# Application limit of the photocentre displacement to fundamental stellar
parameters of fast rotators
\- Illustration on the edge-on fast rotator Regulus

M. Hadjara,1,2,3,4 R. G. Petrov,5 S. Jankov,6 P. Cruzalèbes,5 A. Boskri,5,7 A.
Spang,5 S. Lagarde,5 J. He,8,1,2 X. Chen,9 C. Nitschelm,10 E. S. G. de
Almeida,5 G. Pereira,11 E. A. Michael,11 Q. Gao,1 W. Wang,1,12 I. Reyes,11 C.
Arcos,13 I. Araya,14 M. Curé13
1Chinese Academy of Sciences South America Center for Astronomy, National
Astronomical Observatories, CAS, Beijing 100101, China
2Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Santiago,
Chile
3Instituto de Astronomía, Universidad Católica del Norte, Av. Angamos 0610
Antofagasta, Chile
4Centre de Recherche en Astronomie, Astrophysique et Géophysique (CRAAG),
Route de l’Observatoire, B.P. 63, Bouzareah, 16340,
Alger, Algeria
5Université Côte d’Azur (UCA), Centre National de la Recherche Scientifique
(CNRS), Observatoire de la Côte d’Azur (OCA),
Laboratoire J. L. Lagrange, UMR 7293,Campus Valrose, 06108 Nice Cedex 2,
France
6Astronomical Observatory, Volgina 7, P.O.Box 74 11060 Belgrade, Serbia
7LPHEA Laboratory, Oukaimeden Observatory, Cadi Ayyad University/FSSM, BP 2390
Marrakesh, Morocco,
8Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu
District, Kunming, 650216, China,
9Optical Interferometry Group, Shanghai Astronomical Observatory(SHAO),
Chinese Academy Sciences(CAS) , Shanghai, 200030, China,
10Centro de Astronomía (CITEVA), Universidad de Antofagasta, Avenida Angamos
601, Antofagasta 1270300, Chile
11Radio Astronomical Instrumentation Group (RAIG), Terahertz- and Astro-
Photonics Laboratory, Departamento de Ingeniería Eléctrica,
Universidad de Chile, Avenida Tupper 2007, Santiago, Chile
12CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China
13Instituto de Física y Astronomía, Facultad de Ciencias, Universidad de
Valparaíso, Av. Gran Bretaña 1111, Valparaíso, Chile
14Centro de Investigación DAiTA Lab, Facultad de Estudios Interdisciplinarios,
Universidad Mayor, Alonso de Córdova 5495, Santiago, Chile E-mail:
<EMAIL_ADDRESS>(MH<EMAIL_ADDRESS>(RP) and<EMAIL_ADDRESS>(SJ)

(Accepted 2022 January 11. Received 2022 January 11; in original form 2021
August 16)

###### Abstract

Differential Interferometry allows to obtain the differential visibility and
phase, in addition to the spectrum. The differential phase contains important
information about the structure and motion of stellar photosphere such as
stellar spots and non-radial pulsations, and particularly the rotation. Thus,
this interferometric observable strongly helps to constrain the stellar
fundamental parameters of fast rotators. The spectro-astrometry mainly uses
the photocentre displacements, which is a first approximation of the
differential phase, and is applicable only for unresolved or marginally
objects.
We study here the sensitivity of relevant stellar parameters to the simulated
photocentres using the SCIROCCO code: a semi-analytical algorithm dedicated to
fast rotators, applied to two theoretical modeling stars based on Achernar and
Regulus, in order to classify the importance of these parameters and their
impact on the modeling. We compare our simulations with published VLTI/AMBER
data.
This current work sets the limits of application of photocentre displacements
to fast rotators, and under which conditions we can use the photocentres
and/or the differential phase, through a pre-established physical criterion.
To validate our theoretical study, we apply our method of analysis on observed
data of the edge-on fast rotator Regulus. For unresolved targets, with a
visibility $V\sim 1$, the photocentre can constrain the main stellar
fundamental parameters of fast rotators, whereas from marginally resolved
objects ($0.8\leq V<1$), mainly the rotation axis position angle ($\rm PA_{\rm
rot}$) can be directly deduced from the vectorial photocentre displacement,
which is very important for young cluster studies.

###### keywords:

Methods: observational, numerical – Techniques: interferometric, high angular
resolution – Stars: rotation

††pubyear: 2022††pagerange: Application limit of the photocentre displacement
to fundamental stellar parameters of fast rotators \- Illustration on the
edge-on fast rotator Regulus–LABEL:lastpage

##  1 Introduction

###  1.1 Fundamental parameters of stars

Measuring the fundamental parameters of stars, such as diameter, mass,
rotation, effective temperature or age remains crucial for stellar physics. In
particular this is necessary to properly characterize the host stars and
discuss the evolution of the thousands of extrasolar planets that have been
discovered from radial-velocimetry and transit observations. It is also
important to combine the fundamental parameters with the stellar activity and
asteroseismology through observational breakthroughs offered by the high-
performance interferometers, spectrographs and photometers. The angular
resolution provided by optical long baseline interferometry is crucial to
constrain most of these fundamental parameters, including the proper
separation between local and global velocity fields (Jankov, 2011, and
references therein).

###  1.2 Rapid rotators

In that context, rapid rotators are of specific interest. Rotation velocities
close to the critical velocity are keys to understand mass loss and stellar
winds. The geometrical flattening, coupled with gravity darkening (von Zeipel,
1924a) -known as the von Zeipel effect- and the resulting lower luminosity and
radiation pressure at the stellar equator, is also a key to shape the polar
components of the stellar wind (Kervella & Domiciano de Souza, 2006). The
mechanisms amplified by rapid rotation, such as meridional circulation or
turbulence, may affect the internal structure of the star and its evolution
(Meynet, 2009). The gravity darkening has a profound impact on the physics of
the stars, with important observational consequences. For example, the models
from Collins & Sonneborn (1977) indicate a two-component spectral energy
distribution (SED) for these stars, with an infrared excess due to gravity
darkening. Therefore, it is not easy to include these stars in a single
spectral class, since the observed SED depends on their rotational velocity
and inclination angle.

The most important parameters of a rapid rotator are :

  * •

The equatorial and polar radii $R_{\rm eq}$ and $R_{\rm pol}$, respectively,
which contains information about the size of the star;

  * •

The stellar flattening $R_{\rm eq}/R_{\rm pol}$, which is a key constraint on
the gravity darkening effect and the differential rotation;

  * •

The equatorial velocity $V_{\rm eq}$ , which is connected to the kinematics of
the stellar photosphere;

  * •

The inclination angle $i$, which is the angle between the observation line of
sight and the stellar rotation axis. This parameter provides the true spectral
type due to the gravity darkening effect and influencing the nature of Be
stars (see.g. Porter & Rivinius, 2003);

  * •

The gravitational darkening is mostly described by the coefficient $\beta$
(e.g. Hadjara et al., 2014). This stellar parameter depicts the difference of
temperature, luminosity and surface gravity between the equator and the poles;

  * •

The rotation axis position angle $\rm PA_{\rm rot}$, which is the angle of the
projected rotation axis on the sky, measured from north to east. It can give,
together with the proper motion of the star, information on the protostellar
cloud dynamics. The knowledge of this parameter is very important in multiple
systems, or in the presence of circumstellar material to constrain the
evolution of the system.

###  1.3 Rapidly rotating star modeling for Differential Interferometry

Differential Interferometry (DI) is a technique based on information taken at
different wavelengths. In the case of long-baseline interferometry, it uses
spectrally dispersed fringes. This technique offers two major advantages: i)
getting information beyond the instrumental angular resolution limit, and ii)
measuring simultaneously spatial, spectroscopic and kinematics properties of
the stellar surface, with angular diameter down to $0.1\,\rm mas$, which is
for example 100 parsec for Sun-like stars. Inspired by the early works of
Labeyrie (1975) and Beckers (1982) proposed the Differential Speckle
Interferometry technique, which uses the chromatic displacement of the speckle
photocentre, given by the first-order term of the phase according to the
MacLaurin series (Jankov et al., 2001). This technique has been extended to a
wider range of wavelengths and applied to long-baseline interferometry by
Petrov (1988, 1989) who established the fundamentals of the DI technique. For
the first time, this allowed to separate spatial and spectral information of
two stellar components forming the binary Capella (Petrov & Lagarde, 1992) and
to measure the stellar rotation of the slow rotator Aldebaran using the
photocentre displacement (Lagarde, 1994). The first theoretical general study
of the DI technique was done by Chelli & Petrov (1995) to estimate the angular
diameters, the rotation velocities and the position angles of the rotation
axis of single stars, as well as the angular separations and the radial
velocity differences of close binary systems. The DI offers the possibility to
constrain these parameters simultaneously (Chelli & Petrov, 1995).

It is imperative to mention the spectro-astrometry in this paper, because it
is a method of photocentre measurement (Gravity Collaboration et al., 2018;
Hadjara et al., 2018; Le Bouquin et al., 2009). The potential of this
technique with very large telescopes is obvious, because the precision on the
measurement of a photocentre improves as $B\times D$ (the interferometric
baseline length times the telescope diameter) in the visible and as $B\times
D^{2}$ in the thermal IR or when we are limited by the detector noise (Petrov,
1989). Several E-ELT instruments will provide spectro-astrometry mode like
METIS (Brandl et al., 2008) or possibly HIRES (Maiolino et al., 2013). As well
as two possible future VLTI visitor instruments, in J-band are in being
studied/carried out currently, namely: BIFROST (Kraus, 2018, with a very rich
spectro-astrometry program) and VERMILION (scientific white paper in prep.),
with its new generation fringe tracker (Petrov, 2019). The spectral resolution
is also very important, for further study of the spectroscopic lines.
Currently, at the VLTI, GRAVITY (Gravity Collaboration et al., 2017) offers a
spectral resolution $R\sim 4500$ in K-band, where AMBER (Petrov et al., 2007),
before its decommissioning, was offering $R\sim 12000$ at the same band.
BIFROST and VERMILION expect to offer $R\sim 20000$ to $30000$.

Measuring the stellar rotation is essential : ignoring it can induce spurious
conclusions, such as a mis-classification of the spectral type when the star
rotates rapidly around its rotation axis, which has a strong influence on the
emitted flux (e.g. Maeder & Peytremann, 1972) and on the gravity darkening
(von Zeipel, 1924b), for example the reference star Vega, has been discovered
to be a pole-on rapid rotator (Peterson et al., 2006; Aufdenberg et al.,
2006).

Numerical models of fast rotators, possibly including their close stellar
environment (CSE), are rather rare, very few of them being dedicated to
optical interferometry :

  * •

For the fast rotators only, let us mention: CHARRON (Domiciano de Souza et
al., 2012a) and ESTER (Rieutord et al., 2016; Espinosa Lara & Rieutord, 2013),

  * •

For the CSE only, lets us mention : SIMECA (Stee et al., 2008), BE-DISK (Sigut
et al., 2009), and the numerical model of Kraus et al. (2012) which studies
the photocentre displacement of a Be envelope in local thermodynamic
equilibrium from infrared interferometric measurements,

  * •

For both the rotating stars and their disks, let us mention HDUST (Carciofi &
Bjorkman, 2006), a radiative transfer code which produces spectra and
intensity maps in natural and/or polarized light for CSEs of massive stars
including gas and dust.

In this paper, we further describe in detail the methodology introduced by
Hadjara et al. (2018). We study and discuss the limit and dependence of the
photocentre defined by its angular coordinates
$(\epsilon_{\alpha},\epsilon_{\delta})$ that we deduce from the differential
phase $\phi_{\mathrm{diff}}$, to the relevant physical parameters of our
numerical model SCIROCCO (Hadjara et al., 2012; Hadjara et al., 2013, 2014,
2018), using high spectral resolution interferometric data such as those
obtained with the VLTI-AMBER in the K band (R$\sim$12000) for a given star
with any inclination angle ($0^{\circ}\leq i\leq 90^{\circ}$). As an example,
we chose to use a reference modeling star similar to Achernar, a Be star of
spectral type B6V and an inclination angle $i$ of about $60^{\circ}$, but with
a rotation axis position angle $\rm PA_{\rm rot}=0^{\circ}$ to simplify (as it
is shown in Domiciano de Souza et al., 2012b). We show that, in the case of
fast rotators, the polar photocentre displacement $(\epsilon_{\rm pol})$ is
sensitive to the gravity darkening, where the equatorial one $(\epsilon_{\rm
eq})$ is sensitive to the stellar rotational velocity. This study allows us to
determine the limits of application of photocentre displacements to fast
rotators, and under which conditions we can use the photocentres and/or the
differential phase, through a pre-established physical criterion. Then, we
study and discuss the sensitivity of $(\epsilon_{\alpha},\epsilon_{\delta})$
to relevant stellar parameters for an edge-on star (as an example, a reference
modeling star similar to Regulus, which is a B sub-giant star of spectral type
B8IV), with an $\rm PA_{\rm rot}\neq 0^{\circ}$ and $i\approx 90^{\circ}$
(Hadjara et al., 2018), which allows us to study the impact of the gravity
darkening coefficient $\beta$ on $(\epsilon_{\alpha},\epsilon_{\delta})$. The
present paper is organized as follows:

  * •

In Sect. 2, we introduce the differential phase ($\phi_{\mathrm{diff}}$).

  * •

In Sect. 3, we show how and within which limits the vectorial right-
ascension/declination photocentre $(\epsilon_{\alpha},\epsilon_{\delta})$ can
be deduced from the differential phase $\phi_{\mathrm{diff}}$.

  * •

In Sect. 4, we study the influence of relevant physical parameters on the
photocentre displacement and on the flux, using our model of an Achernar-like
star with an arbitrary inclination.

  * •

In Sect. 5, we show the application of the photocentre displacement method to
real data of Regulus.

  * •

In Sect. 6, we discuss the photocentre displacement sensitivity to the key
stellar parameters; $R_{\rm eq}$, $V_{\rm eq}$, $i$, $\rm PA_{\rm rot}$, and
$\beta$ on our second modeling reference star, through the best results
obtained from Sect. 5.

  * •

In Sect. 7, we analyze and discuss our results.

  * •

In Appendix. A, we explain a few important details about the modeling for fast
rotators with SCIROCCO.

##  2 Simulated differential phase for rapidly rotating stars

The differential phase $\phi_{\mathrm{diff}}$ measured for the on-sky-
projected interferometric baseline $\vec{B}(u,v)$, as function of spatial
frequencies $u$ and $v$, at the reference wavelength $\lambda_{0}$ taken in
the continuum, is defined as :

$\phi_{\mathrm{diff}}(u,v,\lambda)=arg\left(V(u,v,\lambda)\right)-arg\left(V(u,v,\lambda_{0})\right),$
(1)

where $V$ is the object complex visibility. Although the phase
$arg\left(V(u,v,\lambda)\right)$ is not recovering well by interferometer due
to atmospheric turbulence and lack of absolute reference, the differential
phase $\phi_{\mathrm{diff}}$ is well retrieved. This quantity is directly
connected to the chromatic displacement of the photocentre vector according to
:

$\epsilon(\lambda)=-\frac{\phi_{\mathrm{diff}}(u,v,\lambda)}{2\pi\sqrt{u^{2}+v^{2}}},$
(2)

given by the first-order approximation of the McLaurin series of the phase
(the demonstration, as well as the validity limits, of this development have
been handled by Jankov et al., 2001).

In order to study the sensitivity $\epsilon_{\alpha}$ and $\epsilon_{\delta}$
(that we deduce from $\phi_{\mathrm{diff}}$, and which are linked to the right
ascension and the declination respectively) we use the numerical code SCIROCCO
described in Hadjara et al. (2014), to various stellar parameters, as we did
previously (Domiciano de Souza et al., 2012b) for a reference modeling star,
at any inclination angle ($0^{\circ}<i<90^{\circ}$), similar to Achernar with
the CHARRON code described in detail in Domiciano de Souza et al. (2012a);
Domiciano de Souza et al. (2002). Namely, here we use:
$R_{eq}=11\,\mathrm{R}_{\sun}$, $d=50\,pc$, $V_{\mathrm{eq}}\sin
i=250\,\mathrm{km.s}^{-1}$, $i=60^{\circ}$, $M=6.1\,\mathrm{M}_{\sun}$,
$T_{\rm eff}=15000\,\rm K$, $\rm PA_{\rm rot}=0^{\circ}$ (northern direction),
$\beta=0.25$ (theoretical value for pure radiative stellar envelope according
to von Zeipel (1924b)), without taking into account the differential rotation.
In addition, we take into account the limb darkening effects to produce line
profiles with KURUCZ/SYNSPEC stellar atmospheres model. These parameters
correspond to $V_{\rm eq}$ equal to 90% of the critical velocity $V_{\rm
crit}$, $R_{eq}/R_{p}=1.4$ and an equatorial angular diameter
$\Theta=2R_{eq}/d=2\,\rm mas$. Figure 1 shows the monochromatics intensity
maps for a given Doppler shift across 3 wavelengths around the absorption
Br$\gamma$ line, adopting the stellar parameters given above (more details
about features of this kind of figures have been well described by Hadjara et
al., 2014; Hadjara, 2015; Hadjara et al., 2018).

Figure 1: Monochromatic intensity maps simulated by SCIROCCO, with the
physical parameters of the reference model similar to Achernar (see text for
details), for 3 different wavelengths. The red crosses show the
global/astrometric photocentre position (explained in Sect. 3) corresponding
to each wavelength, while the green crosses depict the photocentre position in
the continuum.

In the appendix A, we give more details about how the SCIROCCO code works, in
particular on the flattened shape used of the fast rotating stars as well as
on the line profile.

##  3 FROM $\phi_{\mathrm{diff}}$ TO THE PHOTOCENTRE COORDINATES

###  3.1 Deducing the photocentre coordinates

We deduce the photocentre coordinates $(\epsilon_{\alpha},\epsilon_{\delta})$
from $\phi_{\mathrm{diff}}$ using elementary geometric projections as shown in
Fig. 2.

Figure 2: The photocentre vector $\vec{\epsilon}$ projected on two
interferometric baseline directions $\eta_{\rm i}$ and $\eta_{\rm j}$
($\epsilon_{\rm i}$ and $\epsilon_{\rm j}$) and projected on the right
ascension and declination directions ($\epsilon_{\alpha}$ and
$\epsilon_{\delta}$ respectively).

Let us call $\vec{\epsilon}_{\rm i}$ and $\vec{\epsilon}_{\rm j}$ the
projections of the protocentre vector $\vec{\epsilon}$ on the directions of
the baselines $\rm i$ and $\rm j$ respectively (with the projection angles
$\eta_{\rm i}$ and $\eta_{\rm j}$ respectively). We define the 2 “pseudo-
coordinates” $\epsilon_{\alpha,ij}$ and $\epsilon_{\delta,ij}$ as :

$&\epsilon_{\alpha,\rm ij}=(\epsilon_{\rm i}\sin\eta_{\rm j}-\epsilon_{\rm
j}\sin\eta_{\rm i})/\sin(\eta_{\rm j}-\eta_{\rm i}),\\\ \epsilon_{\delta,\rm
ij}=(\epsilon_{\rm j}\cos\eta_{\rm i}-\epsilon_{\rm i}\cos\eta_{\rm
j})/\sin(\eta_{\rm j}-\eta_{\rm i}),$ (3)

where $\epsilon_{\rm i}$ and $\epsilon_{\rm j}$ are the moduli of the
projected vectors $\vec{\epsilon}_{\rm i}$ and $\vec{\epsilon}_{\rm j}$,
respectively. We can easily check these equations in Fig. 2, where $(\eta_{\rm
i},\eta_{\rm j})=(0,\pi/2)$ $\rightarrow$
$(\epsilon_{\alpha},\epsilon_{\delta})=(\epsilon_{\rm i},\epsilon_{\rm j})$
and $(\eta_{\rm i},\eta_{\rm j})=(\pi/2,\pi)$ $\rightarrow$
$(\epsilon_{\alpha},\epsilon_{\delta})=(-\epsilon_{\rm j},\epsilon_{\rm i})$.
Note that the angle $\eta$ is called the projection angle ($\rm PA$) of the
baseline ($\rm PA=\eta$), and which is defined from north to east. Thus, from
now onwards we will use the notation of $\rm PA$ instead of $\eta$, and
$\Delta\eta=\left|\eta_{\rm i}-\eta_{\rm j}\right|$, will be simply called
$\Delta\rm PA$.

Knowing that for any baseline $\rm n$ at each wavelength $\lambda$, the
relation between the photocentre $\epsilon_{\rm n}(\lambda)$ and differential
phase $\phi_{\rm n}(\lambda)$, which is only satisfied for marginally resolved
and unresolved objects (as demonstrated by Jankov et al., 2001, using the
derivative and definite integral theorem to the Maclaurin expansion of the
complex visibility function), is given by:

$\phi_{\rm n}=-\frac{2\pi}{\lambda}\vec{\epsilon}_{\rm
n}(\lambda)\cdot\vec{B}_{\rm n}.$ (4)

If we use more than 2 independent baselines, we deduce the mean coordinates of
the photocentre displacement $(\epsilon_{\alpha},\epsilon_{\delta})$ thanks to
the weighted average, favoring the least noisy baselines, given by:

$&\epsilon_{\alpha}(\lambda)=\sum_{\rm i,j>i}\frac{\epsilon_{\alpha,\rm
ij}(\lambda)}{Var(\epsilon_{\alpha,\rm ij}(\lambda))}\big{/}\sum_{\rm
i,j>i}\frac{1}{Var(\epsilon_{\alpha,\rm ij}(\lambda))},\\\
\epsilon_{\delta}(\lambda)=\sum_{\rm i,j>i}\frac{\epsilon_{\delta,\rm
ij}(\lambda)}{Var(\epsilon_{\delta,\rm ij}(\lambda))}\big{/}\sum_{\rm
i,j>i}\frac{1}{Var(\epsilon_{\delta,\rm ij}(\lambda))},$ (5)

where Var is the measured variance.

The absolute photocentre coordinates $E_{\alpha}$ and $E_{\delta}$, measured
separately by single-aperture astrometry, is directly deducted from the
monochromatic intensity maps $I(x,y,\lambda)$, using the following formula:

$&E_{\alpha}(\lambda)=\frac{\sum_{\alpha}\sum_{\delta}\alpha
I(\alpha,\delta,\lambda)}{\sum_{\alpha}\sum_{\delta}I(\alpha,\delta,\lambda)},\\\
E_{\delta}(\lambda)=\frac{\sum_{\alpha}\sum_{\delta}\delta
I(\alpha,\delta,\lambda)}{\sum_{\alpha}\sum_{\delta}I(\alpha,\delta,\lambda)}.$
(6)

To keep the same convention as for $\phi_{\mathrm{diff}}$ in Eq. 1, which is
zero on the continuum, $E_{\alpha,\delta}$ becomes:
$E_{\alpha,\delta}(\lambda)=E_{\alpha,\delta}(\lambda)-E_{\alpha,\delta}(\lambda_{0})$.
Unlike the absolute photocentre coordinates $E_{\alpha,\delta}$ which gives
barycentric information on the global stellar image,
$\epsilon_{\alpha,\delta}$ gives the barycentric information depending on the
baselines used. Thus, the (u,v) coverage should be distributed as regularly
and densely as possible, especially for the fast rotators with their flattened
shape. Note that the polar line profile is less deep than the equatorial one,
which plays an important role on the final $E_{\alpha,\delta}(\lambda)$
values. As soon as we measure $\phi_{\mathrm{diff}}$ on two different
baselines $\vec{B}_{\rm i}$ and $\vec{B}_{\rm j}$ ($\rm i\neq j$), it yields
an interferometric vectorial photocentre displacement
$\vec{\epsilon}(\lambda)$. To deduce the equatorial and polar photocentre
displacement $(\epsilon_{\rm eq},\epsilon_{\rm pol})$ from
$(\epsilon_{\alpha},\epsilon_{\delta})$, we must rotate the coordinate system
by the rotation axis position angle $\rm PA_{\rm rot}$ (idem for $(E_{\rm
eq},E_{\rm pol})$ from $(E_{\alpha},E_{\delta})$):

$&\epsilon_{\rm eq}=\epsilon_{\alpha}\cos\rm PA_{\rm
rot}+\epsilon_{\delta}\sin\rm PA_{\rm rot},\\\ \epsilon_{\rm
pol}=-\epsilon_{\alpha}\sin\rm PA_{\rm rot}+\epsilon_{\delta}\cos\rm PA_{\rm
rot},$ (7)

Which means that $\epsilon_{\rm eq}$ and $\epsilon_{\rm pol}$ are completely
independent of $\rm PA_{\rm rot}$. Taking $\rm PA_{\rm rot}=0^{\circ}$,
$\epsilon_{\alpha}$ represents the equatorial photocentre displacement
($\epsilon_{\alpha}=\epsilon_{\rm eq}$) and $\epsilon_{\delta}$ the polar one
($\epsilon_{\delta}=\epsilon_{\rm pol}$).

###  3.2 Discussing the photocentre displacements

Figure 3 shows a comparison between the absolute photocentre displacements
$E_{\rm eq,pol}$ (continuous black line) with the apparent interferometric
photocentre displacements $\epsilon_{\rm eq,pol}$ (discontinuous coloured
lines) for a projection angle $i=60^{\circ}$, for only two baselines
($B=45\,\rm m$), with $\Delta\rm PA$ ranging from $5^{\circ}$ to $90^{\circ}$
by steps of $5^{\circ}$. For simplification reasons, in this subsection we
only use two baselines. Of course, the method can be used and extended to more
baselines, thanks to Eq. 5, as is done in Sec. 5, and as was done by Hadjara
et al. (2018) before.

Fig. 3 (a) shows the wavelength dependency of $\epsilon_{\rm eq}$
$\epsilon_{\rm pol}$. Fig. 3 (b) shows the vectorial photocentre displacement
($\epsilon_{\rm pol}=f(\epsilon_{\rm eq})$), which is in shape of an arrowhead
(further discussion are done about that on Sec. 5 and Appendix B). For
readability reason, we only show the uncertainties of $\epsilon_{\rm eq}$,
because the uncertainties of $\epsilon_{\rm pol}$ cover all the frame, where
both uncertainties are equal to $30\,\mu as$. The error on the photocentre
displacements $(\sigma_{\epsilon}=\pm 30\,\mu as)$ was deduced before by
Hadjara et al. (2018), from RMS of $\phi_{\mathrm{diff}}$ errors of AMBER, in
the continuum, for three VLTI baselines with an average length of $\rm B\sim
75\,\rm m$. The value of $\pm 30\,\mu as$ is considered as a good average
error of both $\epsilon_{\rm pol}$ and $\epsilon_{\rm eq}$ over all
differential VLTI instruments in the IR. Fig. 3 (c) depicts deduced $\rm
PA_{\rm rot}$ values, through a simple linear fit (with $90^{\circ}$ rotation)
of the vectorial photocentre displacements (further details in our previous
paper; Hadjara et al., 2018), which was initially set at $\rm PA_{\rm
rot}=0^{\circ}$ for simplicity. Fig. 3 (c) shows the different differential
visibilities corresponding to each baseline projection angle ($\rm PA$). The
longer baseline length (B) the lower is the visibility. Indeed, the polar
visibility at $\rm PA=0^{\circ}$ is lower than the equatorial one at $\rm
PA=90^{\circ}$. As we can observe, independently of $\rm PA$ value, which
affect single photocentre displacement projections, and the visibilities, the
combination/averaging of photocentre displacement coming form couple(s) is
completely independent form $\rm PA$. The different plots of $\epsilon_{\rm
eq,pol}$ as a function of $\rm PA$ are all superimposed on top of each other,
for unresolved objects.

Figure 3: Absolute photocentres $E_{\rm eq,pol}$ (in black) and photocentres
coming from the (u,v) coverage $\epsilon_{\rm eq,pol}$ (in colours), around
the Br$\gamma$ line, for different position angle differences between two
baselines ($\Delta\rm PA$, where each value corresponds to a specific colour
on the top of each plot). a) $\epsilon_{\rm eq}$ and $\epsilon_{\rm pol}$ as a
function of the wavelength. b) The vectorial photocentre displacement. The
gray transparent box depicts the error of $30\,\mu as$ amplitude related to
the measurement of $\epsilon_{\rm eq,pol}$. For aesthetic reasons we plot here
only $\epsilon_{\rm eq}$ errors. c) Values of $\rm PA_{\rm rot}$, which are
obtained by a simple linear fit of the vectorial photocentre displacements. d)
Corresponding differential visibilities at each baseline projection angle
($\rm PA$).

As we know, Eq. 4 is not at all satisfied for the resolved objects. In other
words, the photocentre method could not be used then, where other terms of
$\phi_{\mathrm{diff}}$ start to alter the interferometric photocentre
displacement. Therefore, one of the main objectives of this study is to
determine the resolution range in which the photocentre method remains
applicable. Indeed, the equality $E_{\rm eq,pol}\approx\epsilon_{\rm eq,pol}$
depends mainly on the angular size of the observed object ($\Theta_{\rm
eq,pol}$), and relatively to the angular resolution $\lambda/B$). Thereby, we
use $\Gamma_{\epsilon}=\Theta\frac{B}{\lambda}$ as criterion for the limits of
use of photocentre displacement in interferometry.

Thus, by varying $\Theta$, $B$, and $\lambda$, allows us to determine the
limit values of $\Gamma_{\epsilon}$, for which the photocentre displacement
method is applicable, i.e. when the equality $E_{\rm
eq,pol}\approx\epsilon_{\rm eq,pol}$ is satisfied. We show, in Appendix B a
few examples related to the case of a star similar to Achernar
($\Theta=2.38\,\rm mas$ Hadjara et al., 2014), but with $\rm PA_{\rm
rot}=90^{\circ}$, and gravity darkening coefficient $\beta=0.25$, observed in
the K-band (at $\lambda=2.16570\,\mu\rm m$), and for several values of $B=15$,
$30$, $45$, $60$, $75$, $90$ until $150\,\rm m$. Also, to study the
sensitivity of $\Gamma_{\epsilon}$ to the inclination angle, we set
$i=30^{\circ}$, $60^{\circ}$, and $90^{\circ}$. Therefore, and according to
all our tests, which are represented by the sample of figures (15-16) gathered
in Appendix B, we observe that the photocentre displacement method works only
for quasi unresolved stars, with $\Gamma_{\epsilon}\leq 0.32$, i.e. when the
angular size of the observed object is approximately 30% less than
$\frac{\lambda}{B}$ ($\Theta\lesssim 0.3\frac{\lambda}{B}$). We add Fig. 16 in
Appendix B, for a fixed baseline length ($B=150\,\rm m$) and three different
inclination angle values ($i=30^{\circ}$, $60^{\circ}$, and $90^{\circ}$), in
order to illustrate the case of a more resolved star.

From averaging couple photocentre projections (Fig. 3 and Figs. 15-16), we
observe that the more the star is edge-on ($i\sim 90^{\circ}$) the higher the
equatorial photocentre amplitude (because of the higher $V_{\mathrm{eq}}\sin
i$), and the lower the polar photocentre amplitude (because of the gradient
contrast variations along the polar axis in function of $i$). Formally, the
more edge-on the star, the wider the arrowhead shape, and in contrary, the
more pole-on the star ($i\sim 0^{\circ}$), the sharper the arrowhead shape.
Moreover, because of the intensity gradient caused by the gravity darkening
effect, we observe that the polar photocentre amplitude is inversely
proportional to the inclination angle.

Therefore, we conclude that the vectorial photocentre displacement method can
be used (i.e. when $E_{\rm eq,pol}\approx\epsilon_{\rm eq,pol}$) for
$\Gamma_{\epsilon}\leq 0.32$, which corresponds to quasi unresolved objects,
with a visibility $V\geq 0.99$. Beyond, i.e. $\Gamma_{\epsilon}>0.32$, only
$\phi_{\mathrm{diff}}$ data are usable. In other words the photocentre
displacement method appears applicable only with unresolved stars. Of course,
we can also use the “classical” differential phase method
($\phi_{\mathrm{diff}}$) in this case, but in terms of time computing, and
when we have many baselines, on unresolved object, it is more efficient to use
the vectorial photocentre displacements method. In contrary, for
$\Gamma_{\epsilon}>0.32$, only the “classical” $\phi_{\mathrm{diff}}$ method
is useful.

The limit of $\Gamma_{\epsilon}$ depends of course on the $\vec{\epsilon}$
uncertainty that we have chosen ($30\,\mu as$). But in practice, with real
data, we must not forget the noise effect (SNR). We could extend the
application of the photocentre displacement method to marginally resolved
stars $V\sim 0.8$ (i.e. $\Gamma_{\epsilon}\sim 0.64$, as done before by
Hadjara et al., 2018) and a difference of the projection angle
($60^{\circ}\leq\Delta\rm PA\leq 90^{\circ}$), especially for edge-on star
(with a very low amplitude of $\epsilon_{\rm pol}$).

##  4 Dependence to model physical parameters

As in Domiciano de Souza et al. (2012b), here we use four interferometric
configurations with two baseline lengths $B_{\rm proj}=75\,\rm m$ and
$150\,\rm m$ and two projection angles $\rm PA=45^{\circ}$ and $90^{\circ}$.
In addition to the visibility modulus used for the reference modeling star,
Fig. 4 shows the four $(u,v)$ coverage points used in our simulation. The
angular diameter of our first modeling reference star (Achernar-like) is of
$2.38\pm 0.10\,\rm mas$ (Hadjara et al., 2014) and the angular resolution
corresponding to our baselines $B_{\rm proj}=75\,\rm m$
($\Gamma_{\epsilon}=0.4$) and $B_{\rm proj}=150\,\rm m$
($\Gamma_{\epsilon}=0.8$) is of $5.96\rm\,mas$ and $2.98\rm\,mas$
respectively.

Figure 4: (u,v) coverage and module of visibility of the reference model star
similar to Achernar.

In this study, we use 4 interferometric configurations (of 2 telescopes), as
shown in Fig. 4, which represents the visibility modulus used for our
reference star (Achernar-like). Our objective is to study the dependence of
the flux and photocentres displacement on relevant physical parameters of our
model, namely: the stellar radius $R$, $V_{\mathrm{eq}}\sin i$, inclination
angle $i$, average effective temperature $\overline{T}_{\mathrm{eff}}$, with
and without limb-darkening, for different line profiles (Voigt, from Kurucz
and Tlusty modeling), fixed and as function of the co-latitude $\theta$. The
dependence of the gravity darkening coefficient $\beta$ will be studied in the
Sec. 6. Note that the West-to-East direction is taken from right to left in
all plots of the current paper, including in the previous Fig. 3. At this
step, we have to remind that we could use a Voigt function as simple and rough
line profile but, and as explained in Appendix. A, we use the pseudo-Voigt
function (of 6 parameters) only to get an analytic expression of the
Kurucz/Tlusty line profiles for faster calculation purpose. So, we study in
the present section the difference between a Voigt line profile and a pseudo-
Voigt fit function of the line profiles from Kurucz/Tlusty model atmospheres.

The influence of each studied parameter on the simulated spectrum and
perpendicular photocentre displacements ($\epsilon_{\alpha}$ and
$\epsilon_{\delta}$) on the $Br_{\gamma}$ line are shown in Figs 17 and 18, in
Appendix C below. We note that, in our case where $\rm PA_{\rm
rot}=0^{\circ}$, $(\epsilon_{\rm eq},\epsilon_{\rm
pol})=(\epsilon_{\alpha},\epsilon_{\delta})$ and $(E_{\rm eq},E_{\rm
pol})=(E_{x},E_{y})$.

For practical reasons, all the figures for this section are gathered in
Appendix C. Thereby, and as we can observe in Figs. 17 and 18, we can classify
the effect of the stellar parameters on photocentre displacements (or on
$\phi_{\mathrm{diff}}$), thanks to $\vec{\epsilon}$ differences of
($\Delta_{\epsilon}$) with respect to reference model, in three categories:

1a) The very sensitive parameters ($\Delta_{\epsilon}>25\%$), which are the
equatorial radius $R_{\rm eq}$, $\rm PA_{\rm rot}$ and the line profile linked
to the analytic/stellar atmosphere modeling (Voigt versus Kurucz).

1b) The moderately sensitive parameters ($10\%\leq\Delta_{\epsilon}\leq
25\%$), which are $V_{\mathrm{eq}}\sin i$, the inclination $i$, and the line
profile linked to the analytic/stellar atmosphere modeling (Tlusty versus
Kurucz, and the line profile linked to the latitudinal versus the regular
aspect).

1c) And finally the less sensitive parameters ($\Delta_{\epsilon}<10\%$),
which are the mean effective temperature $\overline{T}_{\mathrm{eff}}$,
$\beta$,the both darkening (limb and gravitational).

The spectrum, on the other hand, is (the same, thanks to $\Delta_{s}$ with
respect to reference model):

2a) More sensitive (with $\Delta_{s}\geq 5\%$) to the line profile linked to
the analytic/stellar atmosphere modeling aspect; e.g. Voigt-Kurucz.

2b) Moderately sensitive (with 1%$\leq\Delta_{s}<5\%$) to other kind of line
profiles (Tlusty-Kurucz and linked to the latitudinal/regular aspect),
$V_{\mathrm{eq}}\sin i$, $i$.

2c) And finally, less sensitive (with $\Delta_{s}<1\%$) to $R_{\rm eq}$,
$\overline{T}_{\mathrm{eff}}$, $\rm PA_{\rm rot}$.and to the limb-darkening.

It should be noted that the study of the influence of the line profile on the
interferometric measurements allows us to demonstrate the important impact of
this parameter. Indeed we demonstrate the impossibility of using a simple
analytical profile in this kind of simulation (e.g. a simple and rough Voigt
line profile). The comparison of SCIROCCO code with the photocentre
displacements observations via a particular adjustment method (explained in
the section below) to four free parameters ($R_{\rm eq}$, $V_{\rm eq}$, $i$
and $\rm PA_{\rm rot}$), reveal a important dependence of the $R_{\rm eq}$ and
$V_{\rm eq}$ dynamical parameters, which are also strongly correlated and
related in the equation of degree of sphericity $D$ (Domiciano de Souza et
al., 2002):

$D=\frac{R_{\rm pol}}{R_{\rm eq}}=\left(1+\frac{V^{2}_{\rm eq}R_{\rm
eq}}{2GM}\right)^{-1}\approx 1-\frac{V^{2}_{\rm eq}R_{\rm pol}}{2GM},$ (8)

where $G$ the gravitation Cavendish-Newton constant and $M$ the mass of the
star. The parameter $V_{\rm eq}$ is highly sensitive to the type of line
profile (about 13%) of average difference between the Voigt line profile and
the Kurucz/Synspec profile, while the geometrical parameters $R_{\rm eq}$, $i$
and $\rm PA_{\rm rot}$ remain less sensitive to these profiles.

##  5 Application to Regulus

In this section, we discuss in more details our methodology of the photocentre
displacements and its application to real spectro-interferometric data. In a
previous paper about the fast rotator Regulus observed with VLTI/AMBER,
Hadjara et al. (2018) deduced the best fitting parameters, namely: the
equatorial radius $R_{\rm eq}=4.16\pm 0.24\,R_{\odot}$, the equatorial
rotation velocity $V_{\rm eq}=350\pm 22\,\mathrm{km.s}^{-1}$, the rotation-
axis inclination angle $i=86.4\pm 6.3^{\circ}$, and the rotation-axis position
angle $\rm PA_{\rm rot}=251\pm 2^{\circ}$, from differential phase data
($\phi_{\mathrm{diff}}$), using a $\chi^{2}$ minimization and the Markov Chain
Monte Carlo method (MCMC) method (Haario et al., 2006). Hadjara et al. (2018)
showed the final results through corresponding photocentre displacements
$(\epsilon_{\rm eq},\epsilon_{\rm pol})$. They also noticed a slight
dissymmetry on the observed photocentre displacements, between the red and the
blue wings. This dissymmetry was caused by calibration issue in wavelength of
a differential phase measurement over the six ones used. By fixing this issue,
and enhancing the wavelength calibration for the spectrum as well, we obtained
symmetric data and centered around the Br$\gamma$ line, as shown in in Fig. 5
below.

Figure 5: Left: Effect of the wavelength calibration issue on the differential
phase. Before the correction (top) and after the correction (bottom). The
smooth thin green curve superimposed on the observation show the best
$\phi_{\mathrm{diff}}$ model at this (u,v) coordinate. Right: AMBER spectrum
of Regulus in the Br$\gamma$ line. The dashed thick red curve is the raw
spectrum showing two telluric lines. The full thick blue curve is the Regulus
spectrum, with its error bars in green. The thin dark line represents our best
model.

Also, Hadjara et al. (2018) did not take into account the slight slope seen
with the observed $(\epsilon_{\rm eq},\epsilon_{\rm pol})$, which is caused by
an instrumental bias related to the spectrograph of AMBER, and that we treat
and fix in the current section. Thereby, we fix this bias by a simple linear
fit on the continuum, which we remove at the end, as shown in Fig. 6 below.

Figure 6: Left: Observed perpendicular right ascension and declination
photocentre displacements $(\epsilon_{\alpha},\epsilon_{\delta})$ in red thick
curves, and their simple linear fits at the continuum in blue lines. Right:
The same as in left plots, but in addition, corrected
$(\epsilon_{\alpha},\epsilon_{\delta})$ in superimposed black lines (after
removing the left blue lines deduced by a simple linear fit on the continuum).

Thereby, Fig. 7 shows the final perpendicular right ascension and declination
photocentre displacements $(\epsilon_{\alpha},\epsilon_{\delta})$, and their
equivalent equatorial–polar photocentre displacements $(\epsilon_{\rm
eq},\epsilon_{\rm pol})$, deduced by rotation thanks to the rotation-axis
position angle ($\rm PA_{\rm rot}$; Hadjara et al., 2018). The RMS error per
spectral channel has been measured in the continuum outside the spectral line
and found to be $\sim 30\mu as$ on any projection.

Figure 7: Left: The perpendicular right ascension and declination photocentre
displacements $(\epsilon_{\alpha},\epsilon_{\delta})$ as red thick curves for
the observed data. The smooth thin black curves superimposed on the
observations show the best-fitting $\phi_{\mathrm{diff}}$ (new MCMC results
below). The two perpendicular dashed lines represent the zero-point for the
photocentre displacement axis and the central wavelength ($\lambda$ = 2165.7
nm) of the Brγ line. Right: The equivalent equatorial–polar photocentre
displacements $(\epsilon_{\rm eq},\epsilon_{\rm pol})$.

Once these corrections were made, the plot of our best modeling of
equatorial–polar photocentre displacements ($\epsilon_{\rm
pol}=f(\epsilon_{\rm eq})$) is an arrowhead-shaped curve, as shown in the Fig.
8 below, with their new MCMC results. This arrowhead shape characterizes fast
rotators. For edge-on rotators (i.e. $i\sim 90^{\circ}$), the faster the
stellar rotation, the wider the arrowhead shape (as shown previously in Sec.
6).

Figure 8: Best modeling of vectorial photocentre displacement $(\epsilon_{\rm
pol}=f(\epsilon_{\rm eq})$, which is in shape of arrowhead.

Fig. 9 shows the covariance matrix of pairs of parameters, with their
histogram, that were obtained by the MCMC method, with these new corrected
photocentre displacements. We used exactly the same algorithm, configuration
and starting input parameters as in Hadjara et al. (2018).

Figure 9: MCMC covariance matrix distribution results for the four free
parameters ($R_{\rm eq}$, $V_{\rm eq}$, $i$ and $\rm PA_{\rm rot}$) of
Regulus, using six differential phases and spectrum data. The red point and
line show the best recovered parameters, the average of the last MCMC run. The
scatter plots show the projected two-dimensional distributions of the
projected covariance matrix (coloured points) by pair of parameters. The
colour bar represents the distribution of the points around the average,
following the variance $\sigma^{2}$. The histograms show the projected one-
dimensional distributions, with solid red lines representing the best
recovered parameters and dashed red line the uncertainties.

We found the following correlation coefficients (in descending order) :
$\rho(i,\rm PA_{\rm rot})=0.1765$, $\rho(R_{\rm eq},\rm PA_{\rm rot})=0.0128$,
$\rho(R_{\rm eq},i)=-0.0473$, $\rho(V_{\rm eq},\rm PA_{\rm rot})=-0.1486$,
$\rho(R_{\rm eq},V_{\rm eq})=-0.1837$ and $\rho(V_{\rm eq},i)=-0.2349$. When
$\rho(A,B)>0$, the correlation is proportional ($A$ and $B$ increase or
decrease together), and when $\rho(A,B)<0$ the correlation is inversely
proportional ($A$ increases when $B$ decreases and vice versa), e.g. the case
of $R_{\rm eq}$ and $V_{\rm eq}$, because of angular momentum conservation.
Note also that the correlation coefficients of all our free MCMC parameters
are symmetric (i.e. $\rho(A,B)=\rho(B,A)$).

These MCMC results are consistent with those given in a previous paper (see
MCMC section of Hadjara et al., 2018). The equatorial velocity $V_{\rm eq}$ is
more consistent with previously published results, and $R_{\rm eq}$ is
slightly higher but both, within their uncertainties, are closer to previous
results found in the literature, compared to Hadjara et al. (2018), especially
for $V_{\rm eq}$.

These results were obtained by using differential phases
($\phi_{\mathrm{diff}}$) and spectrum data. But dealing with the two
photocentre displacements $(\epsilon_{\alpha},\epsilon_{\delta})$ instead of
the 6 values of $\phi_{\mathrm{diff}}$, in addition of the spectrum values, we
obtain the results which are summarized in Fig. 10.

Figure 10: Same as Fig. 8, but using two photo-centre displacements
$(\epsilon_{\alpha},\epsilon_{\delta})$ and the spectrum data.

The correlation coefficient are: $\rho(V_{\rm eq},i)=0.3594$, $\rho(V_{\rm
eq},\rm PA_{\rm rot})=0.2598$, $\rho(i,\rm PA_{\rm rot})=0.1083$, $\rho(R_{\rm
eq},i)=-0.0567$, $\rho(R_{\rm eq},\rm PA_{\rm rot})=-0.1653$, and $\rho(R_{\rm
eq},V_{\rm eq})=-0.6070$. Relatively to the previous results, these new
results are similar within their uncertainties, including for the inclination
angle $i=81.8\pm 4.8^{\circ}$ (visible south pole), instead of $86.4\pm
2.8^{\circ}$ (visible north pole). The $i$-value of $81.8\pm 4.8^{\circ}$
(south pole-on), is clearly consistent with what we find in the literature
(Hadjara et al., 2018, and references therein). This slight difference of
visible south/north poles could be explained by the fact that Regulus is
marginally resolved (V$\sim$0.8 to 0.9), and the photocentre displacements
method works only for unresolved cases (V$\sim$1). Note also that $\rm PA_{\rm
rot}$-values are the same (i.e. $\sim 252^{\circ}$) in both studies. Figure 11
depicts a comparison between observations and the best MCMC minimization
parameters model, through the two photocentre displacements
$(\epsilon_{\alpha},\epsilon_{\delta})$ and spectrum, over the AMBER spectrum
of Regulus in the Br$\gamma$ line, the equatorial-polar photocentre
displacements $(\epsilon_{\rm eq},\epsilon_{\rm pol})$, and $(\epsilon_{\rm
pol}=f(\epsilon_{\rm eq})$ respectively. Unlike the Fig. 8, where the
arrowhead points down, because of visible north pole, the arrowhead of Fig. 11
points up (visible south pole). For this reason, the $\epsilon_{\rm pol}$
displacement have opposite signs in Fig. 8 (positive) and in Fig. 11
(negative). Because of the von Zeipel effect, the polar photocentre
displacement $(\epsilon_{\rm pol})$ is always higher at the center of the
line. For visible north pole, and as we can observe in Fig. 7, $(\epsilon_{\rm
pol})$ increases from the red wing continuum until the line’s center before
decreasing again to the blue wing continuum. But in the case of visible south
pole, as we can observe in Fig. 11, $(\epsilon_{\rm pol})$ decreases from the
red wing continuum before increasing at the line’s center, then it decreases
before it increases again to the blue wing continuum, which explains the loops
that can be seen on the sides of the arrowhead pointing up of Fig. 11, and
their absences on the arrowhead pointing down of Fig. 8. These “loops”
disappear at an inclination angle $i\geq 10^{\circ}$ (visible south pole),
when the intensity/temperature gradient is more pronounced on $\epsilon_{\rm
pol}$, as we can observe in Fig. 12.

Figure 11: Comparison between observations and the best MCMC minimization
parameters model, through the two photocentre displacements
$(\epsilon_{\alpha},\epsilon_{\delta})$ and spectrum, Top: over the AMBER
spectrum of Regulus in the Br$\gamma$ line. Bottom: The equatorial-polar
photocentre displacements $(\epsilon_{\rm eq},\epsilon_{\rm pol})$. Right:
Best modeling of vectorial photocentre displacement $(\epsilon_{\rm
pol}=f(\epsilon_{\rm eq})$.

In the next section we study the sensitivity of the vectorial photocentre
displacement to the main key parameters of the fast rotators, namely: $R_{\rm
eq}$, $V_{\rm eq}$, $i$ and $\rm PA_{\rm rot}$ for the case of Regulus, where
the line profile varies along the co-latitude $\theta$ according to the
$\beta$-value.

##  6 Sensitivity to $R_{\rm eq}$, $V_{\rm eq}$, $i$, $\rm PA_{\rm rot}$, and
$\beta$

In this section we use the SCIROCCO code to model the almost edge-on fast
rotator Regulus (using the best-fitting parameters given by Hadjara et al.,
2018, with $i=86.4^{\circ}$), in order to study the sensitivity of the
differential photocentre to $R_{\rm eq}$, $V_{\rm eq}$, $i$, $\rm PA_{\rm
rot}$, and especially to the gravitational darkening parameter $\beta$.
Indeed, an edge-on fast rotator allows a better study of the gravity darkening
parameterr on the photocentre displacements. In Fig. 12 we illustrate the
effect of our key parameters ($R_{\rm eq}$, $V_{\rm eq}$, $i$, $\rm PA_{\rm
rot}$, and $\beta$) on the photocentre displacement, for Regulus with a line
profile which varies along the co-latitude $\theta$ according to $\beta$-value
(as we did on Hadjara et al., 2018). An important and complementary section of
discussion (Sec. 5) allows a better comprehension of the results shown below.

Figure 12: Top-Left (Fig. 13a): Sensitivity of the photocentre displacement to
the inclination $i$ with an enhancement of the polar scale, for the parameters
of our best model ($R_{\rm eq}=4.16\,R_{\odot}$, $V_{\rm
eq}=350\,\mathrm{km.s}^{-1}$, $i=86.4^{\circ}$, and $PA_{\rm
rot}=251^{\circ}$; see Sec.5 of our previous paper Hadjara et al., 2018),
which are close of those used by Che et al. (2011), including $\beta\approx
0.16$. The error box with $30\,\mu\rm as$ edges represents the error of each
spectral measure of $\epsilon$. Top-right (Fig. 13b): Sensitivity of the
photocentre displacement to the equatorial velocity $V_{\rm eq}$. Same plots
as in Fig. 13a for $i=86.4^{\circ}$ and various values of $V_{\rm eq}$. The
error box with $2.5\,\mu\rm as$ represents the minimum error on the mean
amplitude of $\epsilon_{\rm eq}$ or $\epsilon_{\rm pol}$ estimated from all
channels in the line. Bottom-Left (Fig. 13c): Sensitivity of the photocentre
displacement to the gravitational darkening parameters $\beta$. Same plot as
in Fig. 13a with $i=86.4^{\circ}$, $V_{\rm eq}=350\,\mathrm{km.s}^{-1}$ and
$\beta$ ranging from 0.05 to 0.25. Bottom-Right (Fig. 13d): The equatorial and
polar photocentre displacements for different $V_{\rm eq}$ with the same
parameters than in Fig. 8b.

In the panels 12a, 12b and 12c we see the effect of the inclination, the
equatorial velocity and the gravitational darkening on a plot of the
photocentre location as a function of wavelength. The vertical axis is in the
polar direction and the horizontal one in the equatorial direction. It is
obvious that all these curves are symmetric with respect to the polar axis.
This confirms that the position angle of the rotation axis can be measured
independently from all other parameters, for example by a linear fit through
all photocentre points as we did in the previous paper (Hadjara et al., 2018).

Note that all photocentre displacement are proportional to the stellar angular
radius. However, panels 12a and 12b show that the total amplitude of
$\epsilon_{\rm eq}$ (the width of the curves) is independent from
$V_{\mathrm{eq}}\sin i$ within a large range of velocities and inclinations.
Panel 12c also shows that the sensitivity of this amplitude $A_{\epsilon_{\rm
eq}}$ to the gravitational darkening parameter $\beta$ is not negligible. We
should therefore have access to an accurate measure of $R_{\rm eq}$ from the
equivalent width of $\epsilon_{\rm eq}(\lambda)$. Here the total amplitude
$A_{\epsilon_{\rm eq}}\approx 400\,\mu as$ and the average width of the figure
12a can be roughly estimated to be $A_{\epsilon_{\rm eq}}/2\approx 400\mu as$.
If we combine all 141 measures in the line with independent errors of
$30\,\mu\rm as$ we get a global error on the half amplitude of $2.5\,\mu\rm
as$ and a relative error of 1.25% on the angular radius, with little influence
from the other parameters discussed here.

Panels 12b and 12a show that the velocity and inclination together, i.e.
$V_{\mathrm{eq}}\sin i$ mainly changes the polar amplitude $A_{\epsilon_{\rm
pol}}$, which increases with $|90^{\circ}-i|$ for $45^{\circ}$ as maximum in
panel 12a. Around $350\,\mathrm{km.s}^{-1}$, the variation $A_{\epsilon_{\rm
pol}}/V_{\mathrm{eq}}\sin i\approx 0.05\,\mu\rm as/\mathrm{km.s}^{-1}$ yields
a very large error on the radial velocity $\sigma_{V_{\mathrm{eq}}\sin
i}\approx 50\,\mathrm{km.s}^{-1}$ in panel 12b. The photocentre measurements
that we use here provide a poor constraint on the equatorial velocity.
However, for a fixed $V_{\mathrm{eq}}\sin i$ constrained by the line profile
broadening, $A_{\epsilon_{\rm pol}}$ varies substantially with the inclination
$i$. Panel 12a shows that for $80^{\circ}<i<90^{\circ}$, the average amplitude
$A_{\epsilon_{\rm pol}}$ changes by typically 1 $\mu\rm as/^{\circ}$ . This
leads to an accuracy $\sigma_{i}\approx 2.5^{\circ}$ in the inclination, if
all other parameters are well constrained.

##  7 Conclusions and discussion

We have established in this paper the general formalism of the photocentre
displacements and studied the impact of some relevant physical parameters for
rapid rotators, using the semi-analytical model (SCIROCCO).

We show that even with a poor $(u,v)$ coverage (e.g. in Hadjara et al., 2018,
and in Sec. 5), our method allows to obtain important information from
photocentre displacements $\epsilon_{\alpha,\delta}$. Indeed, two different
baselines are sufficient for the use of the vectorial photocentre displacement
method. On fully unresolved targets, $\phi_{\mathrm{diff}}$ reduces to the
measurement of the vectorial photocentre displacement $\vec{\epsilon}$, that
is accessible as soon as we have two different baselines, preferably nearly
orthogonal for SNR optimization.

We classify in three categories the stellar parameters impacting the
photocentres displacements (Fig. 17 and 18). The very sensitive parameters
($\Delta_{\epsilon}>25\%$) such the equatorial radius $R_{\rm eq}$, $\rm
PA_{\rm rot}$ and the line profile linked to the analytic/stellar atmosphere
modeling (Voigt versus Kurucz). The moderately sensitive ones
($10\%\leq\Delta_{\epsilon}\leq 25\%$) as the $V_{\mathrm{eq}}\sin i$, the
inclination $i$, and the line profile linked to the analytic/stellar
atmosphere modeling (Tlusty versus Kurucz, and the line profile linked to the
latitudinal versus the regular aspect). And finally the less sensitive
parameters ($\Delta_{\epsilon}<10\%$) which are the mean effective temperature
$\overline{T}_{\mathrm{eff}}$, $\beta$,the both darkening (limb and
gravitational). The spectrum, on the other hand, is more sensitive (with
$\Delta_{s}\geq 5\%$) on the line profile (concerning also the
analytic/stellar atmosphere modeling aspect; Voigt-Kurucz), which was expected
because the line profile approximation affects the spectrum. However, the
spectrum is less sensitive (with $\Delta_{s}<5\%$) to other kind of line
profiles, and to the darkenings, $R_{\rm eq}$, $V_{\mathrm{eq}}\sin i$ ,$i$,
$\rm PA_{\rm rot}$, $\overline{T}_{\mathrm{eff}}$ and $\beta$.

The photocentre displacement is a powerful tool for studying the poorly
resolved critical/fast and slow rotators. Indeed, in the case of critical/fast
rotators, the approach that we propose (for poorly/marginally‐resolved stars)
allows to simultaneously constrain the rotation‐axis position angle $\rm
PA_{\rm rot}$, estimate the range of the rotation‐axis inclination angle $i$,
and under certain conditions the gravity‐darkening coefficient $\beta$
(Hadjara et al., 2018). On the other hand, in the case of slow rotators
($V_{\mathrm{eq}}\sin i\sim 30~{}–~{}50\,\mathrm{km.s}^{-1}$), all fundamental
key parameters, except $i$ (i.e. $R_{\rm eq}$, $V_{\mathrm{eq}}\sin i$, and
$\rm PA_{\rm rot}$) could be determined with acceptable uncertainties
(1-2$\sigma$) by our method. Indeed, without a rapid rotation, there is no
strong gravity darkening (von Zeipel) effect, and we can not estimate
separately $V_{\rm eq}$ and $i$ from the $V_{\mathrm{eq}}\sin i$. Therefore,
the only solution to deduce the exact flatness shape from slow rotators is to
use a dense $(u,v)$ coverage with the differential visibility (which contains
angular size information, as done by Domiciano de Souza et al., 2003). This
method could be applied to AGNs too.

We have computed the limits of accuracy of the radius $R_{\rm eq}$,
inclination angle $i$ and rotation-axis position angle $\rm PA_{\rm rot}$
parameters are that we could achieve with our quality of data and this numbers
are summarized in Table 1 here above.

Table 1: Limiting and achieved accuracy for the parameter $\beta=0.165\pm
0.009$ of Regulus with data used in this study (Hadjara et al., 2018, more
details in Sec. 5). The “best possible accuracy” is given for the estimation
of a parameter from the $\vec{\epsilon}$ alone or with the spectrum
$s(\lambda)$, when we assume that all other parameters are known.

Parameter | Best possible | Accuracy from
---|---|---
| accuracy from | MCMC⋆ fit of
|  $\vec{\epsilon}(\lambda)$ only |  $\vec{\epsilon}(\lambda)$ & $s(\lambda)$
$R_{\rm eq}$ | 1.25% | 5.7%
$V_{\rm eq}$ | $50\,\mathrm{km.s}^{-1}$ | $22\,\mathrm{km.s}^{-1}$
$i$ | 2.5∘ | 6∘
$\rm PA_{\rm rot}$ | 1.4∘ | 1.8∘

  * $(\star)$ The accuracy of MCMC fit (for Markov Chain Monte Carlo method) is that we found from our previous paper of Hadjara et al. (2018),

We conclude that SNR is insufficient to give a significant direct constraint
on $\beta$ from a fit of our data. To hope to constrain $\beta$, we need at
least an SNR 10 to 15 times better.

Finally, we have defined a criterion $\Gamma_{\epsilon}$ which helps to
determine the application limits of the photocentre displacement method on
fast rotating stars with gravity darkening effect. We demonstrated that for
$\Gamma_{\epsilon}\leq 0.32$ (which corresponds to unresolved objects $V\geq
0.99$) this method is applicable, no matter the value of $\Delta\rm PA$ for
couple baselines combination/averaging, and in condition of perpendicular
baselines ($\Delta\rm PA\sim 90$) for single photocentre projections. Beyond,
i.e. $\Gamma_{\epsilon}>0.32$, only $\phi_{\mathrm{diff}}$ data are usable.
Also, we proved using observed spectro-interferometric data (under certain
conditions summarized in Sec. 5, and applied previously by us in Hadjara et
al., 2018), that the application field of this method can be extended to
marginally resolved stars, i.e. $V\sim 0.8$, which corresponds to
$\Gamma_{\epsilon}\sim 0.64$). No matter the value of $\Delta\rm PA$ for
couple baselines combination/averaging and when $60^{\circ}\leq\Delta\rm
PA\leq 90^{\circ}$ for single photocentre projections, especially for edge-on
stars. The only relevant incompatibility that we noticed, with respect the
literature results, corresponds to the inclination angle $i$ (south pole
apparent instead of the north one). Thus, to say that all key parameter
information are entirely contained in the vectorial photocentre displacement,
which can be measured as soon as there are two non-collinear bases is
completely justified, even with a relatively bad SNR. On the other hand, the
use differential phase is suitable only for $\Gamma_{\epsilon}\geq 0.64$ with
uncertainties, (when the object is resolved; i.e. $V\leq 0.99$).

Therefore, the estimate of $\rm PA_{\rm rot}$ is possible with both methods,
which opens a wide perspectives of studies, such as the determination of the
ecliptic plane of targets and thus properly characterize and predict the
evolution trajectory of the thousands of extra solar planets that we discover
from radial-velocimetry and occultation observations, as well as the dynamical
study of clusters evolution by comparing the $\rm PA_{\rm rot}$ of their star.

##  Appendix A modeling fast rotators with SCIROCCO

In contrast to the CHARRON code (Domiciano de Souza et al., 2012a), which uses
the pure Roche model to compute the shape of the rotating star, the SCIROCCO
code uses the simple Jacobi ellipsoidal model, as the HDUST code does
(Carciofi & Bjorkman, 2006), which makes the two last codes much lighter and
faster. Figure 13 shows the differences between the Roche and the ellipsoidal
models. Although replacing the Roche model with that of the Jacobi ellipsoid
does not seem to be theoretically the best approach for early-type stars,
which are all centrally condensed, the angular size results of the two
approaches remain the same in the limits of uncertainties (e.g. The results on
the same observed VLTI/AMBER data of Achernar using CHARRON by Domiciano de
Souza et al., 2012b and through SCIROCCO by Hadjara et al., 2014).

Figure 13: Drawing of both Roche (continuous line) and ellipsoidal models
(discontinuous line), for our reference Achernar-like model, with 4 different
values of equatorial rotation velocities, and their corresponding equatorial
radius $R_{\rm eq}$; polar radius $R_{\rm p}$ and equatorial-to-polar radii
$R_{\rm eq}/R_{\rm p}$. The closer $V_{\rm eq}$ to $V_{\rm
crit}=269\,\mathrm{km.s}^{-1}$ the more obvious the difference in shape
between the two models, while the radii $R_{\rm eq}$ and $R_{\rm p}$ remain
exactly the same.

SCIROCCO uses the pseudo-Voigt analytic function with 6 parameters (PVoigt6).
The mathematical function that best fits the absorption line profiles is given
by :

$\begin{array}[]{l}\rm
PVoigt_{6}(\lambda,\theta)=a_{6}(\theta)(L(\lambda,\theta)+(1-a_{6}(\theta))G(\lambda,\theta)),\end{array}$
(9)

where $\theta$ is the co-latitude, and $G$ and $L$ are the Gaussian and
Lorentzian functions respectively, given by :

$&G(\lambda,\theta)=1+a_{5}(\theta)-a_{1}(\theta)\exp\Big{(}-\pi\frac{a_{1}^{2}(\theta)}{a_{2}^{2}(\theta)}(\lambda-\lambda_{\rm
c})^{2}\Big{)},\\\
L(\lambda,\theta)=1+a_{5}(\theta)-a_{3}(\theta)\left[1+\pi\frac{a_{3}^{2}(\theta)}{a_{4}^{2}(\theta)}(\lambda-\lambda_{\rm
c})^{2}\right]^{-1}.$ (10)

The coefficients $a_{1}$ to $a_{5}$ are those of the Gaussian and the
Lorentzian functions respectively (absorption lines), $a_{6}$ is the
coefficient of the pseudo-Voigt function, and $\lambda_{\rm c}$ is the the
central wavelength of the spectral line.

Figure 14 shows the results of the numerical fit of the 6-parameter pseudo-
Voigt function on the SYNSPEC profile obtained with the KURUCZ photospheric
model for 2 line profiles (at the equator and at the poles). The great
advantage of this method is to get very good line profiles for any wavelength
with no need of interpolation, decreasing notably the computing time.

Figure 14: Top: PVoigt6 functions fitted on the KURUCZ/SYNSPEC model (black
line) for the equatorial line profile $[T_{\rm eff},log\
g]=[10500\,K,30\,cm.s^{-1}]$ (blue line) and the polar profile $[T_{\rm
eff},log\ g]=[19000\,K,35\,cm.s^{-1}]$ (red line). Middle: Difference between
the KURUCZ/SYNSPEC and the PVoigt6 models at the poles. Bottom: The same at
the equator.

Note that the photocentre displacement is roughly proportional to the line-to-
continuum amplitude ratio (Chelli & Petrov, 1995). An error of $1\%$ in the
line profile corresponds to an error of $1\%$ in the photocenter displacement,
which is much lower than the measurement uncertainty.

##  Appendix B Limits of photocentre displacements

We gather here all additional figures discussed in Sec. 3.

Figure 15: Vectorial photocentre displacement, for a star similar to Achernar
(with $\rm PA_{\rm rot}=0^{\circ}$ and $\beta=0.25$), around Br$\gamma$
(K-band), for various inclination angles $i$ from $30^{\circ}$ to
$90^{\circ}$, and several baseline configurations $B$ from $15$ to $90\,\rm
m$, and different values of projection angle differences ($\Delta\rm PA$ from
$5^{\circ}$ to $90^{\circ}$ in different colours). The absolute vectorial
photocentre $E_{\rm eq,pol}$ is represented in black line. In each panel, we
indicate the value of our criterion $\Gamma_{\epsilon}=\frac{\Theta
B}{\lambda}$. The equality $E_{\rm eq,pol}\approx\epsilon_{\rm eq,pol}$ is
clearly satisfied for $\Gamma_{\epsilon}\leq 0.32$, independently from
$\Delta\rm PA$-value. The grey error bars corresponds to the measurement error
of the photocentre displacements $\sim 30\,\mu\rm as$, as deduced before in a
previous work (Hadjara et al., 2018). V-value on each panel represents the
average visibility over all PAs.

Figure 16: Same as Fig. 15, but for a fixed baseline length ($B=150\,\rm m$)
and three different inclination angle values, namely: $i=30^{\circ}$,
$60^{\circ}$, and $90^{\circ}$.

##  Appendix C Dependence on physical parameters

We gather here all figures discussed in Sec. 4. We observe that the
interferometric photocentre displacements (in coulour lines) fit the
astrometric ones (in black line) within the average error bars of
$\sigma_{\epsilon}=\pm 30\,\mu as$, only for $\Gamma_{\epsilon}\leq 0.32$.

Fig. 17 (top) shows the dependence of simulated spectrum and perpendicular
photocentres displacements ($\epsilon_{\alpha}$ and $\epsilon_{\delta}$) on
$R_{\rm eq}$ and $V_{\mathrm{eq}}\sin i$ (solid line: tested model; dashed
line: reference model as described above). The parameters of the tested model
are identical to those of the reference model, except that $R_{\rm
eq}=9\,\mathrm{R}_{\sun}$. The values of the tested and reference models
differ by $(\Delta_{\epsilon}=26\%,\Delta_{s}=0\%)$ for $R_{\rm eq}$. All the
models were calculated across the $Br_{\gamma}$ line with a spectral
resolution of $12000$ at four projected baselines ($B_{\rm proj}=75\,m$ and
$150\,m$, and $\rm PA=45^{\circ}$ and $90^{\circ}$), which are typical values
attained with VLTI/AMBER. These wavelengths and baselines result in visibility
amplitudes between $0.6$ and $0.9$ for the studied stellar models,
corresponding to a partially resolved star. While $V_{\mathrm{eq}}\sin i$
changed from $250\,\mathrm{km.s}^{-1}$ to $200\,\mathrm{km.s}^{-1}$. The
values of the tested and reference models differ by
$(\Delta_{\epsilon}=23\%,\Delta_{s}=1.3\%)$ for $V_{\mathrm{eq}}\sin i$. Fig.
17 (middle) is similar to the upper figures but for the dependence of spectrum
and perpendicular photocentres displacements ($\epsilon_{\alpha}$ and
$\epsilon_{\delta}$) on $i$ and $\overline{T}_{\mathrm{eff}}$. $i$ changed
from $60^{\circ}$ to $90^{\circ}$ and $\rm PA_{\rm rot}$ from $0^{\circ}$ to
$45^{\circ}$. The values of the tested and reference models differ by
$(\Delta_{\epsilon}=9\%,\Delta_{s}=0.5\%)$ for $\i$ and by
$(\Delta_{\epsilon}=95\%,\Delta_{s}=0\%)$ for $\rm PA_{\rm rot}$. Fig. 17
(bottom) is similar to the upper figures but for the dependence of spectrum
and perpendicular photocentres displacements ($\epsilon_{\alpha}$ and
$\epsilon_{\delta}$) on $\overline{T}_{\mathrm{eff}}$ and $\beta$.
$\overline{T}_{\mathrm{eff}}$ changed from $15000\,\rm K$ to $20000\,\rm K$
and $\beta$ from $0$ to $0.25$. The values of the tested and reference models
differ by $(\Delta_{\epsilon}=9\%,\Delta_{s}=0.5\%)$ for both
$\overline{T}_{\mathrm{eff}}$ and $\beta$ .

Figure 17: Top: Dependence of simulated spectrum and perpendicular
photocentres displacements ($\epsilon_{\alpha}$ and $\epsilon_{\delta}$) on
$R_{\rm eq}$, $V_{\mathrm{eq}}\sin i$, $i$, $PA_{\rm rot}$,
$\overline{T}_{\mathrm{eff}}$ and $\beta$ (solid line: tested model; dashed
line: reference model as described above).

Fig. 18 (top) is similar to Fig. 17 but for the dependence of simulated
spectrum and perpendicular photocentres displacements ($\epsilon_{\alpha}$ and
$\epsilon_{\delta}$) with different line profile sets. On the left panel it
shown the dependence on“the line profile type”, where the solid line is
related to line profile obtained with Voigt line profile; and the dashed line
to the line profile obtained with Kurucz/Synspec. On right panel the solid
line represents the line profile obtained with Tlusty/Synspec, and the dashed
line represents the line profile obtained with Kurucz/Synspec. The values of
the tested and reference models differ by
$(\Delta_{\epsilon}=36\%,\Delta_{s}=5\%)$ for first and by
$(\Delta_{\epsilon}=15\%,\Delta_{s}=2\%)$ for the second one. Fig. 18 (bottom-
left) is similar to the upper figures but for the dependence on“the line
profile type” in function of the co-latitude. The solid line represents the
line profile obtained with Krucz/Synspec line profile varying according to the
co-latitude, and the dashed line the line profile obtained with fixed
Kurucz/Synspec line profile (considering the average $[T_{\rm eff},\log g]$ of
the star). While Fig. 18 (bottom-right) depicts the dependence on “darkening
type effect” (solid line: just gravity darkening effect -no limb darkening LD
effect-; dashed line: reference model as described above -gravity darkening
and limb darkening GD+LD effects-). The values of the tested and reference
models differ by $(\Delta_{\epsilon}=23\%,\Delta_{s}=0.5\%)$ for first and by
$(\Delta_{\epsilon}=8\%,\Delta_{s}=0.15\%)$ for the second one. Spectrum,
$\epsilon_{\alpha}$ & $\epsilon_{\delta}$ in $Br_{\gamma}$ are not strongly
sensitive to the darkening effect but are more sensitive to the analytic line
profile used.

Figure 18: Top: Similar to Fig. 17 but for the dependence of simulated
spectrum and perpendicular photocentres displacements ($\epsilon_{\alpha}$ and
$\epsilon_{\delta}$) with different line profile sets and on the “darkening
type effect”.

## ACKNOWLEDGEMENTS

The first author acknowledges the support from the scientific french
association PSTJ 111http://www.pstj.fr/ for its official host agreement, the
Lagrange and OCA for compute servers support. A special thanks to the
project’s grant ALMA-CONICYT N∘ 31150002 and the PI Keiichi Ohnaka who
supported this work. Special thanks go to the project’s grant ESO-MIXTO 2019,
as well as the grants from the Fizeau European interferometry initiative
(I2E). This work is also sponsored by the Chinese Academy of Sciences (CAS),
through a grant to the CAS South America Center for Astronomy (CASSACA) in
Santiago, Chile. WW is supported by the National Key RD Program of China No.
2019YFA0405102, the National Natural Science Foundation of China (NSFC) grant
No. 42075123.

## Data Availability Statements

All observed and reduced data used in the current paper are available in
electronic form at the CDS via https://cdsarc.unistra.fr/ftp/J/MNRAS/511/4724/
or through VizieR via http://vizier.u-strasbg.fr/viz-
bin/VizieR?-source=J/MNRAS/511/4724.

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