# Testing the Interaction Between a Substellar Companion and a Debris Disk in
the HR 2562 System

Stella Yimiao Zhang Center for Astrophysics and Space Sciences, University of
California, San Diego, La Jolla, CA 92093, USA  Gaspard Duchêne Astronomy
Department, University of California, Berkeley; Berkeley, CA 94720, USA  Univ.
Grenoble Alpes/CNRS, IPAG, F-38000 Grenoble, France  Robert J. De Rosa
European Southern Observatory, Alonso de Córdova 3107, Vitacura, Santiago,
Chile  Megan Ansdell NASA Headquarters, 300 E Street SW, Washington, DC 20546,
USA  Quinn Konopacky Center for Astrophysics and Space Sciences, University of
California, San Diego, La Jolla, CA 92093, USA  Thomas Esposito Astronomy
Department, University of California, Berkeley; Berkeley, CA 94720, USA  SETI
Institute, Carl Sagan Center, 189 Bernardo Av, Suite 200, Mountain View CA
94043, USA  Eugene Chiang Department of Astronomy, UC Berkeley, Berkeley CA,
94720, USA  Malena Rice Department of Astronomy, Yale University, New Haven,
CT 06511, USA  Brenda Matthews  Herzberg Astronomy and Astrophysics, National
Research Council of Canada, 5071 West Saanich Rd., Victoria, BC V9E 2E7,
Canada  Paul Kalas Astronomy Department, University of California, Berkeley;
Berkeley, CA 94720, USA  SETI Institute, Carl Sagan Center, 189 Bernardo Av,
Suite 200, Mountain View CA 94043,USA  Institute of Astrophysics, FORTH,
GR-71110 Heraklion, Greece  Bruce Macintosh Kavli Institute for Particle
Astrophysics and Cosmology, Department of Physics, Stanford University,
Stanford, CA, 94305, USA  Franck Marchis SETI Institute, Carl Sagan Center,
189 Bernardo Av, Suite 200, Mountain View CA 94043, USA  Stan Metchev
Department of Physics & Astronomy, Institute for Earth and Space Exploration,
The University of Western Ontario, London, ON N6A 3K7, Canada.  Jenny Patience
School of Earth and Space Exploration, Arizona State University, PO Box
871404, Tempe, AZ 85287, USA  Julien Rameau Univ. Grenoble Alpes/CNRS, IPAG,
F-38000 Grenoble, France  Kimberly Ward-Duong Smith College, 10 Elm Street,
Northampton, MA 01063, USA  Schuyler Wolff Steward Observatory, University of
Arizona, Tucson, AZ 85721, USA  Michael P. Fitzgerald Department of Physics &
Astronomy, University of California, Los Angeles, CA 90095, USA  Vanessa P.
Bailey Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Dr., Pasadena, CA 91109, USA  Travis S. Barman Lunar and Planetary Lab,
University of Arizona, Tucson, AZ 85721, USA  Joanna Bulger  Institute for
Astronomy, University of Hawai’i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
Christine H. Chen Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore MD 21218 USA  Jeffrey K. Chilcotte Department of Physics and
Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame,
IN, 46556, USA  Tara Cotten  Department of Physics and Astronomy, University
of Georgia, Athens, GA 30602, USA  René Doyon Institut de Recherche sur les
Exoplanètes, Département de physique, Université de Montréal, Montréal, QC H3C
3J7, Canada  Katherine B. Follette Physics and Astronomy Department, Amherst
College, 25 East Drive, Amherst, MA 01002, USA  Benjamin L. Gerard University
of Victoria, Department of Physics and Astronomy, 3800 Finnerty Rd, Victoria,
BC V8P 5C2, Canada  National Research Council of Canada Herzberg, 5071 West
Saanich Rd, Victoria, BC, V9E 2E7, Canada  Stephen Goodsell Department of
Physics, Durham University, Stockton Road, Durham DH1, UK  Gemini Observatory,
Casilla 603, La Serena, Chile  James R. Graham  Astronomy Department,
University of California, Berkeley; Berkeley, CA 94720, USA  Alexandra Z.
Greenbaum IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena,
CA 91125, USA  Pascale Hibon European Southern Observatory, Alonso de Córdova
3107, Vitacura, Santiago, Chile  Li-Wei Hung  Natural Sounds and Night Skies
Division, National Park Service, Fort Collins, CO 80525, USA  Patrick Ingraham
Vera C. Rubin Observatory, 950 N Cherry Ave, Tucson AZ, 85719, USA  Jérôme
Maire  Center for Astrophysics and Space Sciences, University of California,
San Diego, La Jolla, CA 92093, USA  Mark S. Marley Space Science Division,
NASA Ames Research Center, Mail Stop 245-3, Moffett Field CA 94035, USA
Christian Marois Herzberg Astronomy and Astrophysics, National Research
Council of Canada, 5071 West Saanich Rd., Victoria, BC V9E 2E7, Canada
University of Victoria, Department of Physics and Astronomy, 3800 Finnerty Rd,
Victoria, BC V8P 5C2, Canada  Maxwell A. Millar-Blanchaer Department of
Physics, University of California, Santa Barbara, Santa Barbara, CA, USA  Eric
L. Nielsen Department of Astronomy, New Mexico State University, 1320 Frenger
Mall, Las Cruces, NM 88003-8001, USA  Rebecca Oppenheimer American Museum of
Natural History, Department of Astrophysics, Central Park West at 79th Street,
New York, NY 10024, USA  David W. Palmer  Lawrence Livermore National
Laboratory, 7000 East Ave, Livermore, CA, 94550, USA  Marshall D. Perrin Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Lisa A. Poyneer  Lawrence Livermore National Laboratory, 7000 East Ave,
Livermore, CA, 94550, USA  Laurent Pueyo  Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA  Abhijith Rajan Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Fredrik T. Rantakyrö Gemini Observatory, Casilla 603, La Serena, Chile  Jean-
Baptiste Ruffio Department of Astronomy, California Institute of Technology,
Pasadena, CA 91125, USA  Dmitry Savransky Sibley School of Mechanical and
Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA  Adam C.
Schneider US Naval Observatory, Flagstaff Station, P.O. Box 1149, Flagstaff,
AZ 86002, USA.  Department of Physics and Astronomy, George Mason University,
MS3F3, 4400 University Drive, Fairfax, VA 22030, USA  Anand Sivaramakrishnan
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218,
USA  Inseok Song Department of Physics and Astronomy, University of Georgia,
Athens, GA 30602, USA  Remi Soummer Space Telescope Science Institute, 3700
San Martin Drive, Baltimore, MD 21218, USA  Sandrine Thomas Vera C. Rubin
Observatory, 950 N Cherry Ave, Tucson AZ, 85719, USA  Jason J. Wang Department
of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA)
and Department of Physics and Astronomy, Northwestern University, Evanston, IL
60208, USA  Sloane J. Wiktorowicz Remote Sensing Department, The Aerospace
Corporation, 2310 E. El Segundo Blvd., El Segundo, CA 90245

###### Abstract

The HR 2562 system is a rare case where a brown dwarf companion resides in a
cleared inner hole of a debris disk, offering invaluable opportunities to
study the dynamical interaction between a substellar companion and a dusty
disk. We present the first ALMA observation of the system as well as the
continued GPI monitoring of the companion’s orbit with 6 new epochs from 2016
to 2018. We update the orbital fit and, in combination with absolute
astrometry from GAIA, place a 3$\sigma$ upper limit of 18.5 $M_{J}$ on the
companion’s mass. To interpret the ALMA observations, we used radiative
transfer modeling to determine the disk properties. We find that the disk is
well resolved and nearly edge on. While the misalignment angle between the
disk and the orbit is weakly constrained due to the short orbital arc
available, the data strongly support a (near) coplanar geometry for the
system. Furthermore, we find that the models that describe the ALMA data best
have an inner radius that is close to the companion’s semi-major axis.
Including a posteriori knowledge of the system’s SED further narrows the
constraints on the disk’s inner radius and place it at a location that is in
reasonable agreement with, possibly interior to, predictions from existing
dynamical models of disk truncation by an interior substellar companion. HR
2562 has the potential over the next few years to become a new testbed for
dynamical interaction between a debris disk and a substellar companion.

debris disks — substellar companion stars — brown dwarfs — orbit determination
— gravitational interaction

##  1 Introduction

Debris disks are the gas poor disk structures surrounding stars as the outcome
of the star formation process. The presence of a debris disk suggests that
there needs to be larger bodies colliding and grinding down the dust grains in
order to sustain the disk. Furthermore, observable features like gaps and
warps in disk morphology can be the results of the dynamical interaction
between a substellar companion and the disk itself. These features provide
another pathway for us to find and study the properties of these potential
planets residing near the disk. The most readily available example of this
would be our solar system where substructures within the Asteroid Belt and the
Kuiper Belt were created from the resonance between the planet orbits and
smaller objects (e.g., Tsiganis, 2010). A great amount of information can be
extracted from studying the dynamical interactions that would create such
features in morphology in disk structures, even in the case where the
companions are not directly detected. Sophisticated dynamical models have been
developed to constrain properties of the companion from the properties of the
disk and the system.

However, it is in fact difficult to find appropriate systems to test these
models. Many of the currently discovered debris disks often do not have the
required resolution to study the disk structures, and in those resolved disks,
the detected companions are often found too far away from the disruption site
to be solely responsible. Out of the systems where a companion is detected
close to the irregular structure of the disk, there often isn’t convincing
evidence for disk-companion interaction. As of today, only a few systems were
discovered to be candidates for disk-companion interaction. The first one of
this kind is $\beta$ Pictoris, a system with an almost perfectly edge on
debris disk with a vertical warp at 85 au, and a planet $\beta$ Pictoris b
that is consistent with this warp (Smith, 1987; Augereau & Beust, 2006;
Lagrange et al., 2009, 2010). It is believed that the inner warp of the disk
is a result of the dynamical interaction between the planet $\beta$ pic b and
the disk, but studying this interaction is difficult due to the radial density
profile being model-dependent (Dawson et al., 2011; Augereau & Beust, 2006).
Only a handful of similar systems were discovered in the decades following the
discovery of $\beta$ Pictoris. Those systems all have potential dynamically
interacting planets/brown dwarf candidates residing within the disks’ rings,
but all of them lack determining pieces of evidence due to observational
limitations or other possible scenarios for the disk structure formation
(Wilner et al., 2018; Wang et al., 2021; Musso Barcucci et al., 2019; Chauvin
et al., 2018; Su et al., 2017).

So far, only two systems have directly imaged brown dwarf companions orbiting
inside the debris ring: HD 206893 (Nederlander et al., 2021; Marino et al.,
2020) and HR 2562. In the former, it is unclear how close the companion is to
the disk and whether it is responsible for truncating it. In this paper, we
focus on analyzing the potential for dynamical interaction in HR 2562. HR 2562
is an F5V star with a mass of 1.3$M_{\odot}$, located at 34pc away from the
Sun (Gray et al., 2006; Casagrande et al., 2011; van Leeuwen, 2007a). Its
debris disk was first imaged by Herschel (Moór et al., 2006). In 2016,
Konopacky and the GPI team were able to directly image and obtain data for the
orbit of the substellar companion, HR 2562 B, residing within the inner hole
of the disk with GPIES. The orbit of HR 2562 B was further monitored by
VLT/SPHERE for 10 months from 2016-2017, providing support for a coplanar
geometry (Maire et al., 2018), and its spectral type was characterized as
T2-T3 by IFS and IRDIS of VLT/SPHERE (Mesa et al., 2018). By comparison to
evolutionary models, Konopacky et al. (2016) estimated a mass of 30$\pm$15
$M_{\text{jup}}$ for the companion, with the uncertainty dominated by the
poorly constrained age of the system. However, concrete evidence of brown
dwarf-disk interaction would require a determination of the inner radius of
the disk, which the initial characterization of the disk was unable to
constrain due to limited angular resolution (Moór et al., 2015). This led to
an uninformative upper limit of $\sim 0.24M_{\odot}$ based solely on dynamical
arguments (Konopacky et al., 2016). Continued monitoring of the orbit as well
as better resolved observation of the disk are required to better characterize
the disk, derive evidence for dynamical interaction and further constrain the
properties of the companion.

In this paper, we present the new Atacama Large Millimeter/submillimeter Array
(ALMA) observation of the HR 2562 system (section 2.1) as well as the updated
GPI observation of the companion (section 2.2). We present the reduced ALMA
image in section 3.1 and discuss the companion orbit fit, the coplanar
scenario as well as the dynamical mass limit in section 3.2. The Monte Carlo
Markov Chain (MCMC) fit for the ALMA image and the analysis are detailed in
section 4, and further discussion on the system geometry, SED selection of the
best fitting models and the dynamical interaction are presented in section 5.
We summarize our results and conclude in section 6.

##  2 Observation

###  2.1 ALMA Observation

We observed the HR 2562 debris disk with the Atacama Large Millimeter Array
(ALMA, project 2016.1.00880.S, PI: G. Duchêne) on May 15th, 2018. To maximize
sensitivity for faint dust thermal emission in the system, we observed the
target in Band 7 for a total on-source integration time of 37 min. Observing
conditions were good with 0.8 mm of precipitable water vapor at zenith. As a
compromise between resolving the Herschel-estimated inner radius of the disk
and minimizing spatial filtering on large scales, we selected configuration
C40-1 with the 12m-array, with 44 antennas providing baselines ranging from 14
to 313 m and an angular resolution of about 1$\farcs$1\. As this is a pure
continuum observation, we used four 2 GHz bandwidth channels, centered at
336.5, 338.5, 348.5 and 350.5 GHz. Observations in all four bands are
ultimately combined in a single continuum map at an effective frequency of
343.5 GHz (870 $\mu$m).

The data were processed with standard routines from the Common Astronomy
Software Applications (CASA, McMullin et al., 2007), version 5.3.0-143.
Specifically, we applied phase, bandpass and flux calibrations using the
provided calibrators. To produce the final continuum map, we used the task
tclean with Briggs weighting with robust=0.5. This results in a beam size of
1$\farcs$10$\times$1$\farcs$06 at a position angle of -132$\fdg$3 and a rms of
0.0323 mJy/beam. The left panel of Figure 1 shows the final ALMA map generated
with an 0$\farcs$2 pixel scale.

Figure 1: Left panel: ALMA data of the HR 2562 system at 870 micron. The (0,0)
of the images correspond to RA= 102.5°, DEC=-60.2°; Right panel:a zoomed in
region of the center of the disk is shown on the top left corner. The black
circle represents the center of the disk determined by 2D Gaussian fit; red
star represents the Gaia DR3 coordinates of the star HR 2562; blue circle
represents the orbit predicted location of the companion HR 2562 B on Jan
31st, 2018. The disk center and the star are well aligned, and the companion
is close to the star as expected.

###  2.2 GPI Observation

Table 1: HR 2562 Gemini/GPI observing log and associated KLIP parameters

UT Date | Filter | $N_{\rm exp}$ | $t_{\rm int}$ |  $\Delta$PAaaTotal field rotation over the duration of the observing sequence. |  $\lambda_{\rm min}$–$\lambda_{\rm max}$bbFull spectral range used in the data reduction. |  $n_{\lambda}$ccNumber of independent spectral channels. |  $m$ddMinimum rotation-induced displacement for inclusion in the PSF subtraction process. |  $n_{\rm KL}$eeNumber of KLIP modes used in the PSF subtraction process.
---|---|---|---|---|---|---|---|---
|  |  | (sec.) | (°) | ($\micron$) |  | (px) |
2016 Jan 25$\dagger$$\dagger$footnotemark:  | $H$ | 33 | $59.6$ | 19.4 | 1.514–1.778 | 35 | 3 | 5
2016 Jan 28$\dagger$$\dagger$footnotemark:  | $K_{1}$ | 21 | $59.6$ | 10.6 | 1.947–2.173 | 26 | 3 | 5
2016 Jan 28$\dagger$$\dagger$footnotemark:  | $K_{2}$ | 20 | $59.6$ | 10.5 | 2.119–2.226 | 15 | 3 | 5
2016 Feb 25$\dagger$$\dagger$footnotemark:  | $K_{2}$ | 43 | $59.6$ | 25.7 | 2.116–2.224 | 15 | 3 | 5
2016 Feb 28$\dagger$$\dagger$footnotemark:  | $J$ | 53 | $59.6$ | 26.6 | 1.137–1.330 | 35 | 3 | 5
2016 Dec 17 | $K_{1}$ | 85 | $59.6$ | 40.7 | 1.940–2.173 | 26 | 4 | 5
2017 Feb 13 | $H$ | 19 | $59.6$ | 12.3 | .507–1.778 | 35 | 3 | 5
2017 Nov 29 | $H$ | 50 | $59.6$ | 30.1 | 1.511–1.773 | 35 | 3 | 5
2018 Jan 31 | $K_{2}$ | 54 | $88.7$ | 32.9 | 2.113–2.228 | 15 | 4 | 5
2018 Mar 10 | $Y$ | 66 | $59.6$ | 42.0 | 0.957–1.132 | 35 | 3 | 5
2018 Nov 19 | $H$ | 24 | $59.6$ | 12.5 | 1.509–1.779 | 35 | 3 | 5

††footnotetext: Re-reduction of observations presented in Konopacky et al.
(2016).

####  2.2.1 Observations & Initial reduction

The Gemini Planet Imager (GPI, Macintosh et al., 2014) is an instrument
equipped with a high-order adaptive optics (AO) system (Poyneer et al., 2014,
2016), an apodized Lyot coronagraph (Soummer et al., 2011), and both a
dispersing and a Wollaston prism for spectroscopic and polarimetric
observations. The instrument was designed to achieve high contrast at small
angular separations, providing sensitivity to substellar companions and
circumstellar material around nearby, bright stars. HR 2562 was observed with
GPI on 11 separate epochs between 2016 and 2018 under program IDs
GS-2015B-Q-501 and GS-2017B-Q-501. The first four epochs were originally
analyzed and published in Konopacky et al. (2016) but are re-reduced and
analyzed here to ensure consistency. The observing strategy was similar for
each dataset. The target was observed with GPI’s coronagraphic mode with the
specific coronagraph optimized for the near-infrared filter being used. After
the coronagraph, a lenslet array and dispersing prism was used to disperse the
light at each point within the $2\farcs 8\times 2\farcs 8$ field into a low-
resolution ($R\sim$35–80) spectrum. The resulting dispersed field was imaged
with GPI’s integral field spectrograph (IFS, Chilcote et al., 2012; Larkin et
al., 2014). In each dataset the star itself was used as the AO guide star. The
observations were timed to be taken close to the meridian passage of the star
over the observatory to maximize field rotation, $\Delta$PA, for angular
differential imaging (ADI, Marois et al., 2006). Observations of an argon lamp
were taken during the target acquisition to measure the instrument flexure
induced by the changing gravity vector as the telescope changes position.
Standard dark and wavelength calibration frames were taken during the daytime
as a part of the observatory’s calibration plan. A summary of the observations
are given in Table 1.

The raw and associated calibration data were reduced through two separate
processes. The raw data were reduced using the GPI Data Reduction Pipeline
(DRP, Perrin et al., 2014) v1.5.0 revision cafd46a. This version of the
pipeline resolved several issues identified with the calculation of the
average parallactic angle during an exposure (De Rosa et al., 2020b). Although
the magnitude of this correction was small for the datasets presented in
Konopacky et al. (2016), between 0$\fdg$09–0$\fdg$12 (see Figure 17 in De Rosa
et al., 2020b), we took the opportunity to reprocess these data along with the
new epochs presented in this work. The data were reduced using the procedure
outlined in (De Rosa et al., 2020c). Briefly, the raw images were dark
subtracted and bad pixels were identified using a combination of a static bad
pixel map and outlier rejection and replaced. The microspectra within each
image were extracted to create a three-dimensional $(x,y,\lambda)$ data cube
containing the low-resolution spectra at each point within the field of view.
Additional bad pixel identification was performed using outlier rejection.
Distortion and anamorphism was corrected using a static distortion map applied
to each slice within the data cube. The position and brightness of the star
within each slice of each reduced data cube was estimated by measuring the
four satellite spots generated by a wire diffraction grid within the pupil
plane (Sivaramakrishnan & Oppenheimer, 2006). The calibration data used to
reduce the science data were not affected by the pipeline changes described in
De Rosa et al. (2020b). We used the dark frames and wavelength calibrations
generated by the GPIES Data Cruncher (Wang et al., 2018) created using v1.4.0
of the pipeline to reduce the science data rather than reprocessing them with
the updated pipeline.

####  2.2.2 PSF subtraction and astrometry

Figure 2: The GPI PSF estimated from the four satellite spots (left column),
BKA forward model (second column), the GPI images of HR 2562 B (third column),
and residuals (fourth column) for each observation. The color scales are
different for each dataset and are denoted by the color bar on the right. The
KLIP parameters used for each reduction are given in Table 1.  Table 2:
Relative astrometry between HR 2562 A and HR 2562 B

UT Date | MJD | Instrument | Filter | Plate scale | North offset | $\rho$ | $\theta$ | Ref.
---|---|---|---|---|---|---|---|---
|  |  |  | (mas px-1) | (deg) | (mas) | (deg) |
2016 Jan 25 | 57412.1335 | Gemini-S/GPI | $H$ | $14.161\pm 0.021$ | $0.21\pm 0.23$ | $615.05\pm 1.76$ | $297.83\pm 0.27$ | 1
2016 Jan 28 | 57415.1731 | Gemini-S/GPI | $K_{1}$ | $14.161\pm 0.021$ | $0.21\pm 0.23$ | $612.56\pm 1.46$ | $297.76\pm 0.25$ | 1
2016 Jan 28 | 57415.2002 | Gemini-S/GPI | $K_{2}$ | $14.161\pm 0.021$ | $0.21\pm 0.23$ | $613.47\pm 1.28$ | $298.09\pm 0.24$ | 1
2016 Feb 25 | 57443.0343 | Gemini-S/GPI | $K_{2}$ | $14.161\pm 0.021$ | $0.21\pm 0.23$ | $616.09\pm 1.23$ | $297.71\pm 0.24$ | 1
2016 Feb 28 | 57446.0951 | Gemini-S/GPI | $J$ | $14.161\pm 0.021$ | $0.21\pm 0.23$ | $616.44\pm 1.37$ | $297.78\pm 0.25$ | 1
2016 Dec 12 | 57734.2647 | VLT/SPH-IRD | $H$ | $12.251\pm 0.009$ | $-1.808\pm 0.043$ | $637.80\pm 6.40$ | $297.81\pm 0.54$ | 2
2016 Dec 17 | 57739.3123 | Gemini-S/GPI | $K_{1}$ | $14.161\pm 0.021$ | $0.32\pm 0.15$ | $639.89\pm 1.27$ | $297.98\pm 0.17$ | 1
2017 Feb 07 | 57791.1114 | VLT/SPH-IRD | $H$ | $12.251\pm 0.009$ | $-1.712\pm 0.058$ | $644.00\pm 2.30$ | $297.82\pm 0.19$ | 2
2017 Feb 13 | 57797.0517 | Gemini-S/GPI | $H$ | $14.161\pm 0.021$ | $0.32\pm 0.15$ | $644.20\pm 1.22$ | $298.23\pm 0.16$ | 1
2017 Sep 29 | 58025.3826 | VLT/SPH-IRD | $K_{1}$ | $12.267\pm 0.009$ | $-1.735\pm 0.043$ | $661.20\pm 1.30$ | $297.97\pm 0.16$ | 2
2017 Sep 29 | 58025.3826 | VLT/SPH-IRD | $K_{2}$ | $12.263\pm 0.009$ | $-1.735\pm 0.043$ | $658.90\pm 1.60$ | $298.08\pm 0.17$ | 2
2017 Nov 29 | 58086.3110 | Gemini-S/GPI | $H$ | $14.161\pm 0.021$ | $0.28\pm 0.19$ | $664.82\pm 2.04$ | $298.37\pm 0.24$ | 1
2018 Jan 31 | 58149.1962 | Gemini-S/GPI | $K_{2}$ | $14.161\pm 0.021$ | $0.28\pm 0.19$ | $669.44\pm 1.24$ | $298.55\pm 0.20$ | 1
2018 Mar 10 | 58187.0475 | Gemini-S/GPI | $Y$ | $14.161\pm 0.021$ | $0.28\pm 0.19$ | $670.84\pm 2.83$ | $298.74\pm 0.26$ | 1
2018 Nov 19 | 58441.3261 | Gemini-S/GPI | $H$ | $14.161\pm 0.021$ | $0.45\pm 0.11$ | $685.76\pm 1.25$ | $298.89\pm 0.13$ | 1

References. — (1) - this work; (2) - Maire et al. (2018).

Although GPI can routinely achieve high Strehl ratios on bright stars, it does
not offer perfect suppression of starlight. Within about $1\arcsec$ the images
are dominated by residual quasi-static speckles caused by AO fitting errors
and non-common path aberrations. This residual light often has a pronounced
azimuthal asymmetry in the direction of the jet stream (Madurowicz et al.,
2018), which can result in a significant azimuthal dependence on the achieved
contrast. We took advantage of both ADI and spectral differential imaging
(SDI, Smith, 1987; Racine et al., 1999)) to subtract these quasi-static
speckles from each slice within each data cube. We used pyKLIP (Wang et al.,
2015), a Python implementation of the Karhunen–Loève image projection
algorithm (KLIP, Soummer et al., 2012; Pueyo et al., 2015)), to model and
subtract the residual starlight within each image. Due to the distorting
effects of this algorithm on the PSF of the companion within the PSF-
subtracted image, we used the forward-model based Bayesian KLIP-FM astrometry
package (BKA, Wang et al., 2016)) to model the effect of KLIP on the PSF of
the companion (see Figure 2). We used the average of the four satellite spots
as the instrumental PSF, and low-throughput channels were excluded.

We used the same approach for the BKA forward-modelling as in De Rosa et al.
(2020c). The PSF subtraction and forward model was calculated within a single
annulus centered on the star with a width of either 16 px ($Y$, $J$, $H$) or
20 px ($K_{1}$, $K_{2}$). The radius of the annulus was selected to center the
companion between the inner and outer edge. We explored the effect of varying
two of the main tunable parameters: the movement criteria, $m$, that defines
the minimum number of pixels an astrophysical source needs to have moved by
before an image can be included within the reference PSF library, and the
number of KL modes, $n_{\rm KL}$, used to reconstruct the stellar PSF. We
measured astrometry using BKA for each combination of these parameters. The
forward model of the PSF was compared to the companion within a small
$11\times 11$ px box ($15\times 15$ px for $K_{1}$ and $K_{2}$). Posterior
distributions of the position and flux of the companion, and the correlation
length scale (a nuisance parameter to marginalize over) were sampled using the
Markov chain Monte Carlo (MCMC) affine-invariant sampler within the emcee
package (Foreman-Mackey et al., 2013). We advanced 100 walkers that were
initialized near the expected parameter values for 1000 steps, discarding the
first 200 as burn-in.

Using this set of measurements we investigated the effect of the KLIP
parameters on the companion astrometry. Small values of $m$ and large values
of $n_{\rm KL}$ can cause significant self-subtraction or over-subtraction of
the companion, an effect most pronounced at the longer wavelengths where the
PSF is larger. We found a significant correlation between these two parameters
and the position of the companion for the two $K_{2}$ datasets, most
pronounced in the 2018 Jan 31 epoch. For the other epochs at shorter
wavelengths no significant correlation was observed. The adopted KLIP
parameters are listed in Table 2.2.2. A low value of $n_{\rm KL}$ was adopted
given the relative brightness of the companion, and a high value of $m$ to
minimize self-subtraction especially at longer wavelengths. The posterior
distributions of the companion position measured using the adopted parameter
set were combined with the instrument plate scale and orientation calibrations
from De Rosa et al. (2020b) to yield on-sky relative astrometric measurements
for each epoch (Table 2.2.2). An image of the instrumental PSF, forward model,
companion, and residuals are shown for each dataset in Figure 2.

##  3 Result

###  3.1 ALMA image

As shown in Figure 1, the disk is clearly detected and well resolved along the
major axis at 870 $\mu$m. Placing a beam-sized aperture around the brightest
pixel in the image, we evaluate a peak signal to noise of $\sim$7 in the ALMA
image. Owing to spatial filtering by the interferometer, the reconstructed
image has negative pixels surrounding the disk. This makes it difficult to
measure the total flux of the disk directly from the ALMA image. To correct
for this, we instead report a total flux of 3.3mJy from integrating the best
fit model image (see section 4.1). These observations extend the system’s SED
to a longer wavelength than all prior observations, most notably beyond the
range of Herschel. With an angular resolution that is $\approx$6 times higher
than Herschel’s, the disk geometry is much more clearly apparent than in
previous studies (Moór et al., 2015). The highly elongated structure reveals
the disk to be nearly edge-on. Fitting a 2D Gaussian to the map results in a
FWHM of 8$\farcs$17 along the major axis and 0$\farcs$97 along the minor axis
of the disk. The aspect ratio of the two corresponds to an inclination of
$\approx$ 83°, assuming a thin disk. We notice that the FWHM of the minor axis
is marginally smaller than the size of the beam, indicating the disk is
unresolved along the minor axis. Therefore, the $\approx$83° inclination we
inferred from the aspect ratio is a strict lower limit.

To verify the alignment between the detected emission and the location of the
star, we use the absolute astrometry information of the ALMA map, which is
known to a precision of about 0$\farcs$1 based on the pointing calibration.
The position of the star is retrieved from the Gaia EDR3 Release, taking into
account its proper motion and the epoch of the ALMA observation (Gaia
Collaboration et al., 2021). The star and the center of the disk, as defined
by our 2D Gaussian fit, are within 0$\farcs$07 of each other, i.e., consistent
with a symmetric disk about the star (see Figure 1). The location of the star
and the center of the disk are shown in right panel of Figure 1.

From the image, a clump-like structure can be observed in the southeast side
of the disk. This clump also manifests in the surface brightness profile of
the disk shown in Figure 3, where we see a peak between 2” and 6”. We show a
horizontally flipped profile in dashed line in Figure 3 , from which we can
see that the peak stands at a marginally significant ($\sim 2.5\sigma$)
difference measuring from the height of the brightness profile at the
horizontally reflected location of the peak. There are a few plausible
explanations for this feature. 1) It is due to the random noise of the image;
2) The trough is a real feature that represents a real physical underdensity,
e.g. a gap or spiral in the disk, thus making the disk asymmetrical; 3) The
trough was the result of a contamination from an unrelated point source,
presumably a background submillimeter galaxy. Whilst a physical disk origin
cannot be ruled out, the signal-to-noise ratio means cannot say with
confidence that the feature is real. In addition, although it is possible that
there might be previously unseen background galaxy, the chance of it aligning
so well with the disk is low. We will therefore model the disk with a
symmetric profile in the analysis that follows, but note that future
observations may better determine the nature of the trough.

Figure 3: 870 $\mu$m brightness profile along the disk major axis. The yellow
shade indicates the 1 $\sigma$ uncertainty as determined from the surrounding
background areas. The dashed line represent a horizontally reflected
brightness profile about the location of the star.

###  3.2 Orbit of the brown dwarf companion

####  3.2.1 Unconstrained fit

The orbit of the companion was fitted to the measured astrometry in Table
2.2.2 using a modified version of the procedure described in De Rosa et al.
(2020c). The visual orbit can be described using the standard Campbell
elements; the period ($P$), semi-major axis ($a$), eccentricity ($e$),
inclination ($i$), argument of periastron ($\omega$), position angle of the
ascending node ($\Omega$), and the time of periastron ($T_{0}$). Here, we use
the standard convention that $\Omega$ refers to the position angle that the
companion passes through the plane tangent to the sky at the location of HR
2562, moving away from the observer. We substituted two of these elements when
fitting the orbit of HR 2562. $P$ was replaced with the total system mass
($M_{\rm total}$) as the period is currently unconstrained given the
measurements, and $T_{0}$ was replaced with the dimensionless parameter $\tau$
describing the time of the next periastron passage in fractions of the orbital
period since MJD 57412.1335, the start of the astrometric record. In addition
to these seven elements we also required the distance to the star, represented
by the parallax ($\varpi$), in order to link the angular semi-major axis to
the system mass to derive the orbital period.

We used the parallel-tempered affine-invariant Markov chain Monte Carlo (MCMC)
sampler within the emcee package (Foreman-Mackey et al., 2013) to sample the
posterior distribution of these eight parameters describing the visual orbit
of HR 2562 B. We performed the sampling using the parameters $\log a$ and
$\cos i$ to enforce a log-uniform prior probability density function (PDF) on
$a$ and a sine prior PDF on $i$ rather than computing the prior probability at
each step. Gaussian priors were used for $M_{\rm total}$ ($1.31\pm 0.13$
$M_{\odot}$; Moór et al., 2015), under the assumption that $M_{2}\ll M_{1}$,
and $\varpi$ ($29.4738\pm 0.0185$ mas; Gaia Collaboration et al., 2021). We
advanced 512 chains at each of 16 different temperatures for one million
steps. In the parallel-tempered framework the lowest temperature chains sample
the posterior probability distribution, while the highest temperature chains
sample the prior probability distribution. We saved every hundredth sample of
each chain to disk, and conservatively discarded the first half of each as a
“burn-in”.

The median and one-sigma confidence intervals for the fitted and derived
parameters are shown in Table 3, along with the set of parameters describing
the maximum likelihood (minimum $\chi^{2}$) and maximum probability orbit. A
random selection of visual orbits consistent with the relative astrometry are
shown in Figure 4. The posterior PDFs for a subset of the fitted and derived
parameters are shown in Figure 5 (black contours). We found a strong anti-
correlation between the eccentricity and the inclination of the orbit; less
eccentric orbits tend to have a more edge-on configuration ($i\sim 84$ °).
Although very high eccentricities are seemingly preferred based on the shape
of the PDF, the eccentricity of the orbit is not yet well constrained and
still dependent on the input priors. We discuss this further in section 3.2.4.

####  3.2.2 Co-planar scenario

The results regarding the orbit of the companion presented so far make no
assumptions regarding the alignment of the orbital plane with that of the
debris disk resolved in the ALMA observations presented in Section 2.1. We
repeated the orbit fitting procedure described above with an additional prior
on the inclination ($i$) and position angle of the ascending node ($\Omega$)
to investigate the properties of orbits consistent with both the measured
astrometry and a near co-planar configuration with respect to the external
debris disk. We used Gaussian kernel density estimation (KDE) to construct a
prior probability density function on the orbital inclination ($i$) and
position angle ($\Omega$) from the disk fitting MCMC samples (Sec. 4). We used
the scipy.gaussian_kde function with Scott’s method for the estimator
bandwidth calculation. The resulting two-dimensional prior distribution is
shown in Figure 6, along with the two marginalized distributions. The Gaussian
KDE is a good match to the posterior distributions describing the disk
geometry from the ALMA observations presented in Section 4. The additional
term in the prior probability was calculated using the Gaussian KDE and the
value of $i$ and $\Omega$ at each step in the MCMC process. We accounted for
the ambiguity in these two parameters when fitting to the ALMA data by
evaluating the Gaussian KDE at four possible combinations of $i$ and $\Omega$:
($i$, $\Omega$), ($i$, $\Omega+\pi$), ($\pi-i$, $\Omega$), and ($\pi-i$,
$\Omega+\pi$). The maximum of these four values was used as the prior
probability at this step for this combination of $i$ and $\Omega$.

The median and 1-sigma credible intervals for the posterior distributions on
the fitted and derived parameters from this analysis are given in Table 3, in
addition to the maximum likelihood and maximum probability orbit. The
posterior distributions of a subset of the fitted and derived parameters are
compared to those calculated without the additional prior in Figure 5. The
full corner plot is presented in Appendix B. The additional prior restricts
the range of orbital eccentricities that are consistent with the measured
astrometry in that very eccentric orbits ($e\gtrsim 0.8$) require increasingly
misaligned configurations with respect to the outer disk when considering the
general fit. The other orbital parameters are not significantly affected by
the prior, in particular the apoastron distance is unchanged due to the anti-
correlation between the orbital eccentricity and semi-major axis.

####  3.2.3 Dynamical mass constraint

We used the relative astrometry between the star and companion presented here
and absolute astrometric measurements of the photocenter from the Hipparcos
and Gaia satellites to constrain the mass of the companion. With a
sufficiently high mass for the companion, the photocenter of the system will
be perturbed from linear motion through space due to the orbit of the star
around the barycenter of the system. For Hipparcos we used the intermediate
astrometric data (IAD), i.e., the residuals along the scan direction from the
best fit astrometric solution (van Leeuwen, 2007b). These astrometric
measurements of HR 2562 were best fit with a non-standard “stochastic”
solution where the uncertainties on the individual measurements are inflated
to reduce the goodness of fit statistic below an acceptable threshold for the
final catalogue (van Leeuwen, 2007a). The need for this type of solution can
be attributed to unresolved orbital motion on a time-scale much shorter than
the three year duration of the mission, or to remaining modelling noise from
the construction of the catalogue (van Leeuwen, 2007b).

For Gaia we used the astrometric parameters and correlation coefficients
reported in the Early Data Release 3 catalogue (EDR3; Gaia Collaboration et
al., 2021). We compared the various goodness of fit metrics reported in the
EDR3 catalogue for HR 2562 to those of stars with a similar brightness. As
illustrated in Appendix A, HR 2562 appears relatively well behaved compared to
these other sources. We also collected the timings of the individual Gaia
measurements using the Gaia Observation Forecast
Tool111https://gaia.esac.esa.int/gost/, excluding those taken during satellite
down-times reported in Lindegren et al. (2021).

The procedure used in De Rosa et al. (2020a) for a joint fit of absolute
astrometry and radial velocity measurements was adapted to fit the absolute
and relative astrometric measurement available for HR 2562, as well as
updating it to account for the change in reference epoch between Gaia DR2 and
EDR3. The model that simultaneously describes the orbit of the companion
around HR 2562 and the orbit of the photocenter around the barycenter of the
system consists of 14 parameters. Seven parameters are used in the orbit fit
described in Section 3.2.1 ($\log a$, $\cos i$, $e$, $\omega$, $\Omega$,
$\tau$, $\varpi$), two describe the masses of the components ($M_{1}$, $\log
M_{2}$), four describe offsets between the Hipparcos catalogue astrometry and
the barycenter position and proper motion at 1991.25 ($\Delta\alpha^{\star}$,
$\Delta\delta$, $\Delta\mu_{\alpha^{\star}}$, $\Delta\mu_{\delta}$), and one
is an error inflation term applied in quadrature to the Hipparcos IAD
($\epsilon_{\Lambda}$). To calculate the semi-major axis of the photocenter
orbit we used an empirical mass-magnitude relationship to determine the
relative fluxes of the two components in the Hipparcos and Gaia passbands when
$M_{2}>0.077$ $M_{\odot}$ (Pecaut & Mamajek, 2013). For lower masses we
assumed that the companion is emitting no flux, and that the photocenter is
coincident with the location of the host star.

We used the same MCMC sampler as in Section 3.2.1 to sample the posterior
distributions of the fourteen parameters in this model. We used Gaussian
priors for the parallax ($\varpi=29.4738\pm 0.0185$ mas) and for the mass of
the primary ($M_{1}=1.31\pm 0.13$ $M_{\odot}$), uniform priors were used for
the remaining parameters. We advanced 512 chains at each of 16 different
temperatures for one million steps, saving every tenth sample to disk. The
first half of each chain was discarded as a “burn-in”.

The median and one-sigma confidence intervals for the fitted and derived
parameters are shown in Table 3, along with the set of parameters describing
the maximum likelihood and maximum probability orbit. The covariance between
the eccentricity, inclination, and mass of the companion is shown in Figure 7.
The eccentricity and inclination exhibit a similar anti-correlation that is
seen for the visual orbit fit (Figure 5), albeit with a slight enhancement of
orbits with a moderate eccentricity ($e\sim 0.5$). The shape of the $M_{2}$
posterior probability distribution (Figure 7, lower right panel) clearly
demonstrates that we do not yet have a statistically significant measurement
of the mass of the companion. Instead, we can only place an upper limit of
$M_{2}<18.5$ $M_{\rm Jup}$ at the 99% confidence level given our assumptions
regarding the prior probability distribution. It should be noted that the
exact number, while prior-dependent, is expected to be rather precise given
the sharp truncation of the posterior. Below this value the posterior
probability distribution is not significantly different from that of the prior
probability distribution that was uniform in $\log M_{2}$. The proper motion
of the HR 2562 photocenter between the Hipparcos and Gaia missions is shown in
Figure 8. The proper motion measurements are consistent with companion masses
$\lesssim 20$ $M_{\rm Jup}$; higher-mass companions would have caused a larger
amplitude astrometric reflex motion between the two epochs, inconsistent with
the observations.

####  3.2.4 The case for low eccentricity solution

As shown here, the limited coverage of the companion’s orbit still leaves a
broad range of possible orbits. In short, the data are consistent with both
low- to moderate-eccentricity orbits that are roughly coplanar with the debris
disk and high-eccentricity orbits at a significantly different inclination. It
is worth noting that, with such limited orbital phase coverage, Keplerian fits
are notoriously subject to eccentricity and inclination biases (e.g., Lucy,
2014; Ferrer-Chávez et al., 2021) and they remain highly sensitive to the MCMC
priors (Pearce et al., 2015; O’Neil et al., 2019). These effects are highly
sensitive to the exact orbital phase coverage and system viewing geometry, so
they are hard to evaluate and correct for. While only more extensive
astrometric coverage will solve both issues, it is worth taking an hollistic
approach of the results presented here. In particular, a high-eccentric and
misaligned companion would most likely result in an eccentric disk, through
apsidal alignment, and/or a significant warp in the system. The ALMA map
presented here does not provide strong evidence for either phenomenon,
suggesting that a lower eccentricity and near coplanarity should be preferred.

As an additional test, we performed an orbital fit where we forced $e=0$. The
resulting fit is slightly poorer, with $\chi^{2}=21.9$ (to be compared with
$\chi^{2}=20.0$ for the eccentric fit). Given 15 two-dimensional datapoints
and 6 (8) free parameters for the circular (eccentric) fit, both the Bayesian
Information Criterion and the Aikake Information Criterion indicate that the
data are sufficiently well fit by the circular fit. Since it is unlikely that
the orbital is exactly circular, we will use the result of the full Keplerian
fit in the remainder of this study, but we consider that both physical
arguments and the circular fit provide suggestive evidence against the high-
eccentricity orbital solutions.

Table 3: Orbital elements for HR 2562 B

Parameter | Unit | Visual | Visual + Prior | Visual + Absolute
---|---|---|---|---
|  | Range | max. $\mathcal{L}$ | max. $\mathcal{P}$ | Range | max. $\mathcal{L}$ | max. $\mathcal{P}$ | Range | max. $\mathcal{L}$ | max. $\mathcal{P}$
Fitted parameters
$\log a$ | [arc sec] | $-0.305_{-0.078}^{+0.180}$ | $-0.286$ | $-0.401$ | $-0.170_{-0.099}^{+0.112}$ | $-0.343$ | $-0.260$ | $-0.252_{-0.116}^{+0.115}$ | $-0.234$ | $-0.204$
$\cos i$ |  $\cdots$ | $0.165_{-0.073}^{+0.306}$ | $0.110$ | $0.531$ | $0.098_{-0.017}^{+0.020}$ | $0.131$ | $0.114$ | $0.125_{-0.036}^{+0.211}$ | $0.089$ | $0.103$
$e$ |  $\cdots$ | $0.79_{-0.35}^{+0.18}$ | $0.58$ | $0.98$ | $0.29_{-0.21}^{+0.28}$ | $0.67$ | $0.52$ | $0.63_{-0.23}^{+0.32}$ | $0.45$ | $0.55$
$\omega$ | deg | $202.2_{-26.6}^{+42.7}$ | $210.4$ | $186.2$ | $179.8_{-17.3}^{+69.1}$ | $182.5$ | $170.9$ | $207.2_{-30.1}^{+34.6}$ | $216.5$ | $230.9$
$\Omega$ | deg | $299.2_{-3.8}^{+3.1}$ | $298.7$ | $298.0$ | $302.1_{-0.8}^{+0.8}$ | $300.5$ | $302.3$ | $299.5_{-2.7}^{+2.5}$ | $299.3$ | $298.5$
$\tau$ |  $\cdots$ | $0.794_{-0.040}^{+0.079}$ | $0.780$ | $0.785$ | $0.696_{-0.089}^{+0.078}$ | $0.716$ | $0.685$ | $0.795_{-0.057}^{+0.075}$ | $0.785$ | $0.841$
$\varpi$ | mas | $29.474_{-0.018}^{+0.018}$ | $29.464$ | $29.476$ | $29.474_{-0.018}^{+0.018}$ | $29.463$ | $29.473$ | $29.474_{-0.018}^{+0.018}$ | $29.507$ | $29.474$
$M_{\rm Total}$ | $M_{\odot}$ | $1.34_{-0.12}^{+0.13}$ | $1.67$ | $1.33$ | $1.34_{-0.12}^{+0.12}$ | $1.67$ | $1.37$ |  $\cdots$ |  $\cdots$ |  $\cdots$
$M_{1}$ | $M_{\odot}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $1.34_{-0.12}^{+0.13}$ | $1.44$ | $1.36$
$\log M_{2}$ | [$M_{\rm Jup}]$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $-0.40_{-1.07}^{+1.14}$ | $1.07$ | $1.01$
$\Delta\alpha^{\star}$ | mas |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $0.59_{-0.43}^{+0.66}$ | $1.90$ | $1.75$
$\Delta\delta$ | mas |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $0.06_{-0.51}^{+0.39}$ | $-0.72$ | $-0.48$
$\Delta\mu_{\alpha^{\star}}$ | mas yr-1 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $0.56_{-0.11}^{+0.02}$ | $0.35$ | $0.37$
$\Delta\mu_{\delta}$ | mas yr-1 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $-0.46_{-0.02}^{+0.06}$ | $-0.35$ | $-0.36$
$\sigma_{\Lambda}$ | mas |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $2.35_{-0.24}^{+0.26}$ | $2.23$ | $2.25$
Derived parameters
$P$ | yr | $60.1_{-14.7}^{+51.2}$ | $56.8$ | $42.9$ | $95.4_{-28.6}^{+45.1}$ | $46.8$ | $68.8$ | $71.5_{-23.2}^{+35.7}$ | $72.9$ | $83.5$
$a$ | arc sec | $0.496_{-0.082}^{+0.255}$ | $0.517$ | $0.397$ | $0.676_{-0.138}^{+0.198}$ | $0.454$ | $0.549$ | $0.560_{-0.131}^{+0.169}$ | $0.583$ | $0.625$
$a$ | au | $16.8_{-2.8}^{+8.7}$ | $17.6$ | $13.5$ | $22.9_{-4.7}^{+6.7}$ | $15.4$ | $18.6$ | $19.0_{-4.4}^{+5.7}$ | $19.8$ | $21.2$
$r_{\rm apo}$ | au | $28.8_{-1.9}^{+8.4}$ | $27.7$ | $26.6$ | $29.3_{-1.5}^{+3.3}$ | $25.7$ | $28.4$ | $29.2_{-2.1}^{+7.2}$ | $28.6$ | $32.9$
$i$ | deg | $80.5_{-18.6}^{+4.2}$ | $83.7$ | $57.9$ | $84.4_{-1.2}^{+1.0}$ | $82.5$ | $83.5$ | $82.8_{-12.5}^{+2.0}$ | $84.9$ | $84.1$
$T_{0}$ | yr | $2005.0_{-4.9}^{+1.6}$ | $2003.5$ | $2006.8$ | $1991.0_{-20.2}^{+7.6}$ | $2002.8$ | $1994.4$ | $2003.3_{-6.1}^{+2.9}$ | $2000.4$ | $2002.8$
$M_{2}$ | $M_{\rm Jup}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $<18.5$ | $11.67$ | $10.28$
Goodness of fit
$\chi^{2}_{\rho}$ |  $\cdots$ | $6.79_{-1.66}^{+2.96}$ | $3.54$ | $4.34$ | $10.61_{-2.49}^{+3.19}$ | $3.92$ | $7.91$ | $6.88_{-1.71}^{+3.00}$ | $4.23$ | $4.21$
$\chi^{2}_{\theta}$ |  $\cdots$ | $17.84_{-1.03}^{+2.28}$ | $16.48$ | $16.46$ | $17.80_{-1.00}^{+2.17}$ | $16.70$ | $16.47$ | $17.83_{-1.03}^{+2.27}$ | $17.07$ | $17.49$
$\chi^{2}_{\rm HIP}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $108.4_{-13.6}^{+14.8}$ | $111.1$ | $110.2$
$\chi^{2}_{\rm Gaia-Pos.}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $1.84_{-1.20}^{+1.85}$ | $0.10$ | $0.01$
$\chi^{2}_{\rm Gaia-PM}$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $1.40_{-1.05}^{+2.30}$ | $0.01$ | $0.08$
$\chi^{2}_{\nu}$ |  $\cdots$ | $1.15_{-0.11}^{+0.17}$ | $0.91$ | $0.95$ | $1.32_{-0.13}^{+0.17}$ | $0.94$ | $1.11$ | $1.11_{-0.11}^{+0.12}$ | $1.06$ | $1.06$

Note. — Orbits with $\Omega<180$ deg were wrapped by $\omega+180$,
$\Omega+180$, $\tau$ in fractions of the orbital period since MJD 57412.1335.
Reduced $\chi^{2}$ calculated using 22 degrees of freedom for the visual orbit
fit, and 125 for the combined fit.

Figure 4: (left panel) Relative astrometry of HR 2562 B relative to HR 2562
from GPI (red) and SPHERE (blue). The location of the host star is indicated
by the orange cross. Two hundred orbits drawn from the MCMC analysis (black
curves) are also shown. (right panel) Offset between HR 2562 and the
companion, and the residual compared with the median orbit, in the right
ascension (top) and declination (bottom) directions. Figure 5: Posterior
distributions and associated covariances of the semi-major axis ($a$),
periastron distance ($r_{\rm apo}$), eccentricity ($e$) and inclination ($i$)
from an orbit fit with (red) and without (black) a prior probability on the
inclination and position angle ($\Omega$) derived from the ALMA disk fitting.
Contours represent the 1, 2, and 3 $\sigma$ confidence levels. Limiting the
orbit to a co-planar configuration removes the tail of extremely high
eccentricity orbits.  Figure 6: Normalized joint prior probability on the
orbital inclination $i$ and position angle $\Omega$ estimated using a Gaussian
KDE for the co-planar scenario (main panel). Solid contours are spaced in 0.2
dex increments from $-1.8$ to $-0.2$, the inner dashed contour is at $-0.01$.
Marginalized distributions are shown in the top and right panels for the
posterior distribution estimated from the ALMA data (black solid histogram,
Section 4) and for the prior distribution used in the orbit fit (red dashed
histogram). Figure 7: Posterior distributions and associated covariance of the
orbital eccentricity ($e$), inclination ($i$), and companion mass ($M_{2}$)
from a joint fit to the relative and absolute astrometry. Contours represent
the 1, 2, and 3 $\sigma$ confidence intervals. Figure 8: Apparent proper
motion of the photocenter of the HR 2562 system in the right ascension (top)
and declination (bottom) directions for two hundred orbits drawn from the MCMC
analysis presented in Section 3.2.3 that are consistent with the absolute and
relative astrometry. The tracks are colored by the mass of the companion for
each orbit. The astrometric measurements from the Hipparcos and Gaia
catalogues are plotted.

##  4 Disk Modeling

###  4.1 Modeling Setup

We model the submillimeter emission from the disk with MCFOST, a radiative
transfer code for circumstellar modeling (Pinte et al., 2006). In short, the
system is set up as a central star surrounded by a circumstellar disk where
the stellar radiation is propagated through the disk using a Monte Carlo
process to evaluate the dust temperature in all locations, and the resulting
thermal emission map is generated using a ray tracing method. We assume that
the disk is axisymmetric, and that its dust population can be represented by a
uniform composition (astronomical silicates from Draine & Lee, 1984) and power
law size distribution. We assume an $N(a)\propto a^{-3.5}$ distribution
(Dohnanyi, 1969) that extends from 3 $\mu$m, the approximate blowout size for
a mid-F star (e.g., Pawellek & Krivov, 2015), to 1 mm. As is applicable for
optically thin disks, the dust grains are assumed to be in radiative
equilibrium with the stellar radiation field but local thermal equilibrium is
not enforced. This leads to the smaller dust grains being super-heated
compared to large grains and to the blackbody approximation. The star emission
is simulated using a 6600 K stellar spectrum generated by the PHOENIX grid
with a total luminosity of 3.1 $L_{\odot}$ (see eg., Moór et al., 2015).

Following standard practice for debris disks (e.g., Augereau et al., 1999;
Esposito et al., 2020), we select the following prescription for the disk
density as a function of position between the disk’s inner($R_{in}$) and outer
radius($R_{out}$):

$\displaystyle\rho(r,z)\propto\left(\left(\frac{r}{R_{c}}\right)^{-2p_{in}}+\left(\frac{r}{R_{c}}\right)^{-2p_{out}}\right)^{-1/2}\times\text{exp}\left(-\left(\frac{|z|}{h(r)}\right)^{\gamma_{\text{vert}}}\right).$
(1)

We fix $\gamma_{\text{vert}}=1$ for an exponential vertical profile. The
corresponding scale height, $h(r)$, is assumed to be a linear function of
radius, i.e., the disk has a bow-tie shape. In this case,
$h=h_{0}\frac{r}{r_{0}}$ where $r_{0}$ is an arbitrary reference radius. We
selected $r_{0}=100$ au. The surface density profile is a smoothly connected
pair of power law regimes, with exponents $p_{in}$ and $p_{out}$ at radii
$r\ll R_{c}$ and $r\gg R_{c}$ if $p_{in}>0$ and $p_{out}<0$, as is usually the
case. The peak surface density occurs near, but not exactly at, $r=R_{c}$,
depending on the value of $p_{in}$ and $p_{out}$ (Augereau et al., 1999). The
total dust mass, $M_{d}$, is obtained by integrating Eq. 1 from the disk’s
inner and outer radii, $R_{\text{in}}$ and $R_{\text{out}}$, respectively.

The disk image is then produced for a combination of inclination ($i$) and
position angle (PA) with a pixel scale of 0$\farcs$2/pixel, convolved with a
2D Gaussian beam constructed with the major and minor axes and the position
angle from the ALMA beam. To reduce the issue of correlated noise in the
interferometric map, we rebin the observed and model images to a 1″/pixel
scale so that each pixel can be reasonably considered as independent of its
neighbors. The two images are aligned based on the 2D Gaussian fit to the
observed image and a $\chi^{2}$ goodness-of-fit metric is computed based on a
20″$\times$20″ field of view centered on the disk.

With this routine established, we proceed to perform a Monte Carlo Markov
Chain (MCMC) run to determine the best fit parameter for the disk. We vary a
total of 9 parameters: $i$, PA, $M_{d}$, $R_{c}$, $R_{\text{in}}$,
$R_{\text{out}}$, $p_{\text{in}}$, $p_{\text{out}}$, $h_{0}$. The values of
the priors are given in Table 4. The ensemble initializes the walkers with
flat priors for every parameter except for the dust mass, which was sampled
from a log uniform prior. With these priors, we set up the two-temperature
MCMC with 100 walkers both for the hot and cold chains. We ran the chain for
2000 iterations with a script adapted from diskmc (Esposito et al., 2018) and
verified that the chain had reached convergence by the end of the run. The
first 600 iterations were rejected as burn-in steps.

###  4.2 Results

Table 4: Disk physical parameters from ALMA map fitting

Parameter | Unit | Prior Range | Best Fit-ALMA | Confidence Interval | Best Fit-SED
---|---|---|---|---|---
$M_{d}$ | $M_{\oplus}$ | ($4.2\times\,10^{-5}$, $3.3\times\,10^{5}$) | 0.0182 | $0.0192^{+0.0016}_{-0.0015}$ | 0.0158
PA | ° | (-60.,120.) | 32.7 | $32.7^{+0.7}_{-0.8}$ | 33.6
$i$ | ° | (0., 90.) | 84.7 | $\geq 79.3$ | 87.1
$p_{\text{out}}$ |  | (-10., 0.) | -1.2 | $-1.2^{+0.3}_{-0.4}$ | -0.8
$p_{\text{in}}$ |  | (0,10.) | 3.05 | $\leq 10.2$ | 7.6
$R_{c}$ | au | (20.,150.), | 41.0 | $\leq 114.4$ | 47.6
$R_{\text{in}}$ | au | (0.1, 75.) | 20.3 | $\leq 62.3$ | 34.0
$R_{\text{out}}$ | au | (200.,401.) | 260.0 | $258.8^{+26.8}_{-21.2}$ | 234.3
$h_{0}$ | au | (0.5., 20.) | 1.58 | $\leq 11.0$ | 3.91

Note. — We report the prior ranges, best fit image parameter, the 1$\sigma$
confidence interval and the best fit SED parameter. For less constrained
parameters, 3$\sigma$ upper or lower limits are reported instead. The
reference radius is set to $r_{0}=100$ au.

Figure 9: From left to right: ALMA map of the system binned to 1″ pixel; best
fit model to the ALMA data (top) and the model from the converged portion of
the chain that best matches the SED (bottom, see Section 5.2); residuals for
both models. The ALMA maps uses the same colorbar as the models. The ALMA fit
agrees well with the model image while the SED-selected model has an
acceptable geometry but is significantly dimmer.  Figure 10: Results of the
MCMC fit of the ALMA image of the HR 2562 disk properties. The cornerplot,
shown as grey dots and black density contours, is created from the last 1400
iterations of the chain, i.e., after convergence was reached. Contours
represent the 1, 2, and 3 $\sigma$ confidence levels. The 100 models selected
from the converging part of the chain that best fit the observed SED of the
disk (see Section 5.2) are represented in red dots.

The best fit model and the associated residuals are presented in the top row
of Figure 9. This model describes the data reasonably well
($\chi_{red}^{2}=1.50$) with nearly random residuals and, especially, no
structured residuals at the location of the disk emission. Considering the
entirety of the MCMC chain (Figure 10), we find that several parameters (PA,
$M_{d}$, $R_{\text{out}}$ and $p_{\text{out}}$) are well constrained and we
report their $1\sigma$ confidence intervals in Table 4. For the remainder of
the parameters we obtained 3$\sigma$ upper ($R_{\text{in}}$, $p_{\text{in}}$,
$R_{c}$ and $h_{0}$) or lower ($i$) limits from our posteriors. As expected
given that the disk is well resolved in our ALMA observations, several of the
geometric parameters are well constrained. In addition, the disk mass is well
constrained by the integrated flux of the disk.

The corner plot reveals important correlations between parameters. First of
all, there is some ambiguity between the disk inclination and its vertical
extent. Specifically, the ALMA map is consistent with a vertically thin disk
inclined at about 84°,but a thicker disk almost exactly edge-on is also
consistent with the observations. This is unsurprising given the $\approx 35$
au linear resolution of these observations; higher resolution mapping will be
necessary to tighten the constraints on these parameters. The second
correlation connects the disk radial extent and its surface density profile.
For instance, higher values of $R_{c}$ are associated to a steeper outer
density profile. Furthermore, the inner disk radius and inner density profile
slope have broad ranges of allowed value since the resolution of our ALMA
observations does not allow us to fully resolve the disk inner hole. We place
a 3$\sigma$ upper limit on both parameters from the ALMA map and will further
explore their relations with other available information in the next section.

##  5 Analysis

###  5.1 System geometry

Since the discovery of the HR 2562 B substellar companion and the initial
mapping of the disk structure with Herschel, the observations presented here
allow us to further our understanding of the system: the continued imaging of
the companion provides tighter constraints on its orbit, the Gaia DR3 releases
opens the door to a dynamical mass estimate on the brown dwarf via the
absolute astrometry of the star, and the better resolved ALMA data further
clarifies the geometric structure of the disk. Most importantly, with the
updated companion orbit and the resolved image of the disk, we are in a good
position to quantify the system geometry and obtain concrete evidence of
direct interaction between HR2562B and the debris disk.

We first focus on our updated constraints on the disk properties. A similar
analysis was performed in Moór et al. (2015) to derive the properties of the
disk. The acceptable ranges for the geometrical disk parameters are consistent
between their study and our fit result, although $R_{\text{in}}$ and
$R_{\text{out}}$ are less constrained by the Herschel observations due to
their lower resolution, which leads to the disk being only marginally
resolved. To further contrast the ALMA observation with the Herschel
observation, we recreate the Herschel best fit model using MCFOST with the
single power law density profile and the best fit parameters from Moór et al.
(2015), and adjusting the dust mass so that the total flux of the disk model
matches the measured SED of the system at 70$\mu$m. The resulting image is
then convolved with the PACS beam. We also produce the 70 $\mu$m image of the
best fit model to the ALMA data, again convolving it with the PACS beam. The
comparison between observed and synthetic Herschel observations is presented
in Figure 11. The Herschel disk looks geometrically similar to our best fit
result as we expected, and we observe an overall weaker integrated brightness
in the Herschel model compared to the ALMA model. The derived FWHM of the two
models are consistent with each other with the two models both having a weaker
integrated brightness than the observation. Overall, the best fitting model
for the ALMA map is also consistent with the Herschel image at 70 micron.

Figure 11: Observed (left panel, Moór et al., 2015) and synthetic (center and
right panels) 70 $\mu$m Herschel image of HR 2562. The two models represent
the model using the ALMA best fit parameters from this study and the Herschel
best fit model from Moor et al.

Combining the updated orbit fit and disk analysis, we can study the alignment
between the disk and the plane of the orbit of the companion. To do this, we
calculate the misalignment angle with equation 1 in Czekala et al. (2019) and
the unconstrained relative astrometry fit from section 3.2.1. The result is
shown in Figure 12. The blue and yellow lines represent the front and back
ambiguity between the disk and the orbit. This is because it is impossible to
disambiguate between the orbital angular momentum vector pointing out of, or
into, the plane of the sky. In either case, we find that the system is likely
in a near-coplanar situation. Specifically, in the “front” case, where both
angular momentum vectors points the same way relative to the plane of the sky,
we report a 1 $\sigma$ confidence interval of $7.0^{+17.1}_{-3.7}\,\arcdeg$.
In the other case, we report a 1 $\sigma$ confidence interval of
$15.2^{+17.7}_{-4.8}\,\arcdeg$. The low-probability tail of significant
misalignment in both cases can be traced back to the uncertainty in the
companion’s orbit. We further note that the more misaligned solutions also
have higher eccentricity, which we consider to be less likely. Continuous
astrometric monitoring will make this upper limit much more stringent if the
disk and orbit are indeed close to coplanar.

Figure 12: Relative inclination of the disk (as seen by ALMA) and the
companion’s orbit. The two histograms represent the two cases regarding the
front/back ambiguity in both disk and orbital planes.

We continue to explore the density structure of the disk from our findings.
The density profiles of 100 fitting models selected from the converged part of
the MCMC chain are plotted in the left panel of Figure 13. Interestingly, a
number of models that satisfyingly fit the ALMA map have density profiles that
extend down (even interior) to the current separation of the brown dwarf
companion. This is physically implausible, as we expect the massive companion
to clear out dust out to at least its apoastron distance. However, the fit to
the ALMA map was not informed by the location and orbit of the companion, so
it is reassuring to note that at least some of the models have peak surface
density radii that are exterior to the companion’s orbit. Indeed, many of
these models have significant amounts of dust close to the apoastron distance,
suggesting that the brown dwarf may be directly interacting with it. Our SED
selection in the following subsection provides additional support to this
conclusion.

Figure 13: Left: density profiles of 100 randomly selected models from the
converged part of the MCMC chain; right: density profiles of SED-selected
models (see Section 5.2). The density profiles in both panels are normalized
to their maximum value for ease of comparison. The blue line marks the
separation of the brown dwarf at the time of the ALMA observations while the
red bar represent the 1 $\sigma$ range of apoastron distance of its orbit. The
restriction of SED alone is enough to eliminate most models with $R_{in}$
smaller than the current separation of the brown dwarf, which are physically
implausible.

###  5.2 Best fitting SED

So far we have only considered the ALMA disk image in our analysis, but a
holistic approach needs to consider multiple types of observation. An
important piece of information is the spectral energy distribution (SED) of
the disk. In their Herschel-based study, Moór et al. (2015) fit the SED and
the image of the disk separately. The SED fit in Moór et al. (2015) yields a
radius of $64\pm 6$ au, which is about twice as large as the image fit (and
also consistent with our fit results). A likely explanation for this
discrepancy is the assumption in the SED fit that the dust is at blackbody
equilibrium temperature. In debris disks, the smallest grains tend to super-
heat to a temperature that is significantly hotter. As a result, SED-informed
disk radii are significantly smaller than those obtained from resolved
imaging. This discrepancy has also been studied quantitatively with previous
Herschel observations of entire samples of debris disks and a clear trend with
stellar luminosity has been shown (e.g., Morales et al., 2016). Given the
luminosity of HR 2562, we conclude that a factor of 2 discrepancy between SED-
and image-fit disk radii is consistent with previous literature.

Nonetheless, our radiative transfer model, which self-consistently treats the
cooling inefficiency of small dust grains, also predicts a system’s SED so
that we can in principle incorporate this observable in our analysis. This
allows us to consider the SED of the models in the MCMC chain to assess
whether more stringent constraints on the disk properties can be inferred.
Unfortunately, we did not compute the SED during the chain and this analysis
must be conducted a posteriori. To examine this, we computed the SEDs of a
selection of models that were included in the MCMC chain. Specifically, we
randomly selected 1000 models from all the walkers in the lower temperature
chain between steps 1900 to 2000, where the chain was well converged. We then
computed the $\chi^{2}$ between the models and the observed SED (including the
IRAS, Herschel and ALMA fluxes) and selected the 100 best fit models. To put
the result of this SED selection in context, we also randomly selected 100
models from the same part of the chains and plot their SEDs. The results are
shown in Figure 14. Many of the models selected based exclusively on the ALMA
image fit are poor fits to the SED, generally underestimating the 70–160
$\mu$m emission. Nonetheless, some models are good fits to the SED even though
our model fitting was only informed by the ALMA image.

To gauge how well the best fit SED model describes the ALMA image, we compare
it with the best fit image model and the data in Figure 10. The best fit SED
model does not fit the data as well, with a reduced $\chi^{2}$ of 1.68 and the
clump-like structure around the location of the star in the disk, indicating
that the SED-fitting model produces a less centrally peaked surface brightness
profile. To see the effect of SED rejection on the disk structures, we
overplot the models selected by lower SED $\chi^{2}$ with red dots in Figure
10. The main difference between the SED-selected models and the overall MCMC
posteriors is that the former show a preference for larger values of
$R_{\text{in}}$. To illustrate this, we again compare the density profiles of
100 randomly selected models with those of the 100 best fit SED models as seen
in the right panel of Figure 13. The best fit SED models almost universally
have a peak surface density radius that is exterior to the apoastron distance
of the companion. In other words, the best SED models prefer the family of
models with an inner hole potentially consistent with truncation by the brown
dwarf. This shows that the SED indeed provides useful information for us to
narrow down the result from the ALMA image MCMC.

Overall, although our SED-selected models do not fit the ALMA image as well,
they are consistent with strong dynamical interaction between the substellar
companion and the debris disk, providing a new testbed for such interactions.
Future work will require a simultaneous fit to the SED and resolved images of
the disk to provide tighter and more consistent constraints of the exact
architecture of the system.

Figure 14: SED of HR 2562 (orange circles) compared to subsets of the models
from the converged part of the ALMA map fitting MCMC chain. The left panel
shows 100 models randomly selected from the chain without considering the SED
while the right panel shows the 100 models from the same portion of the chain
with the least $\chi^{2}$ with respect to the SED. The green solid line in the
left panel represents the model with the best fit model to the ALMA image, and
the red solid line in the right panel represents the lowest $\chi^{2}$ with
respect to the measured SED.

###  5.3 Dynamical Interaction

A number of theoretical and numerical studies have tackled the problem of disk
truncation by an interior substellar object, but none was tailored to the
exact configuration of the HR 2562 system. The companion mass, for which we
only have a dynamical upper limit and a model-dependent estimate, the
companion semimajor axis and the orbital eccentricity, which is only modestly
constrained at this point, are three main factors that set the location of the
disk’s inner radius. Here we will explore the predictions of several such
models when applied to the parameter values estimated in this study. Relative
inclination likely also plays a role, although most studies assume (near)
coplanarity. Since this is assumption is consistent with all the data at hand
for HR 2562, we can thus use these theoretical predictions to compare them to
the geometry of the HR 2562 system and assess the likelihood that the brown
dwarf companion is actively truncating the debris disk. It is worth pointing
out that disk truncation studies sometimes rely on different criteria to
define the inner edge, such as the “chaotic zone” (Quillen & Faber, 2006;
Chiang et al., 2009) or the “Hill sphere argument” (Pearce & Wyatt, 2014), so
that for a given set of parameters, there is a range of predicted inner radii
for the truncated disk. Nonetheless, we can compare the range of predicted
inner disk radii to the range derived from fitting our ALMA observations.
Although there is a possibility that the companion’s orbit may be highly
eccentric, which may produce the clump we see in Figure 3 as well as an
asymmetric disk, we do not think this is the likely scenario due to the clump
still being marginally consistent with a symmetric disk in brightness profile
as well as the fact that the disk and the orbit are shown to align with each
other (Figure 12). Thus, we only consider the case for low to moderate
eccentricity orbits in the following analysis.

Given our limited constraint on the companion’s orbit, we select three
different values of eccentricity and derive the corresponding disk inner edge
location. We obtain the corresponding semimajor axis from the orbit posterior
distribution in Figure 7 and conservatively assume the upper limit of 18.5
M${}_{\text{Jup}}$ as the companion mass in our predictions.We first use an
eccentricity of $e\sim 0.1$ (and a semi-major axis of 28 au) and adopt
equation 16 from the Fomalhaut-tailored simulations of Chiang et al. (2009)
that is designed for low eccentricity systems. This yields an estimate of the
disk’s inner radius of 44 au. We also considered the analysis of Pearce &
Wyatt (2014) and applied their Equations 9 and 10, assuming a semi-major axis
of 28 au and an eccentricity of $e\sim 0.1$, to predict an inner radius of 54
au. Finally, we consider the result from the N-body disk simulations in
Rodigas et al. (2014) for a more general framework using the same methodology,
exploring explicitly a broader range of perturber mass and orbital
eccentricity (up to $e=0.2$). From their study, we adopt the 10 $M_{J}$ case
as it is the closest to the likely mass of HR 2562 B. Taking results from
their Table 2 and adopting semi-major axes of 30 au and 24 au for the circular
and $e=0.2$ cases, we obtain a predicted location of the disk inner radius of
50.8 and 51.1 au, respectively. To compare the estimates with our models, we
define the effective inner radius of our models as the half peak point of
their surface density. We compute the effective inner radii for 1000 models
from the converged portion of the ALMA MCMC chain as well as the 100 best
fitting models by SED, and show the results alongside the above estimates on
Figure 15.

All in all, existing dynamical models predict that the HR 2562 B mass and
orbit should result in the exterior debris disk being truncated at about 45–55
au. Our ALMA observations suggest that disk extends at least that close to the
brown dwarf, possibly even closer (see Figure 15). It is therefore extremely
likely that the brown dwarf is directly responsible for the disk truncation,
without the need for additional perturbers, as has been suggested for HD
206893. Given the limited orbital coverage and modest resolution of our ALMA
observations, it is currently impossible to discriminate between the different
models discussed above, but it is likely that further observations in the next
few years can produce observational constraints that allow for such a study.
HR 2562 is therefore bound to become a new testbed for dynamical studies of
debris disk truncation by an interior substellar companion.

Figure 15: Distribution of inner disk radius for the 1000 randomly selected
models from the converged part of the ALMA MCMC fit and the 100 best fitting
models by SED are plotted (solid and hatched histogram, respectively). The red
arrow and lines mark the range of predicted disk inner edge locations using
the dynamical models from Chiang et al. (2009), Pearce & Wyatt (2014) and
Rodigas et al. (2014).

##  6 Conclusion

HR 2562 is an exciting system with a brown dwarf companion located inside the
inner hole of a cold debris disk, making it a very likely candidate for
dynamical sculpting between the disk and the companion. In this paper we
report the first ALMA 870 $\mu$m observations of the disk around HR 2562, as
well as the continued GPI monitoring on the substellar companion’s orbital
motion, which is close to edge-on. Coupled with Gaia EDR3 observations of the
star we obtain improved constraints on the substellar companion orbital
elements. In particular, the combined analysis of the absolute and relative
astrometry of the system places a $3\sigma$ upper limit of 18.5
$M_{\text{jup}}$ on the companion.

The new ALMA observations with an angular resolution roughly 6 times higher
than previous Herschel observations of the system achieved a peak signal to
noise ratio of $\sim 7$, enabling a more detailed study of the disk structure.
Using radiative transfer modeling, we perform an MCMC fit of the disk key
parameters and confirm the disk is see at high inclination ($3\sigma$ lower
limit of $i>79$$\fdg$3), consistent with previous studies of the system. We
further compute the misalignment angle between the disk and the orbit and find
that it is either $7^{+17}_{-4}$ ° or $15^{+18}_{-5}$ °, depending on
unresolved ambiguities due to projection effects. This provides further
evidence that the disk and the orbit are close to a coplanar configuration.

To test dynamical models of disk truncation by a low-mass companion, we focus
on the location of the inner edge of the debris disk. Modelling of the ALMA
map yields a $3\sigma$ upper limit of 62.3 au, as the resolution is
insufficient to fully resolve the inner parts of the disk. We note, however,
that consideration of the SED in the analysis further narrows the allowable
range of inner disk radii and locate the latter at $\approx 30$ au, albeit
with a significant uncertainty. This is close to the apoastron distance
derived from our orbital fit, providing further evidence for the companion
dynamically sculpting the disk. We further test three disk truncation models,
which predict the location of the disk inner radius to be in the 45–55 au
range based on the companion’s orbit and estimated mass. The image-derived
inner radius is even closer to the companion, possibly pointing to a lower
mass estimate for the companion and/or to shortcoming in the models.

HR 2562 presents a unique opportunity to quantitatively test disk truncation
models, provided the system’s architecture can be further constrained.
Continued monitoring of the companion’s orbit with high-contrast imaging
instruments and of the reflex motion of the host star with Gaia will yield
increasingly precise estimates of the orbit geometry as well as the first
dynamical measurement of the companion’s mass. Deeper and higher-resolution
images with ALMA, as well as a simultaneous fit to the sub-millimeter map and
the system’s SED, will provided a better defined view of the disk’s inner
regions, directly testing dynamical models.

This work is based on observations obtained at the Gemini Observatory, which
is operated by the Association of Universities for Research in Astronomy, Inc.
(AURA), under a cooperative agreement with the National Science Foundation
(NSF) on behalf of the Gemini partnership: the NSF (United States), the
National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia,
Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência,
Tecnologia e Inovação (Brazil). This work made use of data from the European
Space Agency 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 research made use of
the SIMBAD and VizieR databases, operated at CDS, Strasbourg, France. We thank
support from NSF AST-1518332, NASA NNX15AC89G, NNX15AD95G/NEXSS, and award
SOSPA4-006 through NSF from the NRAO.

##  Appendix A GAIA goodness of fit metric

The GAIA EDR3 catalog provides a number of statistical tools that can be used
to analyze the quality of the astrometric fit. Since the GAIA precision is
sensitive to the brightness of the star, in Figure 16, we present key
diagnostic metrics to evaluate the quality of the HR 2562 EDR3 entry. Overall,
the star appears well behaved, albeit with a small but significant astrometric
excess noise of 0.158 mas. It is not clear if this excess noise is
astrophysical in nature, but the excess noise is significantly lower than
reported in the Hipparcos catalogue (2.4 mas). Overall, we consider the GAIA
data to be of good quality, i.e., the data show no strong evidence for
departure from a linear motion over the course of the GAIA observations.

Figure 16: Goodness of fit metrics for HR 2562 (red cross) compared with other
Gaia sources of a similar $G$ magnitude (dark points). The top two panels give
the $\chi^{2}$ and a goodness of fit metric, while the bottom panel gives the
amplitude of the residuals to a five-parameter fit that assumes linear space
motion. The distribution of these three parameters are shown on the right.
Large values for any of these parameters can be used as an indication of non-
linear motion between mid-2015 and mid-2017 caused by an orbiting companion.

##  Appendix B Full Orbital Fit

We present the full corner plot to the HR 2562 Borbit MCMC run in section
3.2.2.

Figure 17: Full corner plot of HR2562 orbit with and without the new
constraints from the ALMA fit. Contours represent the 1, 2, and 3 $\sigma$
confidence levels.

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