# Long-Term X-Ray/UV Variability in ULXs

Norman Khan,1 Matthew. J. Middleton,1
1School of Physics & Astronomy, University of Southampton, Southampton,
Southampton SO17 1BJ, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The focus of NASA’s Swift telescope has been transients and target-of-
opportunity observing, resulting in many observations of ultraluminous X-ray
sources (ULXs) over the last $\sim 20$ years. For the vast majority of these
observations, simultaneous data has been obtained using both the X-ray
telescope (XRT) and the ultraviolet and optical telescope (UVOT), providing a
unique opportunity to study coupled variability between these bands. Using a
sample of $\sim$40 ULXs with numerous repeat observations, we extract stacked
images to characterise the spatial extent of the UV-Optical emission and
extract long-term light curves to search for first-order linear correlations
between the UV and X-ray emission. We find that a small subset may show weakly
correlated joint variability, while other sources appear to display non-linear
relationships between the bands. We discuss these observations in the context
of several theoretical models: precession, irradiation of the outer accretion
disc and irradiation of the companion star. We conclude that more complicated
analysis or higher quality data may be required to accurately constrain the
nature of the joint X-ray and UV/optical emission in these sources.

###### keywords:

stars: black holes – stars: neutron

††pubyear: 2023††pagerange: Long-Term X-Ray/UV Variability in ULXs–A

##  1 Introduction

Ultraluminous X-ray sources (ULXs) are defined as off-nuclear, point-like
sources with inferred isotropic luminosities exceeding $L>10^{39}\mathrm{erg}\
\mathrm{s}^{-1}$ (see the reviews of Roberts 2007; Kaaret et al. 2017; King et
al. 2023). Pulsations unambiguously indicating the presence of neutron star
(NS) accretors have been located in several ULXs (Bachetti et al. 2014; Fürst
et al. 2016; Israel et al. 2017; Tsygankov et al. 2017; Doroshenko et al.
2018; Carpano et al. 2018; Sathyaprakash et al. 2019; Rodríguez Castillo et
al. 2020) whilst the presence of a cyclotron resonant scattering feature in
M51 ULX-8 (Brightman et al., 2018) and possible detection in NGC300 ULX1
(Walton et al., 2018b) also suggest the presence of neutron stars. Given the
luminosities of these sources and the known accretor mass, it is inescapable
that the accretion flow in such systems is super-critical, either in the flow
or onto the star itself.

Should the Eddington limit be reached in the disc, then standard theory (i.e.
Shakura & Sunyaev 1973; Poutanen et al. 2007) would predict powerful, mass-
loaded winds which have been located in several ULXs to-date, both as imprints
in the CCD spectra (Middleton et al. 2014; Middleton et al. 2015b; Walton et
al. 2016) and using higher energy-resolution detectors (Pinto et al. 2016,
2017; Kosec et al. 2018; Kosec et al. 2021. A corollary of such accretion
flows is that, providing the wind is optically thick, there should be some
degree of collimation and the assumption of isotropy breaks down (King et al.,
2001). The resulting spectrum (and timing properties) of a given ULX then
depends on both the accretion rate and inclination of the source (Poutanen et
al., 2007; Middleton et al., 2015a).

Should the accretion rate be high, we would naturally expect high inclination
ULXs to be dim in the X-rays and instead peak at lower frequencies (Poutanen
et al., 2007). With emission from the wind photosphere being $\sim$Eddington,
these may be prime candidates for detection by next-generation deep surveys
(e.g. LSST: Middleton et al. in prep). A prime example of such an edge-on ULX
is the Galactic source SS433 ($i=78^{\circ}$), which, despite having an X-ray
luminosity of only $\sim 10^{36}\ \mathrm{erg}\ \mathrm{s}^{-1}$ appears to
share many of the same characteristics of ULXs and is inferred to have
considerably brighter, face-on X-ray luminosities (Cherepashchuk, 2002;
Fabrika, 2004; Khabibullin & Sazonov, 2016; Liu et al., 2015; Middleton et
al., 2021) and emit at $\sim 10^{40}\ \mathrm{erg}\ \mathrm{s}^{-1}$ in the UV
(Dolan et al., 1997).

A change in the intrinsic X-ray/UV emission from a given ULX may be driven by
either a change in mass accretion rate at larger radius or a change in our
line-of-sight inclination to the inner-regions. The former may be the result
of mass loss at large radii (Middleton et al., 2022), whilst the latter may be
a result of disc warping due to irradiation (Pringle 1996; Pasham & Strohmayer
2013b) or precession of the super-critical disc and wind (Pasham & Strohmayer
2013a; Middleton et al. 2018; Middleton et al. 2019b).

Whilst the intrinsic emission from high inclination ULXs may peak at low
frequencies, in the optical/UV band there will also be emission from the
secondary star which can be amplified if effectively irradiated, as well as
emission from the outer disc (again, if effectively irradiated). Whilst there
has been a great deal of study of ULXs in the optical – specifically to
elucidate the nature of the companion star (Heida et al., 2014) – there has
been limited exploration in the UV bands. However, it has been observed that
in one ULX, the UV emission appears extremely bright ($>10^{39}$ erg/s: Kaaret
et al. 2017) and in the case of the PULX, NGC 7793 P13, an optical/UV super-
orbital period is seen, out of phase with the X-ray super-orbital period
(Fürst et al., 2016). Although not well explored, the correlation between
X-ray and low frequency emission should provide invaluable insight into the
geometry and nature of the accretion flow. In this paper, we explore the
general shapes of the X-ray/UV correlations we might expect for various
scenarios and search for these within the Swift lightcurves of several
prominent ULXs.

##  2 Predictions

Figure 1: Black body spectrum (bbody in xspec) set at $kT=0.005\
\mathrm{keV}$. Coloured are the effective energy ranges of the UVOT bands used
for the flux calculation shown in figure 2.

In this section, we consider the theoretical impact on the coupled X-ray/UV
properties of ULXs when ascribing the low frequency emission to various
locations in the system. To simplify this picture we make the explicit
assumption that ULXs which contain neutron stars all have dipole fields weak
enough (or accretion rates high enough) that the classical super-critical
picture of disc accretion (Poutanen et al., 2007) holds (i.e. that the
magnetospheric radius, $R_{\mathrm{M}}$, is much smaller than the spherisation
radius, $R_{\mathrm{sph}}$). Currently, measurements only exist for a handful
of pulsating ULXs (see Walton et al. 2018b; Brightman et al. 2018; Middleton
et al. 2019a; Fürst et al. 2023) but suggest field strengths of order $\sim
10^{11}-10^{13}\ \mathrm{G}$ which according to the canonical picture of PULXs
should tend to lead to the condition that $R_{\mathrm{M}}<R_{\mathrm{sph}}$
(King & Lasota, 2020).

###  2.1 Emission from the outer wind photosphere

Following the super-critical model of Poutanen et al. 2007, we assume that the
wind photosphere extends out to some radius $r_{\rm out}$ and reprocesses the
flux from the disc below the wind where the inflow starts to become locally
super-critical, and some fraction of the radiation produced interior to
$r_{\rm out}$ (the latter with a luminosity greater than or equal to the
Eddington luminosity). A full understanding of the shape and intensity of the
emergent SED requires full radiative magnetohydrodynamic (RMHD) simulations
and extensive post-processing, which has not yet been performed in the case of
ULXs (although see the work by Narayan et al. 2017 and Dai et al. 2018). In
the absence of numerical studies, simple qualitative predictions are still
possible. For a fixed inclination, increasing the mass accretion rate pushes
$r_{\rm out}$ to a larger radius due to increased mass loading of the wind and
the larger radial location of $r_{\mathrm{sph}}$ (Poutanen et al., 2007).
Should the opening angle of the wind be connected to the accretion rate at
large radius (as it would seem to be cf Jiang et al. 2014, 2019), then an
increase in mass accretion rate will increasingly collimate the X-ray emission
from within. What follows depends on the orientation of the observer. Should
the observer be able to see into the wind cone, then the X-ray luminosity at
all energies will increase, and the characteristic temperature associated with
the spherization radius, $T_{\mathrm{sph}}$ will decrease. The expansion of
the wind photosphere to larger radius will lead to a reduction in its
temperature according to the formula of Poutanen et al. 2007 (eq 1):

$T_{\mathrm{sph}}\approx
0.8f_{\mathrm{col}}\left(\frac{\zeta\beta}{\epsilon_{\mathrm{w}}}\right)^{1/2}m^{-1/4}\dot{m_{0}}^{-3/4}\
\mathrm{keV}$ (1)

Figure 2: Flux in each of the UVOT bands as a function of the mass accretion
rate $\dot{m}_{0}$. The peak flux is reached at $\dot{m}_{0}\sim 7000$ for the
highest energy band (UVW2), much higher than is expected for ULXs; this means
our UV view of high inclination ULXs, where the observed emission is from the
wind photosphere, exists entirely within the region of UV emission on the
left-hand side of the plot where the flux in a single band increases
monotonically with $\dot{m}_{0}$. The plot was made using the xspec model
bbody with a normalized luminosity and a temperature set to $T_{\mathrm{ph}}$
(eq 1), $m=1.4\ M_{\odot}$, $\beta=\zeta=1$, $f_{\mathrm{col}}=1.7$ and
$\epsilon_{\mathrm{w}}=0.95$.

where $f_{\rm col}$ is a colour temperature correction factor, $\zeta$,
$\beta$ and $\epsilon_{\rm w}$ are constants relating to the wind-cone opening
angle, outflow velocity and fractional energy content in powering the wind,
$m$ is the accretor mass in M⊙, and $\dot{m}_{0}$ is the Eddington-scaled
accretion rate.

By assuming the wind photosphere radiates as a blackbody at the characteristic
temperature $T_{\mathrm{sph}}$, that $\zeta\beta/\epsilon_{\rm w}\approx 1$ (A
simplification that can be made as all parameters can be considered of order
unity), $f_{\mathrm{col}}\approx 2$ (see Shimura & Takahara 1995), and
accretor masses of 10 M⊙ and 1.4 M⊙, the UV emission in the highest energy
band (UVW2) (taking the form of a blackbody, peaking around 3 kT) will
increase in brightness until accretion rates in excess of $\dot{m}_{0}$
$\approx$ 5000 $\times$ Eddington are reached for a 10 M⊙ black hole and
$\dot{m}_{0}$ $\approx$ 7000 $\times$ Eddington M⊙ for a 1.4 M⊙ neutron star
(see figure 2). Such rates are safely above those inferred for known ULXs
(Vasilopoulos et al., 2020). Below these limits, and for an orientation
permitting a view into the wind-cone, one would expect a positive correlation
between the X-rays and UV. It is conceivable that an observer could be
oriented such that the closing of the wind cone inhibits their ability to see
the collimated emission. In this case, there would be a change in the ratio
between hard and soft X-ray emission (the soft being relatively more visible)
accompanying a drop in $T_{\rm sph}$. This would result in an anti-correlation
between $T_{\rm sph}$ and the UV brightness and a more complex correlation
with spectral hardness.

For a fixed accretion rate, a change in the inclination of the disc/wind,
driven by precession (e.g. Middleton et al. 2018; Middleton et al. 2019b)
would result in changes to the X-ray spectral colours similar to the case of
the wind-cone closing (see Middleton et al. 2015b). In short, when the cone
precesses to higher inclinations to our line-of-sight, the X-ray emission
should diminish, and the low frequency emission should become brighter,
leading to an anti-correlation between the X-ray and UV emission (assuming the
accretion rate is high enough for the wind photosphere to emit substantially
at such low energies).

###  2.2 Irradiated outer disc

It has been suggested that the outer accretion disc could be irradiated by
X-ray emission from the inner regions after scattering by the wind (Sutton et
al., 2013). As long as this irradiating SED has sufficient intensity above 2
keV, down-scattering of these photons can produce a UV-shoulder (Gierliński et
al., 2008). Exploring irradiation requires RMHD simulations and radiative
transfer calculations to follow the photons from the inner regions to the
outer disc. However, such calculations have yet to be performed, and so we
instead base our reasoning on a simplified picture (see figure 3).

Figure 3: Schematic for the geometry of an irradiated outer disc, an increase
in the mass accretion rate results in the spherization radius moving outwards
and decreasing in temperature. High energy photons produced in the inner parts
of the accretion flow are scattered by the large scale-height wind cone.

Should the accretor be a black hole or a low dipole field neutron star
($\lesssim 10^{9}G$), then the spectrum from the innermost regions is
insensitive to accretion rate (Poutanen et al., 2007) due to mass loss at
larger radii. If the ULX contains a high dipole field-strength neutron star
(up to $\sim$1013G), the emergent spectrum above 2 keV is predicted to be
dominated by emission from the accretion column, with photons scattering to
escape the magnetosphere (Mushtukov et al., 2017). Assuming the magnetospheric
radius lies within $r_{\rm sph}$, then the accretion rate through the
magnetosphere is mostly insensitive to the accretion rate at larger radii (see
e.g. King & Lasota 2020 although see also Chashkina et al. 2019). Assuming the
intrinsic spectrum from the accretion column above 2 keV (see Brightman et al.
2016; Walton et al. 2018a) does not change, we need only consider the changes
in the scattering medium between the outer disc and inner regions. As the
accretion rate at large radius increases, the wind cone closes and the optical
depth of the wind increases (see figure 3). There are then more scatterings
within the wind cone, which reduces the energy of photons created in the inner
regions; any escaping photons are therefore likely to be at lower energy and
less likely to thermalise in the outer disc (Gierliński et al., 2008). The
converse is true for a drop in accretion rate.

For a fixed observer inclination but a varying accretion rate, the presence
(or lack of) a correlation between the X-rays and UV resulting from an
irradiated outer disc once again depends on whether an observer can see into
the wind cone. Should an observer be able to view the innermost regions
directly, then an increase in accretion rate will lead to an increase in X-ray
flux (and decrease in Tsph) and a drop in UV flux, as fewer hard X-rays
impinge on the outer disc. Should an observer instead view the inner flow at
higher inclinations, then a more complex change in spectral hardness will
result (as described in Middleton et al. 2015b). For a fixed accretion rate, a
change in inclination of the inner disc and wind will not change the UV
emission (even if the wind were to tilt away from us, it would still irradiate
the far side of the disc) and the UV and X-rays will be uncorrelated.

|  |  | Flux
---|---|---|---
Prediction | $\dot{m}$ | Inclination | X-ray | UV
Wind Photosphere | Low | Low (face-on) |  |
Wind Photosphere | Low | Intermediate |  |
Wind Photosphere | Low | High (Edge-on) |  |
Wind Photosphere | High | Low (face-on) |  |
Wind Photosphere | High | Intermediate |  |
Wind Photosphere | High | High (Edge-on) |  |
Irradiated outer disc | Low | Low (face-on) |  |
Irradiated outer disc | Low | Intermediate |  |
Irradiated outer disc | Low | High (Edge-on) |  |
Irradiated outer disc | High | Low (face-on) |  |
Irradiated outer disc | High | Intermediate |  |
Irradiated outer disc | High | High (Edge-on) |  |
Table 1: Predictions of the relative X-ray to UV emission originating from the
wind photosphere and irradiation of the outer disc. The blue and red bars
provide a visual representation of the relative observed intensity of the
emitted radiation in the X-ray and UV bands. These estimates are approximate
and quantized to four discrete lengths to serve as a qualitative indicator.

###  2.3 Irradiated companion star

A third option, distinct from the above two – and proposed to explain the
anti-phase optical super-orbital period seen in NGC 7793 P13 (Fürst et al.,
2021) – is that the X-ray cone sweeps over the companion star and hard X-rays
thermalise in the outer layers of the star leading to enhanced low frequency
emission (Motch et al., 2014). To explore this scenario, we simulate a cone of
X-ray emission irradiating a star at different orbital phases and different
orientations relative to each other and the observer. Figure 4 shows a
schematic of how the irradiated area (shown in light blue) from a cone of
X-ray emission may vary as it sweeps over the star. The irradiated projected
area seen by an observer is subject to many variables, including the size and
shape of the orbit, star and cone, as well as the position of the observer.
For further elaboration on the specific implementations of the irradiated
companion model, please refer to Appendix A.

Figure 5 shows how the irradiated fraction, defined as the projected area seen
by the observer divided by the maximum possible projected area
$A_{\mathrm{proj}}/\pi R_{\star}^{2}$, varies as a function of the (circular)
orbital phase. The transit is simulated for a cone sweeping directly through
the centre of the star and is shown for three different observer inclinations.
It is worth noting that the inclination parameter alone is not sufficient to
fully describe the configuration of the system and the resulting irradiated
area, this is as the entire system can be rotated while the inclination
remains unchanged resulting in different projected areas for the same
inclination value. Never the less, for illustrative purposes figure 5 shows
how the profile of a transit may be significantly altered by the observer’s
position.

If we were to assume that the axis of the wind-cone is perpendicular to the
orbital axis, only wind-cone half-opening angles ($\theta/2$) which satisfy
$\theta/2>\mathrm{arccos}(R_{\star}/a)$ would be able to irradiate the star
(where $R_{\star}$ is the radius of the companion star and $a$ is the average
semi-major axis of the orbit). For example, a half-opening angle of
$10^{\circ}$ would require a $R_{\star}/a\approx$ 1 which poses issues for
stable accretion. Allowing for larger opening angles will of course lower the
value of the ratio of $R_{\star}/a$ although, as super-critical accretion
demands a half-opening angle of at least $45^{\circ}$ (where the scale-height
of the disc is approximately unity), this requires $R_{\star}/a>$ 0.7. Thus,
large amounts of irradiation likely require an offset between the axis of the
wind cone and that of the orbit (which may naturally lead to precession of the
wind cone itself Middleton et al. 2018; Middleton et al. 2019b).

Estimates for the eccentricity of ULX orbits have been obtained for a few
pulsating sources, where the pulse arrival time allows for the modelling of
the orbital parameters of the system. Based on modelling of the X-ray
pulsations in NGC 7793 P13, (Fürst et al., 2018) find an eccentricity of
$e\leq 0.15$, whilst (Bachetti et al., 2014) found that M82 X-2 had a near
circular orbit of $e=0.003$. In the absence of any detailed orbital
information, we therefore assume a simple circular orbit in our model.

We have investigated the potential impact of an irradiated companion,
illuminated by a cone of X-ray emission, on the observed UV/optical emission
in ULXs. Our model is subject to limitations and uncertainties, and it is
possible that specific conditions may be required for its geometry to be a
reality in these sources. However, despite these challenges, it may be
possible in the future to determine the ephemeris based on the observed
profiles seen in the UV/optical light curves of these sources. High-resolution
spectroscopy can also be used to determine whether significant heating of the
star’s outer layers is present from specific spectral lines. In the next
section, we will describe our approach to obtaining observations and reducing
data to create long-term light curves for a sample of ULXs.

Figure 4: Schematic of irradiation of the secondary star as a tight cone of
X-ray emission travelling on an elliptical orbit. Figure 5: How the irradiated
fraction, defined as $A_{\mathrm{proj}}/\pi R_{\star}^{2}$, varies as a
function of orbital phase transiting through a cone of emission with opening
angle $\theta=20^{\circ}$. The transit is shown for three different observer
inclinations, $i=0,22.5$ and $45$ degrees. A ratio of $R_{\star}/a=0.5$ is
assumed.

##  3 Observations and Data Reduction

Source Name | RA | DEC | $\lambda$ | Position reference | $D$ |  $D$ meth. |  $D$ reference | UV
---|---|---|---|---|---|---|---|---
| "h:m:s" | "d:m:s" |  |  | $\mathrm{Mpc}$ |  |  | Emis.
V404 Cygni | 20 24 03.8254 | +33 52 01.962 | O | Gaia Collaboration (2018) | 0.0023 | parallax | Miller-Jones et al. (2009) | PL
Swift J0243.6+6124 | 02 43 40.4252 | +61 26 03.757 | O | Gaia Collaboration (2020) | 0.0055 | parallax | Gaia Collaboration (2020) | PL
SS433 | 19 11 49.5647 | +04 58 57.827 | O | Gaia Collaboration (2020) | 0.0055 |  | Blundell & Bowler (2004) | PL
SMC X-3 | 00 52 05.6251 | -72 26 04.228 | O | Gaia Collaboration (2020) | 0.0600 | cepheid | Karachentsev et al. (2017) | PL
IC10 X-1 | 00 20 29.09 | +59 16 51.9 | X | Bauer & Brandt (2004) | 0.7943 | redshift | Lianou et al. (2019) | N
M31 ULX-1 | 00 42 53.15 | +41 14 22.9 | X | Kaur et al. (2012) | 0.8200 | T-RDB | Karachentsev et al. (2017) | N
M33 ULX-1 | 01 33 50.8965 | +30 39 36.630 | O | Gaia Collaboration (2020) | 0.9300 | T-RDB | Karachentsev et al. (2017) | PL
NGC300 ULX-1 | 00 55 04.86 | -37 41 43.7 | O | Barbon et al. (2008) | 2.0230 | redshift | Lianou et al. (2019) | PL
NGC55 ULX | 00 15 28.89 | -39 13 18.8 | X | Lin et al. (2012) | 2.1100 | T-RDB | Karachentsev et al. (2017) | Edg
IC342 ULX-1 | 03 45 55.612 | +68 04 55.29 | O | Feng & Kaaret (2008) | 3.4356 | redshift | Lianou et al. (2019) | N
IC342 ULX-2 | 03 46 15.61 | +68 11 12.8 | X | Heida et al. (2014) | 3.4356 | redshift | Lianou et al. (2019) | N
NGC4945 XMM-1 | 13 05 32.89 | -49 27 34.1 | X | Swartz et al. (2004) | 3.4674 | redshift | Lianou et al. (2019) | E
Holmberg II X-1 | 08 19 28.99 | +70 42 19.4 | X | Heida et al. (2014) | 3.4674 | redshift | Lianou et al. (2019) | E
M81 X-6 | 09 55 32.95 | +69 00 33.6 | X | Heida et al. (2014) | 3.5975 | redshift | Lianou et al. (2019) | E
M82 X-2 | 09 55 51.040 | +69 40 45.49 | X | Kaaret et al. (2006) | 3.6141 | redshift | Lianou et al. (2019) | E
NGC253 X-2 | 00 47 32.97 | -25 17 50.2 | X | Liu & Bregman (2005) | 3.6983 | redshift | Lianou et al. (2019) | E
NGC253 X-9 | 00 47 22.59 | -25 20 50.9 | X | Heida et al. (2014) | 3.6983 | redshift | Lianou et al. (2019) | N
NGC247 ULX-1 | 00 47 04.00 | -20 47 45.7 | X | Liu & Bregman (2005) | 3.7200 | T-RDB | Karachentsev et al. (2017) | N
NGC7793 P13 | 23 57 50.90 | -32 37 26.6 | X | Pannuti et al. (2011) | 3.7325 | redshift | Lianou et al. (2019) | PL
Holmberg IX X-1 | 09 57 53.290 | +69 03 48.20 | O | Abazajian et al. (2009) | 3.8500 | T-RDB | Karachentsev et al. (2017) | E
NGC1313 X-1 | 03 18 20.00 | -66 29 10.9 | X | Heida et al. (2014) | 4.2500 |  | Tully et al. (2016) | N
NGC1313 X-2 | 03 18 22.00 | -66 36 04.3 | X | Liu & Bregman (2005) | 4.2500 |  | Tully et al. (2016) | E
NGC5204 ULX-1 | 13 29 38.62 | +58 25 05.6 | X | Heida et al. (2014) | 4.5900 | T-RDB | Karachentsev et al. (2017) | E
UGC6456 ULX | 11 28 03.000 | +78 59 53.41 | O | Vinokurov et al. (2020) | 4.6300 | T-RDB | Karachentsev et al. (2017) | E
NGC4395 ULX-1 | 12 26 01.53 | +33 31 30.6 | X | Heida et al. (2014) | 4.7600 |  | Tully et al. (2016) | E
M83 ULX-1 | 13 37 05.13 | -29 52 07.1 | X | Long et al. (2014) | 4.8978 | redshift | Lianou et al. (2019) | N
M83 ULX-2 | 13 37 20.12 | -29 53 47.7 | X | Liu & Bregman (2005) | 4.8978 | redshift | Lianou et al. (2019) | E
NGC5408 ULX-1 | 14 03 19.63 | -41 22 58.7 | X | Heida et al. (2014) | 5.3211 | redshift | Lianou et al. (2019) | N
NGC6946 ULX-1 | 20 35 00.11 | +60 09 08.5 | X | Lin et al. (2012) | 6.7298 | redshift | Lianou et al. (2019) | E
NGC6946 ULX-3 | 20 35 00.74 | +60 11 30.6 | X | Swartz et al. (2004) | 6.7298 | redshift | Lianou et al. (2019) | PL
M101 ULX-1 | 14 03 32.38 | +54 21 03.0 | X | Heida et al. (2014) | 7.1121 | redshift | Lianou et al. (2019) | N
NGC4559 ULX-1 | 12 35 51.71 | +27 56 04.1 | X | Heida et al. (2014) | 7.1450 | redshift | Lianou et al. (2019) | E
M51 ULX-7 | 13 30 01.01 | +47 13 43.9 | X | Heida et al. (2014) | 7.6000 | redshift | Cappellari et al. (2011) | E
NGC5585 ULX | 14 19 39.39 | +56 41 37.8 | X | Heida et al. (2014) | 7.8300 |  | Tully et al. (2016) | PL
NGC925 ULX-1 | 02 27 27.53 | +33 34 42.9 | X | Heida et al. (2014) | 9.2045 | redshift | Lianou et al. (2019) | E
NGC925 ULX-2 | 02 27 21.52 | +33 35 00.8 | X | Heida et al. (2014) | 9.2045 | redshift | Lianou et al. (2019) | E
NGC7090 ULX-3 | 21 36 31.94 | -54 33 57.2 | X | Lin et al. (2012) | 9.5060 | redshift | Lianou et al. (2019) | Edg
NGC5907 ULX | 15 15 58.60 | +56 18 10.0 | X | Swartz et al. (2011) | 17.2187 | redshift | Lianou et al. (2019) | Edg
NGC1365 X-1 | 03 33 34.60 | -36 09 35.0 | X | Liu & Bregman (2005) | 17.2982 | redshift | Lianou et al. (2019) | N
NGC1365 X-2 | 03 33 41.85 | -36 07 31.4 | X/O | Strateva & Komossa (2009) | 17.2982 | redshift | Lianou et al. (2019) | PL
NGC1042 ULX-1 | 02 40 25.62 | -08 24 28.9 | X | Lin et al. (2012) | 19.2000 |  | Oey et al. (2007) | N
ESO 243-49 HLX-1 | 01 10 28.30 | -46 04 22.3 | X | Webb et al. (2010) | 115.3500 |  | Tully et al. (2016) | N
Table 2: Source information for those ULXs investigated in this paper, sorted
by host galaxy distance. $\lambda$ corresponds to the wavelength (X-ray or
optical) that the position was acquired in. The ‘D meth.’ column denotes the
method used for determining the distance (where T-RDB is Tip of the red-giant
branch). The ‘UV Emis.’ column corresponds to the spatial extent of the
UV/optical emission based on visual inspection of the stacked UVOT images.

The Neil Gehrels Swift Observatory (Swift) (Gehrels et al., 2004) is capable
of observing simultaneously in both the UV and X-ray bands via its UVOT
(Roming et al., 2005) and XRT (Burrows et al., 2005) cameras, and so provides
a unique opportunity to explore whether observations of ULXs are consistent
with any of the predictions discussed above. We begin by creating our ULX
sample by crossmatching several ULX catalogues, (Earnshaw et al., 2019;
Kovlakas et al., 2020; Bernadich et al., 2022; Walton et al., 2022) with the
Swift Master Catalogue (SWIFTMASTR), accessible via
HEASARC111https://heasarc.gsfc.nasa.gov. For the sake of comparison to nearby
sources where data quality is high, we also include three extensively studied
Galactic sources which reach super-Eddington rates of accretion, Swift
J0243.6+6124, SS433 and V404 Cygni. Swift J0243.6+6124 is known to contain a
magnetised neutron star and appear as a ULX (van den Eijnden et al., 2020),
SS433 is widely considered to be an edge-on ULX (Fabrika 2004; Middleton et
al. 2021) and V404 Cygni is a LMXB which reached around or just in excess of
its Eddington luminosity during its 2015 outburst (Motta et al. 2017).

We locate all observations where the source lies within the nominal (23.6’)
XRT field-of-view. Due to the differences between the XRT and UVOT field-of-
view, there is a mismatch between the number of observations in both bands. We
set the requirement that there be 20 observations in both bands for a source
to appear in our sample. For each source, we manually cross-matched with
SIMBAD to obtain distances and positions. The final sample with relevant
information is shown in Table 2.

###  3.1 XRT Data Reduction

XRT data was extracted using the standard Swift/XRT processing pipeline (Evans
et al., 2009), using the SIMBAD coordinates of the source, and the ‘simple’
centroid method with a positional error of 1" (see Evans et al. 2009). We
extract light curves in three bands, full $(0.3-10.0\ \mathrm{keV})$, soft
$(0.3-1.5\mathrm{keV})$ and hard $(1.5-10.0\mathrm{keV})$, with the time
binning set at a single point per observation. Using the full band data, we
set the minimum source significance – defined as the number of counts in the
source region divided by the square root of the counts in the source and
background ($\mathrm{C_{src}}/\sqrt{\mathrm{C_{src}}+\mathrm{C_{bkg}}}$) – to
the default value of 3; observations where the source is not detected at or
above this threshold yield upper limits. The pipeline additionally calculates
the hardness ratio, defined as the ratio of hard to soft count rates
$HR=\mathrm{C_{hard}}/\mathrm{C_{soft}}$. $1\sigma$ errors are provided for
each of the above measurements. The pipeline also includes GOOD and BAD
values, which indicates whether the pipeline was able to obtain a centroid in
a given snapshot, (such that BAD values are potentially unreliable).

Figure 6: Mean slope $\bar{m}$ (x) plotted against the inverse coefficient of
variation of the Pearson correlation coefficient $\hat{\gamma}_{r}$ for all
simulations.

###  3.2 UVOT Data Reduction

UVOT data were processed locally using HEASARC v6.29 and the November 8, 2021
CALDB. We stacked all UVOT and XRT observations, combining them into a single
image and manually inspected to identify any clear counterparts.

For each source, a circular extraction region with a radius of 5" was centred
on the Swift XRT position, while background regions were manually positioned
in a contaminant-free location with a size of 15" as recommended by the UVOT
data analysis manual222
https://swift.gsfc.nasa.gov/analysis/UVOT_swguide_v2_2.pdf. The distances
between the centres of the UVOT and XRT (SIMBAD) source regions as well as the
position of the UVOT background regions are provided in the supplementary
material.

Level 2 UVOT images were processed locally using the uvotimsum to combine all
snapshot extensions, then uvotsource with a signal-to-noise threshold of 3 was
used to obtain photometric magnitudes in a given observation. We then
determined whether the source was detected using the NSIGMA column provided as
output from uvotsource. All observations of a given source were combined to
produce a long term light-curve in all of the UVOT bands: V (5469 Å), B (4392
Å), U (3465 Å), UVW1 (2600 Å), UVM2 (2246 Å), UVW2 (1928 Å).

##  4 Analysis and Results

###  4.1 UV Counterparts

The‘UV Emis.’ column in Table 2 indicates the spatial nature of the UV
emission of the source based upon the stacked images. Sources with point-like
UV counterparts and emission regions comparable to Swift’s PSF are labelled as
‘PL’, sources in or near regions of extended UV emission as ‘E’, those that do
not display strong UV emission in the images are labelled as ‘N’, and three in
edge-on galaxies ‘Edg’ where point-sources in the UV/optical are difficult to
disentangle from the galactic emission.

###  4.2 Testing for linear X-ray/UV correlations

Figure 7: Distribution of the correlation coefficient for 10 of the sources
investigated. Error bars correspond to 1$\sigma$, calculated from 10,000 Monte
Carlo iterations. Colours correspond to the different UVOT filters used (See
legend in figures 1 & 2) and are displayed in decreasing energy from top to
bottom. The marker on the datapoint denotes which of the three XRT band with
which the correlation was carried out with: The full band is denoted with a
dot ($\bullet$), the hard band by an upwards triangle ($\blacktriangle$) and
the soft band with a downwards triangle ($\blacktriangledown$), the
correlations against the hardness ratio are not shown here. The same colour
and marker may appear multiple times as separate simulations were conducted
with the inclusion or exclusion of upper limits and/or bad data points.

Figure 8: See caption for figure 7.

Figure 9: See caption for figure 7.

Figure 10: See caption for figure 7.

A cut of $\pm 5\sigma$ around the mean count rate was applied to both the UVOT
and XRT data prior to simulating. This cut resulted in only 5 observations
being excluded due to UVOT outliers, which were due to observations where the
source was close to the edge of the detector or the background region was
located within the fits image but outside the detector plate. Around $\sim 30$
observations were excluded due to XRT outliers, some of these are instrumental
however others may be real and be due to source flaring, we find that around
0-3 observations are excluded for each lightcurve that may be composed of
several hundred data points, the removal of these outliers results in
marginally weaker detected correlations meaning that our results consistute a
slightly more conservative estimate. To search for linear correlations in the
clean X-ray and UVOT light curves, we employ the following method: for each
observation data point we sampled the $1\sigma$ error on the count rate in
both the XRT and UVOT bands assuming Gaussian distributions; if the
observation contained an upper limit, the data point was sampled assuming a
uniform distribution between 0 and this value. This sampling allows us to
obtain realisations of the light curve with the same time-sampling as the
original. We then performed a least-squares fit with a straight line of the
form $y=mx+c$ to the re-sampled data points and calculated the Pearson
correlation coefficient $r$ given by:

$r=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}}}$
(2)

where $x_{i}$ and $y_{i}$ are the ith values in the sample, while $\bar{x}$
and $\bar{y}$ are the means over all the $n$ data points. This resampling and
fitting process was repeated 10,000 times to obtain posterior distributions
for $r$, $m$ and $c$, from which the mean and standard deviation were
calculated.

To assess how well constrained are the posterior distributions for the fit
parameters, we calculate the inverse coefficient of variation (ICV)
($\hat{\gamma}_{\mathrm{par}}$) by dividing the mean of the fit parameters
($\overline{\mathrm{par}}$) by their standard deviation
($\sigma_{\mathrm{par}}$), i.e.
$\hat{\gamma}_{\mathrm{par}}=\overline{\mathrm{par}}/\sigma_{\mathrm{par}}$.
The absolute value of $\hat{\gamma}_{\mathrm{par}}$ may be interpreted as a
significance value, with higher values corresponding to more strongly peaked
distributions (i.e. the distribution has less scatter).

Simulations were carried out on all possible combinations of the four X-ray
bands (full, soft, hard, HR), and the six UVOT filters. Simulations were
additionally carried out including and excluding observations labelled as BAD
in the XRT pipeline, as well as data points labelled as upper limits in the
full XRT band. The simulation grid implies that a single source may have a
total of 24 possible simulation combinations for the full XRT band, and 36 for
the remaining XRT bands (hard, soft and HR), giving a maximum of 60
correlations per source, assuming that the source has been visited in all UVOT
bands and its XRT light curves contain both BAD and upper limit data points.
In practice this is not the case and the total number of sets of simulations
for each source is almost always lower than 60.

Figure 6 shows the mean value of the slope ($m$) and the ICV of the Pearson
correlation coefficient ($\hat{\gamma}_{r}$) across all of our simulations.
Sources appear multiple times on the plot due to the aforementioned grid in
simulation space. The diagonal distance from the centre in this parameter
space might provide an indication that a correlation exists – the upper right
corresponding to positive correlations, the lower left to negative. However as
we will demonstrate, figure 6 may not encode sufficient information to locate
more physical correlations.

Figures 7, 8, 9 and 10 show the correlation coefficient obtained from our
simulations for different X-ray bands and UVOT filters. Each errorbar
corresponds to a discrete simulation run and is coloured based on the UVOT
filter used in the correlation and has a marker corresponding to the XRT band
it was correlated against (see caption in figure 7). If the errorbars for a
single source consistently deviate far from the dotted line (zero) and are
well constrained, then it indicates a consistent correlation between the UVOT
filter and the X-ray bands. The sources V404 Cygni, Swift J0243.6+6124, SS433,
SMC X-3, NGC300 ULX-1, NGC 7793 P13 and Holmberg IX X-1 show the clearest
cases of such correlation. Some errorbars, such as those seen for the V band
in M51 ULX-7, have large sizes due to non-Gaussian distributions for $r$ over
the 10,000 Monte-Carlo simulations.

##  5 Discussion & Conclusions

Although ULXs are defined empirically by their X-ray luminosity, they are
well-known to emit over a broad energy range. Indeed, bright optical/UV
emission (in excess of 10${}^{39}\ \mathrm{erg}\ \mathrm{s}^{-1}$) is observed
to originate in both Galactic super-critical accretors (SS433: Dolan et al.
1997) and well studied ULXs (NGC 6946 ULX-3: Kaaret et al. 2010) with a
mixture of potential origins. In this paper, we have made predictions about
how the UV and X-ray emission might correlate (or anti-correlate) depending on
the dominant mechanism for the low frequency radiation: the irradiated donor
star, irradiated outer disc or wind photosphere.

Based on simple arguments, we predict a lack of any correlation between the UV
and X-ray emission where the star is irradiated by a wind-cone, as, with the
exception of an orientation where the central regions are eclipsed, the X-ray
emission should remain constant (in the absence of precession or mass
accretion rate changes). In the case of disc irradiation or precession of the
super-critical disc/wind, the exact nature of the correlation depends mostly
on the observer inclination and any changes in accretion rate at large radius.
Certainly for a fixed inclination angle (again, in the absence of precession),
a negative correlation would be expected for disc irradiation as the X-ray
emission (assumed here to originate within the wind-cone) increases with
increasing $\dot{m}$, but the optical depth to the outer regions also
increases. A negative correlation must also result in the case of precession,
but can deviate and even become positive when changes in $\dot{m}$ are
invoked. In the absence of precession, the emission in both bands is a
sensitive function of inclination (see Poutanen et al. 2007; Middleton et al.
2015a).

NASA’s Swift satellite offers an unrivalled opportunity to explore the long
timescale changes in both the X-ray and UV bands through simultaneous
observing by the XRT and UVOT instruments. For a sample of $\sim 40$ ULXs, we
have extracted the UV and X-ray light curves and searched for correlations,
placing constraints on the Pearson coefficient via Monte Carlo simulations.

Our sample contains three Galactic sources, V404 Cygni, Swift J0243.6+6124 and
SS433. Strictly speaking, Swift J0243.6+6124, is a ULX under the classical
empirical definition, however the other two sources are indirectly identified
as supercritical accretors and share many of the same properties of extra-
galactic ULXs. V404 Cygni and Swift J0243.6+6124 display among the strongest
positive correlations in the sample; the strength of these correlations can be
ascribed to large outbursts which subsequently decay over time in both the
X-ray and UV/optical bands. The outbursts from both of these sources have been
previously studied (see Oates et al. 2019 and Liu et al. 2022), however, to
our knowledge this is the first time UVOT data has been used in a study for
Swift J0243.6+6124. SMC X-3 is another source that displayed an X-ray outburst
in 2016 (Weng et al. 2017; Tsygankov et al. 2017; Townsend et al. 2017);
before dimming below the Swift detection threshold, many of the observations
were taken in windowed timing mode without simultaneous observations in the
UV. The combination of the large dynamic range in X-ray count rate and the low
number of datapoints may well bias our inferred correlations for this source.

NGC 300 ULX-1 has a bright counterpart and the system has been identified as a
ULX pulsar with a supergiant companion (Carpano et al., 2018; Heida et al.,
2019); it is also the closest of the ULXs in our sample to consistently
display correlated variability. Initially thought to be supernova due to its
transient nature in the optical (Monard, 2010), the source has been studied
using the UVOT by (Villar et al., 2016). We observe that the B band appears to
display a negative correlation. From the lightcurves (available in the
supplementary material), we observe that the source was bright in the B band
around 2010 with count rates peaking at $\sim 10-15\mathrm{ct\ s^{-1}}$ and
X-ray count rates of $\sim 0.05\mathrm{ct\ s^{-1}}$ (in the full X-ray band).
In 2017 the opposite behaviour occurred with X-ray count rates peaking at
$\sim 0.12\mathrm{ct\ s^{-1}}$ and B band count rates of $\sim 4\mathrm{ct\
s^{-1}}$.

There are clearly several objects where the correlation between bands is
negative. This can result from precession at fixed accretion rate, a changing
accretion rate without precession but at viewer inclinations not into the
wind-cone, or from irradiation of the outer disc for viewer inclinations into
the wind-cone. This appears to be the case for NGC 300 ULX-1 and Holmberg IX
X-1 (see bottom left figure 6), both of which are systems where precession has
been suggested by previous authors (Weng & Feng, 2018; Vasilopoulos et al.,
2019). It is also apparent that in some cases, the nature of the correlation
changes for a single source between bands (see Holmberg IX X-1 in figure 8.
This likely indicates that each band is affected to a differing degree by one
of the different processes mentioned above.

Figure 11: Superimposed light curves with different phases (top row) obtained
from the the combination of a sigmoid-like light curve (black) from ULXLC, and
a sinusoidal profile (dotted) predicted for the UV emission. The bottom row
shows the resulting correlation shapes and the associated $r$ value. The
parameters for the ULXLC used were $\theta=5^{\circ}$, $\Delta i=9^{\circ}$
and $i=15^{\circ}$ (see Dauser et al. 2017).

In our analysis, we have assumed the simplest case of a linear correlation.
However, numerous sources show clear patterns of behaviour where an ‘L’ shape
is mapped out (see figure 12), while others show non-linear shapes; a simple
linear correlation test (Pearson) is naturally less sensitive to detecting
such behaviours. Intriguingly, an ‘L’ shape naturally results from precession
of the wind-cone; using ULXLC (Dauser et al., 2017) we create an example X-ray
lightcurve using the parameters $\theta=5^{\circ}$, $\Delta i=9^{\circ}$ and
$i=15^{\circ}$, which provides a quasi-sigmoid profile (see the first row of
figure 11). We assume that the UV/optical profile is sinusoidal (assuming a
quasi-spherical geometry of the outer photosphere of the wind), and plot the
subsequent profile of the correlation that would result for different phases
of precession. In the case of $\phi=0$, the two curves produce a triangular
shape which results in a Pearson correlation coefficient of $r=0$, which
demonstrates that even though the two curves are correlated, the value of $r$
does not reveal this. As the phase is changed to $\pi/2$ the curves produce a
characteristic ‘L’ shape which is similar to some of those seen in our light-
curves (such as Holmberg IX X-1). This preliminary result may suggest that
more complicated (physical) models may be required to understand and search
for correlated variability between bands, and that systems which show such ‘L’
shaped correlations, may be our clearest indication of precession.

Figure 12: Row 1 & 2: The full band XRT and UVW1 light curves for Holmberg IX
X-1 (plotted sequentially) with simultaneous observations. Row 3: The same
light curves overlaid. Row 4: The correlated data points, with and without
errors, plotted with the best fitting linear model in red (1 and 2$\sigma$
contours are shaded in grey). Row 5: The distribution of the fit parameters
$r$, $m$ and $c$ for these correlations.

## Acknowledgements

NK acknowledges support via STFC studentship project reference: 2115300.

MM is supported by STFC through grant ST/V001000/1.

This work made use of data supplied by the UK Swift Science Data Centre at the
University of Leicester.

The authors thank the anonymous referee for useful suggestions.

## Data Availability

The data underlying this article are available in the supplementary material
(online).

The data includes the XRT and UVOT lightcurves, Source tables with extraction
regions, mean count rates and number of observations, the supplementary
material also includes the full set of simulation results in this paper.

The source code for this paper may be found at:
https://github.com/nx1/anticorr_data/

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##  Appendix A Irradiated Companion Model

We model a cone of emission centred at the origin $(0,0,0)$ with opening
parameter $c=\mathrm{sin}(\theta/2)$ where $\theta$ is the full opening angle
of the cone. We then place a companion star, modelled as a sphere placed at
$(x_{0},y_{0},z_{0})$ with radius $r$.

The equations and the cone and sphere in Cartesian coordinates are given by
equations 3 and 4 respectively, this geometry allows us to define the $z$ axis
as being in-line with the radiation of the beamed cone of emission and adjust
the position of the star relative to this axis.

$\frac{x^{2}+y^{2}}{c^{2}}=z^{2}$ (3)

$(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}$ (4)

We then solve 3 and 4 simultaneously for $x$, $y$ using the SymPy Python
package (Meurer et al., 2017), this provides four solutions, positive and
negative for both $x(z)$ and $y(z)$.

Figure 13: The intersection of a cone with opening angle $\theta=15^{\circ}$
and a star at position $x_{0}=2$, $y_{0}=0$, $z_{0}=10$ and a radius of
$r=2.2$. These set of parameters make it such that the cone is grazing the
star. This figure is an intermediate step in our calculations where we have
yet to cull the path due to obscuration of the star itself.

These equations allow us to calculate the $x$ and $y$ coordinates for a given
$z$ that corresponds intersection between the cone of emission and the sphere,
by ignoring the $z$ coordinate and plotting only the $x$ and $y$ values, we
may obtain the projection that would be observed by an observer situated down
the line-of-sight of the $z$ axis. Since the cone is centred at the origin, by
applying a rotation $R_{x}(i)$ or $R_{y}(i)$ and re-projecting down the z
axis, we can simulate the effect of an observation at a specified inclination
(with respect to the jet axis).

We then cut any values of the intersection that are above the hemisphere of
the star ($z>z_{0}$) as these parts would not be illuminated by the beam. If
the configuration is such that the star passes through one wall of the cone as
seen in figure 13, then we consider this as an edge case and explicitly
calculate the new path by joining the culled path with the arc created from
the circle bounded by the hemisphere of the star.

The result is a closed path, defined by x, y coordinates of the area that
would be seen by an observer, the area is calculated via two methods, one
using the shoelace formula and the other via triangulation both of
consistently produce the same result.