# Intensity Interferometry observations of the H$\alpha$ envelope of $\gamma$
Cas with MéO and a portable telescope

Nolan Matthews  Université Côte d’Azur, CNRS, Institut de Physique de France,
France  Jean-Pierre Rivet  Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Laboratoire Lagrange, France  David Vernet  Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, UMS Galilée, France  Mathilde
Hugbart  Université Côte d’Azur, CNRS, Institut de Physique de France, France
Guillaume Labeyrie  Université Côte d’Azur, CNRS, Institut de Physique de
France, France  Robin Kaiser  Université Côte d’Azur, CNRS, Institut de
Physique de France, France  Julien Chabé  Université Côte d’Azur, Observatoire
de la Côte d’Azur, CNRS, Laboratoire Géoazur, France  Clément Courde
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Géoazur, France  Olivier Lai  Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Laboratoire Lagrange, France  Farrokh Vakili  Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
Olivier Garde  2SPOT (Southern Spectrocopic Project Observatory Team)
Observatoire de la Tourbière - 38690 Châbons - France  William Guerin
Université Côte d’Azur, CNRS, Institut de Physique de France, France

###### Abstract

We report on observations of the extended environment of the bright Be star
$\gamma$-Cas performed using intensity interferometry measurements within its
H$\alpha$ emission line. These observations were performed using a modified
version of the I2C intensity interferometry instrument installed onto the 1.54
meter MéO optical metrology telescope and a portable 1-meter telescope (T1M).
In order to better constrain the extent of the H$\,\alpha$ envelope,
observations were performed for two different positions of the T1M telescope,
corresponding to an intermediate and long baselines in which the extended
region was partially and fully resolved. We find that the observed data are
consistent with past interferometric observations of $\gamma$-Cas. These
observations demonstrate the capability to equip optical telescopes of
different optical designs with intensity interferometry capabilities and
illustrate the potential to scale a similar system onto many additional
telescopes.

stars: emission-line, Be — instrumentation: high angular resolution —
techniques: interferometric

##  1 Introduction

Recognized as the first stellar object displaying emission line spectra
(Secchi A., 1867), $\gamma$-Cas is the prototype of the Be stellar class. The
emission line features originate from radiative processes with up to X-ray
energies (Smith et al., 2012) occurring in an extended disc surrounding the
star. The disc formation is primarily attributed to mass ejection from the
central star enabled from a combination of strong radiative pressure, and low
effective surface gravity near the equatorial latitudes. The latter is a
consequence of the extremely high rotation rate that is nearly critical, in
which the outward centrifugal force is equal to the inward gravitational
force.

Due to its bright stellar magnitude and characteristic stellar size, optical
interferometry has been extensively used to study the disk emission of
$\gamma$-Cas. The extended atmosphere of $\gamma$-Cas was first resolved with
the I2T interferometer (Thom et al., 1986) and subsequently by the GI2T
interferometer showing that the $H\alpha$ region could be fit by a disk model
and was in Keplerian motion (Mourard et al., 1989). Observations by
Quirrenbach A. et al. (1997) with the MkIII interferometer demonstrated that
the emission-line region were not compatible with circularly symmetric models
and required the assumption of an elongated profile. Density and velocity
relationships in the equatorial plane were constrained and accounted for by a
radiative wind driven model in Stee et al. (1995). Subsequently, spectro-
interferometric measurements of the envelope size were performed across both
the H$\alpha$ and H$\beta$ lines, as well as in the near-by continuum emission
(Stee et al., 1998) leading to a measurement of the disk mass and opening
angle (Stee et al., 2003). In addition, the Navy Precision Optical
Interferometer (NPOI) was used to characterize the disc geometry and further
confirmed the oblateness of the disc (Tycner et al., 2006). The CHARA
interferometric array measured the disk extent in the K’ photometric band for
the first time, found to be slightly smaller than previous observations in
H$\alpha$ (Gies. D. R. et al., 2007). Finally, high sensitivity spectro-
interferometric measurements with CHARA were performed in the near-infrared,
as well across the H$\alpha$ line and near-by continuum suggesting a larger
disk size than prior measurements and linking the origin of X-ray emission to
a compact binary companion due to the absence of one-armed spiral structures
or secondary star (Stee et al., 2012).

In this work we present the first known intensity interferometry (II)
measurements of the extended atmosphere of $\gamma$-Cas using a modified
version of our intensity interferometry instrument (I2C) installed onto the
1.54-meter telescope of the MéO laser ranging facility and a mobile 1-meter
telescope (hereafter T1M), both located on the Calern Plateau site of the
Observatoire de la Côte d’Azur. While the I2C instrument shares similarities
with past II observations using the telescopes of the Centre Pédagogique
Planète Univers (C2PU) (Guerin et al., 2017, 2018; Rivet et al., 2020; de
Almeida et al., 2022), there were several modifications required to outfit
these telescopes with II capabilities, and to be compatible with each other in
an interferometric mode. The experimental setup is thus described in Section 2
with additional details also presented in Matthews et al. (2022). The
observations and results are shown in Section 3 with an analysis presented in
Section 4. Finally, we discuss the results and present an outlook for future
intensity interferometry measurements in Section 5.

##  2 Experimental Setup

###  2.1 Principles of Intensity Interferometry

An intensity interferometer correlates the intensity fluctuations of starlight
between separated telescopes in order to measure the squared visibility. For
two telescopes with a projected baseline $r$ between them, the second order
coherence function is

$g^{(2)}(r,\tau)=\frac{\langle I_{1}(t)I_{2}(r,t+\tau)\rangle}{\langle
I_{1}\rangle\langle I_{2}\rangle}$ (1)

where $I_{1}$ and $I_{2}$ are the intensities recorded at each of the two
telescopes, $\tau$ is the relative time-lag between the signals, and the
brackets indicate an average over time $t$. The Siegert relation (Siegert,
1943; Ferreira et al., 2020) relates the second-order coherence function
$g^{(2)}$ to the first order coherence function $g^{(1)}$ by

$g^{(2)}(r,\tau)=1+|g^{(1)}(r,\tau)|^{2}$ (2)

where the first-order coherence function can be separated into spatial and
temporal components

$g^{(1)}(r,\tau)=V(r)g^{(1)}(\tau)$ (3)

where $V(r)$ is the interferometric visibility of the source, given by the
Fourier transform of the source sky brightness distribution. For an unresolved
point-like source $V(r)=1$, and the resulting second order coherence function
will depend only on the temporal component $g^{(1)}(\tau)$ given by the
Fourier transform of the measured light spectral density (Wiener, 1930;
Khintchine, 1934). For linearly polarized thermal light at zero optical path
delay $g^{(1)}(\tau=0)=1$ where for time-lags much greater than the coherence
or correlation time the first order coherence function should be equal to zero
such that there is a “bunching peak” centered about zero optical path delay
with an effective temporal width given by the coherence time. The peak
amplitude at zero time-lag of $g^{(2)}$ thus measures the squared visibility
at some projected baseline assuming that the instrumental resolving time is
shorter than the light coherence time. The coherence time can be defined by
the integral of the squared first-order coherence function (Mandel, L., &
Wolf, E., 1995),

$T_{c}=\int|g^{(1)}(\tau)|^{2}d\tau=\int|s(\nu)|^{2}d\nu,$ (4)

which is equal to the integral of the squared normalized spectral density
$s(\nu)$ by Parseval’s theorem. For visible light with a bandpass of
$\Delta\lambda\sim 1\,$nm the corresponding coherence time is of order 1 ps,
much shorter than what can be achieved with conventional detectors. In this
case, a measurement averages over many coherence times and reduces the value
of the $g^{(2)}$ peak amplitude at $\tau=0$ by a factor of $\sim T_{c}/T_{d}$
where $T_{d}$ is the effective time-resolution of the detector. The amplitude
of the $g^{(2)}$ peak therefore measures this loss of contrast times the
squared visibility. The squared visibility can be extracted by dividing the
value of $g^{(2)}(r)-1$ peak amplitude measured between telescopes to the
$g^{(2)}(r=0)-1$ peak amplitude measured at zero-baseline under the assumption
that the profile of the $g^{(2)}(\tau)$ peak is constant. In practice, we
measure the ratio of the area of the $g^{(2)}(r)-1$ peak to the area of the
$g^{(2)}(r=0)-1$ peak for the squared visibility,

$|V(r)|^{2}=\frac{\int\left(g^{(2)}(r,\tau)-1\right)d\tau}{\int\left(g^{(2)}(r=0,\tau)-1\right)d\tau}.$
(5)

The denominator, or equivalently the area of the $g^{(2)}$ peak at zero
baseline, corresponds to the coherence time that can be calculated from the
measured spectrum as given by Equation 4. The equivalence between the
coherence time from intensity interferometry and spectral measurements assumes
that the spectral resolution is narrower than any spectral lines within the
instrumental bandpass. Since intensity interferometry measurements probe the
intrinsic spectrum it is a useful method for characterizing narrow spectral
lines present in, for example, studies of light scattering off of atomic
clouds (Dussaux et al., 2016), with potential applications in astrophysics
(Tan & Kurtsiefer, 2017).

###  2.2 Telescopes

The II observations presented in this paper were performed by outfitting two
telescopes located on the Calern Plateau site of the Observatoire de la Côte
d’Azur with individual coupling assemblies (CAs). A CA is mounted near the
focus of each telescope, both shown in Figure 1. The first facility was the
1.54 m diameter MéO (Métrologie Optique) telescope primarily used for
satellite laser ranging (Bertrand, B. et al., 2021), lunar ranging
measurements (Bourgoin, A. et al., 2021) and low Earth orbit satellite laser
communication (Giggenbach, D. et al., 2022). The optical design is based upon
a Ritchey-Chrétien configuration on an altitude-azimuth mount with a primary
focal length of 31.0 m giving an approximate focal ratio of f/20.1. In typical
operation the light is brought to a Coudé focus, but for II observations the
light is redirected to the CA along the Nasmyth arm using a removable 45
degree mirror. In addition, a f=150 mm lens is inserted before the CA in order
to decrease the effective focal length.

The second facility is the portable 1 m diameter T1M, a Newtonian telescope on
a Dobson-type fully motorized azimuthal mount. The portability of the
telescope enables configurable baselines to expand the accessible coverage of
the uv-plane where the telescope can be disassembled and moved in just a few
hours. The telescope has a primary focal length of 3 m, and a Barlow lens is
included in order to expand the effective length at the input of the CA.

Figure 1: Photographs of the coupling assemblies mounted on the Nasmyth arm of
the MéO telescope (left) and at the Newtonian focus of the T1M portable
telescope (right).

While both telescopes are azimuthal, there will be a relative field rotation
due to the Newtonian versus Nasmyth optical designs. However, each CA utilizes
polarizing filters that must be aligned with respect to one another, although
not with respect to the target as astrophysical polarization effects are not
investigated. To compensate, the MéO telescope CA is mounted into a rotation
stage that orients the CA such that the polarization axes are aligned. The
stage is actively controlled throughout the observation where the amount of
rotation is determined from the target sky position.

The relative position of both telescopes must be known with a precision less
than a few centimeters for optical path delay corrections. For MéO, the
position was previously determined to millimeter accuracy in terrestrial
coordinates due to its use in geodetic surveys. To determine an absolute
position of the mobile T1M telescope, geodetic markers were installed by the
National Institute of Geographic and Forest Information at the ground level
for several positions and their positions were measured from differential GPS
methods. The T1M was installed above these markers and the offset between the
marker and the T1M reference point was estimated. The estimated cumulative
error on the reference position is $\pm$1.5 cm in all directions.

Figure 2: Schematic of the experimental setup. The light collected by both
telescopes are brought to individualized coupling assemblies (CA), shown in
detail in the right inset. A tip-tilt corrects the beam with respect to
transverse displacements. The converging beam is collimated using a diverging
lens ($L_{1}$). A dichroic ($D$) reflects short wavelengths to a guiding
camera (GC) used in a closed loop with the tip-tilt. The transmitted light
passes through a narrowband filter ($NF$) centered on H$\alpha$. A polarizing
beam splitter (PBS) separates the light into orthogonal polarizations where
each polarization is injected into a graded-index multimode fiber (GRIN-MMF).
A linear polarizer (P) is included on the reflected arm to improve
polarization purity. Not shown is the rotation stage used for the MéO
telescope, and additional focal reducers/extenders, which are described in the
text. The light for each polarization mode for each telescope is split by a
50/50 fiber beamsplitter, and passed to single photon resolving detectors. The
photon arrival times are recorded by the TDC that also produces intensity
correlations.

###  2.3 Instrumental Setup

The primary function of the CAs (shown in Figure 2) is to perform
spectral/polarization filtering, and fiber injection. The current version
includes an automated tip-tilt device that provides stable fiber injection
over several hours without manual intervention (Matthews et al., 2022). The
detector signal output is fed to a time-to-digital converter (TDC) that
measures photon arrival times, and produces the correlation between all
relevant pairs of detectors. Across both telescopes, there are 4 independent
measurements of the zero baseline correlation $g^{(2)}(r=0,\tau)$ enabled by
using fiber splitters in each polarization mode. These allow normalization of
the spatial correlations across telescopes to measure visibilities. Spatial
intensity correlations are obtained by calculating the correlation across
telescopes for all pairs of detectors in the same polarization mode
corresponding to a total of 8 measurements of $g^{(2)}(r,\tau)$. During one
night, a time delay monitoring system was used on the electronic cables
connecting the detectors from the MéO telescope to the TDC. The drift
throughout the whole night was significantly less than the characteristic
jitter of the detectors of $\sim 500$ ps.

##  3 Observations

The observations of $\gamma$-Cas were performed between the nights of January
17th, 2022 to January 21st, 2022 in the short baseline configuration, and then
from January 24, 2022 to January 27th, 2022 in the longer baseline
configuration. At the beginning of each night images were recorded in both
telescopes of the visual binary system $\gamma$-Ari. The measured position
angle of the binary for both telescopes, and thus of the polarization axes,
were found to be always within 5 degrees, corresponding to a loss of
visibility of less than 1%.

###  3.1 Temporal Intensity Correlations

Figure 3: The left plot shows the measured zero-baseline correlation in the
blue points with a Gaussian fit shown by the dashed black line. The right
image displays the measured spectrum of $\gamma$-Cas, along with the
theoretical (black) and measured (blue) spectral transmission of the filter.

The coherence time obtained from zero baseline intensity correlations were
compared to expected values from the spectral throughput. These temporal
intensity correlation functions were computed for each polarization state and
for each telescope by summing all individually acquired correlations acquired
over the entire observation. Each of the resulting correlation functions were
then shifted by the instrumental delay and then co-added together. The
resulting correlation function is displayed on the left in Figure 3. The peak
is fit by a Gaussian with free parameters for the amplitude and width. The
coherence time, given by the integral of the peak, is extracted via the fit
values. Before fitting, there is a choice of the time-lag range to fit over
and the number of time-lags to bin. Here, we fit and show the data over a
range of $\pm$ 20 ns binned into 50 ps bins. Under these parameters, we find
an amplitude of (1.43$\pm$0.05)$\times 10^{-3}$ and a full-width at half-
maximum of 885$\pm$40 ps corresponding to a measured coherence time of
$1.35\pm 0.05\,$ps. Systematics of the fitting process were studied by fitting
the data varying the fit range from $\pm$10 ns to $\pm 40$ ns and additionally
the binning size from 10 ps to 80 ps. Within these parameters we find a
maximum difference of 0.015 ps in the extracted coherence time, notably less
than the measurement error, In the previously quoted coherence time, the
correlations from different polarization states and telescopes were co-added
and subsequently fit. This procedure requires that shape of the bunching peak
in each correlation, given by the temporal response of the detectors, are
similar. To test systematics, each correlation function for both polarization
states and telescopes were fit by a Gaussian to extract the coherence time.
Each individual fit was within 1$\,\sigma$ of the quoted coherence time, and
furthermore the weighted mean of fits (1.35$\pm$0.05 ps) is in perfect
agreement with a single fit of co-added correlations indicating that within
our measurement precision there are no significant systematics that preclude
us from combining individual zero-baseline correlations from separate detector
pairs.

The coherence time measured from the zero-baseline correlations can then be
compared to expectation from the recorded spectrum. The spectral transmission
of the H$\alpha$ filter was measured in the laboratory with a high resolution
spectrograph (Matthews et al., 2022). Spectra of $\gamma$-Cas with resolution
R=25000 were recorded contemporaneously with our observations using a
Whoppshel echelle spectrograph provided through collaboration with the
2SPOT111www.2spot.org association of amateur astronomers. This is especially
important as the width of the temporally variable emission line is narrower
than the filter bandpass thus affecting the coherence time. The right side of
Figure 3 shows the filter transmission, and the emission line spectra. Through
Equation 4 we extract the expected coherence time to be 1.41 ps which is 1.2
$\sigma$ larger than the value from zero-baseline correlations when taking
only the measurement uncertainty and thus in fair agreement. In contrast, the
coherence time that would be expected for a flat stellar spectrum would be
1.16 ps. This is considerably less than the measured value by 3.8$\,\sigma$
illustrating the importance of including the emission line profile in
calculations of the coherence time. The general agreement of the coherence
time measured between intensity interferometry and spectral measurements
indicate that there are no systematic effects arising from the presence of
unidentified narrow spectral lines due to a lack of spectral resolution.

###  3.2 Spatial Intensity Correlations

The spatial intensity correlations correspond to the correlations between all
detector pairs on separate telescopes that are observing in the same
polarization mode corresponding to a total of 8 cross-correlations across
telescopes. All computed correlations are shifted in time by instrumental and
geometrical delays, and then co-added together. The full data set corresponds
to a wide projected baseline and position angle range and so the data was
divided into smaller baseline ranges. For each sub-division, we compute the
averaged correlation function and then fit a Gaussian function to the
resulting bunching peak. The measured areas of the $g^{(2)}$ peak for each
subdivision are presented in Table 1. Squared visibilities are extracted by
computing the ratio of the integral of the Gaussian peak of the cross-
correlation to the computed value from the measured spectrum.

Table 1: Spatial intensity correlation results.  Baseline Range | Peak Area
---|---
m | ps
0.0 | 1.35$\pm$0.05
3.8 $<$ r $\leq$ 13.0 | 1.40$\pm$0.17
13.0 $<$ r $\leq$ 18.0 | 0.83$\pm$0.14
18.0 $<$ r $\leq$ 21.3 | 0.31$\pm$0.08
32.0 $<$ r $\leq$ 37.8 | 0.07$\pm$0.04

##  4 Analysis of Results

The reduced II data resulted in 4 measurements of the squared visibility, each
averaged over a range of baselines required to significantly resolve a
bunching peak. In turn, the limited sampling does not allow any reasonable
independent visibility modeling. Nevertheless, it is interesting to compare
the measured values to past results. Past interferometric observations
generally characterize the angular brightness distribution of $\gamma$-Cas
with a parameterized geometrical model. A common assumption is a two-component
system consisting of a photosphere and disk, with some flux ratio between
them. The photosphere is typically approximated as a uniform disk. This is an
oversimplified model of Be stars as it does not take into strong temperature
gradients and equatorial flattening from the near critical rotation (Domiciano
de Souza et al., 2002). However, to resolve these effects requires an angular
resolution at the characteristic diameter of the photosphere ($\sim$ 0.5 mas
for $\gamma$-Cas (Stee et al., 1998)), whereas the effective resolution for
our observations (1.22$\lambda/D$) at the largest baseline is $\sim 8.3\,$mas.
Furthermore, these observations were conducted within the H$\alpha$ line in
which the disk emission is much stronger, such that photospheric contributions
are significantly minimized. For the disk emission, several geometric models
were tested, including elongated uniform disks, Gaussian disks, and uniform
rings. Tycner et al. (2006); Stee et al. (2012) showed that a Gaussian disk
profile best described the extended emission relative to the other
assumptions.

Table 2: Reported Gaussian disk fit values in prior $\gamma$-Cas observations. $\theta_{GD}$ is the full-width at half-maximum, $\phi$ is the position angle, and $r$ is the axial ratio.  Observatory | Ref. | $\theta_{GD}$ | $\phi$ | r
---|---|---|---|---
|  | (mas) | (∘) |
MkIII | Quirrenbach A. et al. (1997) | 3.47$\pm$0.02 | 19$\pm$2 | 0.70$\pm$0.02
NPOI | Tycner et al. (2006) | 3.59$\pm$0.04 | 31.2$\pm$1.2 | 0.58$\pm$0.03
CHARA | Stee et al. (2012) | 4.4$\pm$0.4 | 19$\pm$5 | 0.74

Figure 4: The left image shows the uv-plane coverage, over-plotted on expected
squared visibilities formed from a Gaussian disk model of $\gamma$-Cas using
reported parameters from Stee et al. (2012). Each of the colors represent the
range of sampled points averaged together in order to measure squared
visibilities, as correspondingly plotted on the right. Additionally, we plot
the expected squared visibilities for each of the models in Table 2 at our
sampled uv-plane points.

This two component model of a uniform disk + elongated Gaussian disk was
applied to $\gamma$-Cas data in three prior reported observations using the
MkIII interferometer (Quirrenbach A. et al., 1997), NPOI (Tycner et al.,
2006), and CHARA (Stee et al., 2012). The reported parameters for these
observations are summarized and presented in Table 2. Figure 4 shows our
squared visibilities, along with expected values obtained from a Gaussian disk
model from the previous observations. The comparison of our results with the
models produced using reported values from literature tends to align with the
values given by Tycner et al. (2006) and Quirrenbach A. et al. (1997) over
Stee et al. (2012) who suggest a smaller extent of the H$\alpha$ region.
Within our measurement precision this is not strongly conclusive and can also
be a result of instrumental differences. Already, Stee et al. (2012) noted the
larger angular extent could be explained in that they used high resolution
spectro-interferometry, in contrast to the other observations including our
own, that utilize narrowband filters. The reasoning is that the filters detect
more of the less resolved continuum resulting in an effectively smaller
angular extent.

##  5 Discussion and Outlook

We reported here on II measurements of the extended H$\alpha$ emitting region
of $\gamma$-Cas. The observed angular extent of the emission was found to be
consistent with past direct interferometry measurements. Following our
previous observations of Rivet et al. (2020) and de Almeida et al. (2022) this
extends the work of II measurements in emission lines to another system and
complements recent on-sky results of other intensity interferometry facilities
(Acciari et al., 2020; Abeysekara et al., 2020; Horch et al., 2022). Future
improvements to the system will aim to improve the sensitivity. The most
significant gain comes from simultaneously performing II correlations in many
spectral channels that can be co-added to improve the signal to noise ratio by
a factor of the square root of the number of channels (Trippe et al., 2014).
One could also imagine recording many independent spectral channels across the
H$\alpha$ line in order to perform intensity spectro-interferometry to test
the discrepancy seen in past observations between those using filtered
bandpasses, and those in dispersed light. Furthermore, higher sensitivity
observations, paired with polarimetric capabilities would allow for better
constraints, if not a direct measure of, radiative processes displaying
polarized emission in the disk, as was attempted by Rousselet-Perraut et al.
(1997).

These observations were performed using two facilities: the T1M and MéO that
had not been used for interferometric observations prior to this report. The
portability of the T1M provides the capability to optimize the baseline
configuration for a given target, or similarly perform multiple configurations
for a given target as done here. This technical accomplishment also
illustrates the potential for performing up to 4-telescope II measurements on
the Calern Plateau enabling 6 simultaneous baselines by including the 2
additional C2PU telescopes.

We acknowledge the financial support of the Région PACA (project I2C), the
French National Research Agency (ANR, project I2C, ANR-20-CE31-0003), OCA,
Doeblin federation, UCA science councils grants, and the LABEX Cluster of
Excellence FIRST-TF (ANR-10-LABX-48-01), within the program Investissements
d’Avenir operated by the ANR. The authors would like to thank Jacques Belin
and Damien Pesce for their installment of reference markers, and to the
members of the MèO team including Hervey Mariey, Mourad Aimar, Hervé Viot,
Grégoire Martinot-Lagarde, Julien Scariot, Nicolas Maurice, Duy-Hà Phung, and
Nils Raymond for their assistance during the observations.

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