{G\mbox{$M_{\star}$}\rho
s}\,,$ (2)

where $\dot{M}_{\star}$ is the stellar mass-loss rate, $V_{\mathrm{SW}}$ is
the stellar wind speed, $C_{\mathrm{D}}$ the dimensionless free molecular drag
coefficient which we take equal to 2 (see e.g. Augereau & Beust 2006, and
references therein), $G$ the gravitational constant, $M_{\star}$ the mass of
the star, and $\rho$ the volumetric mass density of the dust, respectively,
and

$\displaystyle\beta_{\mathrm{RP}}$ $\displaystyle=$
$\displaystyle\frac{3}{16\pi}\frac{\mbox{$L_{\star}$}\langle
Q_{\mathrm{RP}}\rangle}{cG\mbox{$M_{\star}$}\rho s}\,,\mathrm{with}\,\langle
Q_{\mathrm{RP}}\rangle=\frac{\int_{\lambda}Q_{\mathrm{RP}}F_{\lambda}d\lambda}{\int_{\lambda}F_{\lambda}d\lambda}\,,$
(3)

where $L_{\star}$ the stellar luminosity, $Q_{\mathrm{RP}}$ the dimensionless
radiation pressure efficiency, which along the wavelength also depends on the
dust grain size and composition, $F_{\lambda}$ the stellar flux, and $\lambda$
the wavelength. As can be seen from the equations, in general and for
sufficiently large dust grains $\beta$ varies with $s^{-1}$. It should be
noted however that for smaller grain sizes, the relationship between $\beta$
and $s$ becomes more complex and increasingly dependent on the grain
composition, the stellar mass-loss rate $\dot{M}_{\star}$, and stellar wind
speed $V_{\mathrm{SW}}$ (Sezestre et al. 2017).

For a given $\beta$ the code draws $3\,000$ samples from an uninformative
uniform prior distribution for the mean anomaly, to decide where the dust
grain is located upon its release. For each realization, the code then
calculates the “updated”orbital parameter ($a_{n}$, $e_{n}$, and $\omega_{n}$)
using Eq. (2) in Wyatt et al. (1999); Wyatt (2006); Lee & Chiang (2016). In
case the updated eccentricity $e_{\mathrm{n}}$ is larger or equal to zero and
strictly smaller than unity (to avoid hyperbolic orbits), the resulting orbits
are populated with 300 dust particles, uniformly distributed in mean anomaly.
The vertical distribution of the disk is accounted for by drawing from a
normal distribution with a standard deviation $h=0.04\times r$, following
Thébault (2009). This allows us to account for column density effects, as
explained in further detail in Olofsson et al. (2020). The $(x,y,z)$ positions
of each particle are registered, and depending on inclination and position
angle of the disk, the corresponding closest pixel is determined, thus
producing number density maps for each value of $\beta$.

Furthermore, and as discussed in Olofsson et al. (2019), when computing the
number density maps for each $\beta$ value, the contribution of each particle
is also multiplied by a correction factor (Strubbe & Chiang 2006; Lee & Chiang
2016), which is roughly proportional to their total orbital period divided by
the time spent within the birth ring. This correcting “enhancement” factor to
the high-$\beta$ grain number density allows us to at least to the first order
account for the fact that small grains produced inside the belt on high
eccentricity (bound) orbits will spend most of their time in the collision-
free outer regions where they cannot be collisionally destroyed, hence
enhancing significantly their collisional lifetimes, and thus their number
density (Strubbe & Chiang 2006; Thébault & Wu 2008).

Once each of the $3000\times 300$ particle has been launched the code computes
the scattering angle between the central star and the observer for each
particle in the image. The polarized light images per grain size bin are then
computed by multiplying the estimated number density of each pixel by
$S_{12}\times\pi s^{2}\times Q_{\mathrm{sca}}/(4\pi r^{2})$, where $r$ is the
distance from the particle to the star, $Q_{\mathrm{sca}}$ the scattering
efficiency, and $S_{12}$ the polarized phase function. The final image is the
collapse of all individual images for each grain size bin, weighted by the
grain size distribution $n(s)$.

###  3.2 Modeling strategy

####  3.2.1 Caveats and modeling approach

Most likely due to a degeneracy between the minimum grain size and the
porosity of the dust particles or the high inclination of the disk around
GSC~07396-00759 our primary modeling attempts were not converging on a unique
solution (see Sec. 4 for a detailed discussion). To simplify the problem, we
therefore used a different, two-step approach in which we first focused on the
morphology of the disk and then on the dust properties. The procedure can be
summarized as follows.

We first try to alleviate the influence of some of the dust properties by
replacing the polarized phase function in our disk model with an analytical
form, the Henyey-Greenstein approximation (Henyey & Greenstein 1941) described
in Sec. 3.2.2. This form of the phase function is parameterized for all grain
sizes. Thus, the absolute value of $\beta$ for each grain size becomes less
relevant, and the spatial distribution of the dust particles is mostly
determined by the global shape of the $\beta$ function ($\beta(s)\propto
s^{-1}$). By sampling the entire range of $\beta$ values below 0.5, we can
therefore estimate the spatial distribution of the dust particles in the disk,
without having to pay too much attention to their actual sizes and properties.

Outside of the birth ring there is also an over-abundance of small dust
grains, mostly due to the effect of radiation pressure, as discussed for
example in Thébault & Wu (2008). While the grain size distribution is expected
to follow the ”classical” distribution with an exponent $p=3.5$ (see above) in
the birth ring, outside of the birth ring, the smallest dust grains have an
increased collisional lifetime as they are set on highly eccentric orbits and,
thus, spend most of their time near their apoapsis, i.e. they survive longer
and hence contribute even more to the flux observed in scattered light. In
fact, the deviation of the size distribution from -3.5 can actually be quite
significant (see e.g. Fig. 3, Strubbe & Chiang 2006), resulting in the small
dust grains dominating outside of the birth ring. Therefore the phase function
we retrieve is the one for the dominant/most representative grain size, which
we assume to be close to the blow out size of the dust particle.

With the radial distribution and the phase function established by our disk
model, we can then, in the second step, try to conversely infer other
properties of the dust particles. We achieve this by comparing the inferred
analytical polarized phase function with a model dependent polarized phase
function computed by other means such as effective medium theory (EMT), the
discrete dipole approximation (DDA; Purcell & Pennypacker 1973; Draine 1988),
or a distribution of hollow spheres (DHS, see Min et al. 2005). Depending on
the grain composition and porosity we can then try to estimate individual
parameters, such as the effective radius ($s_{\mathrm{min}}$), below which
particles will not stay in the system, or the stellar mass-loss rate
($\dot{M}_{\star}$).

####  3.2.2 Modeling the disk

The free parameters of our disk model are the reference radius of the radial
distribution $r_{0}$, its standard deviation $\delta_{r}$, the inclination
$i$, and the position angle $\phi$. The high inclination of the disk around
GSC~07396-00759, however, makes it hard to reasonably constrain the
eccentricity. We therefore assume a circular disk and fix the eccentricity to
$e=0$, making the argument of periapsis $\omega$ irrelevant for the modeling.
For the stellar parameters we use a bolometric luminosity of 0.13 $L_{\star}$,
a stellar mass of 0.62 $M_{\odot}$, and a distance of 71.43 pc.

We sample the grain size distribution so that the entire range of $\beta$
values from 0.5 down to $10^{-5}$ is probed, i.e. the dust particles remain
gravitationally bound over the entire grain size distribution between
$s_{\mathrm{min}}$ and $s_{\mathrm{max}}$. To determine the polarized phase
function $S_{12}$ as a function of the scattering angle, we use a modification
of the analytical Henyey-Greenstein approximation (HG, Henyey & Greenstein
1941) which is given by

$\displaystyle S_{\mathrm{12,HG}}$ $\displaystyle=$
$\displaystyle\frac{1-g^{2}}{4\pi(1+g^{2}-2g\cos(\theta))^{3/2}}\frac{1-\cos^{2}(\theta)}{1+\cos^{2}(\theta)}\,,$
(4)

where $\theta$ is the scattering angle and $g$ the anisotropic scattering
factor ($-1\leq g\leq 1$), which governs the scattering efficiency as a
function of the scattering angle $\theta$. The first factor in Eq. 4 is the HG
function that describes the scattered flux produced by the photons hitting and
interacting with the dust particles, while the second term in Eq. 4 takes into
account the angle dependence of the linear polarization produced by the
particle scattering, using the Rayleigh scattering function as a simple
approximation. Thus, for $g=0$ (isotropic scattering) the maximum of scattered
polarized flux occurs at $\theta=90\,\mbox{${}^{\circ}$}$, while for $g>0$ the
maximum is shifted to smaller scattering angles resulting in an asymmetry in
the amount of polarized light received from the front and backsides of the
disk (Engler et al. 2017).

The best solution to our model with 5 free parameters ($r_{0}$, $\delta_{r}$,
$i$, $\phi$, and $g$) is determined using an affine invariant ensemble sample
Monte-Carlo Markov Chain ($\mathtt{emcee}$ package, Foreman-Mackey et al.
2013). For the initial conditions of the model we use the disk parameters
($r_{0}$, $i$, and $\phi$) reported by Sissa et al. (2018). We then draw
random samples from uniform priors we also report in Table 3, using a Monte
Carlo Markov chain composed of 30 walkers, a burn-in phase of 1000 models,
followed by the actual modeling using chains of 10000 models for each walker.
To speed up the modeling process, we exclude all image data points located
outside of an elliptical mask defined by a semi-major and semi-minor axis of
2.3 ″and 0.6 ″, and a position angle of 150∘, respectively. We also place a
numerical mask with a radius of 0.3 ″on the center to reduce the influence of
the increased noise around the central star on the modeling (see Fig. 2). The
goodness of fit for each model is estimated as the sum of the squared
residuals divided by the uncertainties. The results of the disk modeling are
presented in Sec. 4.

####  3.2.3 Polarized phase function

In the second step of our analysis we use the parameterized polarized phase
function obtained in the previous step to try to infer further dust properties
by comparing it to a model-dependent phase function. In this study, we use the
$\mathtt{OpacityTool}$, a dedicated Fortran
package666https://dianaproject.wp.st-andrews.ac.uk/data-results-
downloads/fortran-package of the DIANA project (see Woitke et al. 2016; Toon &
Ackerman 1981), to compute dust opacities for the purpose of disc modeling.
The package uses a distribution of hollow spheres (DHS, see Min et al. 2005)
to compute absorption and scattering properties, as well as six different
elements of the scattering matrix, including $S_{11}$, the scattering phase
function, and $S_{12}$, the polarized phase function, respectively. We use the
same grain size distribution as in the previous step to calculate polarized
phase functions.

Preliminary tests, however, suggested that a size distribution between
$s_{\mathrm{min}}$ and $s_{\mathrm{max}}$
($s_{\mathrm{max}}>>s_{\mathrm{min}}$) could not match the phase function
derived from the modeling of the disk, given the limitations of the data and
the observed degeneracies in the model. Therefore, we chose to only compare
our results to models with one representative grain size to find the typical
grain size that can best explain the inferred polarized phase function. For
each grain size we calculate polarized phase functions varying the porosities
from 0.01 to 0.99 in steps of 0.01, and the grain size ($s$) between
$0.01\,\mbox{$\mathrm{\SIUnitSymbolMicro m}$}$ ($s_{\mathrm{min}}$) and 1 mm
($s_{\mathrm{max}}$), respectively. The $\mathtt{OpacityTool}$ calculates the
opacities and other properties using a mixture of 85% amorphous laboratory
silicates (Dorschner et al. 1995, Mg0.7Fe0.3SiO3) with 15% amorphous carbon
(Zubko et al. 1996, BE-sample). The effective refractory index of the porous
material is calculated by applying the Bruggeman (1935) mixing rule. The
maximum hollow volume ratio is set to $f_{\mathrm{max}}=0.8$. This filling
factor $f_{\mathrm{max}}$, computationally, represents the maximum volume
fraction occupied by the central void in the hollow sphere, while in practice,
this parameter represents the more general amount of deviation from a perfect
homogeneous sphere (Min et al. 2016).

Using the grain size and porosity as the only free parameters, we then compute
a grid of models to estimate whether we can reproduce the parameterized phase
function. As a consequence of the high inclination of the disk and the
employed inner mask not all scattering angles between 0 and 180 degrees can be
sampled. To account for this, we use the best-fit parameters determined for
the disk and the applied masks to estimate range of valid scattering angle.
The result of this exercise is shown in Fig. 14, yielding observable
scattering angle from 9.14∘ to 170.86∘
($\theta_{\mathrm{min,max}}=90\mbox{${}^{\circ}$}\pm
80.86\mbox{${}^{\circ}$}$). We also note at this point that due to the method
employed, the uncertainties in our disk model may be underestimated. We
therefore utilize the 3$\sigma$ error bounds on the value of $g$, rather than
the $1\sigma$ values.

Figure 3: Observation and model images for GSC~07396-00759. Shown from left to
right are the measured $Q_{\phi}$, best-fit model and residual image obtained
from our disk model using a linear scaling for each panel. The regions outside
of the ellipse and within the circle (green dashed lines) are excluded from
the $\chi^{2}$ calculation. The coronagraphic mask is indicated by the
(shaded) circular region in each panel. North is to the top and east to left
in each panel.

##  4 Results

###  4.1 Geometric properties of the disk

Table 3: Model details and best-fit parameter for the disk modeling of the
SPHERE/IRDIS observations and the modeling of the polarized phase function.

Parameter | Uniform prior | Best-fit value
---|---|---
$r_{0}$ [au] | [10, 300] | 107$\pm$2
$\delta_{r}$ [au] | [0, 200] | 27$\pm$1
$i$ [∘] | [60, 90] | 84.3$\pm$0.3
$\phi$ [∘] | [100, 180] | 148.7$\pm$0.7
$g$ | [0, 1] | 0.60$\pm$0.03
$s$ [$\mathrm{\SIUnitSymbolMicro m}$] | [0.01, 1000] | ${\sim}0.3-1$
porosity | [0.01, 0.99] | n.d.
777$r_{0}$: Reference radius and center of the radial distribution.
$\delta_{r}$: Standard deviation of the radial distribution. $i$: Inclination.
$\phi$: Position angle. $g$: Scattering efficiency paramater. $s$: Grain size.
The porosity could not be determined (n.d.) due to a degeneracy in the model,
between the minimum grain size and the porosity of the dust particles.

We first used the model described in the previous section to reproduce the
observed disk around GSC~07396-00759, and to determine the most probable
values for each of the five free parameters ($r_{0}$, $\delta_{r}$, $i$,
$\phi$, and $g$). To assess the convergence and stability of the MCMC
solution, we estimated the maximum autocorrelation length among all
parameters, i.e. the average autocorrelation time, after each iteration. The
fitting is considered converged when the number of iterations is larger than
100 times the average autocorrelation time and its changes, between subsequent
iterations, are less than 1%. In Appendix 15 we show the evolution of the
autocorrelation time as a function of the iteration step over the course of
the modeling. At the end of the modeling, the average autocorrelation time was
79 steps and the mean acceptance fraction (Gelman & Rubin 1992) for our best-
fitting model was 0.47. The best-fit model, along with the residuals and the
observations, are presented with the same linear scaling in the center, right
and left panels of Fig. 3, respectively. The most probable parameters and
their uncertainties are summarized in Table 3, while the probability density
functions, plotted with $\mathtt{corner}$ package (Foreman-Mackey 2016), are
shown in Fig. 4.

We found that our observations are best described by an extended disk with a
dust distribution centered at a radius $r_{0}\approx 107\pm 2$ au, with a
standard deviation of the radial distribution of $\delta_{r}\approx 27\pm 1$
au which is highly inclined at an inclination of $i\approx
84.3\,\mbox{${}^{\circ}$}\pm 0.3$ and a position angle $\phi\approx
148.7\,\mbox{${}^{\circ}$}\pm 0.1$. For the analytical polarized phase
function we found a coefficient $g=0.60\pm 0.03$.

The uncertainties for the MCMC result parameters are estimated from the 0.16
and 0.84 quartiles using the $\mathtt{corner}$ package, and are also shown
along with the projected posterior distributions in Fig. 4. These
uncertainties, however, are most likely underestimated and should be taken
with caution. This might be because we computed the goodness of the fit using
a noise map derived from the standard deviation in concentric annulii in the
$U_{\phi}$ image, a strategy commonly employed in direct imaging studies. We
therefore may be underestimating the true uncertainties, resulting in larger
$\chi^{2}$ values. This in turn might also have led to a narrower probability
distribution as the Monte Carlo algorithm samples a smaller range of parameter
values.

###  4.2 Dust properties

Figure 4: Projected posterior distribution and density plots of the free
parameters used in the disk modeling. The plot additionally shows the 50 %, 16
%, and 84 % quartiles (vertical dashed lines), representing the distributions
median and the 1 $\sigma$ uncertainties (lower and upper bound), respectively.
The 1D histograms represent the probability distributions of each parameter
marginalized over the other free parameter. The contour maps represent the
central 68.3%, 95.5%, and 99.73% of the 2D probability distributions of
different combinations of parameters, marginalized over each other.

Figure 5: 2D $\chi^{2}$ map constructed from the computed $\chi^{2}$ at each
point (model) of the parameter grid. The free parameter of the model are the
porosity and the grain size. The contours enclose the regions where the
$\chi^{2}$ is smaller than the 5% (dashed line) and 15% (solid line)
quartiles. To increase the contrast the upper limit of the $\chi^{2}$ color
map is set to the 50% quartiles.

Figure 6: Polarized phase function ($S_{\mathrm{12,\,HG}}$) as function of
scattering angle $\theta$. Shown are the parameterized polarized phase
function from our best-fit disk model for $g=0.60\pm 0.09$ (black line and
gray area) and the range (red area, enclosing values between the 16% and 84%
quartiles) of best-fit model ($s=0.34$ $\mathrm{\SIUnitSymbolMicro m}$,
$\mathrm{porosity}=0.01-0.99$) of this phase function considering a
distribution of hollow spheres. Additionally shown are the polarized phase
functions resulting from the “low”-$\chi^{2}$ regions indicated in Fig. 5. The
phase functions were sampled around grain sizes and porosities of about 1
$\mathrm{\SIUnitSymbolMicro m}$, and 0.80 (blue area), and grains larger than
100 $\mathrm{\SIUnitSymbolMicro m}$ and porosities below 0.6 (green area),
respectively. The range of possible scattering angle ($\theta=9.14$∘ to
$\theta=170.86$∘) observable with the used mask is indicated by the blue
shaded area.

Figure 7: 2D map of the anisotropic scattering factor
$g_{\mathrm{S_{12,\,DHS}}}$ computed at each point of the grain-size, porosity
parameter grid with the DHS model. The contours for
$g_{\mathrm{S_{12,\,HG}}}=0.50$ (blue lines) and
$g_{\mathrm{S_{12,\,HG}}}=0.70$ (red lines) indicate the range of scattering
factors from our best-fit disk model with the HG-approximation. Figure 8: The
disk around GSC~07396-00759 as seen with SPHERE/IRDIS in polarized light (top-
left panel) and total intensity (top-right panel, Sissa et al. 2018) in the
H-band. The bottom panels show the best-fit disk model found in this work
(bottom-left) and found by Sissa et al. (2018) in the same scaling as the
according observations. The dash-dotted line is parallel to the disk semi-
major axis and crosses the central star. The coronagraphic mask is indicated
by the (shaded) circular region in each panel.

Figure 9: Full width at half maximum as a function of the center of the radial
distribution $r_{0}$ (left panel), and FWHM normalized by the radius of the
disk as function of $r_{0}$ (right panel). Objects for which the FWHM was
estimated readily from a radial Gaussian or power-law distribution are shown
as circles, while objects where only the inner and outer radius were given are
marked by diamonds. The spectral types of the sample stars are indicated by
the color bar. The radius and width of the disk around GSC~07396-00759,
estimated by Sissa et al. (2018) and in this work, are marked by the cross and
the asterisk, respectively. To guide the eye, we also show the limits for
FWHM=$r_{0}$ and FWHM=2$r_{0}$ as gray line in the left panel.

Figure 10: Scattered light disk radii as a function of the stellar luminosity.
Shown are the best-fit power-law function as solid (red) line and the
1-$\sigma$ confidence interval drawn from the corresponding probability
distributions. Data points that are also part of the sample analyzed by
Esposito et al. (2020) are marked as diamonds. Similar to Esposito et al.
(2020), we excluded HR 7012 (blue open circle) from the fit as an outlier. The
best-fit result using the radius estimate by Sissa et al. (2018) (black star)
is plotted as solid (black) line. Also shown for comparison are the
radius–luminosity power laws for planetesimal belt central radii from thermal
light imaging by Matrà et al. (2018) (dash-dotted line), and the GPIES-
detected disks in scattered light by Esposito et al. (2020) (dashed line),
respectively.

Figure 11: $\beta$ as a function of grain size. $\beta_{\mathrm{RP}}$ and
$\beta$ are shown as black and solid lines, respectively. The horizontal solid
gray line ($\beta=0.5$) is the upper limit for bound trajectories assuming
zero eccentricity for the parent body, see Sezestre et al. (2017) for further
details. The vertical dashed line indicates the corresponding minimum grain
size ($s=0.34\,\mbox{$\mathrm{\SIUnitSymbolMicro m}$}$) of dust particles
required not to be expelled from the disk, assuming a porosity of 70%.

Next we applied a grid of models, computed with the $\mathtt{OpacityTool}$, as
described in Sec. 3.2.3, to estimate whether we can reproduce the inferred
polarized phase function ($S_{\mathrm{12,\,HG}}$) considering a distribution
of hollow spheres. The resulting $\chi^{2}$ map is shown in Fig. 5. Also shown
are the contours that enclose the regions where the $\chi^{2}$ is smaller than
the lower 5% (dashed line) and 15% (solid line) quartiles. We found that the
analytical form of the polarized phase function ($S_{\mathrm{12,\,HG}}$) is
best reproduced by small micron-sized dust grains with $s{\sim}0.34$
$\mathrm{\SIUnitSymbolMicro m}$. Due to a degeneracy between the grain size
and the dust porosity, however, we cannot readily constrain the porosity for
this grain size (see discussion below).

In Fig. 6, we present the parameterized polarized phase function
($S_{\mathrm{12,\,HG}}$) as a function of the scattering angle $\theta$ from
our disk modeling for $g_{\mathrm{S_{12,\,HG}}}=0.60\pm 0.09$ (black line and
gray area), along with the best-fit model of the polarized phase function
($S_{\mathrm{12,\,DHS}}$, red line). Also shown is the 1 $\sigma$ range of
models (red area, enclosing values between the 16% and 84% quartiles of the
likelihood distribution), estimated from the phase functions for $s=0.32-0.36$
$\mathrm{\SIUnitSymbolMicro m}$, and the full range of modeled porosities
(0.01-0.99). As can be seen, despite being the best solution to our model
approach, there are notable differences between both curves, but in particular
for scattering angle of less than 90∘. The $\chi^{2}$ map, furthermore, not
only shows ”low” $\chi^{2}$ values around 0.34 $\mathrm{\SIUnitSymbolMicro
m}$, but also for particles of 1 $\mathrm{\SIUnitSymbolMicro m}$ with
porosities around 0.8, and for larger particles ($s>100$
$\mathrm{\SIUnitSymbolMicro m}$) with porosities of less than 0.6. The
corresponding range of phase functions, determined similar to the best-fit
results, are shown as well in Fig. 6. These phase functions clearly provide a
worse fit to our approximation (shown in black) then the best solution using a
DHS model (shown in red).

The presence of multiple solutions, on the other hand, could indicate
additional hidden degeneracies beyond the one on the minimum size and porosity
mentioned, restricted only to the second step of our alternative two step
approach. For instance, we use a parameterized approximation
($S_{\mathrm{12,\,HG}}$) of the real phase function ($S_{\mathrm{12}}$) which
effectively limits the examinable parameter space. We then fit the obtained
curve with the phase function computed for a DHS model. However, it is
questionable whether we can reasonably reproduce the analytical HG
approximation of the phase function with the phase function from a DHS model.

To address this question we created a map similar to the $\chi^{2}$ map shown
in Fig. 5, where each point of the grid represents the best g value fitting
the DHS model for a given set of grain size and porosity. The resulting map of
$g$ values (hereafter called $g_{\mathrm{S_{12,\,DHS}}}$) is shown in Fig. 7.
Also shown are the regions where the results overlap with the range of g
values from the HG approximation. As can be seen, for the smaller grains
(0.3–1 $\mathrm{\SIUnitSymbolMicro m}$) the $g_{\mathrm{S_{12,\,DHS}}}$ value
estimates are mostly consistent with the range of $g$ values using the HG
approximation, while the $g$ values for the larger particles appear
systematically higher. The data also indicate that there are possibly isolated
solutions for particles with sizes between 5 $\mathrm{\SIUnitSymbolMicro m}$
and 10 $\mathrm{\SIUnitSymbolMicro m}$. We also summarize these results in the
lower part of Table 3.

##  5 Discussion

###  5.1 Disk properties

Our results from the modeling of the dust distribution around GSC~07396-00759
indicate a radius of $\sim 107$ au for the radial distribution of the grain
number density of the debris disk, and as shown in the residuals image in the
right panel of Fig. 3, most of the signal coming from the disk has been
successfully removed.

The modeling results are globally consistent with the results reported by
Sissa et al. (2018). In particular, we found that the estimated full width at
half maximum (FWHM) of the disk of $2\sqrt{2\ln 2}\,\delta_{r}\approx 64$ au
($\delta_{r}\approx 27$ au) as well as the inclination ($i\approx
84\,\mbox{${}^{\circ}$}$) and position angle ($\theta\approx
149\mbox{${}^{\circ}$}$) for the disk around GSC~07396-00759 are in good
agreement with the results obtained by Sissa et al. (2018, $i\approx
83\,\mbox{${}^{\circ}$}$, $\theta\approx 149\mbox{${}^{\circ}$}$, and FWHM
$\approx 56$ au), although their forward modeling of the angular differential
imaging (ADI) observations of GSC~07396-00759 yielded a smaller disk radius
($r_{0}\approx 70$ au) than our model of the polarized light observations
($r_{0}\approx 107$ au).

Our disk radius estimate is also slightly larger than the radius obtained by
Sissa et al. (2018) using a geometric approach, where the authors measured the
position of the maximum brightness (disk spine) with respect to the apparent
semi-major axis (i.e. the apparent disk radius obtained via forward modeling)
as a function of the separation from the star after the removal of background
stars in the five-mode PCA IRDIS image (see Fig. 2, Sissa et al. 2018). Under
the assumption that the disk is not flared, Sissa et al. (2018) found that a
ring of material with radius $r=1.34\arcsec$ ($\sim 98$ au) and inclination
$i=84.5\mbox{${}^{\circ}$}$ can properly describe the observed spine up to
1.2$\arcsec$ ($\sim 89$ au) in their ADI images. However, as also pointed out,
the presence of at least three very bright stars may have altered the light
distribution of the disk, and thus, the results have to be taken with caution
as well.

It is surprising, nevertheless, that we found $r_{0}$ to be $\approx 107$ au
($\approx$ 1.5$\arcsec$) in our model, while we only detected uniform emission
up to about $\approx 70-80$ au (1–1.1$\arcsec$) in the image. This suggests
that according to our model, we do not fully recover the major axis of the
disk with our observations, because we also found that the maximum of the
polarized phase function (see Fig. 6) is shifted to smaller scattering angle
($\theta\approx 55\,\mbox{${}^{\circ}$}$), while the phase function at the
same time is relatively low at scattering angles of 90 ∘.

We suggest that this is due to a degeneracy between the reference radius of
the density distribution $r_{0}$ and the value of the anisotropic scattering
factor $g$ for the phase function. We emphasize, however, that this degeneracy
is solely the case for highly inclined disks, because significant azimuthal
information is lost in this particular, nearly edge-on configuration (see e.g.
Fig. 8, Engler et al. 2017).

Our value of $g=0.60$ therefore might be too large. Nevertheless, we can
exclude very small $g$ values, because, if $g$ were close to zero, the major
axis of the disk would become much brighter than the minor axis, both due to
the phase function peaking closer to 90 degrees and the column density that is
larger at the major axis of the disk (see e.g. Olofsson et al. 2020, for
discussion). This, however, is not what we observe in the images, because we
see a more or less uniform distribution of polarized light along the disk, as
opposed to two distinct lobes, seen for example in the modeling of the disk
around HD 61005 (Olofsson et al. 2016). The observed degeneracy might also
play a significant role for ADI images, since forward-scattering is even more
pronounced in ADI observations, and therefore, the results of Sissa et al.
(2018) may also be impacted by this, as indicated by the side-by-side
comparison of the two observations shown in Fig. 8.

To explore the possibility that our radius for the disk around GSC~07396-00759
is overestimated, we re-run our grid of models with the reference radius fixed
to $r_{0}=69.9$ au, obtained by Sissa et al. (2018) using forward modeling, to
test whether we can find a solution that fits our observations. The resulting
best-fit model in this scenario is shown in Fig. 16. In particular, we found a
standard deviation of the radial distribution of $\delta_{r}\approx 15.3\pm
0.2$ au, an inclination of $i\approx 78.5\,\mbox{${}^{\circ}$}\pm 0.3$ and a
position angle $\phi\approx 149.2\,\mbox{${}^{\circ}$}\pm 0.1$. For the
analytical polarized phase function we found a coefficient $g=0.33$. Although
the model converged on a solution, the results clearly show that we under-
estimate the reference radius, as in particular the outer regions are not well
reproduced, when using the results from the forward modeling of the ADI
observation by Sissa et al. (2018). On the other hand, the reference radius we
inferred is in fact quite close to the radius of the disk of $r_{0}\approxeq
98$ au obtained by Sissa et al. (2018) from their geometric measurements. We
thus suggest that the estimated reference radius ($r_{0}\approxeq 107$ au),
while not definitive, is probably a better estimate than the one inferred by
Sissa et al. (2018) using forward modeling. However, an independent
confirmation of the disk radius is required to settle the discussion. One
possible way to alleviate this degeneracy and to resolve the discrepancies
between the different measurements is to obtain high angular resolution
observations at longer wavelengths with the Atacama Large Millimeter Array
(ALMA, Cronin-Coltsmann, in prep.). These observations are more sensitive to
larger dust grains, and would therefore allow us to place stronger constraints
on the shape parameters of the debris disk, and consequently allow us to more
reliably estimate the shape of the phase function.

###  5.2 Disk radii

Another important diagnostic of debris disks alongside the phase function and
the size distribution of their dust are the dimensions of the debris disks,
i.e. their extent, and radii. Believed to be the by-product of the planet
formation process, the dust observed in debris disks is thought to be produced
by the grinding down of larger bodies, the planetesimals. In general, theory
would suggest that the presence of a planetesimal belt, be it caused by failed
growth to planets or enhanced planetesimal formation due to complex and
dynamical interaction with already formed planets, should be related to
distance to the central star (Matrà et al. 2018). Regardless of the underlying
processes, it is clear that the radii of debris disks contain valuable
information on the formation processes of planetesimals and planets.

Several studies of the radius-luminosity (R–L) relation have been conducted
over the years on this topic, such as for Herschel PACS-resolved disk radii
between 70 $\mathrm{\SIUnitSymbolMicro m}$ and 160 $\mathrm{\SIUnitSymbolMicro
m}$ (Pawellek et al. 2014), for planetesimal belt central radii from thermal
light imaging (Matrà et al. 2018), and recently on 25 GPIES-detected debris
disks ($\mbox{$L_{\star}$}=2-14$ $L_{\odot}$) in scattered light by Esposito
et al. (2020). These studies found only a marginal correlation between the
luminosity of the star and the radius of the disk, indicating that the
morphology of debris disk is likely a result of temperature-independent
processes, such as for instance, shaping of planetesimal populations by
planets, stirring by various mechanisms, and long-term collisional depletion
(Pawellek et al. 2014).

Despite the interesting results, the studies lack stars with L¡2 $L_{\odot}$
at the lower end of the mass spectrum which may have introduced a selection
bias affecting their results. With the newly available scattered light data
from resolved disks around low-mass stars with $\mbox{$L_{\star}$}<2$
$L_{\odot}$, we therefore compiled a list of stars with resolved debris disks,
and observed in scattered light from the literature, and attempt to test for a
possible correlation between the size of the disk and the properties of the
host star, such as its spectral type or luminosity. We chose to primarily
focus on scattered light observations, because the size and radius of the
observed disk depends also on the wavelength in which they are observed, as
well as the strength of radiation pressure and stellar winds depending on the
evolutionary state of the star.

Our sample contains a total of 46 stars (see Table 4 in the appendix),
including GSC~07396-00759, with spectral types ranging from B9 to M3, ages
between 10 Myr and 2 Gyr, and stellar luminosities reaching from 0.1
$L_{\odot}$ to 25 $L_{\odot}$. In case there are several publications for a
given target, we focused on those publications with the highest S/N
observations, and a similar modeling approach to ours. We then used the
published parameters that describe the radial dust density distribution of the
disk, such as $r_{0}$, $\Delta r$, $\alpha_{\mathrm{in}}$, and
$\alpha_{\mathrm{out}}$, to estimate the FWHM as a function of the “peak”
radius of the radial distribution $r_{0}$ along the mid-plane of the disk. Out
of the total sample presented, for 11 stars only the detected inner and outer
extent of the disk ($r_{\mathrm{min}}$, $r_{\mathrm{max}}$) were given. In
these cases we equate the peak radius $r_{0}$ to the mean of
$r_{\mathrm{min}}$, and $r_{\mathrm{max}}$, and the difference
$r_{\mathrm{max}}-r_{\mathrm{min}}$ as width of the disk, respectively. For
the other 35 stars, the estimated FWHM is based on a radial distribution that
was directly derived either from the standard deviation of the assumed
Gaussian, or a radial distribution $R(r)$ that uses two power laws and is
given by the expression:

$\displaystyle R(r)$ $\displaystyle=$
$\displaystyle\left[\left(\frac{r}{r_{c}}\right)^{-2\alpha_{\mathrm{in}}}+\left(\frac{r}{r_{c}}\right)^{-2\alpha_{\mathrm{out}}}\right]^{-1/2}\,,\;$
(5)

where $r$ is the radial coordinate in the equatorial plane, and $r_{c}$ is a
critical radius that divides the ring into inner and outer regions with
separate density power law indices of $\alpha_{\mathrm{in}}$ and
$\alpha_{\mathrm{out}}$, respectively (Augereau et al. 1999). As also pointed
out by Augereau et al. (1999), the maximum of the dust density does not occur
at $r_{c}$ but at a peak radius calculated via

$\displaystyle r_{0}$ $\displaystyle=$
$\displaystyle\left(\frac{\Gamma_{\mathrm{in}}}{-\Gamma_{\mathrm{out}}}\right)^{\left(2\Gamma_{\mathrm{in}}-2\Gamma_{\mathrm{out}}\right)^{-1}}r_{c}\,,\;$
(6)

where $\Gamma_{\mathrm{in}}=\alpha_{\mathrm{in}}+\gamma$ and
$\Gamma_{\mathrm{out}}=\alpha_{\mathrm{out}}+\gamma$. For this analysis we set
$\gamma=1$, thus assuming the scale height is a constant fraction of the
radius throughout the disk.

To estimate the dependency of the disk radii on the stellar luminosity we
follow the description in Matrà et al. (2018) and use a power-law model where
the belt locations $R_{i}$ (in au) are linked to their host star luminosities
$L_{\star,i}$ (in $L_{\odot}$) through the form
$R_{i}=R_{1\mathrm{L}_{\odot}}L_{\star,i}^{\alpha}+\epsilon_{i}$. The
parameter $\epsilon_{i}$ represents the intrinsic scatter of the distribution,
and is assumed to follow a Gaussian distribution with standard deviation
$\sigma_{\mathrm{intr}}=f_{\Delta R}R_{i}$. The best solution to this power-
law model is determined using an affine invariant ensemble sample Monte-Carlo
Markov Chain ($\mathtt{emcee}$ package, Foreman-Mackey et al. 2013). We then
draw random samples from an uninformative uniform prior on the free parameters
$R_{1\mathrm{L}_{\odot}}$, $\alpha$ and $f_{\Delta R}$, using a likelihood
function described by Equation (24) in Kelly (2007) to determine the posterior
probability distributions for the parameters. To facilitate the comparison
with Esposito et al. (2020), we excluded HR 7012 from the fit as an outlier
too.

In Fig. 9 and Fig. 10 we present the results of this analysis. Figure 9 shows
the estimated FWHM (left panel) and the FWHM divided by $r_{0}$ (right panel),
respectively, as function of the radius of the disk $r_{0}$. To point out that
the sample is not homogeneous, we mark the stars where only the inner and
outer radius of the disk is given with a diamond, while the results obtained
from density distributions using a Gaussian or power laws to describe the
radial profile are marked by circles. The stars are color-coded according to
their spectral type. We additionally indicated the limits for FWHM=$r_{0}$ and
FWHM=2$r_{0}$ to guide the eye. The largest disk contained in our sample is
that of HD 15745, a F2V star whose disk was discovered by Kalas et al. (2007).
The star is one of 10 stars that have a disk with a FWHM larger than the
radius of the respective disk, with the largest width-to-radius ratios being
found for AU Mic (Krist et al. 2005), HD 15745, and $\beta$ Pic (Millar-
Blanchaer et al. 2015). Most of the stars have FWHM smaller than the radius of
the disk, and overall we found neither an obvious correlation, nor clustering
of data, between the radius of the disk, its radial extent, and the spectral
type of the star.

From the analysis of the radius–luminosity relationship in our sample we found
$R_{1\mathrm{L}_{\odot}}=70.7^{+9.6}_{-8.3}$ au, $\alpha=0.02\pm 0.08$, and
$f_{\Delta R}=0.07\pm 0.02$, taking the $50_{-34}^{+34}$ percentiles of the
posterior probability distributions for each parameter. The results are
illustrated in Fig. 10. Shown are the scattered light disk radii as a function
of the stellar luminosity of our sample stars, as well as our best-fit power-
law function as solid (red) line and the 1-$\sigma$ confidence interval
estimated from a randomly drawn sample of the corresponding probability
distributions shown in Fig. 17. The data points that are also contained in the
sample analyzed by Esposito et al. (2020) are marked as diamonds. Because of
the inconsistent radius estimates between this work and the work of Sissa et
al. (2018), we repeated the fit using the radius estimate by Sissa et al.
(2018) and show the result as solid (black) line. However, it is apparent that
the derived relation is not strongly dependent on which estimate of the radius
of the disk around GSC~07396-00759 is included. For comparison, we furthermore
plotted the radius–luminosity power laws for the planetesimal belt central
radii from thermal light imaging by Matrà et al. (2018, $\alpha=0.19\pm 0.04$,
dash-dotted line), and the GPIES-detected disks in scattered light by Esposito
et al. (2020, $\alpha=0.25\pm 0.09$, dashed line), respectively.

As can be seen, our result differs from the results reported by Matrà et al.
(2018) and Esposito et al. (2020), both reporting evidence of a statistical
relation between the radius and the luminosity (although marginal), while our
result suggest that there is no correlation between the stellar luminosity
$L_{\star}$ and the scattered-light radius $r_{0}$. This would be consistent
with the Pawellek et al. (2014) finding of no significant correlation between
$L_{\star}$ and $r_{0}$ detected in the Herschel PACS survey, and further
support for the idea that the dimensions of debris disks are likely set or
influenced by temperature independent processes like collisions, or dynamical
interaction with planetary perturber. However, we want to emphasize that most
of the Herschel detections are marginally resolved and therefore are not
directly comparable to our results. The presented sample was also derived from
different observations, with varying spatial resolution, contrast performance
at short angular separations, sensitivities, and differing modeling
strategies. One should furthermore note that, as it is the case for
GSC~07396-00759, most of the disks resolved, in particular around low-mass
stars, are highly inclined, making them favorable for detection in scattered
light, but also susceptible to projection effects and inaccurately estimated
phase functions. The influence of stellar winds on the shape of the disk, in
particular on the radial distribution of the smaller dust grains, also remains
rather unclear due to the low number of resolved disks around low mass stars
($\mbox{$L_{\star}$}<1\,\mbox{$L_{\odot}$}$). It is thus also possible that
some radii are over- or under-estimated and may not necessarily measure the
“peak” radius of the disk (i.e., the radius of the dust-producing planetesimal
belt).

Because of these potential sources of uncertainty in the scattered-light
radius measurements, the lack of correlation should be taken with caution, and
better constraints on the radii of resolved disk around low mass stars using
independent observation methods are required to better understand the relation
between the luminosity of the star, the peak of the radial dust distribution
of debris disks observed in scattered-light, and its implications for the
formation and evolution of the dust-producing parent belt.

###  5.3 Dust properties

Despite the aforementioned assumptions and caveats about our modeling
strategy, the best fitting model does account for most of the observed signal,
the probability density functions of their parameters appear well constrained,
and the estimated disk properties are compatible with the previous result by
Sissa et al. (2018). We therefore used the inferred parameterized phase
function to further investigate the dust properties of the disk around
GSC~07396-00759, as described in Sec. 3.2.3.

The $\chi^{2}$ map obtained from our fit of the parameterized phase function
is presented in Fig. 5. We found that the parameterized polarized phase
function is relatively well reproduced by small dust grains of 0.34
$\mathrm{\SIUnitSymbolMicro m}$ (see Fig. 6). The plot however also revealed a
degeneracy between the grain size and the porosity, that suggests that we are
most likely only tracing the small dust particles (0.34 –1
$\mathrm{\SIUnitSymbolMicro m}$) that are dispersed throughout the disk.
Composed of sub-$\mathrm{\SIUnitSymbolMicro m}$ sized monomers these particles
can also make up larger aggregates. As explained in Min et al. (2016), the
polarization properties of such aggregates are intimately related to the size
of the individual monomers and not to the overall size of the aggregate
itself. This degeneracy suggests that we cannot fully reconcile all key
aspects (e.g., phase function, porosity and grain size) with a single
scattering theory, a well known problem in the study of debris disks (see e.g.
the review by Hughes et al. 2018, or the studies of HR 4796 A, or HD 191089 by
Milli et al. 2017, 2019; Olofsson et al. 2019, and Ren et al. 2019,
respectively). Since the dust grains can be aggregates composed of smaller
particles, the inferred dust particle size of $s\approx 0.34$
$\mathrm{\SIUnitSymbolMicro m}$ is most likely a lower limit for the dust
grain size in the disk of GSC~07396-00759.

Nevertheless, we can use this information to try to constrain the strength of
the stellar winds. To put very rough constraints on the mass-loss rate of
GSC~07396-00759, we used Eq. 2 and Eq. 3 described in Sect. 3.1 to compute the
total net pressure force acting on a grain
($\beta=\beta_{\mathrm{RP}}+\beta_{\mathrm{SW}}$, Sezestre et al. 2017),
depending on the estimated minimum grain size $s_{\mathrm{min}}$, the blow-out
size of the dust grains, for which $\beta\leq 0.5$ (assuming the parent bodies
are on circular orbit). Consequently, all the grains that are smaller are no
longer bound to the star, and thus would be removed from the system rapidly.
$Q_{\mathrm{RP}}$, the radiation pressure efficiency averaged over the stellar
spectrum in Eq. 3, was calculated using the asymmetry parameter
$g_{\mathrm{sca}}$ which can also be computed with the $\mathtt{OpacityTool}$,
and is equal to $Q_{\mathrm{ext}}(\lambda,s)-g_{\mathrm{sca}}(s)\times
Q_{\mathrm{sca}}(\lambda,s)$.

Assuming a stellar luminosity $\mbox{$L_{\star}$}=0.13$ $L_{\odot}$, a mass of
$\mbox{$M_{\star}$}=0.62$ $M_{\odot}$, a minimum blow-out size of 0.34
$\mathrm{\SIUnitSymbolMicro m}$ for the dust grains, we estimate that the
requirement of having a sufficiently strong stellar wind is obtained for
average stellar mass-loss rates ranging from $\mbox{$\dot{M}_{\star}$}\approx
10$ $\dot{M}_{\odot}$ up to $\mbox{$\dot{M}_{\star}$}\approx 500$
$\dot{M}_{\odot}$, depending on the assumed porosity of the dust particles
with this size. For completeness we show in Fig. 11 the total net pressure
force $\beta$ and the force exerted by the radiation pressure
$\beta_{\mathrm{RP}}$ as a function of the grain size, exemplary for an
assumed minimum blow-out size of $s_{\mathrm{min}}=0.34$
$\mathrm{\SIUnitSymbolMicro m}$, and a porosity of 70%, yielding a mass-loss
rate of $\dot{M}_{\star}$$\approx 250$ $\dot{M}_{\odot}$. Although our results
are consistent with findings for similar-type stars such as AU Mic (see e.g.
Schüppler et al. 2015; Sezestre et al. 2017), with mass-loss rate between
$\mbox{$\dot{M}_{\star}$}\approx 50$ $\dot{M}_{\odot}$ and
$\mbox{$\dot{M}_{\star}$}\approx 300$ $\dot{M}_{\odot}$ for grain sizes
ranging from 0.04 $\mathrm{\SIUnitSymbolMicro m}$ up to 0.34
$\mathrm{\SIUnitSymbolMicro m}$, respectively, we cannot derive a definitive
conclusion due to the degeneracies in our modeling approach.

##  6 Conclusion

In this paper we presented high angular resolution polarimetric observations
of GSC~07396-00759 obtained with SPHERE/IRDIS in the near-infrared employing a
broadband $H$ filter. We have derived the stellar parameters from the SED,
reconstructed using photometric data ranging from the UV to the MIR
wavelengths. From the position in the Hertzsprung-Russell diagram (HRD)
combined with theoretical isochrones and mass tracks (Baraffe et al. 2015) we
estimated an upper limit for the age of $\lesssim$ 20 Myr, a stellar mass of
${\sim}0.62\,{\mbox{$M_{\odot}$}}$, and a stellar radius of
$\mbox{$R_{\star}$}\sim 0.71\,\mbox{$R_{\odot}$}$.

With the newly determined stellar parameters, we then characterized the disk
structure and modeled the dust distribution of the disk around GSC~07396-00759
using a radiative transfer model that takes into account the effects of
radiation and stellar wind pressure which is likely more efficient in low-mass
stars such as GSC~07396-00759. We found that the polarized light observation
is best described by our model by an extended disk with a dust distribution
centered at a radius $r_{0}\approx 107\pm 2$ au, with a standard deviation of
the radial distribution of $\delta_{r}\approx 27\pm 1$ au which is highly
inclined at an angle $i\approx 84.3\,\mbox{${}^{\circ}$}\pm 0.3$ and a
position angle $\phi\approx 148.7\,\mbox{${}^{\circ}$}\pm 0.1$. The derived
scattering asymmetry parameter is $g=0.60\pm 0.03$, and was estimated using a
modified Henyey-Greenstein function that accounts for the angle-dependence of
the fractional polarization, produced by the particle scattering, by adopting
the Rayleigh scattering function as approximation.

We were not able to fully recover the major axis of the disk, because of its
nearly edge-on configuration we observe in the almost uniform emission along
the disk, and measure in the polarized phase function. The anisotropic
scattering factor $g$ and consequently the radius of the disk therefore may be
overestimated, although we argue that the observations are reasonably well
reproduced by our model, and the geometric structure of the disk seen in
polarized light is consistent with total intensity images taken with SPHERE-
IRDIS by Sissa et al. (2018).

We further studied the dust properties such as the shape and size of the dust
particles using a polarized phase function model that we calculated with the
$\mathtt{OpacityTool}$, and assuming a distribution of hollow spheres with a
single grain size to fit the inferred parameterized polarized phase function.
We found that the polarized phase function is reasonably well reproduced by
small micron-sized dust grains with $s\approx 0.34$
$\mathrm{\SIUnitSymbolMicro m}$. However, the modeling results also imply that
the DPI observations may only trace the small monomers that are part of larger
aggregates (Min et al. 2016), if they are present, and therefore the deduced
dust grain size may only be a lower limit on the size of the particles
dispersed throughout the disk. The results, therefore, suggest that we cannot
reconcile all key aspects of the disk using a single scattering theory such as
the DHS theory to explain for example the shape of the phase function, or its
dependence on the dust grain size and porosity. This is a long standing
problem in the analysis of debris disks and further observations are required
to solve this issue. Such observations could be obtained for example using
SPHERE/IRDIS, with the star-hopping mode, which would allow to retrieve both
the polarized and scattered light phase function, and therefore the degree of
polarization, which contains crucial information on the shape of the phase
function, but is typically challenging to measure.

Nonetheless, the observed extent of the disk around GSC~07396-00759 is
comparable to the width of other disks around low-mass stars such as AU Mic or
GJ 581, and appear less well constrained than the dust belts resolved in
scattered light around higher mass stars such as HR 4796 A. However, the
comparison of the width as a function of the radius of the disk for 46 stars
of spectral type A to M-type and with resolved disks in scattered light to a
first approximation does not show any significant correlation between the
extent of the disk, its radius, and the spectral type of the star. As a second
diagnostic we also used the estimated radii, and analyzed their dependency on
the luminosity of the host star. From our sample of 44 disk-bearing stars with
luminosities ranging from 0.1 $L_{\odot}$ to 25 $L_{\odot}$, we found that the
$R-\mbox{$L_{\star}$}$ relation is best explained by a power-law function of
the form $R=71^{+10}_{-8}\mbox{$L_{\star}$}^{0.02\pm 0.08}$. This is
consistent with no statistically significant correlation between belt radii,
where most of the observed dust is released, and host star luminosities and
lends further support to the idea that the dimensions of debris disks are
likely set or influenced by temperature independent processes like collisions,
or dynamical interaction with planetary perturber. However, the peak
distribution of the dust particle, and by extension the location of the
planetesimal belt from which they originate, observed in scattered light may
be overestimated due to projection effects, inaccuracies in the estimated
phase function or in case of young low mass stars, like GSC~07396-00759, the
influence of the stellar activity, especially stellar winds. Thus, further
studies are needed to better understand how and why the planetesimals belts
arise, in particular around low mass stars where the distribution of small
dust particle may be significantly affected by stellar winds, present in
young, low mass stars.

Overall, we conclude that if the disk around GSC~07396-00759 is indeed this
extended ($\delta_{r}\approx 30$ au), and the disk is dominated by small dust
grains of 0.34 $\mathrm{\SIUnitSymbolMicro m}$, then the stellar winds could
be as strong as 500 times the solar mass loss rate in GSC~07396-00759, and,
thus, could play a dominant role in the transport of particle into the outer
disk that would otherwise remain closer to their parent bodies. We note,
however, that the mass-loss rates estimates should be taken with caution,
because of the observed degeneracies in our models. Furthermore, our mass-loss
rate primarily is an averaged one over time, because the considerations above
do not take into account that $\beta_{\mathrm{SW}}$ for a given dust grain
size may be time-variable because of fluctuations in the stellar flux caused
by intense and frequent flares. This is common for young and low-mass stars
such as GSC~07396-00759, although the light curves of GSC~07396-00759 obtained
with TESS and during the ASAS-SN survey do not show signs of stellar flares.
Nevertheless the coronal activity of the star suggested by the observed
variations in the UV and X-ray, may not be negligible, and, thus, should be
taken into account in future studies in order to better understand the
influence of the different pressure forces acting on the dust grains and that
are responsible for the observed morphology of the disk.

###### Acknowledgements.

We are extremely grateful to the anonymous referee for carefully reading the
manuscript, and providing very helpful comments to improve the quality of our
manuscript. C. Adam, J. O., A. B., and M. M. acknowledge support by ANID, –
Millennium Science Initiative Program – NCN19_171. J. O. acknowledges
financial support from Fondecyt (grant 1180395). A. B. acknowledges financial
support from Fondecyt (grant 1190748). A. Z. acknowledges support from the
FONDECYT Iniciación en investigación project number 11190837. T. H.
acknowledges support from the European Research Council under the Horizon 2020
Framework Program via the ERC Advanced Grant Origins 83 24 28. This research
has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This
work has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This publication makes use
of VOSA, developed under the Spanish Virtual Observatory project supported by
the Spanish MINECO through grant AyA2017-84089. VOSA has been partially
updated by using funding from the European Union’s Horizon 2020 Research and
Innovation Programme, under Grant Agreement nº 776403 (EXOPLANETS-A). This
paper includes data collected by the TESS mission, which are publicly
available from the Mikulski Archive for Space Telescopes (MAST). Funding for
the TESS mission is provided by the NASA Explorer Program. This research made
use of Lightkurve, a Python package for Kepler and TESS data analysis
(Lightkurve Collaboration, 2018).

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##  Appendix A Light curves

To measure the stellar rotation period of GSC~07396-00759 we analyze available
light-curves of GSC~07396-00759 observed with the Transiting Exoplanet Survey
Satellite (TESS, Ricker et al. 2015), and during the All-Sky Automated Survey
for Supernovae (ASAS-SN, Shappee et al. 2014; Kochanek et al. 2017),
respectively. GSC~07396-00759 was observed with TESS (600–1000 nm), covering
an observation period of about 27 days, between 2019-06-19 and 2019-07-17 and
shown in Fig. 12. The ASAS-SN light curve, on the other hand, was obtained in
the $V$-band over a time period of 2.5 years between 2016-03-10 and 2018-09-22
and is presented in Fig. 13. The period estimate was obtained using
Lightkurve, a Python package for Kepler and TESS data analysis (Lightkurve
Collaboration et al. 2018). After each light curve was pre-processed, i.e.
outlier removal and normalization was applied, we determine the stellar
rotation period $P_{\mathrm{rot}}$ from the corresponding Lomb–Scargle
periodogram (Lomb 1976; Scargle 1982). The uncertainty associated with each
period was evaluated by re-sampling the light curve using the bootstrap method
(with replacement), and corresponds to the 95% confidence interval of our
sample estimates. It should be noted, however, that these are only statistical
uncertainties and they do not represent the totality of the error budget of
the light curves. From the TESS light curve of GSC~07396-00759, we estimate a
stellar rotation period $P_{\mathrm{rot}}=11.63\pm 0.02$ d, whereas the ASAS-
SN light curve yielded a period $P_{\mathrm{rot}}=12.06\pm 0.02$ d,
respectively.

Figure 12: Left panel: Normalized TESS light curve of GSC~07396-00759,
observed between 2019-06-19 and 2019-07-17. Right panel: Lomb-Scargle
periodogram computed from these data, with an inset showing the light curve
folded over the detected stellar rotation period $P_{\mathrm{rot}}=11.63$ d.
Figure 13: Left panel: Normalized ASAS-SN light curve of GSC~07396-00759,
observed between 2016-03-10 and 2018-9-22. Right panel: Lomb-Scargle
periodogram computed from these data, with an inset showing the light curve
folded over the detected stellar rotation period $P_{\mathrm{rot}}=12.06$ d.

##  Appendix B Scattering angle

As mentioned in Sec. 3.2.3, we compute a grid of models to estimate whether we
can reproduce the parameterized phase function, using the grain size and
porosity as the only free parameters. However, as a consequence of the high
inclination of the disk and the employed inner mask not all scattering angles
between 0 and 180 degrees can be sampled. To account for this, we use the
best-fit parameters determined for the disk, i.e. $r_{0}$, $\delta_{r}$, $i$,
$\phi$, as well as the parameter of the employed masks to estimate range of
valid scattering angle. In Fig. 14 we show the results of this exercise which
yielded observable scattering angle from 9.14∘ to 170.86∘
($\theta_{\mathrm{min,max}}=90\mbox{${}^{\circ}$}\pm
80.86\mbox{${}^{\circ}$}$).

Figure 14: Distribution of possible scattering angle as function of the model
geometry obtained from our best-fit model of the disk. The inner and outer
model boundaries are indicated by the circle and the ellipse in the image.
Values within the circular mask or outside the ellipse are not considered in
the modeling. North is to the top and east to the left.

##  Appendix C Model convergence

The convergence and stability of our MCMC solution was assessed by estimating
the maximum autocorrelation length among all parameters, i.e. the average
autocorrelation time $\hat{\tau}$, after each iteration. The fitting was
considered converged when the number of iterations is larger than 100 times
the average autocorrelation time and its changes, between subsequent
iterations, are less than 1%. The evolution of the autocorrelation time as a
function of the iteration step over the course of the modeling is presented in
Fig. 15. At the end of the modeling, the average autocorrelation time was 79
steps and the mean acceptance fraction (Gelman & Rubin 1992) for our best-
fitting model was 0.47.

Figure 15: Auto-correlation time as function of MCMC iteration steps. The