11institutetext:  Max Planck Institute for extraterrestrial Physics,
Giessenbachstraße 1, 85748 Garching, Germany 22institutetext: LESIA,
Observatoire de Paris, Université PSL, Sorbonne Université, Université Paris
Cité, CNRS, 5 place Jules Janssen, 92195 Meudon, France 33institutetext: Max
Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
44institutetext: $1^{\rm st}$ Institute of Physics, University of Cologne,
Zülpicher Straße 77, 50937 Cologne, Germany 55institutetext: Univ. Grenoble
Alpes, CNRS, IPAG, 38000 Grenoble, France 66institutetext: Universidade de
Lisboa - Faculdade de Ciências, Campo Grande, 1749-016 Lisboa, Portugal
77institutetext: Faculdade de Engenharia, Universidade do Porto, rua Dr.
Roberto Frias, 4200-465 Porto, Portugal 88institutetext: European Southern
Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching, Germany
99institutetext: European Southern Observatory, Casilla 19001, Santiago 19,
Chile 1010institutetext: Sterrewacht Leiden, Leiden University, Postbus 9513,
2300 RA Leiden, The Netherlands 1111institutetext: Departments of Physics and
Astronomy, Le Conte Hall, University of California, Berkeley, CA 94720, USA
1212institutetext: CENTRA - Centro de Astrofísica e Gravitação, IST,
Universidade de Lisboa, 1049-001 Lisboa, Portugal 1313institutetext:
Department of Physics, Technical University Munich, James-Franck-Straße 1,
85748 Garching, Germany 1414institutetext: Max Planck Institute for Radio
Astronomy, Auf dem Hügel 69, 53121 Bonn, Germany 1515institutetext: Department
of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801,
USA 1616institutetext: Hamburger Sternwarte, Universität Hamburg,
Gojenbergsweg 112, 21029 Hamburg, Germany 1717institutetext: School of Physics
& Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
1818institutetext: Max Planck Institute for Astrophysics, Karl-Schwarzschild-
Str. 1, 85741 Garching, Germany 1919institutetext: Institute of Astronomy, KU
Leuven, Celestijnenlaan 200D, B-3001, Leuven, Belgium 2020institutetext:
European Space Agency, European Space Astronomy Centre, Madrid, Spain
2121institutetext: Institute of Astronomy, Madingley Road, Cambridge CB3 0HA,
UK 2222institutetext: Research School of Astronomy and Astrophysics, College
of Science, Australian National University, Canberra, Australia
2323institutetext: Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, France 2424institutetext: Univ. Lyon, Univ. Lyon
1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574,
F-69230, Saint-Genis-Laval, France

# First Light for GRAVITY Wide:
Large Separation Fringe Tracking for the Very Large Telescope Interferometer

GRAVITY+ Collaboration: R. Abuter GRAVITY+ is developed by the Max Planck
Institute for extraterrestrial Physics, the Institute National des Sciences de
l’Univers du CNRS (INSU) with its institutes LESIA / Paris Observatory-PSL,
IPAG / Grenoble Observatory, Lagrange / Côte d’Azur Observatory and CRAL /
Lyon Observatory, the Max Planck Institute for Astronomy, the University of
Cologne, the CENTRA - Centro de Astrofisica e Gravitação, the University of
Southampton, the Katholieke Universiteit Leuven and the European Southern
Observatory.
Corresponding authors: A. Drescher (email<EMAIL_ADDRESS>and J.
Woillez (email<EMAIL_ADDRESS>88 F. Allouche 2323 A. Amorim 661212 C.
Bailet 2323 M. Bauböck 151511 J.-P. Berger 55 P. Berio 2323 A. Bigioli 1919 O.
Boebion 2323 M.L. Bolzer 111313 H. Bonnet 88 G. Bourdarot 11 P. Bourget 99 W.
Brandner 33 Y. Clénet 22 B. Courtney-Barrer 992222 Y. Dallilar 11 R. Davies 11
D. Defrère 1919 A. Delboulbé 55 F. Delplancke 88 R. Dembet 22 P.T. de Zeeuw
101011 A. Drescher 11 A. Eckart 441414 C. Édouard 22 F. Eisenhauer 11 M.
Fabricius 11 H. Feuchtgruber 11 G. Finger 11 N.M. Förster Schreiber 11 E.
Garcia 88 P. Garcia 771212 F. Gao 161611 E. Gendron 22 R. Genzel 111111 J.P.
Gil 99 S. Gillessen 11 T. Gomes 771212 F. Gonté 88 C. Gouvret 2323 P. Guajardo
99 S. Guieu 55 M. Hartl 11 X. Haubois 99 F. Haußmann 11 G. Heißel 22 Th.
Henning 33 S. Hippler 33 S. Hönig 1717 M. Horrobin 44 N. Hubin 88 E. Jacqmart
2323 L. Jochum 99 L. Jocou 55 A. Kaufer 99 P. Kervella 22 H. Korhonen 99 L.
Kreidberg 33 S. Lacour 2288 S. Lagarde 2323 O. Lai 2323 V. Lapeyrère 22 R.
Laugier 1919 J.-B. Le Bouquin 55 J. Leftley 2323 P. Léna 22 D. Lutz 11 F. Mang
111313 A. Marcotto 2323 D. Maurel 55 A. Mérand 88 F. Millour 2323 N. More 11
H. Nowacki 55 M. Nowak 2121 S. Oberti 88 T. Ott 11 L. Pallanca 99 T. Paumard
22 K. Perraut 55 G. Perrin 22 R. Petrov 2323 O. Pfuhl 88 N. Pourré 55 S.
Rabien 11 C. Rau 11 S. Robbe 2323 S. Rochat 55 M. Salman 1919 M. Schöller 88
J. Schubert 11 N. Schuhler 99 J. Shangguan 11 T. Shimizu 11 S. Scheithauer 33
A. Sevin 22 F. Soulez 2424 A. Spang 2323 E. Stadler 55 J. Stadler 181811 C.
Straubmeier 44 E. Sturm 11 L.J. Tacconi 11 K.R.W. Tristram 99 F. Vincent 22 S.
von Fellenberg 141411 S. Uysal 11 F. Widmann 11 E. Wieprecht 11 E. Wiezorrek
11 J. Woillez 88 S. Yazici 11 A. Young 202011 G. Zins 88

(Received May 4, 2022; accepted May 30, 2022)

GRAVITY+ is the upgrade of GRAVITY and the Very Large Telescope Interferometer
(VLTI) with wide-separation fringe tracking, new adaptive optics, and laser
guide stars on all four 8 m Unit Telescopes (UTs), for ever fainter, all-sky,
high contrast, milliarcsecond interferometry. Here we present the design and
first results of the first phase of GRAVITY+, called GRAVITY Wide. GRAVITY
Wide combines the dual-beam capabilities of the VLTI and the GRAVITY
instrument to increase the maximum separation between the science target and
the reference star from 2 arcseconds with the 8 m UTs up to several 10
arcseconds, limited only by the Earth’s turbulent atmosphere. This increases
the sky-coverage of GRAVITY by two orders of magnitude, opening up
milliarcsecond resolution observations of faint objects, and in particular the
extragalactic sky. The first observations in 2019 – 2022 include first
infrared interferometry of two redshift $z\sim 2$ quasars, interferometric
imaging on the binary system HD 105913A, and repeated observations of multiple
star systems in the Orion Trapezium Cluster. We find the coherence loss
between the science object and fringe-tracking reference star well described
by the turbulence of the Earth’s atmosphere. We confirm that the larger
apertures of the UTs result in higher visibilities for a given separation due
to larger overlap of the projected pupils on sky and give predictions for
visibility loss as a function of separation to be used for future planning.

###### Key Words.:

Instrumentation: interferometers – Instrumentation: high angular resolution –
Galaxies: quasars: supermassive black hole – Stars: individual: Orion
Trapezium Cluster

##  1 Introduction

Shao & Colavita (1992) describe for the first time an optical interferometer
observing simultaneously two widely separated targets contained inside the
atmospheric turbulence isopistonic patch. The Palomar Testbed Interferometer
(PTI) (Colavita et al. 1999) represents the first implementation of this dual-
field technique, where star separators are located at the focus of the
telescope and deliver two independent beams allowing to operate simultaneously
two interferometric instuments. At the time however, the emphasis was
primarily on astrometry, i.e. measuring the angular distance between the two
targets, in preparation for NASA’s space astrometry missions for exoplanets.
The possibility to use the technique to observe much fainter targets was
tentatively explored and presented in Lane & Colavita (2003), but remained
within the limiting magnitudes of the PTI around $m_{K}=5$. On the VLTI
(Beckers 1990), the dual-field instrument PRIMA (Delplancke 2008) was foreseen
to deliver astrometric and phase-referencing capabilities. The emphasis
remained on astrometry rather than pushing the sensitivity of the
interferometer, until the project was discontinued, in face of the competition
with Gaia (Perryman et al. 2001), while preserving the dual-field capability
of the infrastructure. The first dual-field phase-referenced observations to
demonstrate a sensitivity improvement were carried out by the ASTRA instrument
(Woillez et al. 2014) of the Keck Interferometer (Colavita et al. 2013),
reaching a magnitude of mK = 12.5, which was about 10 times fainter than
contemporaneous direct observations. The scientific exploitation of this
nascent capability was however cut short by the early demise of this facility
in July 2012.

The sensitivity revolution was finally delivered by the GRAVITY instrument
(Gravity Collaboration et al. 2017) at the VLTI. It has transformed high
angular resolution astronomy with the first interferometric instrument to
routinely offer milliarcsecond (mas) resolution imaging for objects as faint
as $m_{K}=19-20$, a sensitivity increase by more than a factor thousand over
previous interferometers, 30-100 microarcsecond ($\mu$as) astrometry, and
microarcsecond differential spectro-astrometry. The key to success are
technical breakthroughs on several fronts, including the development of quasi-
noiseless infrared detectors (Finger et al. 2019), infrared single-mode
waveguides and integrated optics (Perraut et al. 2018), robust fringe tracking
(Lacour et al. 2019), infrared adaptive optics (Scheithauer et al. 2016), and
laser metrology (Gillessen et al. 2012), as well as performance improvements
all over the VLTI observatory (Woillez et al. 2018). GRAVITY is also the first
interferometer to routinely offer dual-field interferometry, for which a
bright reference star is used to stabilize and phase-reference the
interferogram of the science object.

In the first five years of science operation, GRAVITY brought groundbreaking
results covering a broad range of astrophysical science: It has provided the
strongest experimental evidence that the compact mass in the Galactic Center
(Sgr A*) is a black hole, including the first detection of the gravitational
redshift (Gravity Collaboration et al. 2018a) and the Schwarzschild precession
(Gravity Collaboration et al. 2020a) in the orbit of the star S2 around the
black hole. Further, GRAVITY has detected orbital motion of hot gas close to
the innermost stable orbit of the black hole (Gravity Collaboration et al.
2018b), and performed the most precise measurement of the black hole’s mass
and distance (Gravity Collaboration et al. 2019a), surrounding mass
distribution (Gravity Collaboration et al. 2022b), and tests of the Einstein
equivalence principle (Amorim et al. 2019). GRAVITY has provided high
resolution spectra of the atmosphere of several exoplanets, including HR8799e
(Gravity Collaboration et al. 2019b) and $\beta$ Pic b (Gravity Collaboration
et al. 2020e). On the latter, the measured C/O ratio indicates that this
planet has undergone substantial core accretion and planetesimal enrichment.
Additionally, GRAVITY has delivered the first direct detection of a radial
velocity planet $\beta$ Pic c (Nowak et al. 2020), and the measurement of the
mass of an exoplanet from the astrometry of a second planet (Lacour et al.
2021). GRAVITY was also the first instrument to spatially resolve a quasar
broad line region (BLR) (Gravity Collaboration et al. 2018d), and to image at
milliarcsecond resolution the dust sublimation region around a Seyfert 2
active galactic nucleus (AGN) (Gravity Collaboration et al. 2020f). It has
also provided a comprehensive dataset of spatially resolved disks of young
stellar objects (Gravity Collaboration et al. 2019c, 2021b, 2021c, 2021d,
2021e, 2021f), and spatially resolved the magnetospheric accretion onto a T
Tauri star (Gravity Collaboration et al. 2020d). Further, GRAVITY resolved for
the first time the two images produced by gravitational microlenses (Dong et
al. 2019).

Until now, the number of observable targets with the dual-feed mode is limited
by the requirement that the fringe-tracking (FT) source and the science target
(SC) have to be within the field of view of the VLTI, which is 2 arcseconds
(arcsec) in diameter for the Unit Telescopes (UTs) and 4 arcsec in diameter
for the Auxiliary Telescopes (ATs). This requirement can be overcome by
implementing wide-angle off-axis fringe tracking, where we enlarge the
separation between FT and SC up to about 30 arcsec, limited by the atmospheric
turbulence. To break the limitation in separation between the two fields of
GRAVITY, they are separated at the telescope level and finally overlapped at
the entrance of GRAVITY. This implementation, which we refer to as GRAVITY
Wide, is one of the primary components of the ongoing upgrades to the VLTI and
GRAVITY. This improved instrument is called GRAVITY+ (Eisenhauer 2019).
Besides GRAVITY Wide, GRAVITY+ also includes the implementation of new
deformable mirrors and state-of-the-art adaptive optics (AO) wavefront
sensors, an improved instrument throughput and vibration control, as well as
laser guide stars on all four 8 m UTs. These upgrades will enable fringe
tracking on objects as faint as $m_{K}=13$ and, together with GRAVITY Wide,
enable all-sky interferometry with high resolution imaging at milliarcsec
accuracy with a limiting magnitude of $m_{K}=22$.

The increased FT–SC angular separation will make it possible to observe faint
targets with fringe tracking on a nearby bright source that can be picked from
a much larger area. This will open up observations and discoveries in
different areas of astronomy, such as: to spatially resolve young stellar
objects in their embedded phase, to study the multiplicity of massive stars in
the Small and Large Magellanic Cloud, to constrain for the first time
intermediate mass black holes with accurate motions of stars in globular
clusters, to discover single stellar-mass black holes and free floating
planets via microlensing, and to probe supermassive black holes in active
galactic nuclei out to beyond $z\approx 2$ (Eisenhauer 2019) and in nearby
inactive galaxies with transient tidal disruption events.

The enlargement of the FT–SC separation, however, brings along a challenge. In
ground-based optical and infrared interferometry, atmospheric turbulence plays
an important role (Fried 1966). Local changes in the temperature and humidity
in the atmosphere lead to changes in the refractive index of air. When an
initially flat wavefront from a distant science object enters the atmosphere,
it gets distorted. Adaptive optics and the fringe tracker are able to correct
these distortions. However, when the separation between SC and FT increases,
the correction degrades on the SC due to residual wavefront errors in the
direction of the SC. This effect is called anisoplanatism (Fried 1982) and it
becomes more severe for larger off-axis separations. Thus, it is important to
understand the behaviour of the atmosphere and investigate the influence of
atmospheric turbulence on large separation fringe tracking.

In this paper, we present the first wide-angle interferometric observations
with four telescopes, performed with GRAVITY Wide. We describe the
instrumental changes of GRAVITY to GRAVITY Wide in Sect. 2. Sect. 3, we
present the GRAVITY Wide data, and demonstrate first GRAVITY Wide science.
From observations of multiple star systems in the Orion Trapezium Cluster, we
derive a new orbit for $\theta^{1}$ Ori B, and refine the orbits of
$\theta^{1}$ Ori C and $\theta^{1}$ Ori D. We present the separation and flux
ratio of the binary HD 105913A, and the successful detection of fringes across
the H$\alpha$ line for two $z>2$ quasars, SDSS J161513.84+084914.4 ($z=2.33$,
hereafter SDSS1615) and LAMOST J092034.16+065717.9 ($z=2.46$, hereafter
LAMOST09). In Sect. 4, we discuss the influence of atmospheric turbulence on
the new large separation fringe tracking mode with 17 observed FT–SC pairs
with separations up to 32 arcsec. Finally, we give a summary and an outlook
for future GRAVITY+ science in Sect. 5.

Figure 1: Modifications made to the VLTI switchyard in December 2021 to
implement the first phase of GRAVITY Wide. Located in the VLTI laboratory
underneath the VLTI platform, the switchyard receives the light from the main
VLTI delay lines and directs the light to the various downstream instruments
such as GRAVITY, MATISSE, and PIONIER. Typically, the light first enters the
beam compressors to convert the beam diameters from 80 mm down to 18 mm. To
allow for simultaneous observations of FT targets separated by more than 2
arcsec from the SC we enabled the use of the B beams from the VLTI. We added a
four-fold periscope consisting of eight flat fold mirrors to merge the A
(black) and B (blue) beams with a 2 arcsec separation. For this it was also
necessary to rearrange four of the eight main switchyard mirrors, and
reactivated the original PRIMA differential delay lines.

##  2 The GRAVITY Wide upgrade

###  2.1 Instrumental changes

The original design of the VLTI incorporates dual-field interferometric
capability. Two subsections of the telescopes fields of view, each about 2
arcsec wide (4 arcsec for the ATs) and separated by up to 60 arcsec are picked
up by the star separators (STS) located at the coudé focus of each telescope
and propagated through the VLTI delay lines into the VLTI laboratory. The two
beams are generally referred to as the A and B beams, where commonly only the
A beams are used in non-dual-field applications.

Figure 2: Photographs of the added or modified hardware for the implementation
of GRAVITY Wide. Panel a) The newly added periscopes merge the A and B beams
of the respective four UT or ATs. Due to space constraints they were mounted
hanging down from two motorized bridge structures. Panel b) To compensate for
the differential optical path length between the A and B-beams, we now use the
differential delay lines formerly belonging to PRIMA. Panel c) These DDLs,
however, needed to be modified to relay the pupil to the appropriate location
for GRAVITY. The PRIMA DDLs consist of a three mirror cat’s eye with five
optical reflections. We replaced the tertiary mirrors to adjust the pupil
relay. Our beam path design allowed us to choose identical radii of curvature
for all tertiary mirrors.

To enable wide-mode observations we introduced new optics that merge the A and
B beams on the VLTI switch yard before feeding them to GRAVITY. This is
achieved through four periscopes that pick up the B beams and translate them
laterally to bring them into overlap with the A beams (see Fig. 1). These
periscopes are implemented through flat mirrors hanging upside down from two
bridges, which are motorized and integrated into VLTI’s ARAL (Morel et al.
2004) system for automated removal if not in use. The mirrors all ensure a
peak-to-valley wavefront error of better than $\lambda/20$ ($\lambda$ = 632.8
nm) within the footprint of the beam and were coated simultaneously to
minimize differential polarisation between the beams.

The main optical delay lines do not compensate for the differential optical
path length (OPD) changes that result from the up to several 10 arcsec wide
on-sky separation of the A and B beams. To correct for this, PRIMA (Delplancke
2008) originally introduced the differential delay lines (DDLs) (Launhardt et
al. 2005). After the discontinuation of the PRIMA project the DDLs were turned
off. We reactivated the DDLs for the use in GRAVITY Wide. This however
required a modification to relay the pupil at the correct distance for the
pickup by GRAVITY. The DDLs (see Fig. 2) consist of a three-mirror cat’s eye
system with five optical reflections (it’s primary and secondary mirrors are
passed twice). The tertiary mirror is located in the focus of the system and
its radius of curvature directly controls the distance of the output pupil
plane. We replaced the M3s with new mirrors with appropriately modified radii
of curvature. The differential delay between the SC and FT is stabilized on
the internal laser metrology of the PRIMA DDLs. The current GRAVITY Wide
implementation does not propagate the GRAVITY laser metrology up to the
telescope, and therefore does not yet provide the absolute phase and
astrometry between the SC and FT. The upgrade with a full optical path length
coverage by the GRAVITY metrology is foreseen for the next project phase.

With two beams entering GRAVITY, two sets of pupil beacons would appear in the
acquisition camera pupil tracking images. Hence, in April 2022 we have
installed a narrow band filter blocking the pupil beacon light from the A
beam, such that only one set of pupil beacons appears in the acquisition
camera from which to measure the lateral and longitudinal offsets. These
offsets can then be used to adjust the pupil for both beams A and B. Finally,
the VLTI switchyard itself required a reconfiguration to allow for this new
beam routing. For this four of eight motorized flat mirrors were re-
positioned.

All these modifications were carried out in December 2021 and conclude the
hardware part of the first phase of the GRAVITY Wide implementation. We have
begun to work on the second phase as part of the GRAVITY+ project which will
remove the PRIMA DDLs entirely and save five optical reflections for the B
beams. For this, in the second phase we will motorize the beam compressors of
the VLTI (see again Fig. 1) to take over the differential optical path length
compensation. The updated mechanics will ensure that the original optical
specifications of the DDL system are met.

###  2.2 Software changes

Along with the instrumental changes, a number of software changes had to be
made to bring GRAVITY Wide into operation. One general modification is to use
the VLTI field selector mirrors (FSMs) and variable curvature mirrors (VCMs)
as the actuators in all control loops instead of GRAVITY’s internal mirrors.
In particular, the field tracking and fringe tracker beam optimization loops
are now controlled through FSM A and lateral pupil tracking is controlled
through VCMs A and B. Further, during acquisition of the FT and SC targets
only FSMs A and B are moved to properly align them with the fibres.

We additionally developed and implemented a new “SC Tracking” control loop
which is similar to the current field tracking loop. Here, we measure the SC
target position through Gaussian fitting directly from the acquisition camera
field images. The separation between the target position and SC fibre position
is calculated and translated into the necessary FSM B offsets to bring the SC
target back to the fibre. To help facilitate this for faint SC targets
($m_{H}<17$), we added the option for a longer detector integration time (DIT)
(2.8s) on the acquisition camera such that the SC target would be reliably
detected on the field images.

###  2.3 Metrology OFF mode

Figure 3: Detector noise of GRAVITY for a sky frame with an integration time
of $30\text{\,}\mathrm{s}$. The noise is shown with the metrology laser on (in
grey) and the laser turned off (in red).

When using GRAVITY Wide to observe extremely faint targets it is important to
reduce existing noise sources as much as possible. The dominant instrumental
noise source in GRAVITY is the scattering of the metrology laser in the
instrument. While the metrology laser wavelength lies outside the science
wavelength, Raman scattering and backscattering from rare-earth elements in
the optical fibers create a broad noise peak in the blue part of the K-band,
as well as a constant background flux over the full detector (Lippa et al.
2018). The detector noise of GRAVITY is shown in Fig. 3. The direct back
scattering is the dominant noise source from
$2.0\text{\,}\mathrm{\SIUnitSymbolMicro
m}2.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a diffuse background from the
laser contributes to the noise above $2.1\text{\,}\mathrm{\SIUnitSymbolMicro
m}$.

The metrology system is only necessary for astrometric observations in the
GRAVITY dual-beam mode. For observations where the science target is within a
single beam, such as observations in GRAVITY on-axis or GRAVITY Wide, the
metrology system is not needed. To avoid unnecessary noise in those
observations, we developed a new instrument mode: the Metrology OFF mode. In
this mode the laser amplifier of the metrology laser is turned off during the
observation. This removes the back scattering of the laser on the detector.
The resulting noise for a $30\text{\,}\mathrm{s}$ sky frame is shown in Fig.
3. The noise decreases by a factor of eight in the blue part of the spectrum
(from $2.0\text{\,}\mathrm{\SIUnitSymbolMicro
m}2.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$) and on average by a factor of
2.5. The remaining dominant noise sources are the read-out noise from the
detector as well as the thermal background from the telescopes, beam relay and
sky.

Figure 4: Middle: Orion nebula in the background and zoom on the Orion
Trapezium Cluster ($\theta^{1}$) stars in the inset. Blue arrows mark the
separation between FT and SC. Upper: Orbit of $\theta^{1}$ Ori B6 around the
eclipsing binary $\theta^{1}$ Ori B1,5 at the center. We use $\theta^{1}$ Ori
A at a separation of 8.85 arcsec as the FT. Note that $\theta^{1}$ Ori B6 has
completed more than one orbital revolution between 2017 and 2021. Left: Orbit
of $\theta^{1}$ Ori C2 around the primary star $\theta^{1}$ Ori C1 at the
center. $\theta^{1}$ Ori C2 has completed more than two orbital revolutions.
Literature data points are measurements taken from Weigelt et al. (1999),
Schertl et al. (2003), Kraus et al. (2007), Patience et al. (2008), Kraus et
al. (2009), and Grellmann et al. (2013). The observation was performed with
$\theta^{1}$ Ori A at a separation of 12.86 arcsec as the FT. Right: Orbit of
$\theta^{1}$ Ori D3 around the primary star $\theta^{1}$ Ori D1 at the center.
We use $\theta^{1}$ Ori C at a separation of 13.41 arcsec as the FT.
$\theta^{1}$ Ori D3 has revolved 20 times between the first and last data
point. Background image and zoom from ESO/M. McCaughrean et al. (AIP).

##  3 First GRAVITY Wide observations

###  3.1 Data

The first wide-angle interferometric observations were performed in five runs.
The first two runs were executed between November 2019 and March 2020 with a
prototype implementation of GRAVITY Wide (no proper pupil relay to GRAVITY). A
third and fourth run was performed in December 2021 and January 2022, and a
fifth run in April 2022. We used both the UTs and ATs. The ATs were mounted in
the configuration A0-G1-J2-K0 in November 2019 and December 2021, and on the
stations A0-G1-J2-J3 in March 2020. In November 2019 we carried out
observations with the ATs on the Orion Trapezium Cluster. With the UTs, we
performed GRAVITY Wide observations with the prototype implementation on the
binary star HD 105913A in March 2020, and with the proper implementation of
GRAVITY Wide on the quasars LAMOST09 in December 2021 and January 2022, and
SDSS1615 in April 2022. We provide a detailed list of the observations and
their parameters in Appendix A.1 for the prototype implementation of GRAVITY
Wide, and in Appendix A.2 for the data for the later runs.

In total, we observed 36 FT–SC pairs with angular separations between 2 arcsec
and 32 arcsec. The observations were performed in one of the three spectral
resolutions offered: low with $R=\lambda/\Delta\lambda\approx$ 20, medium with
R $\approx$ 500 and high with R $\approx$ 4500\. Further, the light of the FT
and SC was measured in either combined or split linear polarization. The
integration time on the science spectrometer was between seconds and minutes,
depending on the magnitude of the science target. The data allow us to
demonstrate the performance of GRAVITY Wide on the one side, and evaluate the
coherence loss for increasing off-axis separations due to atmosphere
anisoplanatism on the other side. Further, the FT–SC pairs include faint
objects that were observed to explore the current limit of the new observing
mode. We use the GRAVITY pipeline (Lapeyrere et al. 2014; Gravity
Collaboration et al. 2017) to reduce the data.

###  3.2 Binaries in the Orion Nebula

One of the main targets for science demonstration of GRAVITY Wide with the ATs
is the Orion Trapezium Cluster. The cluster is located in the heart of the
Orion Nebula at a distance of 414 $\pm$ 7 pc (Menten et al. 2007) from Earth.
It is one of the closest regions of massive star formation (Genzel & Stutzki
1989; Hillenbrand 1997; Menten et al. 2007; Muench et al. 2008), and the best-
studied cluster of massive stars. An interferometric study of the cluster
stars has been carried out with GRAVITY in single-field mode between November
2016 and January 2018 and it has revealed that most of the massive stars are
not single stars, but multiple star systems (Gravity Collaboration et al.
2018c). This is expected for massive O-type stars, as they are found more
often in multiple systems than low mass stars (Sana et al. 2014).

These systems are good targets for GRAVITY Wide for several reasons. First,
the binaries are located within the field of view of GRAVITY. Additionally,
many of the main components of the cluster can be used as FT targets, since
they all are separated by $<$ 20 arcsec from each other. And lastly, the
former GRAVITY observations provide orbits for some of the stars. This gives
the unique opportunity to test and verify GRAVITY’s new wide-field mode. We
observed three principal components of the cluster, $\theta^{1}$ Ori B,
$\theta^{1}$ Ori C, and $\theta^{1}$ Ori D. Compared to GRAVITY single-field
mode, where one multiple system serves as both FT and SC, in GRAVITY Wide we
use one multiple system as the FT and another multiple system as the SC. This
is shown in the lower middle panel in Fig. 4. For the observation of
$\theta^{1}$ Ori B and $\theta^{1}$ Ori C, we used $\theta^{1}$ Ori A at a
separation of 8.85 arcsec and 12.86 arcsec, respectively, as the FT. For the
observation of $\theta^{1}$ Ori D, the FT was $\theta^{1}$ Ori C at a
separation of 13.41 arcsec. For details about each multiple system we refer to
Gravity Collaboration et al. (2018c). Here, we present the results from the
observations with GRAVITY Wide, i.e. measured separation and flux ratio of the
binary components, as well as orbital parameters for each of the systems,
which were derived as described in Appendix B.

Table 1: Binary separation and flux ratio for $\theta^{1}$ Ori B, $\theta^{1}$
Ori C and $\theta^{1}$ Ori D.

Object | Date | Sep. [mas] | $f$
---|---|---|---
$\theta^{1}$ Ori B | Nov 1, 2019 | 4.65 $\pm$ 0.07 | 0.3224 $\pm$ 0.0006
| Dec 15, 2021 | 15.66 $\pm$ 0.01 | 0.3504 $\pm$ 0.0022
$\theta^{1}$ Ori C | Nov 1, 2019 | 23.39 $\pm$ 0.14 | 0.3099 $\pm$ 0.0021
$\theta^{1}$ Ori D | Nov 2, 2019 | 1.41 $\pm$ 0.02 | 0.3419 $\pm$ 0.0064
111 Here, given are the binary separation (sep.) and flux ratio ($f$) for the
multiple star systems observed with GRAVITY Wide in November 2019 and December
2021.

Table 2: Orbital parameters for $\theta^{1}$ Ori B, $\theta^{1}$ Ori C and
$\theta^{1}$ Ori D.

| This work | (1)
---|---|---
|  $\theta^{1}$ Ori B
$a$ [mas] | 12.982 $\pm$ 0.018 | –
$e$ | 0.67326 $\pm$ 0.00037 | –
$i$ [∘] | 69.233 $\pm$ 0.063 | –
$\omega$ [∘] | 133.48 $\pm$ 0.13 | –
$\Omega$ [∘] | 282.860 $\pm$ 0.028 | –
$P$ [yr] | 3.22571 $\pm$ 0.00056 | –
$t_{\mathrm{P}}$ [yr] | 2019.3935 $\pm$ 0.0019 | –
$M_{\mathrm{tot}}$ [M⊙] | 14.928 $\pm$ 0.057 | –
|  $\theta^{1}$ Ori C
$a$ [mas] | 44.47 $\pm$ 0.19 | 45 $\pm$ 2
$e$ | 0.5908 $\pm$ 0.0034 | 0.59 $\pm$ 0.04
$i$ [∘] | 98.702 $\pm$ 0.050 | 98.6 $\pm$ 0.6
$\omega$ [∘] | 283.70 $\pm$ 0.14 | 283 $\pm$ 2
$\Omega$ [∘] | 27.516 $\pm$ 0.055 | 27.9 $\pm$ 0.7
$P$ [yr] | 11.4426 $\pm$ 0.0097 | 11.4 $\pm$ 0.2
$t_{\mathrm{P}}$ [yr] | 2002.304 $\pm$ 0.019 | 2002.2 $\pm$ 0.2
$M_{\mathrm{tot}}$ [M⊙] | 47.70 $\pm$ 0.61 | 46.4 $\pm$ 5.9
|  $\theta^{1}$ Ori D
$a$ [mas] | 1.94 $\pm$ 0.014 | 1.86 $\pm$ 0.06
$e$ | 0.391 $\pm$ 0.011 | 0.43 $\pm$ 0.03
$i$ [∘] | 155.19 $\pm$ 0.70 | 160 $\pm$ 12
$\omega$ [∘] | 156.40 $\pm$ 12.03 | 166 $\pm$ 27
$\Omega$ [∘] | 339.56 $\pm$ 9.52 | 346 $\pm$ 24
$P$ [yr] | 0.1449059 $\pm$ 0.0000086 | 0.1452 $\pm$ 0.0002
$t_{\mathrm{P}}$ [yr] | 2017.1004 $\pm$ 0.0015 | 2017.101 $\pm$ 0.001
$M_{\mathrm{tot}}$ [M⊙] | 24.91 $\pm$ 0.54 | 21.68 $\pm$ 0.05

(1) Gravity Collaboration et al. (2018c). 222 $a$ is the semi-major axis, $e$
the eccentricity, $i$ the inclination, $\omega$ the argument of periastron of
the secondary’s orbit, $\Omega$ the longitude of ascending node, $P$ the
period, $t_{\mathrm{P}}$ the time of periastron passage, and
$M_{\mathrm{tot}}$ the total system mass assuming a parallax of 2.415 $\pm$
0.040 mas (Menten et al. 2007).

We observed the multiple system $\theta^{1}$ Ori B in the nights of November
1st, 2019 and December 15th, 2021 with GRAVITY Wide. The measured separation
and flux ratio between $\theta^{1}$ Ori B6 and $\theta^{1}$ Ori B1,5 is given
in Tab. 1. We use the measurements from GRAVITY single-field mode in 2017/18
(Gravity Collaboration et al. 2018c) and the new measurements from GRAVITY
Wide to determine orbital parameters. Fig. 4 presents the new orbit. The
corresponding orbital parameters are shown in Tab. 2. We find a total system
mass of M${}_{B_{1,5}B_{6}}$ = 14.93 $\pm$ 0.06 M⊙. This result is consistent
with the sum of the masses of M${}_{B_{1}}$ = 7 M⊙ (Weigelt et al. 1999),
M${}_{B_{5}}\sim$ 2 M⊙ (Gravity Collaboration et al. 2018c) and the upper
limit of Gravity Collaboration et al. (2018c), which is M${}_{B_{6}}\sim$ 6
M⊙.

In a second observation with GRAVITY Wide, we observed $\theta^{1}$ Ori C in
the night of November 1st, 2019. Tab. 1 provides the measured separation and
flux ratio of the binary components $\theta^{1}$ Ori C1 and $\theta^{1}$ Ori
C2. We fit the orbit of $\theta^{1}$ Ori C with the new GRAVITY Wide
measurement and with measurements from GRAVITY single-field and the
literature. The orbit is presented in Fig. 4. In Tab. 2 we compare our results
with the results from Gravity Collaboration et al. (2018c). We find that they
agree with each other and that our fit result supports the large mass of the
binary $\theta^{1}$ Ori C. Further, we obtain more constrained values for the
orbital parameters, and a slightly larger total system mass, which is however
within the error bars of Gravity Collaboration et al. (2018c).

Finally, we observed $\theta^{1}$ Ori D in the night of November 2nd, 2019
with GRAVITY Wide. We give the measured separation and flux ratio between
$\theta^{1}$ Ori D1 and $\theta^{1}$ Ori D3 in Tab. 1. We use the measurements
from GRAVITY single-field (Gravity Collaboration et al. 2018c) and the new
measurement with GRAVITY Wide and fit the orbit of $\theta^{1}$ Ori D3 around
the primary $\theta^{1}$ Ori D1. We present the orbit in Fig. 4. We compare
the fit results with the results from Gravity Collaboration et al. (2018c) in
Tab. 2. We notice that the uncertainties are smaller for $a$, $e$, $i$,
$\omega$ and $\Omega$. Additionally, we find a larger total system mass.

The above presented results improve the results of previous observations of
multiple star systems in the Trapezium. This verifies the functionality of
GRAVITY Wide with the ATs, and proves that the new data points are of equal
quality than the GRAVITY single-field points.

###  3.3 HD 105913A

We highlight the first GRAVITY Wide observation with the 8 m UTs. The binary
HD 105913A was observed with GRAVITY Wide in the night of March 9th, 2020. A
third companion star, HD 105913B, is located at a separation of 5.11 arcsec
from HD 105913A (Gaia Collaboration et al. 2018) and was used as the FT. The
binary components in HD 105913A, Aa and Ab, have a period of 211.59 days and a
mass ratio of $q$ = 0.874. The triple system HD 105913 Aa,Ab,B is of spectral
type K1 and located at 34 pc from the Sun (Tokovinin 2019).

In Fig. 5 we show the observed and fitted visibility squared and closure
phases, respectively. We find the position of the secondary star at (dRA,
dDec) = (-5.83 $\pm$ 0.002, 24.11 $\pm$ 0.002) mas with respect to the primary
star fixed at the center, therefore the measured binary separation is 24.8
$\pm$ 0.002 mas. The error on the position, thus separation, is statistical
and does not include systematics. We observe a flux ratio $f$ = 0.64 $\pm$
0.039, thus Aa at the center is the brighter star. We show an image of the
binary reconstructed with the imaging code GR (Gravity Collaboration et al.
2022a) in Fig. 6, which is obtained from imaging with closure phases and
visibility amplitudes.

Figure 5: Upper: Observed (color) and fitted (black) visibility squared, and
Lower: observed (color) and fitted (black) closure phases for the observation
of HD 105913A on March 9th, 2020. The binary system Aa,Ab was observed with
the UTs. The separation to the fringe tracker, HD 105913B, is 5.11 arcsec.
Figure 6: Image of the binary HD 105913A, reconstructed with the imaging code
GR (Gravity Collaboration et al. 2022a). We indicate the FWHM of the
corresponding dirty beam in the bottom left corner. The flux is normalized to
the star Aa at the center.

###  3.4 Broad line region of redshift two quasars

GRAVITY has spatially resolved the broad-line region of low-redshift AGNs
(Gravity Collaboration et al. 2018d, 2020b, 2021a). This is achieved by
measuring the differential phase of the BLR referenced to the continuum
emission from the hot dust closely surrounding the BLR. Before GRAVITY Wide,
however, only the brightest and therefore nearby AGN could be observed with
GRAVITY because it is nearly impossible to find a bright FT star within 2
arcsec of extragalactic objects. Therefore, only on-axis GRAVITY observations
were possible which imposed a $m_{K}<10.5$ limit to enable fringe tracking on
the AGN.

With GRAVITY Wide, we are able to observe fainter quasars at higher redshift
given the increased sky coverage and ability to find off-axis FT stars. In
particular, quasars at $z\approx 2-3$ are ideal targets, as the bright
H$\alpha$ line is redshifted into the $K$ band. This line is very strong
compared to the continuum which boosts the observed differential phase by a
factor of $\gtrsim 10$ compared to the $z\lesssim 0.1$ AGN. $z\approx 2$ is
often called “cosmic noon”, the epoch in galaxy evolution when both star
formation and SMBH accretion peaked (Madau & Dickinson 2014). Since SMBHs and
their host galaxies are thought to co-evolve together (e.g. Heckman & Best
2014), measuring SMBH masses throughout the history of the Universe and
especially at cosmic noon is critical for understanding galaxy evolution.

Figure 7: Left: Acquisition camera field image during observations of
SDSS1615, showing successful acquisition and detection of the quasar. Middle:
Average total flux spectrum (red) over all four telescopes showing the
detection of the broad H$\alpha$ line at 2.28 $\mu$m consistent with a
redshift of $z=2.46$. In blue is the average coherent flux spectrum using the
UT4-UT1 and UT3-UT1 baseline and the post-processing explained in the text.
Coherent flux is significantly detected across the entire spectrum. Right:
Average differential phase spectrum using the UT4-UT1 and UT3-UT1 baselines
overlaid on the normalized total flux spectrum. With an RMS of $\approx
1-2$∘within the FWHM of the line (gray shaded region), we tentatively detect
the rotating disk signal of the BLR.

We selected two quasars, LAMOST09 ($z=2.33$, m${}_{K}=15.1$) and SDSS1615
($z=2.46$, m${}_{K}=15.6$), to be observed with the UT array and GRAVITY Wide.
LAMOST09 has a FT star at a separation of 12.8 arcsec with m${}_{K}=10.4$
while SDSS1615’s FT star is 7.5 arcsec away with m${}_{K}=10.45$. Observations
took place on 18 December 2021, 19 January 2022, and 25 January 2022 for
LAMOST09 and 17 April 2022 for SDSS1615 in MEDIUM spectral resolution.
Atmospheric conditions ranged from poor to moderate over the three nights for
LAMOST09 resulting in difficulty maintaining fringe tracking especially on 18
December. Still, each night we were able to acquire, fringe-track, and detect
fringes with the QSO-star pair resulting in 96 min of useable data. For
SDSS1615, we had very good weather conditions for the single night of
observing and were able to consistently fringe-track and integrate on the QSO
over 1 hour. The left panel of Fig. 7 shows an example acquisition camera
field image containing both the FT star and SDSS1615 demonstrating successful
acquisition.

The raw data were reduced through the GRAVITY pipeline to produce complex
visibilities for every DIT (60s for LAMOST09 and 100s for SDSS1615). Due to
the low flux level, significant fringe jumps, and a slow drift in the OPD, we
chose not to use the standard coherent integration within the pipeline and
instead manually ran our own post-processing. This involved first running a 2D
Discrete Fourier Transform on each individual night’s dataset to determine an
overall group delay accounting for the OPD drifts for each baseline. The
drifts were removed for each DIT individually with the self-reference method
from Tatulli et al. (2007); Millour et al. (2008), whereby the phase reference
for each spectral channel is constructed from all other spectral channels.
This post-processing significantly increased the coherent flux for each
baseline. We note that once the cause of the drift can be identified and
potentially fixed, even longer DITs will be possible to further increase the
sensitivity of QSO observations.

In the middle panel of Fig. 7, we plot the average coherent flux overlaid on
the total flux in the wavelength range of the expected H$\alpha$ line for
SDSS1615. The detected H$\alpha$ line averaged over all four telescopes peaks
around $2.28\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and has a FWHM of 4300 km
s-1. The coherent flux is averaged over two of the longest baselines, UT4-UT1
and UT3-UT1 where we expect the strongest differential phase signal. We
clearly detect coherent flux across the entire spectrum, which represents the
first near-infrared interferometric fringes of a high redshift object. The
right panel shows the average differential phase for the same two baselines.
Within the FWHM of the H$\alpha$ line (gray shaded region), we measure an RMS
noise of approximately 1–2∘and observe a tentative “S-shape” signal indicative
of a rotating BLR. SDSS1615 is a high luminosity quasar with an estimated BLR
size of $\sim 1.1$ pc and SMBH mass of $10^{9.9}$ M⊙ based on the [CIV] line
profile (Rakshit et al. 2020). Using these values, the differential phase peak
should be between 3–6∘depending on the inclination and position angle on-sky
of the BLR which matches well the emerging signal in Fig. 7.

##  4 The role of atmospheric seeing for wide-angle fringe tracking

###  4.1 Atmospheric coherence loss

Similar to classical ground-based observations in the optical and near-
infrared, atmospheric turbulence also plays an important role in near-infrared
interferometry. Turbulence in the atmosphere of Earth leads to a blurred image
of an astronomical object with a typical diameter of around 1 arcsec. The full
width at half maximum (FWHM) of the blurred image is the so-called seeing,
$\epsilon$. It depends on the individual conditions at the observational site,
and is given by (Kornilov et al. 2007)

$\epsilon=0.98\frac{\lambda}{\text{r}_{0}}\ ,$ (1)

where r0 is the Fried parameter, and $\lambda$ the observed wavelength.

For interferometry, especially with off-axis fringe tracking as it is done in
GRAVITY Wide, atmospheric effects are crucial. In this context, the
isoplanatic angle becomes an important parameter, which defines how far from
the SC the FT can be without losing coherence on the SC. Angular
anisoplanatism occurs when the light from two targets separated by an angle
$\theta$ experiences different phase variations as it travels through
different parts of the atmosphere (Quirrenbach 2000). The disturbed wavefront
of the on-axis star can be corrected by AO, whereas the off-axis star has
residual wavefront errors. Thus, the wavefront correction degrades if the
science target is further away from the fringe tracking star, which leads to a
loss in coherence, thus SNR. We therefore try to understand the effects of the
isoplanatic angle on the observations to estimate which atmospheric conditions
are best suited for observations with GRAVITY Wide.

The isoplanatic angle is given by (Quirrenbach 2000)

$\theta_{0}=\text{0.314(cos }z)\frac{r_{0}}{H}\ ,$ (2)

where $z$ is the zenith angle, and $H$ the mean effective turbulence height
which can be expressed as

$H\equiv\left(\frac{\int dh\ C^{2}_{N}(h)h^{5/3}}{\int
dhC^{2}_{N}(h)}\right)^{3/5}\ ,$ (3)

where $C_{N}^{2}$ is the strength of refractive index fluctuations, and $h$
the height in the atmosphere.

Elhalkouj et al. (2008) and Esposito et al. (2000) developed a model that
describes visibility reduction for off-axis fringe tracking as expected from
atmosphere anisoplanatism. Fig. 8 illustrates the geometrical elements
considered for the calculation. Consider a two-aperture interferometer, each
with an aperture diameter $D$, with a baseline length $\Delta$. The science
object and the phase-reference star are separated by $\theta$, and $P_{1}$,
$P_{2}$ and $P_{1}^{{}^{\prime}}$, $P_{2}^{{}^{\prime}}$ are the pupils
projected onto a single turbulent layer at height $h$. Further, $d_{12}$ and
$d_{21}$ are the distances between the pupils $P_{1}$ and
$P_{2}^{{}^{\prime}}$, and $P_{2}$ and $P_{1}^{{}^{\prime}}$, respectively.

Assuming that angular anisoplanatism is the only effect that reduces the
visibility, the loss of visibility as a function of FT–SC separation can be
approximated (following Maréchal) as

$V_{\text{average}}(\theta)=V\
\text{exp}\Big{[}-\frac{2\pi^{2}}{\lambda^{2}}\sigma_{p}^{2}(\theta)\Big{]}\
,$ (4)

where $V$ is the instant visibility for a delay time $\tau$ between the two
optical paths, and $\sigma_{p}^{2}(\theta)$ is the anisopistonic error
variance, which is the variance of the differential piston error. The
expression for $\sigma_{p}^{2}(\theta)$ is derived in Elhalkouj et al. (2008)
under the following assumptions:

  * •

In long-baseline interferometry, the product $h\theta$ is in general much
smaller than the baseline length $\Delta$: for $h\simeq$ 10 km, $\theta\simeq$
15 arcsec, the product results in $h\theta\ \simeq$ 0.75 m, which is small
compared to $\Delta_{UT1-UT4}$ = 130 m. Thus, the distances $d_{12}$ and
$d_{21}$ can be approximated by $\Delta\simeq d_{12}\simeq d_{21}$.

  * •

Further, the outer scale of turbulence $L_{0}$ is assumed to be infinity
(Buscher & Longair 2015), following the Kolmogorov model. A full treatment
taking into account the effect and statistics of the outer scale of turbulence
can be found in Boskri et al. (2021).

  * •

Elhalkouj et al. (2008) consider two different scenarios for deriving the
isopistonic angle error. The decisive parameter for this is the fraction $\pi
D/L_{0}$, which in our case is much smaller than one. Thus, we follow the
”small-aperture case”, which assumes that the apertures are small compared to
the outer scale of turbulence $L_{0}$.

  * •

Finally, the product ($h\theta$)max is equivalent to $h_{max}\theta_{0}$

The expression for the isopistonic angle error in Eq. (4) is given by

$\sigma_{p}(\theta)\sim
0.12\pi^{1/3}\lambda\left(\frac{D}{r_{0}}\right)^{-1/6}\frac{\theta}{\theta_{0}}\hskip
28.45274pt\text{for}\ L_{0}\rightarrow\infty\ .$ (5)

Under the assumptions described above, the anisopistonic error variance
$\sigma_{p}(\theta)$ in Eq. (5) only depends on two geometrical elements,
aperture diameter $D$ and FT–SC separation $\theta$, on two atmospheric
parameters, the height of the turbulent layer $H$ and the seeing $\epsilon$,
as well as on the wavelength $\lambda$ when $r_{0}$ and $\theta_{0}$ are
replaced by Eq. (1) and Eq. (2). For observations with the VLTI the atmosphere
profiler MASS-DIMM (Haguenauer et al. 2020) at the Paranal observatory
measures $\theta_{0}$ and $\epsilon$, while $D$, $\theta$, and $\lambda$ are
given by the telescopes, the FT–SC pair used, as well as the K-band wavelength
observed with GRAVITY. We can therefore calculate the expected visibility of
the SC for a given observation with GRAVITY Wide with Eq. (5).

Figure 8: Two-aperture interferometer with baseline length $\Delta$, and
projected pupils onto a single turbulent layer at altitude $h$. The aperture
diameter is $D$ and science object and phase-reference star form an angle
$\theta$. Image adapted from Esposito et al. (2000).

###  4.2 Visibility as a function of FT–SC separation

To show how well the model for the visibilities matches the observed data we
provide an example in Fig. 9 for the science target HD 48543B observed with
the ATs. We compare the measurement to the modelled visibilities from Eq. (4)
and Eq. (5). At the time of observation, the seeing is 0.41 arcsec (at 500
nm), and the isoplanatic angle is 1.7 arcsec (at 500 nm), which corresponds to
10.0 arcsec in K-band. We use HD 48543A at a separation of 7.90 arcsec as the
FT.

Figure 9: Observed (color) and modelled (black) visibility for the star HD
48543B. The FT, HD 48543A, is located at 7.90 arcsec from the SC. During the
observation, the MASS-DIMM measured a seeing $\epsilon$ of 0.41 arcsec and an
isoplanatic angle $\theta_{0}$ of 1.7 arcsec (at 500 nm), which is 10.0 arcsec
in K-band. AT1-AT2-AT3-AT4 correspond to the stations A0-G1-J2-K0. The model
by Elhalkouj et al. (2008) is able to explain the visibility loss from
atmosphere anisoplanatism. Figure 10: Observed contrast loss at 2.2 $\mu$m
versus off-axis separation for the ATs. The three curves indicate the typical
contrast loss for different seeing and isoplanatic angle values as expected
from atmospheric turbulence following Elhalkouj et al. (2008). Crosses
represent data from GRAVITY Wide observations with the prototype
implementation in November 2019, while circles represent data from GRAVITY
Wide commissioning in December 2021. Seeing and isoplanatic angle values are
measured by the MASS-DIMM at the Paranal observatory, and are defined at 500
nm.

Fig. 9 shows that the modelled visibility from atmosphere anisoplanatism
matches the observed visibility very well. The visibility loss ranges from
approximately 0.2 to 0.4, depending on the wavelength, in accordance with what
is predicted by the model. The data presented in Fig. 9 are calibrated for
instrumental effects inside GRAVITY, but not for atmospheric coherence loss in
the main delay lines and coherence loss of optics in the beam train from the
telescopes to the VLTI lab. Therefore, we can say that the visibility loss
observed in this exposure comes mostly from atmosphere anisoplanatism. Just as
in Fig. 9, we compute the atmospheric visibility loss at a wavelength of 2.2
$\mu$m of 60 GRAVITY Wide AT observations. In Fig. 10, we display each
observation by either a cross (for observations from 2019) or a filled circle
(for observations from 2021). Further, we sort the data in three groups based
on their isoplanatic angle measured. The first group contains the highest
values of $\theta_{0}$ with 3.12 arcsec $<\theta_{0}<$ 3.3 arcsec. The second
group spans 2.02 arcsec $<\theta_{0}<$ 2.7 arcsec, and the third group 1.23
arcsec $<\theta_{0}<$ 1.93 arcsec. Per group, we determine the mean value for
both isoplanatic angle and seeing from the exposures and calculate the
visibility loss with Eq. (4) for off-axis separations up to 30 arcsec,
represented by the solid orange, blue and green curve, respectively.
Additionally, we color the area of the minimum and maximum visibility
reduction, given by the minimum and maximum seeing and isoplanatic angle per
group, respectively.

We find that the coherence loss with increased FT–SC separation is well
described by atmosphere anisoplanatism for a large outer scale of turbulence
following Elhalkouj et al. (2008). We note two important aspects. Firstly, we
find that the visibility is higher for larger isoplanatic angles. The reason
for this is that a larger isoplanatic angle means a lower turbulent layer, and
therefore a larger overlap of the projected pupils from the SC and FT target.
This results in a better correction for wavefront aberrations by AO, as well
as correction for the fringe motion of the SC. Secondly, we find that the
model is more sensitive to the isoplanatic angle than to the seeing. A large
isoplanatic angle is crucial for being able to observe at large off-axis
separations. Until now, seeing and coherence time are taken into account for
scheduling service mode observations. Based on our findings, we point out that
the isoplanatic angle should be taken into account as well when executing
GRAVITY Wide observations.

###  4.3 Atmospheric conditions on Paranal

In Fig. 10 we see that while the overall trend is well matched by the model by
Elhalkouj et al. (2008), only few observations match the expected visibility
exactly. We discuss possible reasons in the following.

First, the model assumes the outer scale of turbulence to be infinity,
following the Kolmogorov model. However, this scale ranges from 12 to 50 m at
all major astronomical sites (Ziad 2016), and is about 22 m (Martinez et al.
2010) in the atmosphere model of Paranal. This leads to an underestimation of
the maximum visibility in our calculations for large telescopes (Elhalkouj et
al. 2008; Boskri et al. 2021), but does not affect much the visibility
estimates for the comparably small ATs (Fig. 10). Another point is that the
model might be too simple to describe the full effects. For example, it does
not take parameters such as DIT, total exposure time, coherence time, airmass
or magnitude of the SC and FT into account. The target HD 10257 at 19.77
arcsec separation in Fig. 10 was observed with a sequence of DITs between 0.13
s and 10 s to investigate the influence of the DIT on the visibility. In the
analysis we omitted the shortest DITs of 0.13 s and 0.3 s, because they might
be short enough to ”freeze” the turbulence and thus artificially increase the
measured visibilities. The 60 GRAVITY Wide observations presented in Fig. 10
were performed with the ATs. For observations with the UTs we expect a higher
visibility due to a larger overlap of the projected pupils on sky for given
FT–SC separation. Tab. 3 presents the seeing categories for the median seeing
and isoplanatic angle, respectively, for the percentiles 10$\%$, 25$\%$ and
50$\%$ measured by the MASS-DIMM at the Paranal platform. Based on these
values, we compute the visibility reduction for both UTs and ATs for
separations up to 30 arcsec, and present the result in Fig. 11. We can see
that for the same values of seeing and $\theta_{0}$, the UTs provide a higher
visibility.

Table 3: Statistics of seeing and isoplanatic angle $\theta_{0}$ at Paranal.

Percentile | 10$\%$ | 25$\%$ | 50$\%$
---|---|---|---
Seeing [arcsec] at 500 nm | 0.52 | 0.62 | 0.76
$\theta_{0}$ [arcsec] at 500 nm | 3.01 | 2.48 | 1.96
$\theta_{0}$ [arcsec] at 2.2 $\mu$m | 17.81 | 14.68 | 11.60
333 The seeing is given at 500 nm and the isoplanatic angle $\theta_{0}$ at
500 nm and at 2.2 $\mu$m. Both parameters are measured by the MASS-DIMM at the
Paranal observatory.

We conclude that the predictions by the model of Elhalkouj et al. (2008)
overall are in good agreement with the coherence loss we observe with GRAVITY
Wide. It gives a good starting point to plan and execute observations. It also
shows that it is not enough to check the seeing conditions and coherence time,
but that one also needs to take the isoplanatic angle into account.
Considering the performance as theoretically predicted and observationally
confirmed (see Fig. 9 and Fig. 10), for GRAVITY Wide operations the offered
separation will be limited to 30 arcsec, as for larger distances the coherence
loss is considered too large. Fig. 10 also demonstrates that especially
observations at the largest separations should be done at atmospheric
conditions with a large isoplanatic angle and with a small zenith angle.

##  5 Summary and outlook

GRAVITY Wide has provided another breakthrough in near-infrared interferometry
with the first observations using wide-angle separation fringe tracking up to
about 30 arcsec across four telescopes. GRAVITY Wide significantly expands the
sky coverage of GRAVITY and opens up near-infrared interferometry to new
fields. In particular we demonstrate first near-infrared fringes of a $z=2.46$
quasar. At m${}_{K}=15.6$, this is now the faintest extragalactic object
observed by a factor of hundred along with the highest redshift. In addition,
we demonstrated interferometric imaging with this new observing mode on the
binary system HD 105913A, and derived new and updated orbits for several
binary stars in the Orion Trapezium Cluster.

Figure 11: Visibility at a wavelength of 2.2$\,\mu$m versus off-axis
separation for different seeing and isoplanatic angle values given in Tab. 3.
Solid lines represent the visibility loss for the UTs, and dashed lines for
the ATs, respectively.

We investigate the influence of atmospheric turbulence on the new wide-angle
fringe tracking mode, and find that atmosphere anisoplanatism well describes
the contrast loss in the GRAVITY Wide observations following Elhalkouj et al.
(2008). In particular we note higher visibilities for observations with the 8
m UTs compared to the 1.8 m ATs due to a larger overlap of the projected
pupils on sky. Because wide-angle separation observations are more sensitive
to the isoplanatic angle than to the seeing, we propose to include the
isoplanatic angle in future planning for GRAVITY Wide observations.

GRAVITY Wide is just the beginning of the full GRAVITY+ upgrade (Eisenhauer
2019). The main limitation now is the performance of the adaptive optics which
has a two-fold effect in both reducing the SC target light and preventing
fringe tracking on fainter stars. The next phases of GRAVITY+ therefore will
install a new state-of-the-art adaptive optics system in 2024 and laser guide
stars on all four UTs in 2025. The combination will allow us to push to even
fainter targets across the whole sky (Fig. 12). Together with enhanced
vibration control for the telescopes and performance improvements of the
GRAVITY instrument itself, we then expect fringe tracking on stars as faint as
m${}_{K}=13$ both on-axis and off-axis, and observations of objects with a
magnitude up to m${}_{K}\approx 22$. The performance improvements from
GRAVITY+ will open up key advances in many fields of astrophysics: e.g. the
possibility to measure the spin of the Galactic Center black hole, to study
SMBH growth and coevolution with galaxies over cosmic time, to directly detect
exoplanets that are out of reach for traditional coronographs, to measure
their atmospheric composition and orbital architecture to unprecedented
precision, and to spatially resolve stars and planetary systems in formation.
The leaps and bounds with regard to near-infrared interferometric AGN science
can be best seen in the numbers of AGN that GRAVITY can observe at each step.
GRAVITY with its original performance could observe $\approx 10$ AGN at $z\sim
0$ (Gravity Collaboration et al. 2020c). GRAVITY Wide provides the same
$10-20$ but now at $z=2$. GRAVITY+, with its full expanded capabilities, will
make the jump to over 1000 AGN across cosmic time.

Figure 12: Sky coverage for laser guide star adaptive optics supported off-
axis fringe tracking with a fringe tracking star as faint as $m_{K}=13$, and a
maximum allowed separation of 30 arcsec.

The first GRAVITY Wide results presented in this paper prove the functionality
of the new large separation fringe tracking mode, which will be offered to the
community through ESO from October 2022 on 111The JMMC tech group has
developed a tool for finding FT targets in order to assist in GRAVITY Wide
proposal preparations. Available at https://searchftt.jmmc.fr/ and give a
glimpse to the exciting future of near-infrared interferometry with GRAVITY+.

###### Acknowledgements.

We are very grateful to our funding agencies (MPG, DFG, BMBF, ERC, CNRS (CSAA,
ASHRA), Ile-de-France region (DIM ACAV+), Paris Observatory-PSL, Observatoire
des Sciences de l’Univers de Grenoble, Université Grenoble Alpes, Observatoire
de la Côte d’Azur, Université Côte d’Azur, and the Fundação para a Ciência e
Tecnologia) and the generous support from the Max Planck Foundation - an
independent, non-profit organization of private supporters of top research in
the Max Planck Society. A.A. and P.G. acknowledge support from grants
UIDB/00099/2020 and PTDC/FIS-AST/7002/2020. We also thank ESO and the
ESO/Paranal staff, and the many scientific and technical staff members in our
institutions, who helped to make GRAVITY Wide a reality. This work was
supported by the European Union through ERC grant Nos. 866070 (AB, DD, RL, and
MS), and has made use of the Jean-Marie Mariotti Center Aspro and LITpro
services (http://www.jmmc.fr/) and of the CDS astronomical Data Centers SIMBAD
and VIZIER.

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##  Appendix A GRAVITY Wide data

###  A.1 Data from GRAVITY Wide prototype implementation

We list observations performed with the prototype implementation of GRAVITY
Wide, i.e. no pupil relay, in the runs in November 2019 and March 2020.

Table 4: Observations with the GRAVITY Wide prototype implementation in
November 2019 and March 2020.

SC | FT | Sep. [arcsec] | Date [UTC] | Pol. | Res. | Baseline Conf. | DIT [s] | NDIT
---|---|---|---|---|---|---|---|---
GJ 65B | GJ 65A | 2.08 | 2019 Nov 01 | S | MED | A0 G1 J2 K0 | 5.0 | 30
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 0.3 | 300
HD 105913A | HD 105913B | 5.11 | 2020 Mar 09 | S | MED | U1 U2 U3 U4 | 1.0 | 128
HD 24071 | HD 24072 | 8.16 | 2019 Nov 03 | C | MED | A0 G1 J2 K0 | 5.0 | 16
HD 218268 | HD 218269 | 8.83 | 2019 Nov 01 | S | MED | A0 G1 J2 K0 | 1.0 | 32
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 0.3 | 300
|  |  | 2019 Nov 03 | S | MED | A0 G1 J2 K0 | 1.0 | 64
|  |  | 2019 Nov 03 | C | MED | A0 G1 J2 K0 | 1.0 | 64
HD 10257 | HD 10268 | 19.77 | 2019 Nov 01 | S | MED | A0 G1 J2 K0 | 5.0 | 30
|  |  | 2019 Nov 01 | C | MED | A0 G1 J2 K0 | 0.13 | 500
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 0.3 | 300
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 1.0 | 64
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 2.0 | 32
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 5.0 | 32
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32
GJ1048B | HD 16270 | 12.06 | 2019 Nov 03 | S | MED | A0 G1 J2 K0 | 10.0 | 32
LEDA 1264801 | TYC 274-754-1 | 22.14 | 2020 Mar 04 | S | MED | U1 U2 U3 U4 | 30.0 | 10
$[$VV96$]$ J095516.0 | 2MASS 09551512 | 23.93 | 2020 Mar 04 | S | MED | U1 U2 U3 U4 | 30.0 | 10
-251732 | -2517108 |  |  |  |  |  |  |
QSO B0435-300 | HD 29514 | 25.99 | 2019 Nov 01 | C | MED | A0 G1 J2 K0 | 10.0 | 32
|  |  | 2019 Nov 01 | C | MED | A0 G1 J2 K0 | 30.0 | 16
TYC 8071-854-1 | HD 26404 | 29.89 | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 30
TCC 59 |  $\theta^{1}$ Ori A | 4.23 | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32
|  |  | 2019 Nov 03 | S | MED | A0 G1 J2 K0 | 10.0 | 16
$\theta^{1}$ Ori E |  $\theta^{1}$ Ori A | 4.51 | 2019 Nov 01 | C | MED | A0 G1 J2 K0 | 5.0 | 32
|  |  | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32
|  |  | 2019 Nov 03 | S | MED | A0 G1 J2 K0 | 10.0 | 16
$\theta^{1}$ Ori F |  $\theta^{1}$ Ori C | 4.52 | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32
|  |  | 2019 Nov 03 | S | MED | A0 G1 J2 K0 | 10.0 | 16
TCC 43 |  $\theta^{1}$ Ori B | 6.82 | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32
$\theta^{1}$ Ori G |  $\theta^{1}$ Ori C | 7.38 | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32
$\theta^{1}$ Ori H |  $\theta^{1}$ Ori A | 8.19 | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32
$\theta^{1}$ Ori B |  $\theta^{1}$ Ori A | 8.85 | 2019 Nov 01 | C | MED | A0 G1 J2 K0 | 5.0 | 32
|  |  | 2020 Mar 03 | S | MED | A0 G1 J2 J3 | 10.0 | 16
|  |  | 2020 Mar 03 | S | MED | A0 G1 J2 J3 | 10.0 | 32
$\theta^{1}$ Ori C |  $\theta^{1}$ Ori A | 12.86 | 2019 Nov 01 | C | MED | A0 G1 J2 K0 | 5.0 | 32
|  |  | 2020 Mar 03 | S | MED | A0 G1 J2 J3 | 10.0 | 32
$\theta^{1}$ Ori D |  $\theta^{1}$ Ori C | 13.41 | 2019 Nov 02 | C | MED | A0 G1 J2 K0 | 10.0 | 32

444 The first block of SC–FT pairs are single and binary stars, the second
block indicates a brown dwarf, three AGN, and a faint star with $m_{K}=9.91$
(Cutri et al. 2003) (from top to bottom). Finally, the third block lists
single and binary stars within the Orion Trapezium Cluster. Columns from left
to right: name of the SC, name of the FT, separation (Sep.) between SC and FT
in arcsec, date of observation night in UTC, polarization (Pol.) mode in
combined (C) or split (S) linear polarization, resolution (Res.) in low (LOW),
medium (MED) or high (HIGH), baseline configuration (Baseline Conf.), detector
integration time (DIT) in seconds and number of DITs (NDIT).

###  A.2 Data from GRAVITY Wide commissioning

We list observations during GRAVITY Wide commissioning in December 2021 and
January 2022.

Table 5: Observations from GRAVITY Wide commissioning in December 2021,
January 2022 and April 2022.

SC | FT | Sep. [arcsec] | Date [UTC] | Pol. | Res. | Baseline Conf. | DIT [s] | NDIT
---|---|---|---|---|---|---|---|---
HD 83368A | HD 83368B | 3.28 | 2021 Dec 16 | S | MED | A0 G1 J2 K0 | 10.0 | 12
|  |  | 2021 Dec 16 | C | MED | A0 G1 J2 K0 | 10.0 | 12
TYC6018-535-1 | TYC6018-1742-1 | 3.91 | 2021 Dec 18 | C | HIGH | U1 U2 U3 U4 | 1.0 | 200
|  |  | 2021 Dec 19 | C | HIGH | U1 U2 U3 U4 | 3.0 | 30
2MASS 09325130 | IRAS 09312-5534 | 4.83 | 2021 Dec 17 | S | MED | A0 G1 J2 K0 | 30.0 | 4
-5548206 |  |  |  |  |  |  |  |
HD 28255A | HD 28255B | 5.52 | 2021 Dec 13 | S | MED | A0 G1 J2 K0 | 10.0 | 12
SDSS J161513.84 | 2MASS 16151430 | 7.50 | 2022 Apr 17 | C | MED | U1 U2 U3 U4 | 100.0 | 6
+084914.4 | +0849167 |  |  |  |  |  |  |
HD 48543B | HD 48543A | 7.90 | 2021 Dec 15 | C | MED | A0 G1 J2 K0 | 10.0 | 12
|  |  | 2021 Dec 15 | S | MED | A0 G1 J2 K0 | 10.0 | 12
∗ f Eri A |  ∗ f Eri B | 8.19 | 2021 Dec 16 | S | MED | A0 G1 J2 K0 | 10.0 | 4
|  |  | 2021 Dec 16 | S | MED | A0 G1 J2 K0 | 10.0 | 12
|  |  | 2021 Dec 16 | S | MED | A0 G1 J2 K0 | 3.0 | 12
$\theta^{1}$ Ori B |  $\theta^{1}$ Ori A | 8.85 | 2021 Dec 15 | S | MED | A0 G1 J2 K0 | 10.0 | 32
|  |  | 2021 Dec 15 | S | MED | A0 G1 J2 K0 | 30.0 | 12
HD 34750 | HD 34749 | 10.65 | 2021 Dec 14 | C | MED | A0 G1 J2 K0 | 30.0 | 4
∗ p Eri A |  ∗ p Eri B | 11.46 | 2021 Dec 15 | C | HIGH | A0 G1 J2 K0 | 3.0 | 32
∗ p Eri B |  ∗ p Eri A | 11.46 | 2021 Dec 14 | C | HIGH | A0 G1 J2 K0 | 10.0 | 16
|  |  | 2021 Dec 14 | S | MED | A0 G1 J2 K0 | 3.0 | 32
|  |  | 2021 Dec 15 | C | HIGH | A0 G1 J2 K0 | 3.0 | 32
$\theta^{1}$ Ori C |  $\theta^{1}$ Ori A | 12.85 | 2021 Dec 14 | C | MED | A0 G1 J2 K0 | 10.0 | 32
LAMOST | Gaia 586750639 | 12.86 | 2021 Dec 18 | S | MED | U1 U2 U3 U4 | 30.0 | 12
J092034.16 +065 | 2466652 |  | 2022 Jan 19 | S | MED | U1 U2 U3 U4 | 30.0 | 12
|  |  | 2022 Jan 25 | S | MED | U1 U2 U3 U4 | 30.0 | 12
2MASS 05165352 | HD 34708 | 13.80 | 2021 Dec 14 | S | MED | A0 G1 J2 K0 | 30.0 | 8
-4811412 019 |  |  |  |  |  |  |  |
|  |  | 2021 Dec 16 | S | MED | A0 G1 J2 K0 | 30.0 | 12
|  |  | 2021 Dec 16 | S | MED | A0 G1 J2 K0 | 3.0 | 4
∗ gam01 Vol |  ∗ gam02 Vol | 14.14 | 2021 Dec 13 | S | HIGH | A0 G1 J2 K0 | 10.0 | 12
HD 10268 | HD 10257 | 19.77 | 2021 Dec 14 | S | MED | A0 G1 J2 K0 | 10.0 | 12
RMC 141 | W61 7-8 | 21.58 | 2021 Dec 15 | C | MED | A0 G1 J2 K0 | 30.0 | 12
PKS0435-300 | HD 29514 | 25.99 | 2021 Dec 14 | S | MED | A0 G1 J2 K0 | 30.0 | 12
|  |  | 2021 Dec 15 | C | MED | A0 G1 J2 K0 | 30.0 | 12
2MASS J07300800 | HD 59867 | 29.11 | 2021 Dec 15 | S | MED | A0 G1 J2 K0 | 3.0 | 32
-3939086 |  |  |  |  |  |  |  |
HD 42092 | HD 42111 | 29.17 | 2021 Dec 15 | S | MED | A0 G1 J2 K0 | 10.0 | 12
BAT99 113 | W61 7-8 | 31.92 | 2021 Dec 15 | S | LOW | A0 G1 J2 K0 | 30.0 | 12
|  |  | 2021 Dec 15 | C | MED | A0 G1 J2 K0 | 30.0 | 12

555 Columns from left to right: name of the SC, name of the FT, separation
(Sep.) between SC and FT in arcsec, date of observation night in UTC,
polarization (Pol.) mode in combined (C) or split (S) linear polarization,
resolution (Res.) in low (LOW), medium (MED) or high (HIGH), baseline
configuration (Baseline Conf.), detector integration time (DIT) in seconds and
number of DITs (NDIT).

##  Appendix B Analysis of binary star systems

The analysis of binary star systems in Sect. 3.2 proceeds in two steps. First,
we use LITpro (Tallon-Bosc et al. 2007) to fit a binary model to our data. In
the second step, we use orbitize! (Blunt et al. 2020) for $\theta^{1}$ Ori B
and $\theta^{1}$ Ori C, and the fitting code from Gillessen et al. (2009) for
$\theta^{1}$ Ori D to determine orbital parameters from our separation
measurements and measurements from the literature.

###  B.1 Binary Fitting with LITpro

For fitting a binary model to our data, we use the model fitting software
LITpro which is based on the modified Levenberg–Marquardt Algorithm (Dennis &
Schnable 1983) and includes a Trust Region Method (Moré & Sorensen 1983). It
was developed by the Jean-Marie Mariotti Center (JMMC) 222Available at
https://www.jmmc.fr/english/tools/data-analysis/litpro/, the french center for
optical interferometry. LITpro uses the same OI Exchange format as the GRAVITY
pipeline, thus we can directly upload our data and fit observables, such as
squared visibilities, closure phases or visibility amplitudes.

We can choose the types of data to be fitted and take the interferometric
observables visibility squared, visibility amplitude and/or closure phase. The
visibility squared for a binary model is

$\begin{split}\nu_{\text{binary}}^{2}&=\nu_{\text{binary}}^{*}\nu_{\text{binary}}\\\
&=\frac{\nu_{a}^{2}+f^{2}\nu_{b}^{2}+2f|\nu_{a}||\nu_{b}|\text{cos}(2\pi(u\Delta\alpha+v\Delta\beta))}{(1+f)^{2}}\\\
&=\frac{\nu_{a}^{2}+f^{2}\nu_{b}^{2}+2f|\nu_{a}||\nu_{b}|\text{cos}(2\pi\textbf{B}\cdot\textbf{s}_{\text{binary}}/\lambda)}{(1+f)^{2}}\
,\end{split}$ (6)

where $\nu_{a}$ and $\nu_{b}$ are the primary star and companion star
visibility, respectively, $\lambda$, is the wavelength, $f=f_{b}/f_{a}$, is
the flux ratio between both stars, B, is the baseline and
s${}_{\text{binary}}$ $\equiv(\Delta\alpha,\Delta\beta)$ is the position of
the companion star with respect to the primary star in dRA and dDEC.

The closure phase is determined by taking the argument from the bispectrum

$\begin{split}B_{123}&=\nu_{12}^{measured}\nu_{23}^{measured}\nu_{31}^{measured}\\\
&=|F_{1}|\ |F_{2}|e^{i(\Phi_{1}-\Phi_{2})}\nu_{12}^{true}\cdot|F_{2}|\
|F_{3}|e^{i(\Phi_{2}-\Phi_{3})}\nu_{23}^{true}\cdot\\\ &\indent|F_{3}|\
|F_{1}|e^{i(\Phi_{3}-\Phi_{1})}\nu_{31}^{true}\\\ &=|F_{1}|^{2}\ |F_{2}|^{2}\
|F_{3}|^{2}\ \nu_{12}^{true}\cdot\nu_{23}^{true}\cdot\nu_{31}^{true}\
,\end{split}$ (7)

between three telescopes 1,2 and 3:

$\phi_{1-2-3}=\text{arg}(\nu_{\text{binary,1-2}}\ \nu_{\text{binary,2-3}}\
\nu_{\text{binary,3-1}})\ ,$ (8)

where $\nu_{\text{binary}}$ is computed with Eq. (6).

From Eq. (6) and (8), the parameters $\Delta\alpha$ and $\Delta\beta$ give the
distance from the primary star located at (dRA, dDec) = (0, 0) mas to the
companion star, and are together with the flux ratio, $f$, and a background,
$f_{\text{background}}$, the parameters to be fitted. The latter variable
represents the effect of visibility loss from atmospheric turbulence. We give
a starting value for the companion star position, and let LITpro’s fitting
engine iteratively find the minimum $\chi^{2}$ value. At the end of the
fitting procedure, we obtain the final $\chi^{2}$, values and standard
deviations for the fitted parameters, and covariance and correlation matrices
for outlining possible parameter degeneracies.

###  B.2 Orbit Modelling with orbitize!

orbitize! is an open-source Python package for orbit fitting of directly
imaged exoplanets, which can also be applied to binary star systems. orbitize!
is comprised of a parallel-tempered Affine-invariant Markov Chain Monte Carlo
(MCMC) algorithm (Vousden et al. 2016; Foreman-Mackey et al. 2013), and is
based on Bayesian statistics. In orbitize! the input data used in the orbit-
fitting includes the date of the observation in MJD, and the dRA and dDec
offsets from the primary star at (0,0) mas with corresponding uncertainties of
the secondary star in mas. The posterior distribution over the orbital
parameters, are then computed using Bayes’ theorem.

The MCMC algorithm in orbitize! represents an efficient sampling method to
explore the likelihood over a multidimensional parameter space and to
determine expectation values of the model parameters, i.e. the orbital
parameters. Further, it is possible to use several Markov chains, called
walkers, in parallel to fully explore the parameter space at different
”temperatures”. A higher temperature helps the walkers not to become stuck in
regions of local minima. A larger number of walkers increases the samples
used, but also increases the computation time.

We use 20 temperatures, 1000 walkers, and 1 million steps to ensure
convergence. Further, we impose a Gaussian prior on the measurement for the
mass, and we use the parallax of 2.415 $\pm$ 0.040 mas from Menten et al.
(2007), which is fixed in our analysis. If we have an initial guess for best-
fit orbital parameters, for example from the literature, we use lower and
upper bounds of a uniform prior around this value. This was done for the orbit
of $\theta^{1}$ Ori C (see Sect. 3.2), where we set bounds around orbital
parameters that were already determined in the literature (Gravity
Collaboration et al. 2018c). In the case of $\theta^{1}$ Ori B (see Sect.
3.2), there are no previously determined orbital parameters, and therefore we
let orbitize! find the best-fit orbit without giving an initial guess.