# 4U 1210$-$64: a new member of the rare intermediate-mass X-ray binary
subclass

I.M. Monageng1,2, V.A. McBride1,2,3, J. Alfonso-Garzon4, L. J. Townsend1,5,
J.B. Coley6,7, B. Montesinos4, R.H.D. Corbet8,9, K. Pottschmidt7,9,
1South African Astronomical Observatory, P.O Box 9, Observatory, 7935, Cape
Town, South Africa
2Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa
3International Astronomical Union’s Office of Astronomy for Development, c/o
SAAO
4Centro de Astrobiologia (CAB), INTA-CSIC, Camino Bajo del Castillo s/n,
E-28692 Villanueva de la Cañada, Madrid, Spain
5Southern African Large Telescope, P.O Box 9, Observatory, 7935, Cape Town,
South Africa
6Department of Physics and Astronomy, Howard University, Washington, DC 20059,
USA
7CRESST/Mail Code 661, Astroparticle Physics Laboratory, NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA
8CRESST/Mail Code 662, X-ray Astrophysics Laboratory, NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA
9Center for Space Sciences and Technology, University of Maryland, Baltimore
County, Baltimore, MD 21250, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

4U 1210$-$64 is a peculiar X-ray binary that was first discovered in 1978 by
the Uhuru satellite. The analysis of the X-ray data revealed a 6.7-day orbital
period and an additional long-term modulation that is manifested as low and
high flux states. Based on the previous classification of the donor star from
the analysis of its optical spectra, the system has been suggested to be a
high-mass X-ray binary. We re-visit the optical classification where, based on
the spectra from the Southern African Large Telescope (SALT), we conclude that
the donor star is of spectral class A8 III-IV, making it a member of the rare
intermediate-mass X-ray binaries. We perform radial velocity analysis using
the SALT spectra where we consider circular and eccentric orbits. From the
mass function derived and the mass constraints of the donor star, we
demonstrate that a neutron star is favoured as the compact object in the
binary system. We show, for the first time, the folded optical lightcurves,
whose shape is interpreted to be due to a combination of ellipsoidal
variations, irradiation of the donor star, and mutual eclipses of the star and
accretion disk.

###### keywords:

keyword1 – keyword2 – keyword3

††pubyear: 2015††pagerange: 4U 1210$-$64: a new member of the rare
intermediate-mass X-ray binary subclass–A

##  1 Introduction

X-ray binary stars are interacting stellar systems comprising a donor star and
a compact object (either a neutron star or a black hole) that is accreting
matter from its companion. Depending on the nature of the donor star, X-ray
binaries are generally classified into low-mass X-ray binaries (for systems
that have a late-type donor, with mass $\lesssim 1$ M⊙) and high-mass X-ray
binaries (for cases where the donor is an early-type with mass $\gtrsim 10$
M⊙). Companions with masses in between these extremes, the intermediate-mass
X-ray binaries, are rare (Tauris & van den Heuvel, 2006). It has been
suggested that the intermediate-mass X-ray binary phase represents a stage
just before the low-mass X-ray binary phase, where most of the mass from the
donor star has not yet been lost/transferred (van den Heuvel, 1995;
Podsiadlowski et al., 2002; Pfahl et al., 2003). Two of the most well-known
intermediate-mass X-ray binary systems are Cyg X-2 (neutron star and a
companion of spectral-type A9; Casares et al. 1998) and Her X-1 (neutron star
with an A7 spectral-type companion; Leahy & Abdallah 2014).

The interaction between the donor star and the compact object can result in an
increase in X-ray emission due to the accretion of matter from the donor onto
the compact object. Depending on the nature of the two objects and their
orbits, the process of accretion can take place through different mechanisms.
Wind and/or circumstellar disc accretion is the most dominant form of mass
transfer in high-mass X-ray binaries, while in the case of low-mass X-ray
binaries matter is transferred via Roche-lobe overflow.

4U 1210$-$64, the subject of this paper, was first discovered by the _Uhuru_
satellite in 1978 (Forman et al., 1978), and has suffered a few identity
crises since. It has been reported both as a cataclysmic variable (Revnivtsev
et al., 2007) and a high-mass X-ray binary (Masetti et al., 2010) in the past.
Using the lightcurve from the Rossi X-ray Timing Explorer (RXTE) All-sky
Monitor (ASM) Corbet & Mukai (2008) found a modulation of $\sim$6.7 days,
which they interpreted as the orbital period. From the analysis of the long-
term ASM and MAXI light curves, Coley et al. (2014) found an X-ray eclipse
with an eclipse half-angle of 11${}^{\circ}_{.}$2$\pm$0${}^{\circ}_{.}$6,
which they estimated the spectral type of the mass donor in the binary to be a
B5 III or earlier. In addition to the orbital modulation, the X-ray behaviour
of 4U 1210$-$64 shows a long-term periodicity on the order of 5083 days that
is indicative of a variable accretion rate, where it undergoes high and low
flux states (Coley et al., 2014; Nakajima et al., 2022). The analysis of the
X-ray data shows that the eclipse profile is not observed during the low flux
state. The nature of the compact object in this system remains unclear, as no
pulsations and cyclotron features have been detected. A detailed background of
past X-ray observations is provided by Coley et al. (2014). In this paper, we
use the same ephemeris that is used in Coley et al. (2014): $P=6.7101\pm
0.0005$ and $T_{0}=\text{MJD}55553\pm 0.002$. In this work, we present optical
spectroscopy and photometry of the optical counterpart of 4U 1210$-$64, where
we re-visit the spectral type and the implications of the variability.

This paper is structured as follows: we present the observations in Section 2,
and discuss the characterization of the donor star in Section 3.1. In Section
3.2 we present the radial velocity analysis and the implications of the mass
of the compact object. In Section 3.3 we present the photometric variability.
A discussion of the results is presented in Section 4, where we make
inferences from the observational data, and summarise the paper in Section 5.

##  2 Observations

###  2.1 Spectroscopy

We observed the optical counterpart of 4U 1210$-$64 in two separate programmes
in 2013 and 2022, where we collected 19 spectra between 2013 January and 2023
February using the Robert Stobie Spectrograph (Burgh et al., 2003; Kobulnicky
et al., 2003) on the Southern African Large Telescope (Buckley et al., 2006).
The spectrograph was used in long-slit mode, with a slit width of
$1.5^{\prime\prime}$, and employing grating PG0900 at an angle of
$14.375^{\circ}$, with the camera station at $28.75^{\circ}$. This resulted in
a wavelength range of 3884 – 6987 Å and resolution of 6.6 Å at 4300 Å.
Exposure times of 180 s and 600 s were used for data obtained in 2013 and
2022/3, respectively.
The PG3000 grating was used in one observation (2023-02-12) at a grating angle
of $39.8725^{\circ}$ (camera station angle = $79.745^{\circ}$) which resulted
in a wavelength range of 3875 – 4610 ÅÅ (resolution of 1.4 Å at 4300 Å). An
exposure time of 1200 s was used for the observation.

The observations were corrected for gain and amplifier cross-talk by the RSS
pipeline (Crawford et al., 2010). Spectra were extracted using the standard
longslit tools available in iraf 111Image Reduction and Analysis Facility:
iraf.noao.edu, and extracted spectra had signal-to-noise ratios between 23 and
99 per pixel, dependent on the observing conditions. Further details on our
observations are presented in Table 1.

###  2.2 Photometry

We have compiled a range of photometric archival data.

####  2.2.1 Gaia

We found the Gaia counterpart of 4U 1210$-$64, which is identified as Gaia
6053076566300433920. The source was classified as a Cepheid in Gaia data
release 2 (Gaia DR2, Gaia Collaboration & et al. 2018) providing an estimation
of the period of 3.35560938 d, which is half the orbital period of the system.
We downloaded the Gaia G ($3300-10500$ Å), BP ($3300-6800$ Å), and RP
($6300-10500$ Å) light curves from the catalogue I/345 through Vizier
catalogue access tool (Ochsenbein et al., 2000).

####  2.2.2 Las Cumbres Observatory

We obtained photometric observations of 4U 1210$-$64 using the Las Cumbres
Observatory (LCO) telescope network (Brown et al., 2013). The data were
obtained using Johnson-Cousins $B$ ($3915-4806$ Å) and $V$ ($5028-5868$ Å)
filters in an epoch between 11 April 2017 to 03 May 2017. Exposure times of 3
s and 5 s were used for the $V$ and $B$ filters, respectively. The raw images
were reduced using the BANZAI pipeline which, in summary, performs bad-pixel
masking, bias subtraction, dark subtraction and flat-field correction (Harbeck
et al., 2018). Aperture photometry was used to extract the instrumental
magnitudes using the iraf package apphot. To calibrate the instrumental
magnitudes to standard photometric systems, we used the known $B$ and $V$
magnitudes of the stars in the field of 4U 1210$-$64 from the APASS catalogue
(Høg et al., 2000; Henden et al., 2016) and used the transformation equations
described in Da Costa (1992).

####  2.2.3 Universitatsternwarte Bochum

4U 1210$-$64 was observed using the Sloan $i$ ($6900-8400$ Å) and $r$
($5500-6900$ Å) filters as part of the Bochum Survey of the Southern Galactic
Disk catalog (Hackstein et al. 2015; labelled GDS 1213148$-$645230) between 12
May 2011 and 30 April 2015. The survey utilizes a robotic 15-cm twin telescope
of the Universitatsternwarte Bochum in Chile. Several standard star fields
were obtained each night and were used to perform the absolute calibration.
The full details of the survey are described in Haas et al. (2012).

####  2.2.4 Swift/UVOT

We have also looked for available archival observations from the Ultra-
Violet/Optical Telescope (UVOT, Roming et al. 2005) on board the Neil Gehrels
Swift Observatory (hereafter Swift; Gehrels et al., 2004) instrument. In this
work, we have used the photometric observations in the three UW filters from
the Swift Archive Download Portal.

##  3 Results

Figure 1: The normalized spectrum of 4U 1210-64 corresponding to orbital phase
zero (black), obtained on 12 Feb 2023 (MJD59988.02), and the Kurucz synthetic
model computed with $T\\!=\\!7250$ K, $\log g\\!=\\!3.25$, solar metallicity
and $v\sin i\\!=\\!50$ km/s (green), also normalized, shifted 0.4 units
downwards for clarity (green). Some line identifications are included; the red
crosses mark the position at which the He i lines would appear if the object
was a B-type star. See text for details.

###  3.1 Characterization of the donor star

In this section we carry out a thorough analysis of the spectra and the
spectral energy distribution (SED) of the system in order to quantify the
parameters and the spectral classification of the donor star. For doing that,
observations of the system, both spectroscopic and photometric, containing the
minimum possible contribution of the accretion disk, have been used. From the
spectroscopy, we used the mid-resolution spectrum (PG3000) close to phase zero
–when the star is eclipsing the compact object. The observation was taken on
12 February 2023 (MJD 59988.02), corresponding to an orbital phase 0.95. The
deep Ca ii K, and its ratio with the Ca ii H + H$\epsilon$ blend, together
with the absence of He I absorption lines, and the large number of metallic
lines, suggest a spectral type around A8–9 (Evans & Howarth, 2003; Gray &
Corbally, 2009), much later than the classification as a B-type star by
Masetti et al. (2009). The normalized spectrum is shown in Fig. 1 (lower
panel) together with some line identifications; the red crosses mark the
position where the He i lines would appear, some of them very prominent,
should the object be a B-type star. Note that the wavelengths of some of the
strong absorptions in the spectrum match those of the He i lines, however,
according to the results of the spectral synthesis, the depths of the helium
features are negligible as compared with those of the metals that appear in
the labels.

Due to the complex characteristics of the system, we have obtained the
fundamental parameters of the donor star through an iterative procedure in
which the observed spectrum and the spectral energy distribution (SED), built
with photometric data, are fitted in parallel; the combination of the results
of each fit provides feedback for next step of the iteration until a self-
consistent solution is reached for both sets of data.

The solution to the problem for this particular star is more complex than the
case of a single object, since, as we will see below, the contribution of the
disk and the close environment of the compact object to the total output of
energy from the system, at all bands, is not negligible. Moreover, this
contribution varies with the orbital phase and the state of the X-ray activity
of the source.

####  3.1.1 Spectral fitting

To quantify the spectral classification and the stellar parameters, we first
synthesized a set of high-resolution theoretical spectra, using the suite of
programmes synthe (Kurucz, 1993), which takes as input the list of the atomic
and molecular transitions and the spectral range to be synthesized, together
with the corresponding model atmosphere for a given temperature, gravity and
metallicity, which contains, among other ingredients, the atomic fractions of
the different elements. The spectral resolution of the spectra collected does
not allow to determine accurately the value of $v\sin i$, therefore we
considered a standard value of 50 km/s, typical of late-A stars. Solar
metallicity was assumed.

Figure 1 shows the observed spectrum (12 Feb 2023, MJD 59988.02). Although at
this epoch the source was in an active X-ray state (see Fig. 3), we selected
this spectrum because it has the highest available resolution in our dataset.
We explored the parameter space with $T_{\rm eff}$ between 7000 and 8000 K and
$\log g$ between 2.5 and 4.5, and concluded that a model with $T_{\rm
eff}$=7250 K, $\log g$=3.25 provides a good fit. It is apparent that bluewards
below $\sim\\!4300$ Å there is a gradual decrease in the depth of the Balmer
lines, the metal lines, and the Ca ii K line, when compared with those of the
synthetic spectrum. This veiling is most probably caused by a non-stellar
contribution which, when added to the stellar spectrum, has the effect of
decreasing both the relative depth and the equivalent widths of the absorption
lines. This hypothesis is reinforced by the fact that the He ii 4686 Å is
observed in emission at all orbital phases (even at phase 0) during the X-ray
active epoch (2022/2023 spectra). Unfortunately, that wavelength does not fall
within the interval covered by the observation obtained on 12 Feb 2023, but
two lower-resolution spectra taken on 24 Jan 2023 (phase 0.10) and 31 Mar 2023
(phase 0.91) clearly show that He ii 4686 Å line in emission, which implies
that it must originate in a region with hot material. Therefore a plausible
scenario is that even when the compact object is eclipsed by the donor star
during the X-ray active state, a small fraction of the disc, probably the hot-
spot, with material at a temperature larger than $\sim\\!30,000$ K, could
still be visible, contributing to increasing the blue continuum.

####  3.1.2 Spectral energy distribution (SED) fitting

The spectral energy distribution (SED) has been built using photometric
measurements obtained during the least active states and/or close to zero
orbital phase, when a minimum contribution from the emission of the accretion
disk is expected (see Fig. 2). We selected the Gaia $BP$, $G$, and $RP$
observations at MJD 56887 (phase 0.80), the Bochum $r$ and $i$ observations
from MJD 56827.97 (phase 0.99), and the $B$ and $V$ LCO observations from MJD
57867.83 (phase 0.96). We also added the 2MASS $J$, $H$, and $K_{\rm s}$
measurements from the 2MASS All Sky Catalog of point sources (Cutri et al.,
2003) for which we do not know the dates of observations, although all the
2MASS observations were taken from June 1997 to February 2001, and the W1 and
W2 measurements taken on MJD 57418.446 (phase 0.99) from the NEOWISE project
(Mainzer et al., 2011).Additionaly, Swift/UVOT observations in the three UW
and U filters obtained from MJD 54580.32 to 54580.99 are also plotted for
reference, although since they were taken during an X-ray active phase, they
were not used in the SED fit. The epochs of the photometric data used for the
SED fit are indicated alongside the long-term X-ray lightcurve in Fig. 3.

We have used the distance from Gaia DR2 obtained by Bailer-Jones et al. 2018,
who gave a value of $d\\!=\\!3.777$ kpc (16th percentile $d\\!=\\!3.530$ kpc,
84th percentile $d\\!=\\!4.064$ kpc); this value was preferred instead of that
from Gaia eDR3 (Bailer-Jones et al., 2021), because it has a slightly smaller
value of the RUWE (Renormalised Unit Weight Error) associated to each Gaia
source (Gaia Collaboration et al., 2021) (1.017 in DR2, against 1.046 in
eDR3). The best SED fit is obtained for R∗=6.4$\pm$0.2 R⊙ and
$E(B\\!-\\!V)$=0.60$\pm$0.05 mag (see Fig. 2). This estimation of
$E(B\\!-\\!V)$ is in agreement with the value $E(B\\!-\\!V)$=0.66$\pm$0.06
mag, obtained introducing the coordinates and distance in the 3D interstellar
dust reddening maps of the Galactic plane by Chen et al. 2019.

Fig. 2 shows the final fit to the SED. The photometric observations are
plotted as black stars, the stellar model multiplied by 4$\pi$R∗ (being R∗=6.4
R⊙ the value providing the best fit), divided by 4$\pi$Rd (being $d$=3.77 kpc
the distance from Gaia DR2), and reddened with the value of E(B-V) that
provided the best fit ($E(B\\!-\\!V)$=0.60 mag) is plotted in red.

####  3.1.3 Stellar mass and final parameters

The pair ($T_{\rm eff}$, R∗) –or, which is equivalent, ($T_{\rm eff}$, L∗/L⊙)–
was then superimposed on the set of PAdova and tRieste Stellar Evolution Code
(PARSEC; Bressan et al. 2012) tracks and isochrones. Fig. 4 shows the final
position of the star in the HR diagrams, with the black triangle displaying
the results from the best SED fit. The location of the star using the stellar
parameters obtained in Sect. 3.1.1, leads to an estimation of the mass
M∗=2.55$\pm$0.05 M⊙. Using the radius obtained from the SED fitting, the
gravity implied for this mass is $\log g\\!\simeq\\!3.25$, i.e., which is in
agreement with the spectroscopic estimation. We conclude that the final set of
parameters for the donor star, with their corresponding conservative
uncertainties is $T_{\rm eff}\\!=\\!7250\pm 100$ K and $\log g\\!=\\!3.25\pm
0.25$. Incidentally, from the orbital period of the system and the estimated
value of the radius, under the assumption of synchronous rotation, the value
of the equatorial stellar velocity would be $\sim$50 km/s, which is in
agreement with what we considered from the very beginning.

Figure 2: Spectral Energy Distribution fit. Observations taken close to phase
zero and in a low X-ray phase (see text), are plotted as black stars; the
stellar theoretical flux reddened with the value which gives the best SED fit
of $E(B\\!-\\!V)\\!=\\!0.60$ mag is plotted in red. The Swift/UVOT UW2, UM2,
UW1, and U mean fluxes from an archival observation from MJD 54580 are plotted
as purple triangles for comparison.

Figure 3: Long-term 3-12 keV RXTE ASM and MAXI lightcurve of 4U 1210$-$64\.
The lightcurve is binned to a time resolution of 21 days. The different
symbols in the plot indicate the epochs of the photometric data that was used
to perform the Spectral Energy Distribution fit.

Figure 4: Evolutionary tracks for different masses in a R∗ vs $\log\,T_{\rm
eff}$ diagram (top) and a $\log\,(L/L_{\odot}$) vs $\log\,T_{\rm eff}$
(bottom) with the estimations from the SED fit marked with black triangles.

Despite of the fact that both the spectral and SED fittings provide reasonable
results, we are conscious that a deeper analysis, mainly focused on the
modeling of the disk and the environment of the compact object, is required.
That work, as well as a detailed study of the spectra and SED at different
orbital phases is deferred to a follow-up paper. In conclusion, the spectral
type of the optical companion can be classified as $\sim$A8, and the location
in the HR diagram, implies a luminosity class around IV, i.e. the star seems
to have left the main sequence, but not entered into the giant branch yet.

###  3.2 Radial velocity analysis

The relative radial velocities between the spectra of 4U 1210$-$64 were
determined through cross-correlation analysis (Tonry & Davis, 1979). The
method used is described by Foellmi et al. (2003), Manick et al. (2015) and
Monageng et al. (2017) which uses the xcsao task in iraf. Heliocentric
corrections were applied to the spectra using iraf tasks rvcorr and dopcor.

Figure 5: The measured radial velocity values folded on the orbital period
with a Keplerian fit overplotted. The radial velocities plotted here are
presented in Table 1. Top panel, Case 1: a fixed orbital period and free
eccentricity. Bottom panel, Case 2: a fixed orbital period but fixed
eccentricity (circular orbit).

We used an iterative process to generate a high signal-noise ratio (SNR) zero
velocity template. We firstly rectified and subtracted the continuum level of
the spectra, after which they were converted to a logarithmic wavelength
scale. The spectra were then grouped in descending order according to their
SNR. We then measured the relative radial velocity shifts of each spectrum to
the highest SNR spectrum. From the results of the first cross-correlation
iteration, we shifted the individual spectra to the same wavelength as the
high SNR template. The shifted spectra were then co-added, creating a mean
spectrum which was used as a final template to compute the RV shifts shown in
Table 1 and Fig. 5.

Table 1: SALT observations of the optical counterpart of 4U 1210$-$64 with the measured radial velocities. Date | JD | Orbital phase | RV (km/s) | SNR
---|---|---|---|---
2013-01-24 | 2456317.515 | 0.846 | -103.0 $\pm$ 7.5 | 49
2013-02-06 | 2456330.602 | 0.796 | -127.5 $\pm$ 9.3 | 42
2013-02-26 | 2456350.614 | 0.779 | -87.4 $\pm$ 4.5 | 59
2013-03-22 | 2456374.356 | 0.317 | 31.3 $\pm$ 9.0 | 43
2013-04-12 | 2456395.324 | 0.442 | -47.5 $\pm$ 10.6 | 38
2013-04-24 | 2456407.253 | 0.219 | -2.6 $\pm$ 9.4 | 62
2014-01-18 | 2456676.515 | 0.347 | 90.3 $\pm$ 12.5 | 26
2014-02-10 | 2456699.441 | 0.764 | -156.5 $\pm$ 9.9 | 41
2022-06-07 | 2459738.294 | 0.641 | -125.2 $\pm$ 14.5 | 71
2022-06-19 | 2459750.328 | 0.435 | 19.1 $\pm$ 12.2 | 61
2022-06-25 | 2459756.273 | 0.321 | 66.0 $\pm$ 6.7 | 82
2022-06-26 | 2459757.282 | 0.471 | -50.0 $\pm$ 17.9 | 94
2022-06-29 | 2459760.233 | 0.911 | -102.1 $\pm$ 6.4 | 62
2022-06-30 | 2459761.246 | 0.062 | -20.8 $\pm$ 5.5 | 53
2022-07-13 | 2459774.260 | 0.001 | -110.5 $\pm$ 5.1 | 54
2022-07-28 | 2459789.232 | 0.233 | 44.2 $\pm$ 4.2 | 77
2023-01-24 | 2459969.504 | 0.098 | -16.0 $\pm$ 6.2 | 74
2023-02-11 | 2459987.460 | 0.774 | -210.6 $\pm$ 10.3 | 99
2023-02-12 | 2459988.602 | 0.945 | -131.6 $\pm$ 8.3 | 23
Table 2: Orbital parameters derived from Keplerian fits to the radial velocities for the two cases studied. Parameter | Value | Value
---|---|---
| Case 1 | Case 2
Period (days) | 6.7101 (fixed) | 6.7101 (fixed)
Eccentricity | 0.27 $\pm$ 0.23 | 0.00 (fixed)
$\omega$ (degrees) | 25 $\pm$ 13 | 51 $\pm$ 12
K (km/s) | 91 $\pm$ 20 | 86 $\pm$ 15
$\gamma$ (km/s) | -54 $\pm$ 12 | -42 $\pm$ 11

Due to the modest orbital phase coverage we performed Keplerian fits to the
radial velocity measurements with different settings. In Case 1 we fixed the
period to that obtained from X-ray analysis (Coley et al., 2014) but with the
eccentricity left as a free parameter. In Case 2, both the period and
eccentricity were fixed to perform a sinusoidal fit. Fig. 5 shows the fits for
the two cases, with the orbital parameters listed in Table 2, where $\omega,K$
and $\gamma$ are the longitude of periastron, semi-amplitude and systemic
velocity of the binary, respectively. In our notation, the longitude of
periastron is the sum of the argument of periastron and the longitude of the
ascending node. The mass function can then be calculated using

$f(M)=\frac{M_{1}^{3}}{(M_{1}+M_{2})^{2}}\sin^{3}i=\frac{P}{2\pi
G}K_{2}^{3}(1-e^{2})^{3/2}$ (1)

where:

  * •

$M_{1}=$ compact object mass

  * •

$M_{2}=$ donor star mass

  * •

$K_{2}=$ semi-amplitude velocity of the donor star

  * •

$P=6.7101$d (Coley et al., 2014)

For the parameters listed in Table 2, a mass function of $f(M)=0.47\pm 0.32$
M⊙ and $f(M)=0.44\pm 0.22$ M⊙ is obtained for the case of the eccentricity
left as a free parameter (Case 1) and the eccentricity fixed (Case 2),
respectively.

###  3.3 Folded light curves

Fig. 6 shows the optical light curves from the observations described in
Sect.2.2 folded over the orbital period using the ephemeris described in Coley
et al. (2014). Although the various optical datasets were taken at different
epochs, the modulation of the lightcurves shows similar behaviour. The folded
lightcurves show sinusoidal modulation with a dip in flux between orbital
phases $\phi\sim 0.4$ and $\phi\sim 0.6$, similar to that observed in the
X-ray lightcurve between phases $\phi\sim-0.04$ and $\phi\sim 0.03$ (Coley et
al., 2014). Analogous to the X-rays, the dip in flux seen in the optical
lightcurves is suggestive of an eclipse. The shape of the optical lightcurves
is possibly due to a combination of ellipsoidal variations, irradiation
effects and eclipsing of the two stellar components of the binary. The minimum
at $\phi\sim 0.5$ is noticeably deeper than that at $\phi\sim 0$. At $\phi\sim
0.5$ the accretion disc eclipses the hotter, irradiated side of the donor star
while at $\phi\sim 0$ the observer views the cooler, non-irradiated side of
the star when the compact object and accretion disc are eclipsed. In Fig. 7 an
illustration of the binary system of 4U 1210$-$64 is provided.

Figure 6: Folded optical light curves. On the left, from top to bottom: Gaia
RP, Gaia G, Gaia BP; on the right, from top to bottom: Bochum r, Bochum i, LCO
V, LCO B. The folded light-curves are colour-coded with time (see colorbar).
Blue points are from an X-ray active phase (see Coley et al. 2014).

##  4 Discussion

###  4.1 An intermediate-mass X-ray binary

The firm classification of the optical counterpart of 4U 1210$-$64 as an A8
III-IV star means that this object is a rare intermediate-mass X-ray binary
(IMXB). These systems are rare because their mass transfer timescales are
short. As explained by Tauris & van den Heuvel (2006), the significantly
higher mass ratio in IMXBs means that when Roche lobe overflow does occur in
the system, it happens rapidly and is either over on a sub-thermal timescale
or only in a 1000 years. The weak stellar wind outflow from the donor star is
also insufficient to power the compact object for strong X-ray emission to be
detected. This presents a selection bias against detecting IMXBs as X-ray
sources.

Figure 7: A schematic illustrating a face-on view of the binary system with
the components labeled. The binary phases are indicated.

###  4.2 Accretion disc corona

The X-ray lightcurve of 4U 1210$-$64 shows a smooth modulation with a minimum
occurring $\sim 0.2$ cycles before the eclipse (Coley et al., 2014). Similar
modulation has been observed in the low-mass X-ray binaries, 4U 1822$-$37 and
4U 2129$+$47 (White & Holt, 1982; Parmar et al., 2000). The eclipse can be
interpreted as an X-ray emission region that is extended due to the X-rays
being scattered in the accretion disc corona (ADC). The smooth variation is
caused by X-rays from ADC being obscured by the structure at the rim of the
accretion disc. According to Coley et al. (2014), the X-ray lightcurve shows
an eclipse that spans an orbital phase range between $\phi\sim-0.04$ and
$\phi\sim 0.03$. The optical photometric lightcurve shows an eclipse almost
half an orbital phase later spanning orbital phase range $\phi\sim 0.4$ and
$\phi\sim 0.6$. During the X-ray eclipse the donor star completely obscures
the compact object and the extended corona as it is projected onto our line of
sight (phase 0.0 in Fig. 7). The X-ray eclipse has a shorter time span than
the optical eclipse. This is possibly due to the hot spot, which is
responsible for significant X-ray emission, being compact. In this picture,
when the donor star eclipses the compact object and accretion disc during the
X-ray eclipse it blocks out the hot spot briefly while the accretion disc
blocks out a large portion of the optical light from the donor star for a
longer period.

###  4.3 Nature of the compact object

Fig. 8 shows the mass-mass plots obtained from the orbital parameters listed
in Table 2 and equation 1, with the mass constraints of the two stars for
different inclination angles for the calculated mass function. The results are
similar for both cases of the eccentricity left as a free parameter (Case 1)
and the sinusoidal fit (Case 2). Using the donor star mass obtained in Section
3.1.3 of 2.55 M⊙ and a range of compact object masses of 1.4 M⊙ (canonical
neutron star mass) and 2.35 M⊙ (the most massive neutron star known; Romani et
al. (2022)), we find that a neutron star is possible for inclination angles in
the range $\sim 50^{\circ}\lesssim i\lesssim 73^{\circ}$. For the same donor
star mass, a black hole ($M_{X}\gtrsim 3.0$ M⊙) is possible for an inclination
angle range $i\lesssim 54^{\circ}$. The eclipses that are seen in the optical
and X-ray lightcurves suggest that the plane of the orbit is viewed at large
inclination angles close to $90^{\circ}$, which favours a neutron star as the
compact object in the binary system. It is also possible to use the mass
functions obtained for Case 1 and Case 2 with an inclination angle of
$i=90^{\circ}$ to estimate the mass range of the donor star. For Case 1, a
lower limit canonical neutron star mass of 1.4 M⊙ results in a donor star mass
of $M_{\ast}=1.0$ M⊙, while the upper limit neutron star mass of 2.35 M⊙
yields $M_{\ast}=2.9$ M⊙. Similarly for Case 2 the lower and upper limits of
the neutron star mass result in donor star masses of $M_{\ast}=1.1$ M⊙ and
$M_{\ast}=3.1$ M⊙, respectively. A black hole ($M_{X}\gtrsim 3.0$ M⊙) results
in donor star mass range $M_{\ast}\gtrsim 4.6$ M⊙.

Binary stellar evolution models indicate that donor stars in binary systems
exhibit different properties to isolated stars of the same spectral type as a
result of mass transfer to the companion (Rappaport & Joss, 1983). This
results in undermassive and undersized donor stars for a given spectral type.
The mass of the donor star obtained in Section 3.1.3 uses isochrone fitting
models that do not take into account binary evolution. Consequently, the mass
estimate obtained from isochrone fitting could be lower, which would make the
neutron star even more probable.

Figure 8: Mass-mass plot from the calculated mass-function showing the mass
constraints of the two objects as a function of inclination angle for the
sinusoidal Keplerian fit to the radial velocity measurements. The vertical
line in both plots indicates the mass of the donor star of 2.55 M⊙. Top panel,
Case 1: a fixed orbital period and free eccentricity. Bottom panel, Case 2: a
fixed orbital period but fixed eccentricity (circular orbit).

##  5 Conclusions

We have presented optical spectroscopic and photometric observations of 4U
1210$-$64 obtained with SALT and LCO/Bochum/Gaia, respectively. From the
analysis of the spectra, we demonstrate that the donor star is an A8 III-IV
spectral type, making the system a member of the rare intermediate-mass binary
sub-class. The shape of the phase-folded optical photometric lightcurves
reveals behaviour that is possibly due to ellipsoidal variations, irradiation
of the donor star, and mutual eclipses of the star and accretion disk. We have
measured radial velocities from the spectra and fitted a Keplerian model where
we have considered eccentric and sinusoidal fits. Based on the results of the
fits, we suggest that a neutron star is strongly favoured as the compact
object.

## Acknowledgements

LJT is supported by the SALT Foundation and the South African NRF. IMM and VAM
are supported by the South African NRF. JAG is funded by Spanish
MCIN/AEI/10.13039/501100011033 grant PID2019-107061GB-C61. BM is supported by
grant PID2021-127289-NB-I00 by the Spanish Ministry of Science and Innovation
(MCIN) / State Agency of Research (AEI). The material is based upon work
supported by NASA under award number 80GSFC21M0002. Some of the observations
reported in this paper were obtained with the Southern African Large Telescope
(SALT), under the science programmes: 2012-2-RSA_OTH_UKSC-003,
2013-2-RSA_OTH_UKSC-003 (PI: McBride) and 2021-2-MLT-005 (PI: Townsend).

## Data Availability

The X-ray data are freely available from the RXTE and MAXI archives. The SALT
optical data in this article will be shared on any reasonable request to
Vanessa McBride and Lee Townsend. The LCO data will be shared on any
reasonable request to Itumeleng Monageng. The Gaia and Bochum data are freely
available from the missions’ respective webpages.

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##  Appendix A Dates for X-ray and optical observations

Table 3: A log of the Bochum photometric observations. Date (MJD) | Orbital Phase | i band mag | r band mag
---|---|---|---
55693.1991200 | 0.8791404 | 13.780 $\pm$ 0.019 | 13.529 $\pm$ 0.033
55983.1720000 | 0.0935306 | 13.871 $\pm$ 0.020 | 13.686 $\pm$ 0.037
55986.1487000 | 0.5371455 | 14.078 $\pm$ 0.023 | 13.883 $\pm$ 0.044
56721.3698000 | 0.1064664 | 14.028 $\pm$ 0.022 | -
56750.2800800 | 0.4149387 | 14.078 $\pm$ 0.023 | 13.944 $\pm$ 0.046
56751.2750900 | 0.5632241 | 14.202 $\pm$ 0.025 | -
56755.3153100 | 0.1653343 | 13.929 $\pm$ 0.021 | 13.747 $\pm$ 0.039
56756.2745300 | 0.3082860 | 13.861 $\pm$ 0.020 | -
56783.0629500 | 0.3005395 | 13.888 $\pm$ 0.020 | 13.701 $\pm$ 0.038
56797.0383900 | 0.3832864 | 14.004 $\pm$ 0.022 | -
56806.0334500 | 0.7238119 | 13.850 $\pm$ 0.020 | 13.714 $\pm$ 0.038
56821.9745700 | 0.0995022 | 14.024 $\pm$ 0.022 | 13.797 $\pm$ 0.041
56822.9715500 | 0.2480813 | 13.823 $\pm$ 0.019 | 13.695 $\pm$ 0.038
56824.9718300 | 0.5461811 | 14.286 $\pm$ 0.026 | 13.992 $\pm$ 0.048
56825.9719800 | 0.6952326 | 13.870 $\pm$ 0.020 | 13.730 $\pm$ 0.039
56827.9720000 | 0.9932937 | 14.081 $\pm$ 0.023 | 13.911 $\pm$ 0.045
56828.9721200 | 0.1423407 | 13.973 $\pm$ 0.021 | 13.765 $\pm$ 0.040
56829.9723000 | 0.2913966 | 13.843 $\pm$ 0.020 | 13.649 $\pm$ 0.036
56830.9726700 | 0.4404808 | 14.153 $\pm$ 0.024 | 13.941 $\pm$ 0.046
57138.1357100 | 0.2167047 | 13.860 $\pm$ 0.020 | 13.668 $\pm$ 0.037
57141.1161700 | 0.6608799 | 13.931 $\pm$ 0.021 | 13.716 $\pm$ 0.038
57142.1317500 | 0.8122308 | 13.893 $\pm$ 0.020 | 13.690 $\pm$ 0.037
Table 4: A log of the LCO photometric observations. Date (MJD) | Orbital Phase | V band mag | B band mag
---|---|---|---
57855.1830477 | 0.0776065 | 14.133 $\pm$ 0.083 | 14.828 $\pm$ 0.132
57858.5004219 | 0.5719918 | 14.354 $\pm$ 0.109 | 15.071 $\pm$ 0.177
57862.5733099 | 0.1789705 | 14.125 $\pm$ 0.125 | 14.776 $\pm$ 0.224
57860.5255085 | 0.8737885 | 14.152 $\pm$ 0.107 | 14.707 $\pm$ 0.129
57857.1669306 | 0.3732628 | 13.915 $\pm$ 0.111 | 14.594 $\pm$ 0.182
57859.3580015 | 0.6997961 | 14.016 $\pm$ 0.111 | 14.614 $\pm$ 0.157
57864.5277377 | 0.4702371 | 14.431 $\pm$ 0.108 | 15.155 $\pm$ 0.186
57866.6387006 | 0.7848319 | 13.956 $\pm$ 0.111 | 14.691 $\pm$ 0.156
57861.8337478 | 0.0687542 | 14.174 $\pm$ 0.102 | 14.928 $\pm$ 0.136
57853.9592288 | 0.8952219 | 14.105 $\pm$ 0.109 | 14.793 $\pm$ 0.181
57859.8337043 | 0.7706896 | 13.937 $\pm$ 0.096 | 14.616 $\pm$ 0.134
57867.8337022 | 0.9629219 | 14.270 $\pm$ 0.131 | 14.943 $\pm$ 0.165
57876.7023838 | 0.2846133 | 14.003 $\pm$ 0.080 | 14.487 $\pm$ 0.104
57873.1677888 | 0.7578559 | 14.086 $\pm$ 0.142 | 14.736 $\pm$ 0.168
57873.9694145 | 0.8773214 | 14.208 $\pm$ 0.110 | 14.865 $\pm$ 0.133
57875.9742466 | 0.1760997 | 14.447 $\pm$ 0.131 | 15.085 $\pm$ 0.182
57865.1830831 | 0.5679026 | 14.435 $\pm$ 0.092 | 15.025 $\pm$ 0.114
57867.1670566 | 0.8635723 | 14.125 $\pm$ 0.113 | 14.778 $\pm$ 0.149
57855.1837082 | 0.0777050 | 14.150 $\pm$ 0.094 | 14.869 $\pm$ 0.137
57870.0454548 | 0.2925373 | 13.995 $\pm$ 0.095 | 14.653 $\pm$ 0.156
57871.9758040 | 0.5802155 | 14.371 $\pm$ 0.101 | 15.187 $\pm$ 0.183
57871.3967696 | 0.4939225 | 14.396 $\pm$ 0.104 | 15.130 $\pm$ 0.148


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  *[IP]: Intraluminal Procedures
  *[EI]: Endovascular Interventions
  *[IPEI]: Intraluminal Procedures and Endovascular Interventions
  *[MIS]: Minimally Invasive Surgical
  *[MP]: Motion Planning
  *[LoA]: Levels of Autonomy
  *[DoFs]: Degrees-of-Freedom
  *[CRC]: Colorectal Cancer
  *[fURS]: Robotic Flexible Ureteroscopy
  *[TOE]: Transoral Endoscopy
  *[TNE]: Transnasal Endoscopy
  *[CT]: Computed Tomography
  *[MRI]: Magnetic Resonance Imaging
  *[ACO]: Ant Colony Optimization
  *[CBS]: Centerline-based Structure
  *[DFS]: Depth First Search
  *[BFS]: Breadth First Search
  *[LPA*]: Lifelong Planning A*
  *[ROI]: Region of Interest
  *[CTRs]: Concentric Tube Robots
  *[RRT]: Rapidly-exploring Random Tree
  *[PRM*]: Probabilistic RoadMap*
  *[RRM]: Rapidly-exploring RoadMap
  *[RRG]: Rapidly-exploring Random Graph
  *[L-BFGS]: Limited-memory Broyden-Fletcher-Goldfarb-Shanno
  *[GA]: Genetic Algorithm
  *[RL]: Reinforcement Learning
  *[HMMs]: Hidden Markov models
  *[LfD]: Learning from Demonstrations
  *[GMM]: Gaussian Mixture Model
  *[DMPs]: Dynamical Movement Primitives
  *[GAIL]: Generative Adversarial Imitation Learning
  *[GAN]: Generative Adversarial Network
  *[CNN]: Convolutional Neural Network
  *[LSTM]: Long Short Term Memory
  *[DNN]: Deep Neural Network
  *[DRL]: Deep Reinforcement Learning
  *[DQN]: Deep Q-Network
  *[FBG]: Fiber Bragg Gratings
  *[EM]: Electromagnetic
  *[IVUS]: IntraVascular UltraSound
  *[SLAM]: Simultaneous Localisation And Mapping