11institutetext:  Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge CB3 0HA, United Kingdom 11email<EMAIL_ADDRESS>22institutetext: Kavli Institute for Cosmology, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, UK 33institutetext: Centre for Exoplanets
and Habitability, University of Warwick, Coventry CV4 7AL, UK 44institutetext:
Department of Physics, University of Warwick, Coventry CV4 7AL, UK
55institutetext: School of Physics and Astronomy, University of Leicester,
Leicester LE1 7RH, UK 66institutetext: LESIA, Observatoire de Paris, PSL,
CNRS, Sorbonne Université, Université de Paris, 5 place Janssen, 92195 Meudon,
France 77institutetext: European Southern Observatory, Karl-Schwarzschild-
Straße 2, 85748 Garching, Germany 88institutetext: School of Physics and
Astronomy, Monash University, Clayton Vic 3800, Australia

# The orbit of HD 142527 B is too compact to explain many of the disc features

M. Nowak The orbit of HD 142527 B is too compact to explain many of the disc
featuresThe orbit of HD 142527 B is too compact to explain many of the disc
featuresThe orbit of HD 142527 B is too compact to explain many of the disc
featuresThe orbit of HD 142527 B is too compact to explain many of the disc
features S. Rowther The orbit of HD 142527 B is too compact to explain many of
the disc featuresThe orbit of HD 142527 B is too compact to explain many of
the disc featuresThe orbit of HD 142527 B is too compact to explain many of
the disc featuresThe orbit of HD 142527 B is too compact to explain many of
the disc featuresThe orbit of HD 142527 B is too compact to explain many of
the disc featuresThe orbit of HD 142527 B is too compact to explain many of
the disc features S. Lacour The orbit of HD 142527 B is too compact to explain
many of the disc featuresThe orbit of HD 142527 B is too compact to explain
many of the disc featuresThe orbit of HD 142527 B is too compact to explain
many of the disc featuresThe orbit of HD 142527 B is too compact to explain
many of the disc features F. Meru The orbit of HD 142527 B is too compact to
explain many of the disc featuresThe orbit of HD 142527 B is too compact to
explain many of the disc featuresThe orbit of HD 142527 B is too compact to
explain many of the disc featuresThe orbit of HD 142527 B is too compact to
explain many of the disc features R. Nealon The orbit of HD 142527 B is too
compact to explain many of the disc featuresThe orbit of HD 142527 B is too
compact to explain many of the disc featuresThe orbit of HD 142527 B is too
compact to explain many of the disc featuresThe orbit of HD 142527 B is too
compact to explain many of the disc features D. J. Price The orbit of HD
142527 B is too compact to explain many of the disc featuresThe orbit of HD
142527 B is too compact to explain many of the disc features

###### Abstract

Context. HD 142527 A is a young and massive Herbig Ae/Be star surrounded by a
highly structured disc. The disc shows numerous morphological structures, such
as spiral arms, a horseshoe region of dust emission, a set of shadows cast by
an inner disc on the outer disc, and a large cavity extending from
$\simeq{}30\,\mathrm{au}$ to $\simeq{}130\,\mathrm{au}$. HD 142527 A also has
a lower mass companion, HD 142527 B ($M=0.13\pm{}0.03\,M_{\odot}$), which is
thought to be responsible for most of the structures observed in the
surrounding disc.

Aims. We aim to fully constrain the orbit of HD 142527 B and determine whether
the binary alone is truly responsible for the observed morphology of the HD
142527 disc.

Methods. We gathered VLTI/GRAVITY observations of HD 142527, either from our
own programmes or from the ESO archive. We used this inhomogeneous set of data
to extract a total of seven high-precision measurements of the relative
astrometry between HD 142527 A and B, spread from mid-2017 to early 2021.
Combined with what is available in the literature, these new measurements
offer a total of 9 yr of astrometric monitoring on HD 142527. We used orbit
fitting tools to determine the orbital parameters of HD 142527 B, and used
them as inputs for a 3D hydrodynamical model of the disc to determine whether
or not the binary is able to create the structures observed in the disc.

Results. Our VLTI/GRAVITY astrometry gives excellent constraints on the orbit
of HD 142527 B. We show that the secondary is following an orbit of semi-major
axis $a=10.80\pm 0.22\,\mathrm{au}$, with moderate eccentricity
($e=0.47\pm{}0.01$), and has recently passed its periapsis ($\tau=2020.42$).
With such a compact orbit, we show that HD 142527 B can only generate a gap
and spiral arms of $\sim{}30\,\mathrm{au}$ in the disc, which is much smaller
than what is revealed by observations.

Conclusions. Even from a theoretical standpoint, the observed cavity size of
$\sim{}100\,\mathrm{au}$ far exceeds even the most generous predictions for a
companion like HD 142527 B on such a compact orbit. Thus, we conclude that the
low-mass companion cannot be solely responsible for the observed morphology of
the disc surrounding the system.

###### Key Words.:

binaries: visual – protoplanetary disks – circumstellar matter – techniques:
high angular resolution – technique: interferometric

##  1 Introduction

HD 142527 A is a $5.0\pm{}1.5\,\mathrm{Myr}$ old Herbig Ae/Be star of type
F6-IIIV, with a mass of $2.0\pm{}0.3\,M_{\odot}$ (Mendigutía et al., 2014),
located at a distance of $159.3\pm{}0.7\,\mathrm{pc}$ (GaiaCollaboration et
al., 2021). The system has been extensively studied for its spectacular
circumstellar disc, which harbours several enigmatic morphological features.
Observations in the near-infrared with Subaru/CIAO have revealed the presence
of a large spiral arm at $>100\,\mathrm{au}$ (Fukagawa et al., 2006), with a
counterpart in CO later detected with the Atacama Large
Millimetre/submillimetre Array (ALMA, Christiaens et al., 2014). Smaller
spiral arms were also detected in the near-IR (Casassus et al., 2012; Canovas
et al., 2013; Avenhaus et al., 2014, 2017), as well as other larger ones,
extending up to $\simeq{}500\,\mathrm{au}$ (Christiaens et al., 2014).

The main feature of the HD 142527 disc is arguably its large central cavity,
which extends from $\simeq{}30$ to $\simeq{}130\,\mathrm{au}$. Its presence
was first suggested by Fukagawa et al. (2006), before being confirmed with
ALMA by Casassus et al. (2013), who also detected a horseshoe structure of
continuum emission surrounding this central cavity.

Early observations with the MIDI instrument at the Very Large Telescope
Interferometer (VLTI) revealed the presence of an inner disc, which has since
been detected in visible light with SPHERE/Zimpol (Avenhaus et al., 2017).
This inner disc is thought to be inclined at $\simeq{}70\,\mathrm{deg}$
relative to the outer disc, thus casting a narrow shadow on it, and explaining
some of the azimuthal asymmetries observed in the near-IR (Casassus et al.,
2012; Marino et al., 2015).

Its highly structured morphology, eccentric outer disc, and high fraction of
crystalline silicates (van Boekel et al., 2004; Avenhaus et al., 2017), have
made HD 142527 a prime taget for planet searches. The low-mass companion HD
142527 B was discovered by Biller et al. (2012) using the Sparse Aperture
Masking (SAM) mode of SPHERE/NACO, and this new object quickly became the
prime suspect to explain the complex disc structures. Lacour et al. (2016)
used the SAM mode of SPHERE/NACO and Gemini/GPI to further characterise the
companion and its orbit. They showed that HD 142527 B was likely an M dwarf
with a mass of $0.13\pm{}0.03\,M_{\odot}{}$ and a temperature of
$T=3000\pm{}100\,\mathrm{K}$ (i.e. with a contrast ratio of $\simeq{}1/75$ and
a mass ratio of $\simeq{}1/10$ with respect to the primary). Their astrometric
measurements covered epochs ranging from early 2012 to mid-2014, and suggested
two possible families of orbits: one that had passed periapsis, and another
approaching periapsis. Both families had moderate eccentricities
($e\simeq{}0.5$), and semi-major axes ranging from $70$ to $260\,\mathrm{au}$.

The first comprehensive study linking the low-mass companion HD 142527 B to
the morphological structure of the disc was conducted by Price et al. (2018a).
With the use of a series of 3D hydrodynamical models of HD 142527, they showed
that a low-mass companion, with characteristics compatible with that reported
by Lacour et al. (2016), could explain most of the observed structures of the
disc. Their study also strongly favoured the family of orbit approaching
periapsis, as only this one was compatible with the disc features. Claudi et
al. (2019) further refined the orbital parameters of the binary with a set of
new astrometric measurements with SPHERE, and reinforced the suggestion of
Price et al. (2018a) that HD 142527 B was approaching its periapsis.

More recently, Balmer et al. (2022) has presented the most extensive set of
astrometric measurements to date on HD 142527 B, with epochs ranging from
early 2012 to mid-2018. They were able to give good constraints on most of the
orbital parameters, and concluded that the low-mass companion was indeed
approaching its periapsis over this period. They reported a semi-major axis of
$\simeq{}15\,\mathrm{au}$, an eccentricity of $\simeq{}0.3$, and an orbit
close to perpendicular to the outer disc (mutual inclination of
$\simeq{}80\,\mathrm{deg}$). They concluded that this high mutual inclination
supported the idea that the binary was sculpting the disc, as the nearly
perpendicular orbit was one of their main assumptions needed by Price et al.
(2018a) to explain the disc features.

We note, however, that both the semi-major axis and the eccentricity derived
by Balmer et al. ($a\simeq{}15\,\mathrm{au}$ and $e\simeq{}0.3$) are
significantly smaller than the values used by Price et al.
($a\simeq{}30\,\mathrm{au}$ and $e\simeq{}$0.6-0.7). As the lower semi-major
axis and eccentricity are expected to result in different disc structures,
this raises the question of whether the companion can really explain the
complex morphological structures observed in the disc.

In this work we attempt to answer this question with a set of VLTI/GRAVITY
observations gathered both from the ESO archive and from our own observing
programmes. These long-baseline interferometric obervations offer a
significant gain in terms of astrometric precision compared to single-dish
observations, resulting in better constraints on the orbital parameters of the
binary.

The paper is structured as follows. In Section 2 we introduce our
observations, and in Section 3 we give some details on how we extracted
astrometric measurements from these data, which were obtained in a variety of
modes and configurations. In Section 4 we focus on the orbit fit, and present
our new constraints on the orbital parameters of the binary, and on its mutual
inclination with the inner and outer discs. We use these orbital parameters as
an input in a set of 3D hydrodynamical simulations of the disc, and discuss
the results in Section 5. Finally, we give our conclusions in Section 6.

##  2 Observations

We gathered a number of observations of HD 142527, obtained with VLTI/GRAVITY
between 2017 and 2021. The observations are spread over ten different nights.
Some of them are part of regular programmes targeting HD 142527, whereas
others were acquired using HD 142527 either as a backup target in case of bad
weather, or as a target for technical tests with the instrument. As a
consequence, the set of observations presented in this paper is inhomogeneous,
with a mixture of VLTI configurations (auxiliary telescopes, ATs, or unit
telescopes, UTs), spectral resolutions (medium or high), polarisation settings
(combined or split), and observing modes (dual-field or single-field). The
full observing log is presented in Table 1.

In principle, single-field observations with GRAVITY are used when the field
of view of the fibre ($\sim{}300\,\mathrm{mas}$ in diameter at half-maximum
with the ATs, $\sim{}60\,\mathrm{mas}$ with the UTs) is larger than the
separation of the observed binary. In this case the light from both components
is injected into the fibre, and the data are analysed using the squared
visibilities and closure phases formalism (see Section 3). However, when the
separation is larger than the field of view of the fibre, the instrument is
set to dual-field mode, where the two components can be observed sequentially
by moving the fibre back and forth between the two. This strategy, used in
particular for exoplanet observations, is described in (GravityCollaboration
et al., 2019, 2020).

In the case of HD 142527, the separation is typically
$\sim{}50\,\mathrm{mas}$, and the observations were carried out in single-
field when observing with the ATs, and in dual-field when observing with the
UTs (see Table 1). However, we found that in all the dual-field observations,
when the fibre is centred on HD 142527 B, it also collects a significant
amount of flux from HD 142527 A, due to the small separation and high contrast
ratio ($\sim{}1/75$ in K-band) between the two components. As a consequence,
we decided to treat these observations as single-field rather than dual-field,
and to discard the dual-field data in which the fibre is centred on the star,
as they are too largely dominated by HD 142527 A. We report only the
observations obtained at the location of HD 142527 B in Table 1. Finally, when
the data obtained are in split polarisation, we treat the polarisations
separately, as if they were independent exposures.

Epoch |  Configuration1 |  Mode2 | $N_{\mathrm{exp}}\times{}N_{\mathrm{DIT}}\times{}\mathrm{DIT}$ | Airmass | Seeing (”) | Programme
---|---|---|---|---|---|---
2017-03-19† | ATs Astro | Single,High,Combined | $7\times{}10\times{}30\,\mathrm{s}$ | 1.05-1.07 | 0.6-0.81 | 098.D-0488(A)
2017-05-03 | ATs Large | Single,Medium,Combined | $2\times{}30\times{}10\,\mathrm{s}$ | 1.22-1.25 | 0.33-0.43 | Technical
3-6[0.5pt/3pt] | Single,Medium,Split | $3\times{}20\times{}30\,\mathrm{s}$ | 1.14-1.67 | 0.41-0.65
2018-02-17 | ATs Large | Single,Medium,Combined | $2\times{}30\times{}0.3\,\mathrm{s}$ | 1.86-2.28 | 0.48-0.67 | 60.A-9800(U)
$4\times{}30\times{}1\,\mathrm{s}$
3-6[0.5pt/3pt] | Single,High,Combined | $2\times{}20\times{}10\,\mathrm{s}$ | 1.71-1.78 | 0.46-0.48
2018-07-25 | UTs | Dual,High,Split | $2\times{}15\times{}30\,\mathrm{s}$ | 1.05-1.05 | 1.10-1.54 | 60.A-9800(U)
3-6[0.5pt/3pt] | Dual,Medium,Split | $2\times{}20\times{}10\,\mathrm{s}$ | 1.06-1.07 | 0.96-1.39
| $2\times{}10\times{}10\,\mathrm{s}$
2018-07-28 | UTs | Dual,Medium,Split | $2\times{}15\times{}10\,\mathrm{s}$ | 1.09-1.11 | 1.13-1.46 | 60.A-9800(U)
2019-04-27† | ATs Medium | Single,Medium,Split | $2\times{}10\times{}30\leavevmode\nobreak\ \mathrm{s}$ | 1.10-1.13 | 0.66-0.73 | 60.A-9132(A)
2019-04-28 | ATs Medium | Single,Medium,Split | $2\times{}10\times{}30\,\mathrm{s}$ | 1.12-1.15 | 0.69-0.88 | 60.A-9132(A)
2019-05-01 | ATs Medium | Single,Medium,Split | $4\times{}10\times{}30\,\mathrm{s}$ | 1.06-1.10 | 1.21-2.06 | 60.A-9132(A)
2019-05-10† | ATs Small | Single,Medium,Split | $4\times{}10\times{}30\,\mathrm{s}$ | 1.06-1.25 | 0.64-1.09 | 60.A-9132(A)
2021-01-08 | UTs | Dual,Medium,Combined | $8\times{}32\times{}1\,\mathrm{s}$ | 1.70-1.80 | 0.59-0.81 | 1104.C-0651(C)
Table 1: Observing log presenting all data acquired with VLTI/GRAVITY on HD
142527 and analysed in this paper.
1 ATs: auxiliary telescopes; UTs: unit telescopes. A description of the AT
configurations is available on the ESO website.
2 Single: single-field mode; Dual: dual-field mode; Medium or high: spectral
resolution; Combined or split: polarisation setting.
† We were not able to extract an astrometry from these datasets (see Section
3). They are listed here only for completeness, as they are available on the
ESO archive.

##  3 Data reduction

All the observations were initially reduced using the GRAVITY pipeline
(Lapeyrere et al., 2014). We generated scivis files, which were averaged over
all DITs of each exposure, and which contained the squared modulus of the
interferometric visibilities, $|V^{2}|$, and the closure phases,
$\mathcal{C}_{\phi}$, with the associated error bars. The squared visibilities
and the closure phases both encode information about the separation of the two
components of the binary, and we treated them as independent quantities.

The coherent flux received at time $t$, along the baseline $b$, and at
wavelength $\lambda$ can be written

$\mathcal{F}(t,b,\lambda)=S_{A}(\lambda)+S_{B}(\lambda)\times{}e^{-j\frac{2\pi}{\lambda}\mathrm{OPD}(t,b,\lambda)},$
(1)

where $S_{A}$ and $S_{B}$ are the spectra of the two components of the binary,
and $\mathrm{OPD}$ is the optical path difference between the two components
of the binary along baseline $b$.

Using the coordinates of the baseline in the _uv_ -plane $b=(u,v)$, and the
separation vector $(\Delta\alpha,\Delta\delta)$ of the binary in RA/DEC, the
OPD can be written as

$\mathrm{OPD}=\Delta{}\alpha\times{}u+\Delta{}\delta{}\times{}v.$ (2)

Equation 1 thus becomes

$\mathcal{F}=S_{A}(\lambda)+S_{B}(\lambda)\times{}e^{-j\frac{2\pi}{\lambda}\left(\Delta{}\alpha\times{}u+\Delta{}\delta{}\times{}v\right)}.$
(3)

Equation 3 presents an idealised case, as it does not take into account the
phase distortions introduced by the atmosphere. The Fringe Tracker of GRAVITY
(Lacour et al., 2019) corrects this atmospheric phase distortion up to a near-
constant term, which affects the telescopes individually and can thus be
represented by a set of four terms:
$\phi_{T_{1}},\phi_{T_{2}},\phi_{T_{3}},\phi_{T_{4}}$. For a baseline $b$
linking telescope $k$ to telescope $l$, the post fringe-tracker atmospheric
phase distortion is $\Delta\phi=\phi_{T_{l}}-\phi_{T_{k}}$. This term affects
both components of the binary in the same way, and Equation 3 thus becomes

$\mathcal{F}_{T_{k}T_{l}}=\left[S_{A}+S_{B}\times{}e^{-j\frac{2\pi}{\lambda}\left(\Delta{}\alpha\times{}u+\Delta{}\delta{}\times{}v\right)}\right]e^{j\left(\phi_{T_{l}}-\phi_{T_{k}}\right)},$
(4)

which shows that Equation 4 encodes information about the separation of the
binary in two forms: in the amplitude of the observed visibility and in the
phase.

The flux received at each telescope $T$ of the array is simply
$F_{T}=S_{A}+S_{B}$, and the interferometric visibility is therefore given by

$\displaystyle V_{T_{k}T_{l}}$
$\displaystyle=\frac{\mathcal{F}_{T_{k}T_{l}}}{\sqrt{F_{T_{k}}F_{T_{l}}}}$
$\displaystyle=\frac{{\left[1+Ce^{-j\frac{2\pi}{\lambda}\left(\Delta{}\alpha\times{}u+\Delta{}\delta{}\times{}v\right)}\right]e^{j\left(\phi_{T_{l}}-\phi_{T_{k}}\right)}}}{1+C},$

where $C=S_{B}/S_{A}$ is the contrast of the binary.

The squared modulus of the visibility for a baseline, $T_{k}T_{l}$, and the
closure phase for a triangle, $T_{k}T_{l}T_{m}$, are defined by

$\displaystyle|V_{T_{k}T_{l}}|^{2}$
$\displaystyle=V_{T_{k}T_{l}}V^{*}_{T_{k}T_{l}}$
$\displaystyle\mathcal{C}_{T_{k}T_{l}T_{m}}$
$\displaystyle=\arg{(V_{T_{k}T_{l}}V_{T_{l}T_{m}}V^{*}_{T_{m}T_{k}})}.$

Using Equation 4 in combination with these definitions, we see that the two
quantities are independent of the residual atmospheric distortion term, and
are given by

$\displaystyle|V_{T_{k}T_{l}}|^{2}$
$\displaystyle=\frac{1+C^{2}+2C\cos{(\Phi_{T_{k}T_{l}})}}{(1+C)^{2}},$ (5)
$\displaystyle\mathcal{C}_{T_{k}T_{l}T_{m}}$
$\displaystyle=\arg{(1+Ce^{-j\Phi_{T_{k}T_{l}}})}+\arg{(1+Ce^{-j\Phi_{T_{l}T_{m}}})}$
$\displaystyle\qquad-\arg{(1+Ce^{-j\Phi_{T_{k}T_{m}}}),}$ (6)

where $\Phi_{T_{i}T_{j}}$ is the phase corresponding the baseline
$(u_{ij},v_{ij})$ linking telescope $T_{i}$ to $T_{j}$:

$\displaystyle\Phi_{T_{i}T_{j}}=\frac{2\pi}{\lambda}\left(\Delta{}\alpha\times{}u_{ij}+\Delta{}\delta{}\times{}v_{ij}\right).$

In principle, since the $(u,v)$ coordinates of the different baselines are
known, Equations 5 and 6 can be used to model the GRAVITY data with only three
parameters: the binary separation vector $(\Delta{}\alpha,\Delta{}\delta)$,
and the contrast ratio $C$ (under assumption of a constant contrast along the
GRAVITY wavelength range, from $\sim{}1.9\,\mu\mathrm{m}$ to
$2.4\,\mu\mathrm{m}$).

In the absence of proper calibration, however, both the squared visibilities
and the closure phases are affected by additional low-order terms, which we
modelled with the use of an additional polynomial in wavelength of order 2. In
Equation 5 the constant terms can be absorbed in this low-order polynomial.
Denoting $a_{0},a_{1},a_{2}$ the coefficients of the additional polynomial
term, and using $\beta=2C/(1+C)^{2}$ as a parameter instead of $C$, our model
for the squared visibility becomes linear in all parameters but
$(\Delta\alpha,\Delta\delta)$. Our model for the closure phase remains non-
linear in three parameters ($(\Delta\alpha,\Delta\delta)$, and $C$) and is
written as

$|V_{T_{k}T_{l}}|^{2}_{\mathrm{model}}=a_{0}+a_{1}\lambda+a_{2}\lambda^{2}+\beta\cos{[\Phi_{T_{k}T_{l}}(\Delta\alpha,\Delta\delta)]},$
(7) $\displaystyle\mathcal{C}_{T_{k}T_{l}T_{m},\mathrm{model}}$
$\displaystyle=a_{0}+a_{1}\lambda+a_{2}\lambda^{2}+\arg{(1+Ce^{-j\Phi_{T_{k}T_{l}}})}{}$
$\displaystyle\qquad+\arg{(1+Ce^{-j\Phi_{T_{l}T_{m}}})}$
$\displaystyle\qquad-\arg{(1+Ce^{-j\Phi_{T_{k}T_{m}}}).}$ (8)

The GRAVITY pipeline reports the squared visibilities and the closure phases
separately for each baseline and wavelength, with the associated error bars.
Therefore, we were able to define two $\chi^{2}$ quantities for each file of
each epoch of our single-field observations (see Table 1) by summing the
$\chi^{2}$ obtained using the models defined in Equations 7 and 8 over the six
baselines (for $|V|^{2}$) and four triangles (for $\mathcal{C}$):

$\displaystyle\chi_{|V|^{2}}^{2}$
$\displaystyle=\sum_{T_{i}T_{j},\lambda}\left(\frac{|V|^{2}_{\mathrm{data}}(T_{i}T_{j},\lambda)-|V|^{2}_{\mathrm{model}}(T_{i}T_{j},\lambda)}{\sigma_{|V^{2}|}(T_{i}T_{j},\lambda)}\right)^{2},$
$\displaystyle\chi_{\mathcal{C}}^{2}$
$\displaystyle=\sum_{T_{i}T_{j}T_{k},\lambda}\left(\frac{\mathcal{C}_{\mathrm{data}}(T_{i}T_{j}T_{k},\lambda)-\mathcal{C}_{\mathrm{model}}(T_{i}T_{j}T_{k},\lambda)}{\sigma_{\mathcal{C}}(T_{i}T_{j}T_{k},\lambda)}\right)^{2}.$

From this we calculated, for each file, a set of two $\chi^{2}$ maps as a
function of $\Delta\alpha,\Delta\delta$, by minimising the two $\chi^{2}$ over
all the nuisance parameters. This represents a minimisation over 19 linear
parameters for the squared visibilities (a set of $a_{0},a_{1},a_{2}$ for each
of the six baselines, plus a value for $\beta$), and 12 linear parameters
($a_{0},a_{1},a_{2}$ for each of the four triangles) plus one non-linear
parameter (the contrast value $C$) for the closure phases:

$\displaystyle\chi^{2}_{|V|^{2},\mathrm{map}}({\Delta{}\alpha},{\Delta{}\delta})$
$\displaystyle=\mathrm{min}_{a_{0},a_{1},a_{2},\beta}\left(\chi^{2}_{|V|^{2}}\right),$
(9)
$\displaystyle\chi^{2}_{\mathcal{C},\mathrm{map}}({\Delta{}\alpha},{\Delta{}\delta})$
$\displaystyle=\mathrm{min}_{a_{0},a_{1},a_{2},C}\left(\chi^{2}_{\mathcal{C}}\right).$
(10)

In Equation 9 we used the closed-form solution of the minimisation problem
since all parameters are linear. For Equation 10 we used the closed-form
solution to minimise over the linear parameters, and the Powel algorithm to
perform the minimisation over the non-linear parameter $C$.

An example of a typical $\chi^{2}$ map is given in Figure 1. It shows a
structure with multiple adjacent minimums, which is characteristic of
interferometric observations with a limited number of baselines. To identify
the correct minimum, we make use of prior information on the orbit of the
binary. We analysed our data sequentially, one epoch at a time, starting from
the first one. For each epoch $t$, we took all the astrometric data available
in the literature (see Table 2), to which we added our astrometric
measurements obtained on the previous GRAVITY epochs (with $t^{\prime}<t$). We
then used `orbitize!` (Blunt et al., 2020) to perform an orbit fit for the
binary. We gathered a total of 100 000 posterior values for the orbital
parameters of the binary, which we used to calculate the expected astrometry
at epoch $t$ (see Figure 1). For all of our epochs, this a priori estimate was
good enough to unambiguously determine the correct minimum in our $\chi^{2}$
maps.

Once we had identified the rough location of the correct minimum in each map,
we took the location of the closest minimum to be our best estimate of the
astrometry, making sure that the $\chi^{2}$ is calculated with a resolution
that is much smaller than our final error bars.

For each epoch, we therefore obtained a set a $2\times{}N_{\mathrm{exp}}$
estimates for the astrometry of the binary. Our final best estimate for this
epoch was taken as the mean of all these measurements, and the error ellipse
was calculated using an empirical estimate of the covariance matrix.

We were not able to obtain a proper astrometric measurement for three of our
ten epochs. All these difficult datasets were obtained in single-field with
the ATs.

The data obtained on March 19, 2017, appeared corrupted even on simple visual
inspection, and were very badly fitted by our models, which makes us suspect a
fringe-tracker problem during the observations. On April 27, 2019, the
baselines were aligned in a very unfortunate way, such that only one of them
produces a signal significantly different from the second-order polynomial
used to detrend the data. With only a single useful baseline, a measurement of
the astrometry is not possible, and thus the data were discarded. Finally, on
May 10, 2019, the observations were performed using the small set-up with the
ATs, which was unable to produce a signal significantly different from our
second-order detrending polynomial due to the small separation of the target.
The results obtained for the three epochs of dual-field UT observations, and
the four epochs of single-field AT observations for which we were able to
obtain an astrometric measurement are given in Table 2.

Figure 1: Typical example of a $\chi^{2}$ map obtained when reducing our data (in this case, for the observation on 2019 April 28). The left panel shows the structure of the $\chi^{2}$ map on a large scale, where the multiple minimums (in purple) are clearly visible. The right panel is a zoomed-in version. In each panel the red dots are generated by performing an orbit fit with all the astrometric measurement data available (including all GRAVITY epochs up to 2019 April 28), and using the posterior distributions of orbital parameters to calculate the estimated position at epoch 2019 April 28. The purple ellipse corresponds to the 3$\sigma$ contour for the astrometric solution. The blue dots are obtained in a similar way, but using a new orbit fit that includes the astrometry extracted from this epoch. In other words, the red dots represents the prediction using the a priori orbital solution, the purple ellipse represents the measurement obtained at this epoch, and the blue dots represent the prediction using the a posteriori updated orbital solution. Epoch | MJD |  Separation∗ |  Position Angle∗ | $\Delta{}\mathrm{RA}$ | $\Delta{}\mathrm{DEC}$ | $\rho^{**}$ | Instrument | Reference∗∗∗
---|---|---|---|---|---|---|---|---
(mas) | (deg) | (mas) | (mas) | -
2012-03-11 | 55997 | $89.70\pm 2.60$ | $133.10\pm 1.90$ | $65.50\pm 10.79$ | $-61.29\pm 11.48$ | 0.92 | NACO | Bl12
2013-03-17 | 56368 | $82.00\pm 2.10$ | $126.30\pm 1.60$ | $66.09\pm 2.16$ | $-48.55\pm 2.22$ | 0.08 | NACO | L16
2013-04-11 | 56393 | $82.50\pm 1.01$ | $127.98\pm 0.46$ | $65.03\pm 0.89$ | $-50.77\pm 0.81$ | -0.39 | VisAO | Ba22
2013-07-14 | 56487 | $82.50\pm 1.10$ | $123.80\pm 1.20$ | $68.56\pm 1.32$ | $-45.89\pm 1.56$ | 0.40 | NACO | L16
2014-04-08 | 56755 | $77.73\pm 1.63$ | $117.87\pm 0.96$ | $68.71\pm 1.56$ | $-36.34\pm 1.39$ | -0.18 | VisAO | Ba22
2014-04-25 | 56772 | $88.25\pm 10.10$ | $123.00\pm 9.20$ | $74.01\pm 11.44$ | $-48.06\pm 13.03$ | 0.30 | GPI | R14
2014-05-10 | 56787 | $78.00\pm 1.80$ | $119.10\pm 1.00$ | $68.15\pm 1.71$ | $-37.93\pm 1.48$ | -0.23 | SINFONI | Ch18
2014-05-12 | 56789 | $77.20\pm 0.60$ | $116.60\pm 0.50$ | $69.03\pm 0.62$ | $-34.57\pm 0.66$ | 0.09 | GPI | L16
2015-05-13 | 57155 | - | - | $65.00\pm 2.00$ | $-24.00\pm 2.00$ | - | SPHERE | Cl18
2015-05-15 | 57157 | $70.16\pm 1.19$ | $110.56\pm 0.80$ | $65.69\pm 1.17$ | $-24.64\pm 1.01$ | -0.12 | VisAO | Ba22
2015-05-16 | 57158 | $72.19\pm 2.02$ | $107.84\pm 0.97$ | $68.72\pm 1.95$ | $-22.12\pm 1.32$ | -0.29 | VisAO | Ba22
2015-05-18 | 57160 | $70.00\pm 1.35$ | $110.12\pm 0.72$ | $65.73\pm 1.30$ | $-24.08\pm 0.95$ | -0.28 | VisAO | Ba22
2015-07-03 | 57206 | - | - | $62.90\pm 0.40$ | $-18.30\pm 0.50$ | - | SPHERE/SAM | Cl18
2016-03-26 | 57473 | - | - | $60.00\pm 2.00$ | $-7.00\pm 2.00$ | - | SPHERE | Cl18
2016-03-31 | 57478 | $62.80\pm 2.70$ | $98.70\pm 1.80$ | $62.08\pm 2.69$ | $-9.50\pm 2.00$ | -0.10 | ZIMPOL | Cu19
2016-06-13 | 57552 | - | - | $61.00\pm 2.00$ | $-7.00\pm 2.00$ | - | SPHERE | Cl18
2017-05-03 | 57876 | - | - | $49.67\pm 0.04$ | $11.33\pm 0.08$ | 0.50 | GRAVITY | This work
2017-05-16 | 57889 | - | - | $47.40\pm 0.50$ | $10.30\pm 0.20$ | - | SPHERE/SAM | Cl18
2018-02-17 | 58166 | - | - | $38.07\pm 0.50$ | $23.49\pm 0.28$ | -0.79 | GRAVITY | This work
2018-04-14 | 58222 | - | - | $36.00\pm 1.00$ | $25.00\pm 1.00$ | - | SPHERE | Cl18
2018-04-27 | 58235 | $43.69\pm 2.01$ | $58.28\pm 1.76$ | $37.16\pm 1.85$ | $22.97\pm 1.55$ | 0.35 | VisAO | Ba22
2018-07-25 | 58324 | - | - | $29.70\pm 0.04$ | $29.65\pm 0.08$ | -0.43 | GRAVITY | This work
2018-07-28 | 58327 | - | - | $29.48\pm 0.02$ | $29.51\pm 0.08$ | -0.30 | GRAVITY | This work
2019-04-28 | 58601 | - | - | $12.59\pm 0.16$ | $36.44\pm 0.10$ | -0.93 | GRAVITY | This work
2019-05-01 | 58604 | - | - | $12.48\pm 0.15$ | $36.53\pm 0.09$ | -0.90 | GRAVITY | This work
2021-01-08 | 59222 | - | - | $-27.77\pm 0.22$ | $23.98\pm 0.05$ | -0.49 | GRAVITY | This work
Table 2: Astrometry on HD 142527 B.
∗ Different measurements made with different techniques and instruments are
reported on two different bases: separation and position angle, or right-
ascension (RA) and declination (DEC). For convenience, we have converted all
values reported in separation/position angle to RA/DEC using an MCMC approach,
but these values are treated on their original bases during our orbit fits.
∗∗ When available, the correlation coefficient $\rho$ is reported in addition
to the error along the RA and DEC axes.
∗∗∗ Sources: Ba22: Balmer 2022; L16: Lacour 16; Bl12: Biller 2012; R14: Radius
2014; Ch18: Christiaens 2018; Cl19: Claudi 2019; Cu19: Cugno 2019.

##  4 The orbit of HD 142527

###  4.1 Orbital parameters

Including both AT and UT observations, we report in this work a set of seven
new astrometric measurements of HD 142527 B, with typical precisions ranging
from a few $100\,\mu\mathrm{as}$ down to $\simeq{}10\,\mu\mathrm{as}$,
obtained between 2017 and early 2021. To this we added a total of 19 epochs
compiled from the literature to create a dataset that covers almost one half
of the orbit of the binary, from 2012 to 2021.

We used `orbitize!` (Blunt et al., 2020) to fit the relative motion of HD
142527 B around HD 142527 A, and to derive estimates of its orbital
parameters. We used the parallel-tempering affine-invariant (Foreman-Mackey et
al., 2013; Vousden et al., 2016) Markov chain Monte Carlo (MCMC) sampler
available in `orbitize!` to determine the posterior distributions of the eight
orbital parameters reported in Table 3. Except for the total mass of the
system, $M_{\mathrm{tot}}$, and the parallax, $\pi$, we used the default
priors in `orbitize!`. Our priors on the total mass and parallax of the system
are set to Gaussian priors, with $M_{\mathrm{tot}}=2.49\pm{}0.27$ (Balmer et
al., 2022) and $\pi=6.356\pm{}0.047$ (GaiaCollaboration et al., 2021). The
MCMC run is set up with 100 walkers and 20 temperatures for a burn-in phase of
$50\,000$ steps, followed by an additional $100\,000$ steps to approximate the
posterior distributions, keeping only one-tenth of the samples. The use of
multiple temperatures allows the walkers to explore different minimums (at
high temperatures), and to properly sample each available minimum (at lower
temperatures).

Our best estimates for each orbital parameter are reported in Table 3, and a
corner plot is given in Figure 2. In Figure 3 we show a sample of 100 orbits
randomly chosen in our posterior distributions, overplotted on the astrometric
measurements presented in Table 2.

Figure 2: Corner plot for the posterior distributions of the orbital elements
of HD 142527 B. The distribution of $\omega_{\mathrm{B}}$ and
$\Omega_{\mathrm{B}}$ is actually a bimodal distribution, with two peaks
spaced 180 deg apart. This is a known degeneracy when radial velocity data are
missing. For readability, only one peak, chosen arbitrarily, is shown here.

Table 3 and Figure 3 show that the orbit of HD 142527 B is now very well
constrained, with a gain of one order of magnitude on the error bars of most
orbital parameters compared to previous studies (Lacour et al., 2016; Claudi
et al., 2019; Balmer et al., 2022). The companion is on an eccentric orbit
($e=0.47\pm{0.01}$), with a semi-major axis of
$a=10.86\pm{}0.26\,\mathrm{au}$, about 2$\sigma$ from the results of Balmer et
al. (2022). The total mass of the system is
$M_{\mathrm{tot}}=2.24\pm{}0.06\,M_{\odot}{}$, resulting in an orbital period
of $P=23.95\pm{}0.99\,\mathrm{yr}$. We also find that the perihelion passage
occurred on June 02, 2020 (between May 15 and June 21, 2020, at 1$\sigma$).
Therefore, our last data point, from January 2021, was obtained after the
perihelion passage. This, in combination with the excellent precision of our
VLTI/GRAVITY measurements, explains the significant gain in precision on the
orbit estimate.

We note that among all the measurements gathered from the literature, the two
epochs of SPHERE/SAM data are the only ones that seem to have a systematic and
significant disagreement with our orbit fit results. Each of these astrometric
measurements is more than 3$\sigma$ away from the best fit in terms of
separation. However, running a second orbit fit with these two data points
excluded produced similar results, with all estimates of the orbital
parameters within the 1$\sigma$ range of what is obtained with the SPHERE/SAM
points. For completeness, we therefore decided to keep these points in our
final fit.

###  4.2 Mutual inclination relative to the discs

Contradictory values have been reported for the mutual inclination between the
orbital plane of the binary and the disc, which is a quantity of particular
interest to understand disc–binary interactions. This apparent contradiction
seems to stem from a misunderstanding of the multiple conventions used to
report disc parameters.

Czekala et al. (2019) reported an inclination relative to the outer disc of
$\theta_{\mathrm{out}}=35\pm{}5\,\mathrm{deg}$ (or
$80\pm{}10\,\mathrm{deg}$111In absence of radial velocity data, there is a
degeneracy in the orbit of the binary, which also leads to a degeneracy in the
values of the mutual inclination.), which they obtained using an inclination
$i_{\mathrm{out}}=153\,\mathrm{deg}$ and a longitude of the ascending node
(LAN) of $\Omega_{\mathrm{out}}=160.9\,\mathrm{deg}$ for the disc. Balmer et
al. (2022) used the values $i_{\mathrm{out}}=38.21\leavevmode\nobreak\
\mathrm{deg}$ and $\Omega_{\mathrm{out}}=162.72\,\mathrm{deg}$ from Bohn et
al. (2022) to calculate the mutual inclination, and found
$\theta_{\mathrm{out}}=89.84^{+2.30}_{-1.64}\,\mathrm{deg}$ (or
$158.82^{+2.76}_{-2.81}\,\mathrm{deg}$).

The mutual inclination $\theta_{\mathrm{out}}$ between the binary orbit and
the outer disc can be calculated using the equation (Czekala et al., 2019)

$\displaystyle\cos(\theta_{\mathrm{out}})$
$\displaystyle=\cos(i_{\mathrm{out}})\cos(i)+\sin(i_{\mathrm{out}})\sin(i)\cos(\Omega_{\mathrm{out}}-\Omega),$
(11)

where $i_{\mathrm{out}}$ and $\Omega_{\mathrm{out}}$ are the inclination and
the LAN of the outer disc.

It is important to note, though, that in order to be used in Equation 11, the
parameters of the binary and of the disc must both follow the conventions used
in Czekala et al. (2019). In this convention the inclination is the angle
between the axis perpendicular to the plane of sky going towards the observer
and the momentum vector of the disc or binary (see their Fig. 1). The LAN is
defined as the angle (positive towards the east) between the north and the
ascending node, which itself is defined as the point where the disc crosses
the plane of sky receding away from us (i.e. redshifted).

This convention is widely accepted in the visual binary and exoplanet direct
imaging fields, and is the one used in `orbitize!`. However, it seems that the
disc parameters reported by Bohn et al. (2022) follow a different convention,
in which the inclination is defined as the angle between the plane of sky and
the orbital plane.

Two conventions seem to be in use to define disc parameters as utterly
different values have been reported for the outer disc of HD 142527. For
example, Casassus et al. (2015) use an inclination of $160\,\mathrm{deg}$ in
their modelling, similar to the value of $153\,\mathrm{deg}$ reported in
Czekala et al. (2019). These values correspond respectively to an angle
between the plane of sky and the orbital plane of 30 deg and 27 deg, more
similar to the values reported by Fukagawa et al. (2013) (26.9 deg) and Bohn
et al. (2022) ($38.21\leavevmode\nobreak\ \mathrm{deg}$).

Therefore, in order to be used in Equation 11, the disc parameters need to be
converted to the proper convention. For the outer disc, the parameters
inferred by Bohn et al. (2022) from ALMA data correspond to

$\displaystyle\begin{cases}i_{\mathrm{out}}=180-38.21\pm{}1.38=141.79\pm{}1.38\,\mathrm{deg}\\\
\Omega_{\mathrm{out}}=162.72\pm{}1.38\,\mathrm{deg}\end{cases},$ (12)

which corresponds to a disc inclined 38 deg relative to the plane of the sky
where the east side is the far side of the disc (see section 4.2 in Casassus
et al. 2015 for a discussion), rotating clockwise on the sky, where the
redshifted node is slightly east of south, in agreement with the velocity maps
presented in Bohn et al. (2022).

To determine the values of the mutual inclination of the orbit of HD 142527 B
with respect to the two different discs, we use the values of the disc
parameters from Equation 12 together with our posterior distribution of
orbits. For each of our $10^{6}$ sets of posterior orbit ($100\,000$ samples
for 100 walkers with a 1/10 thinning), we randomly draw a value for
$i_{\mathrm{out}},\Omega_{\mathrm{out}}$ using Gaussian distributions
corresponding to Equation 12, and we use Equation 11 to calculate
$\theta_{\mathrm{out}}$. The resulting posterior distributions are bimodal,
due to the $180\,\mathrm{deg}$ degeneracy in $\Omega$, and the values are
reported in Table 3.

We note that the inclination reported by Bohn et al. (2022) for the outer disc
is significantly different from previously reported values, which range from
20 to 28 deg (Verhoeff et al., 2011; Perez et al., 2014; Boehler et al., 2017;
Garg et al., 2021). This raises the question of whether this is related to the
modelling, or if this is a sign of a true physical effect, such as a
potentially warped disc. In any case, assuming an inclination of 20 deg (i.e.
160 deg in the reference system of Equation 11) and a longitude of the
ascending node of 160 deg (as in Pinte et al. 2018) results in a mutual
inclination of 10.5 deg (or 50.5 deg, due to the degeneracy in $\Omega$)
relative to the outer disc.

Contrary to what was presented in Price et al. (2018a), and more recently in
Balmer et al. (2022), who both suggested that the presence HD 142527 B is
enough to explain the numerous features observed in the disc (central cavity,
spiral arms, asymmetries), our new VLTI/GRAVITY data paint a very different
picture. With an eccentric ($e=0.47\pm{}0.01$) but smaller
($a=10.86\pm{}0.26\,\mathrm{au}$) orbit, a maximum separation at aphelion of
$15.85\pm{}0.43\,\mathrm{au}$, our new orbit once more raises the question of
whether HD 142527 B can really be responsible for the morphology of the disc,
and in particular for the large $\simeq{}100\,\mathrm{au}$ observed cavity.

Figure 3: Illustration of the orbit fit. The left panel shows 100 samples drawn randomly from our posterior distribution, overplotted on the astrometric measurements of Table 2. The two panels on the right present the same orbits in separation (upper panel) and position angle (lower panel). For each panel, we show the 100 orbits and the astrometric measurements in the upper graph, and the difference (in units of $\sigma$, the error bars on the measurements) in the lower plot. The two dotted lines show the 3$\sigma$ interval. Parameter | Value | Unit
---|---|---
Semi-major axis, $a$ | $10.80\pm 0.22$ | au
Eccentricity, $e$ | $0.47\pm 0.01$ | -
Inclination, $i$ | $149.47\pm 0.71$ | deg
1Argument of periapsis, $\omega$ | $186.45\pm 0.48$ | deg
1Long. of ascending node, $\Omega$ | $161.51\pm 2.01$ | deg
Time at periapsis, $\tau$ | $2020.42\pm 0.05$ | yr
Parallax, $\pi$ | $6.35\pm 0.05$ | mas
Total mass, $M_{\mathrm{tot}}$ ($M_{\odot}{}$) | $2.29\pm 0.06$ | $M_{\odot}{}$
2Period, $P$ | $23.50\pm 0.85$ | yr
2Distance at apoapsis, $R_{\mathrm{apoapsis}}$ | $15.85\pm{}0.43$ | au
1-3[0.5pt/3pt] 1,2Incl. w.r.t. outer disc, $\theta_{\mathrm{out}}$ |  3$8.80\pm 1.54$ | deg
4$2.6-10.5$
Table 3: Orbital elements of HD 142527 B obtained from a fit to the
astrometric measurements presented in Table 2.
1 The orbital period and mutual inclinations are not direct results of the
orbit fit, but are calculated from the posterior distribution.
2 In absence of any radial velocity data, there is a degeneracy in $\Omega$
and $\omega$, and a second, equally valid, solution can be obtained by adding
180 deg to both $\omega$ and $\Omega$. This also results in a degeneracy in
the mutual inclinations, with $\theta_{\mathrm{out}}=68.45\pm
1.54\,\mathrm{deg}$ or $50.5-58.5\,\mathrm{deg}$ being a second set of valid
solutions.
3 Using the parameters reported in Bohn et al. (2022) for the outer disc.
4 Using a range of 20 to 28 deg for the inclination of the outer disc, and
$\Omega_{\mathrm{out}}=160\,\mathrm{deg}$ (Verhoeff et al., 2011; Perez et
al., 2014; Boehler et al., 2017; Price et al., 2018a; Garg et al., 2021).

##  5 Disc–binary interaction

###  5.1 Hydrodynamical simulation

To explore whether the new orbit of the companion HD 142527 B is still able to
explain the numerous features observed in the disc, we run three 3D gas only
hydrodynamical simulations using Phantom, a 3D Smoothed Particle Hydrodynamics
(SPH) code (Price et al., 2018b).

The first simulation recreates one of the simulations in Price et al. (2018a)
(the R2 orbit). This serves as our control simulation to study the effect the
new companion orbit has on the disc. The second simulation uses the same disc
parameters as that of Price et al. (2018a), but the updated orbital
measurements of the companion as given in Table 3. This allows us to easily
demonstrate the difference in disc evolution between our new orbit and that
used in Price et al. (2018a). In this set-up, the mutual inclination is
$\theta_{\mathrm{out}}=50.5\,\mathrm{deg}$, obtained by adding 180 to $\omega$
and $\Omega$ in Table 3. The true anomaly is set as $74.31\,\mathrm{deg}$. The
third simulation is identical to the second simulation, except for a reduced
inner boundary of the disc $R_{\mathrm{in}}$, such that it matches the ratio
of $R_{\mathrm{in}}/R_{\mathrm{apoapsis}}$ in the first simulation.

All three simulations model a $0.01M_{\odot}$ disc with
${R_{\mathrm{out}}=350\,\mathrm{au}}$. The surface density profile is given by
${\Sigma\propto R^{-1}}$. The temperature profile is given by ${T\propto
R^{-0.3}}$, with ${H/R=0.11}$ at $R=R_{\mathrm{out}}$. The inner boundary of
the first and second simulations is $R_{\mathrm{in}}=50\,\mathrm{au}$, whereas
for the third simulation it is $R_{\mathrm{in}}=12.67\,\mathrm{au}$. We set a
mean Shakura–Sunyaev disc viscosity $\alpha_{\mathrm{SS}}\approx 0.005$. This
corresponds to an artificial viscosity $\alpha_{\mathrm{AV}}=0.22$ for the
first and second simulation, and $\alpha_{\mathrm{AV}}=0.18$ for the third
simulation. The artificial viscosity coefficient $\beta_{\text{AV}}$ is set to
2 (see Price et al. 2018b; Nealon et al. 2015).

Sink particles are used to model the binary (Bate et al., 1995). The mass and
orbital parameters of the binary for the first simulation are identical to the
R2 orbit in Price et al. (2018a), whereas for the second and third simulations
the mass of the primary and secondary is $2.03M_{\odot}$ and $0.26M_{\odot}$,
respectively, consistent with Claudi et al. (2019) and the total mass in Table
3. The accretion radius for both sinks is set to $1\textsc{au}$, as in Price
et al. (2018a). The simulation recreating Price et al. (2018a) is shown at 20
orbital periods (3272 yr) corresponding to their Figure 2. We present all of
our results at the same time. For the updated companion orbit, this
corresponds to 141 orbital periods.

Figure 4 shows all three of the simulations at the same moment in time. The
left panel shows the simulation using the R2 orbit set-up from Price et al.
(2018a), which was able to explain the observed features of the disc. In this
orbit set-up, the companion is responsible for the large cavity
(${\sim}100\,\mathrm{au}$), the large spiral arms, and the misaligned inner
disc casting a shadow. The middle panel shows the simulation where the only
change is the companion’s orbit. As we have reproduced the results of Price et
al. (2018a), we can attribute all differences in the disc structure to the
change in the orbital measurements. The new orbit is unable to match any of
the disc features mentioned above. The large separation between the inner edge
of the disc and the companion results in little impact on the disc structure,
resulting in a much smaller cavity (Bohn et al., 2022). Finally, the right
panel shows the simulation where the inner boundary of the disc is closer in,
such that the ratio $R_{\mathrm{in}}/R_{\mathrm{apoapsis}}$ is equivalent to
the original simulation from Price et al. (2018a) (left panel). While there
are improvements, the companion is still unable to carve open a large enough
cavity. Furthermore, the lack of a pressure maximum to trap dust at
${\sim}100\,\mathrm{au}$ makes it difficult to explain the observed dust
structures (Casassus et al., 2015).

The inability of the companion to clear a large enough cavity is unsurprising.
From theoretical predictions (Artymowicz & Lubow, 1994), a circular binary
with $\mu=0.1$ as in this system, is expected to result in a disc inner edge
of $r=1.74a$, where $\mu=M_{\mathrm{companion}}/M_{\mathrm{tot}}$ is the mass
fraction of the perturber. Although eccentric companions result in larger
cavities, there is no prediction where the companion is able to carve open a
big enough cavity to match observations. Figure 4 in Artymowicz & Lubow (1994)
shows that for a binary with $\mu=0.1$, an eccentricity of 0.47, and an
assumed disc viscosity of $5\times 10^{-3}$, the disc inner edge is expected
to be between $r=2.7a$ and $3a$. For a semi-major axis of
$a=10.8\,\mathrm{au}$, this corresponds to a disc inner edge at $r\approx
30\,\mathrm{au}$, which agrees well with the simulations in the middle and
right panels of Figure 4. Additionally, the disc is inclined relative to the
orbital plane of the binary, which decreases the truncation radius (Lubow et
al., 2015; Facchini et al., 2018). The observed cavity size of
${\sim}100\,\mathrm{au}$ far exceeds even the most generous predictions.
Hence, the simulations here show that with the new orbital parameters, the
companion cannot be solely responsible for the observed disc features.

Figure 4: Surface density rendered plots of the HD 142527 protoplanetary disc.
Each panel shows a different simulation at the same moment in time, but with
different disc and binary parameters. The left panel reproduces the results of
the R2 orbit in Price et al. (2018a) where the companion was shown to be
responsible for the observed disc features: a large cavity, prominent spiral
arms, and a misaligned inner disc. The middle panel uses the same disc
parameters as the left panel, but uses the updated binary parameters in Table
3. The right panel is set up with the second solution of the updated binary
parameters, and a smaller inner disc boundary to match the ratio of
$R_{\mathrm{in}}/R_{\mathrm{apoapsis}}$ of the simulation in the left panel.
In the middle and right panels we find that, due to the close configuration of
the updated binary orbit, the companion cannot be solely responsible for the
observed disc features.

For completeness, we also simulated the solution of the binary orbit using the
values of $\Omega$ and $\omega$ listed in Table 3 resulting in
$\theta_{\mathrm{out}}=10.5\,\mathrm{deg}$. Figure 5 shows that neither
solution is able to reproduce the disc features. Since both solutions produce
similar results, we only use the simulation with the second solution (left
panel) for the comparisons with the observations below.

Figure 5: Surface density rendered plots of the HD 142527 protoplanetary disc.
Each panel shows a simulation at the same moment in time, but with different
disc and binary parameters. The right panel is set up with the updated binary
parameters as listed in Table 3. The left panel uses the second solution of
the values in 3 obtained by adding $180\,\mathrm{deg}$ to both $\Omega$ and
$\omega$. We find that neither solution reproduces the observed disc features.

###  5.2 Radiative transfer

Figure 6: Comparison between observed and simulated 12CO and 13CO maps for the
HD142527 disc. The top and bottom panels show the 12CO and 13CO $\text{J}=2-1$
emission, respectively. The left panels are the observations (Garg et al.,
2021). The middle and right panels are synthetic emissions predicted from the
simulations. The middle panels are from the simulation shown in the left panel
of Figure 4 which uses the disc and binary set-up as the R2 orbit in Price et
al. (2018a). The right panels are from the simulations shown in the right
panel of Figure 4, which uses the updated binary parameters in Table 3. In
general, the new binary orbit results in a lack of substructure in either
molecule. Additionally, the lack of a cavity in 13CO further highlights that
the companion alone cannot reproduce the disc features.

For comparison with observations of the 12CO and 13CO ${\text{J}=2-1}$ line
emission, we use the radiative transfer code mcfost (Pinte et al., 2006,
2009). As it uses a Voronoi tesselation where each cell in mcfost corresponds
to an individual SPH particle, mcfost is well suited for post-processing SPH
simulations.

A subset of the simulations above are post-processed, in particular the
simulations using the binary orbit from Price et al. (2018a) and the updated
binary orbit from this work. The sink particles are used to irradiate the disc
using stellar spectral models based on their mass, and assuming a 3Myr
isochrone. For the former simulation, this corresponds to the luminosity and
effective temperature of ${L=2.69L_{\odot}}$ and ${T_{\text{eff}}=4813K}$,
respectively, for the primary, and ${L=0.23}L_{\odot}$ and
${T_{\text{eff}}=3607K}$, respectively, for the secondary. For the latter
simulation, the luminosity and effective temperature is ${L=3.89L_{\odot}}$
and ${T_{\text{eff}}=4991K}$, respectively, for the primary, and
${L=0.14L_{\odot}}$ and ${T_{\text{eff}}=3315K}$, respectively, for the
secondary.

We use $10^{8}$ photon packets to calculate the temperatures in the disc. As
these are gas-only simulations, the dust is assumed to be $1\%$ of the gas
mass, and perfectly coupled to the gas. The dust sizes vary between 0.3 and
1000$\mu m$, and are distributed across 100 grain sizes with a power-law
exponent of $-3.5$. We further assume that the dust particles are all
spherical and homogeneous, and made up of astronomical silicates. The dust
properties are computed using Mie theory. To predict the line emission of 12CO
and 13CO, we assume constant abundances of $5\times 10^{-5}$ and $7\times
10^{-7}$, respectively.

Figure 6 compares the observations (left panels) with the predictions from
simulations (middle and right panels). The top and bottom panels show the 12CO
and 13CO ${\text{J}=2-1}$ line emission, respectively. In general, the new
binary orbit results in a lack of substructure with either molecule as seen by
the right panels. In 13CO, the new orbit is unable to deplete the gas in the
inner regions, highlighted by the lack of any cavity compared to either the
observation (bottom left panel) or the simulation from Price et al. (2018a)
(bottom middle panel). Although the 12CO emission is lacking any substructure
with the new binary orbit, it is more similar to the observation (top left
panel) and the simulation Price et al. (2018a) (top middle panel); all have an
optically thick inner region. While there is a lot of gas in the cavity, this
could clear out if the simulation evolved for longer. Additionally, the beam
size washed out the cavity seen in the surface density.

##  6 Conclusion

Based on a VLTI/GRAVITY observations, we presented a set of new astrometric
measurements of HD 142527 B relative to HD 142527 B obtained at seven epochs
ranging from 2017 to early 2021. Combined with all the measurements available
in the literature, this constitutes an astrometric monitoring of
$\sim{}9\,\mathrm{yr}$, which covers almost half of the orbital period of the
binary.

From these measurements, we derived excellent estimates of the orbital
parameters for the binary. We find that HD 142527 B is on a moderately
eccentric orbit ($e=0.47\pm{}0.01$), with a semi-major axis of
$10.80\pm{}0.22\,\mathrm{au}$. The orbital period is therefore
$P=23.50\pm{}0.85\,\mathrm{yr}$, and we estimate that the low-mass companion
passed its perihelion on June 02, 2020. We also report a total mass for the
system of $M_{\mathrm{tot}}=2.29\pm{}0.06M_{\odot}{}$.

Considering our constraints on the orbital parameters of HD 142527 B, we
calculate that its maximum separation with respect to the central star HD
142527 A is $R_{\mathrm{apoapsis}}=15.85\pm{}0.43\,\mathrm{au}$. Contrary to
what has been previously suggested (Price et al., 2018a) based on different
orbital estimates, this makes it highly unlikely that this low-mass companion
can be solely responsible for the large $\sim{}100\,\mathrm{au}$ cavity
observed in the near- to mid-infrared (Fukagawa et al., 2006; Verhoeff et al.,
2011) and more recently with ALMA (Casassus et al., 2013).

To confirm this, we ran a set of three gas-only 3D hydrodynamical simulations
of the disc around HD 142527 AB. We used the first simulation set-up as a
control to demonstrate that we were able to reproduce the results obtained by
Price et al. (2018a) under the same set of assumption regarding the orbital
parameters of the binary. In the second simulation, we switched the orbit of
the binary for our new estimate, based on the newly available VLTI/GRAVITY
observations. We were unable to reproduce most of the features observed in the
disc with this set-up: the low-mass companion HD 142527 B could only carve a
$\sim{}30\,\mathrm{au}$ cavity, and did not interact with the disc. In the
third set-up, we reduced the inner radius of the disc, allowing the companion
to have a greater impact on the disc structure. However, the companion alone
was still unable to explain the disc features.

A larger theoretical follow-up will be necessary to determine what else is
required to match this new orbit of HD 142527 B with the observed disc
structures. Additional observations could also prove useful as they could
potentially reveal the presence of an additional companion in the system,
which would help explain the disc structures and/or the presence of additional
material extending further in the disc around HD 142527.

###### Acknowledgements.

The authors thank Grant Kennedy for a helpful discussion on the calculation of
mutual inclinations and the conventions used for expressing disc parameters.
This work is based on observations collected at the European Southern
Observatory under ESO programmes 098.D-0488, 60.A-9800, 60.A-9132 and
1104.C-0651. Most simulations in this work were performed using the DiRAC Data
Intensive service at Leicester, operated by the University of Leicester IT
Services, which forms part of the STFC DiRAC HPC Facility. The equipment was
funded by BEIS capital funding via STFC capital grants ST/K000373/1 and
ST/R002363/1 and STFC DiRAC Operations grant ST/R001014/1. DiRAC is part of
the National e-Infrastructure. S. R acknowledges support from the Science &
Technology Facilities Council (STFC) through Consolidated Grant ST/W000857/1.
R. N. acknowledges support from UKRI/EPSRC through a Stephen Hawking
Fellowship (EP/T017287/1). S. L. acknowledges the support of the French Agence
Nationale de la Recherche (ANR), under grant ANR-21-CE31-0017 (project
ExoVLTI). D. P. acknowledges Australian Research Council funding via
DP180104235.

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