# The X-ray spectral-timing contribution of the stellar wind in the hard state
of Cyg X-1

E. V. Lai,1 B. De Marco1,2, A. A. Zdziarski1, T. M. Belloni3, S. Mondal1,3, P.
Uttley4, V. Grinberg5, J. Wilms6, A. Różańska1

1Nicolaus Copernicus Astronomical Center, PAN, ul. Bartycka 18, Warsaw 00-716
Poland
2Departament de Física, EEBE, Universitat Politècnica de Catalunya, Av. Eduard
Maristany, 16, Barcelona 08019, Spain
3INAF - Osservatorio Astronomico di Brera Via E. Bianchi 46, I-23807 Merate,
Italy
4Anton Pannekoek Institute, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands
5European Space Agency (ESA), European Space Research and Technology Centre
(ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, the Netherlands
6Dr. Karl Remeis-Observatory, University of Erlangen-Nuremberg, Sternwartstr.
7, 96049 Bamberg, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The clumpy stellar wind from the companion star in high mass X-ray binaries
causes variable, partial absorption of the emission from the X-ray source. We
studied XMM-Newton observations from the 7.22 d–long “Cyg X-1 Hard state
Observations of a Complete Binary Orbit in X-rays” (CHOCBOX) monitoring
campaign, in order to constrain the effects of the stellar wind on the short-
timescale X-ray spectral-timing properties of the source. We find these
properties to change significantly in the presence of the wind. In particular,
the longest sampled timescales (corresponding to temporal frequencies of
$\nu\sim$ 0.1–1 Hz) reveal an enhancement of the fractional variability power,
while on the shortest sampled timescales ($\nu\sim$ 1–10 Hz) the variability
is suppressed. In addition, we observe a reduction (by up to a factor of
$\sim$ 1.8) of the otherwise high coherence between soft and hard band light
curves, as well as of the amplitude of the hard X-ray lags intrinsic to the
X-ray continuum. The observed increase of low frequency variability power can
be explained in terms of variations of the wind column density as a
consequence of motions of the intervening clumps. In this scenario (and
assuming a terminal velocity of $v_{\infty}=2400\ {\rm km\ s^{-1}}$), we
obtain an estimate of $l\sim$ 0.5–1.5 $\times 10^{-4}R_{\ast}$ for the average
radial size of a clump. On the other hand, we suggest the behaviour at high
frequencies to be due to scattering in an optically thicker medium, possibly
formed by collision of the stellar wind with the edge of the disc.

###### keywords:

X-rays: binaries; black hole physics; X-rays: individual (Cyg X-1); stars:
wind, outflows;

††pubyear: 2021††pagerange: The X-ray spectral-timing contribution of the
stellar wind in the hard state of Cyg X-1–C

##  1 Introduction

Black hole X-ray binaries (BHXRBs) are observed to occasionally and
recursively undergo dramatic changes of their X-ray spectral and timing
properties (e.g. Homan & Belloni, 2005; Belloni, 2010; Belloni et al., 2011).
This behaviour is commonly ascribed to changes in the inner structure of the
accretion flow (e.g. Gilfanov, 2010), and associated with hysteresis between
two main accretion states, the so-called soft and hard states (e.g. Zdziarski
& Gierliński, 2004). During the soft state most of the observed X-ray emission
is consistent with being produced in a geometrically thin and optically thick
disc (Shakura & Sunyaev, 1973), and is characterized by a low level of rapid
(from tens of seconds to milliseconds) X-ray variability (root mean square,
rms, amplitude less than $\sim$ 5 percent, e.g. Muñoz-Darias et al. 2011). In
the hard state the X-ray spectrum is dominated by Comptonised emission. This
is due to scattering of thermal photons from the optically thick disc (e.g.
Shapiro et al., 1976; Ichimaru, 1977; Sunyaev & Truemper, 1979; Narayan & Yi,
1994) and/or of synchrotron photons from the inner hot flow (Vurm & Poutanen,
2008; Veledina et al., 2011) in a cloud of hot electrons, commonly referred to
as hot corona, located close to the black hole (BH). This state is
characterized by high levels of rapid X-ray variability (rms even exceeding 30
percent, Muñoz-Darias et al. 2011) and a complex distribution over frequency
of the X-ray variability power (e.g. Belloni et al., 2005). While the
phenomenology of these states is well established, the physical origin is
still largely debated.

Cygnus X-1 (hereafter Cyg X-1) is amongst the best studied X-ray binary
systems. It is composed of a BH with $M_{\odot}=21.2\pm 2.2\,\rm{M_{\odot}}$
and its supergiant $\rm{O}9.7\,\rm{Iab}$ companion star (HDE 226868) with
$M_{\ast}=40.6^{+7.7}_{-7.1}\,\rm{M_{\odot}}$ and $R_{\ast}=22.3\pm
1.8\,\rm{R_{\odot}}$ (Miller-Jones et al., 2021). The system is characterised
by a quasi-circular orbit with an orbital period of $5.599829(16)\,\rm{d}$
(Gies et al., 2003). Due to the size of the companion, this system is
classified as a high mass X-ray binary, where the BH accretes mass via the
strong stellar wind of the supergiant.

Like all other BHXRBs, Cyg X-1 shows complex temporal frequency dependence of
its X-ray variability power, with a power spectral density function (PSD)
characterized by two main broad “humps” in the hard state, that change
drastically to a smoother profile in the soft state (e.g. Nowak et al., 1999;
Pottschmidt et al., 2000; Pottschmidt et al., 2003; Axelsson et al., 2005;
Böck et al., 2011; Grinberg et al., 2014). In addition, highly coherent hard
($E\gtrsim 1$ keV) and soft ($E\lesssim 1$ keV) X-ray band variations are
regularly observed, whereby variable hard photons lag behind soft photons with
a delay of order 1 percent of the variability timescale (hereafter “hard
lags”, e.g. Nowak et al., 1999; Grinberg et al., 2014). Various mechanisms
have been proposed to explain the observed X-ray spectral-timing behaviour of
BHXRBs (Poutanen & Fabian, 1999; Misra, 2000), with models of inward
propagation of mass accretion rate fluctuations currently offering the most
compelling physical explanation (Lyubarskii, 1997; Kotov et al., 2001; Arévalo
& Uttley, 2006; Ingram & van der Klis, 2013; Mushtukov et al., 2018; Mahmoud &
Done, 2018a, b; Bollimpalli et al., 2020).

However, an important additional component of high mass X-ray binary systems
like Cyg X-1 is the stellar wind from the companion, early type, supergiant
star. Evidence for the presence of a stellar wind was soon found in Cyg X-1
(e.g., Li & Clark, 1974; Remillard & Canizares, 1984; Bałucińska-Church et
al., 2000; Miller et al., 2005; Poutanen et al., 2008; Hanke et al., 2009;
Grinberg et al., 2015; Miškovičová et al., 2016). The strong line-driven wind
presents an inhomogeneous structure, with clumps characterized by highly dense
regions (Owocki & Rybicki, 1984; Owocki et al., 1988; Feldmeier, 1995). When a
clump intercepts our line of sight to the X-ray source, the X-ray emission is
partially absorbed by the wind material leading to a dipping event (sudden
drop of the X-ray flux). Such events occur more frequently and are more
intense at superior conjunction, i.e. when the compact object lies behind the
donor star (at orbital phase $\phi_{\mathrm{orb}}=0$ and our line of sight
probes regions of the stellar wind closer to the surface of the companion,
e.g. Bałucińska-Church et al. 2000; Grinberg et al. 2015; Miškovičová et al.
2016).

It is reasonable to expect that wind absorption does not only affect the
observed spectrum, but also the observed X-ray variability properties of the
source (Grinberg et al., 2015, 2020). For example, motions of wind clumps
through our line of sight to the X-ray source are expected to produce X-ray
spectral-timing signatures on the relevant timescales (El Mellah et al.,
2020). Given current estimates of the size of the clumps, such timescales can
be of the order of minutes or shorter (Grinberg et al., 2017; Hirsch et al.,
2019). In addition, the wind is found to be affected by the radiation field
around the BH, resulting in stronger absorption during the hard state than
during the soft state (Boroson & Vrtilek, 2010; Nowak et al., 2011). Given the
strong flux variability characterising the hard state of BHXRBs (Muñoz-Darias
et al., 2011; Heil et al., 2012), additional wind variability due to changes
in the irradiating flux might be expected (Nicastro et al., 1999; Krongold et
al., 2007; Silva et al., 2016).

All this suggests that the observed X-ray spectral-timing characteristics of
Cyg X-1 can be significantly influenced by the stellar wind, thus affecting
any study aimed at constraining the intrinsic properties of the X-ray source.
On the other hand, the X-ray spectral-timing signatures characteristics of the
most wind-absorbed phases can be a powerful tool to determine the physical
properties of the wind (e.g. Grinberg et al., 2015; Silva et al., 2016; De
Marco et al., 2020; El Mellah et al., 2020).

Cyg X-1 has been recently the target of an unprecedented long multiwavelength
monitoring campaign, the “Cyg X-1 Hard state Observations of a Complete Binary
Orbit in X-rays” (CHOCBOX) campaign, with XMM-Newton as the primary
instrument. The XMM-Newton monitoring consists of four observations of the
source in its hard state, for a total exposure of about $\sim$ 572 ks and
covering about one and a half orbital periods (7.22 d). In this paper, we
study this long dataset, focusing on determining how the variability of the
stellar wind influences the X-ray spectral-timing properties of the source.

##  2 Data reduction

For the spectral-timing measurements, which are optimised for high count
rates, we consider only the EPIC-pn data. The four observations analysed were
carried out between 2016 May 27 and June 2, with a total exposure of $\sim$
572 ks (before data screening). The analysed data are acquired in Timing mode.
The log of the observations is reported in Tab.1. Hereafter, we will refer to
each observation using the last three digits of their ObsID (as specified in
Tab.1).

The data reduction has been carried out using the XMM-Newton Science Analysis
System (SAS) software (version 16.9.0), following standard procedures. In
order to identify and filter out time intervals affected by high particle
background, we extracted light curves with a time resolution of 1 s in the
energy range 10–15 keV (where the instrument response drops). We checked the
light curves to identify background flares, but none was found.

The analysed observations are affected by telemetry drop-outs, i.e. periods of
high count rate producing buffer overflow, during which photons are not
collected. These periods result in gaps in the data, which are accounted for
in the selection of Good Time Intervals, GTIs, using the SAS task tabgtigen.
The resulting GTIs are very short during all observations, with an average
length of $\sim$ 10 s.

We extracted source counts in the range RAWX:30–46. Using the SAS task
epatplot, we checked for the presence of pile-up. We found that the data are
affected by some fraction of pile-up, thus we mitigated it by excluding the
central (RAWX:36–39) pixels. Since the observations are in Timing mode, it is
not possible to select a source-free region for extraction of background
events, while subtracting the background from the outer columns (RAWX
intercepting the tails of the PSF) may modify the source spectrum (Ng et al.,
2010). However, the source is very bright (with a 0.5–10 keV average count
rate of $\sim$ 220 counts/s), thus the effects of background are expected to
be negligible (e.g. Ng et al., 2010). Finally, we notice that the most
absorbed periods, thus potentially more affected by the background, are
excluded in our spectral analysis (Sect. 4), while the background is not
expected to contribute to the rapid X-ray variability of BHXRBs (e.g. Uttley
et al., 2011; De Marco et al., 2017). Therefore, we decided not to subtract
the background.

Ancillary response files (ARF) and redistribution matrix files (RMF) were
extracted using the SAS tools arfgen and rmfgen. We generated an ARF for the
full region (RAWX:30–46) and a second ARF for the excluded region
(RAWX:36–39). The ARF of our extraction region was obtained by subtracting the
latter from the former using the command addarf (e.g. Wilkinson & Uttley,
2009). We used calibration files (CCF) as of June 2019. The spectra were re-
binned to have a minimum of 20 counts in each energy bin. For the spectral
fits (Sect. 4) we used Xspec v12.10.1 (Arnaud, 1996), while codes for the
spectral-timing analysis (Sects. 3 and 4) were implemented using Python 3.6.

Table 1: Log of the XMM-Newton EPIC-pn observations analysed. Effective exposures after regular data screening (Total exposure) and after the additional selection of time intervals not affected by wind absorption (NWA exposure) are reported. The orbital phase $\phi_{\mathrm{orb}}$ is computed using the ephemeris reported in Gies et al. (2008). The last three digits of the ObsID used throughout the test to identify each observation are marked in boldface. ObsID | Date | Total | NWA | $\phi_{\mathrm{orb}}$
---|---|---|---|---
|  | exposure | exposure |
| (yyyy–mm–dd) | (ks) | (ks) |
0745250201 | 2016–05–27 | 92.9 | 30.9 | 0.82–0.06
0745250501 | 2016–05–29 | 83.1 | 79.8 | 0.17–0.46
0745250601 | 2016–05–31 | 76.7 | 74.8 | 0.53–0.79
0745250701 | 2016–06–02 | 84.6 | 44.9 | 0.89–0.11
Figure 1: XMM-Newton EPIC-pn light curves of the four analysed observations of
Cyg X-1 (i.e. observations 201, 501, 601, and 701, each separated by the gaps
in the light curve). The reference starting time of the monitoring is 57535.0
MJD. The upper panel reports the 0.5–1.5 keV light curve. The middle panel
reports the 3–10 keV light curve. The bottom panel reports the ratio between
count rates in the 3–10 keV and 0.5–1.5 keV energy bands. The red lines
indicate the two consecutive passages at superior conjunctions
($\phi_{\mathrm{orb}}=0$) occurring during the XMM-Newton monitoring.

###  2.1 Selection of events not affected by wind absorption

Depending on the orbital phase of the compact object the X-ray emission will
be more or less absorbed by the stellar wind. As a consequence of the orbital
modulation of the intervening absorbing column, light curves show recurrent
dips of variable intensity. Due to the spectral dependence of the opacity of
the wind, the soft X-ray band is the most affected.

Fig. 1 shows the EPIC pn light curves of the source extracted (with a time bin
of 10 s) in the two energy bands: 0.5–1.5 keV and 3–10 keV. During the
monitoring, two consecutive passages at superior conjunction (occurring during
observations 201 and 701) are observed. These are marked by red vertical lines
in Fig. 1. The 0.5–1.5 keV light curve (Fig. 1, top panel) shows several dips,
in particular around superior conjunction. Such dips are less intense in the
3–10 keV energy band (Fig. 1, middle panel), resulting in a net increase of
the spectral hardness (defined as the ratio between count rates in the 3–10
keV and 0.5–1.5 keV energy bands; Fig. 1, bottom panel). We note that, though
clustering around superior conjunction (at the beginning and at the end of the
monitoring), dips are sparsely present throughout the entire campaign.

Figure 2: The colour-colour diagram of each XMM-Newton observation, showing
hard and soft colours calculated for soft (0.5–1.5 keV), intermediate (1.5–3
keV) and hard bands (3–10 keV). Each point in the plot corresponds to a 10
s–long data segment. Time intervals in green are those characterised by values
of hard colour $\geqslant$ 0.95 and soft colour $\geqslant$ 0.7, thus
representing our selection of NWA datasets, the least affected by wind
absorption.

We followed the method proposed in Nowak et al. (2011) and Hirsch et al.
(2019) to select time intervals not affected by absorption. The method
consists in constructing a colour-colour diagram which shows the time-resolved
spectral behaviour of the source. The diagram displays ratios between a soft
and an intermediate band (soft colour) and between an intermediate and a hard
band (hard colour). The position of the source in the colour-colour diagram
depends on the amount of absorption along the line of sight (Grinberg et al.,
2020).

In order to build the colour-colour diagram, we used light curves with a time
bin of 10 s so as to uncover also very short dips. We considered the three
energy bands 0.5–1.5 keV, 1.5–3 keV, and 3–10 keV (hereafter soft,
intermediate, and hard). The resulting colour-colour diagram for each XMM-
Newton observation is shown in Fig. 2.

Following Hirsch et al. (2019), time intervals characterised by high values of
both hard and soft colours (upper right corner of the colour-colour diagram)
are the least affected by wind absorption. In order to extract a dataset
relatively free from wind absorption, we thus selected events from that region
of the diagram by defining a threshold for the hard and soft colours. We
selected all the 10 s–long data segments with a hard colour $\geqslant$ 0.95
and a soft colour $\geqslant$ 0.7. We consider the simple model adopted in
Hirsch et al. (2019), which comprises a power law with spectral index $1.7$
partially covered by a neutral absorber (representing the clumpy wind),
modified by Galactic absorption (tbnew $\times$ tbpcf $\times$ powerlaw in
Xspec, with Galactic $\mathrm{N_{H}}=0.7\times 10^{22}\,\mathrm{cm^{-2}}$;
Basak et al. 2017; HI4PI Collaboration et al. 2016; and assuming abundances
from Wilms et al. 2000111https://pulsar.sternwarte.uni-
erlangen.de/wilms/research/tbabs/). For this model, we find that if the
partial covering factor is kept constant and only the wind column density
varies, the theoretical curve that best samples the range of hard and soft
colours covered by the data corresponds to a covering factor of $\sim$ 0.9.
For such a value of the covering factor, the chosen threshold of hard and soft
colours selects data characterised by a wind column density $\mathrm{N_{H,w}}$
$\lesssim 1.08\times 10^{22}\,\mathrm{cm^{-2}}$. The final datasets resulting
from the described filtering process will be hereafter referred to as “NWA”
(standing for “no wind absorption”) as opposed to “Total”, which will be used
to refer to the datasets unfiltered from wind absorption. It is worth noting
that the inferred theoretical curve describing the colour-colour tracks and
the corresponding physical parameters represent just an approximation. Indeed,
the curve does not accurately reproduce the observed tracks (Appendix A). As
discussed in Grinberg et al. (2020), this highlights the need for more complex
absorption models in order to properly describe these tracks. While this is
beyond the scope of this paper, this issue will be addressed in a follow up
paper.

It is worth noting that our filtering choices, together with the use of a 10 s
time bin for the light curves employed to build the colour-colour diagram,
ensure a good filtering of strong wind absorption events of duration
$\geqslant$ 10 s. Nonetheless, residual absorption might still be present.
Wind clumps characterised by lower $\mathrm{N_{H,w}}$ and/or lower covering
factor, and moving faster across the line of sight (thus producing shorter
dips) would be the major contributors to this residual absorption.

##  3 Power Spectra

In order to study the effects of the stellar wind on the X-ray variability of
the source we first extracted the PSD of the NWA and Total datasets of each
observation. The PSD were extracted in the soft, intermediate and hard energy
bands defined in Sect. 2.1. To this aim we used light curves with a time
resolution of 6 ms. The light curves were split into segments, containing 1666
data points each (which correspond to a length of 9.996 s). This allows us to
sample the range of temporal frequencies $\nu\sim$ 0.1–83 Hz (corresponding to
variability timescales between $\sim$ 0.01–10 s). The chosen segment length is
the result of a trade-off between obtaining a sufficiently broad frequency
coverage while retaining a high number of GTIs. We calculated the PSD of each
segment and then averaged over all segments. We adopted the fractional rms
normalization (Belloni & Hasinger, 1990; Miyamoto et al., 1991). We did so for
each observation separately, in order to study the evolution of the wind-
modulated variability as a function of the orbital phase. We estimated the
Poisson noise contribution by fitting the PSD with a constant at frequencies
$>25$ Hz. We corrected the PSD for Poisson noise by subtracting this
contribution. The PSD was geometrically rebinned, using a rebinning factor of
$\nu_{i+1}=1.2\,\nu_{i}$.

In Fig. 3, we compare the PSDs obtained from the NWA and Total datasets. The
PSDs are shown separately for each observation and for the soft, intermediate,
and hard energy bands. We observe that the wind contributes significantly to
the short-timescale X-ray variability. The main effect of the wind is that of
smoothing out the typical double-hump shape of the PSD of Cyg X-1 in the hard
state (e.g. Pottschmidt et al., 2003; Axelsson & Done, 2018; Mahmoud & Done,
2018a, b). This shape is recovered when considering the NWA datasets and when
the source is not at superior conjuction (thus it is less affected by wind
absorption). More specifically, wind absorption tends to reduce (by up to a
factor of $\sim$ 2) the fractional variability at high frequencies ($\gtrsim$
1 Hz), and to increase it at low frequencies ($\lesssim$ 1 Hz), as most
clearly seen in observation 201. However, given the limited bandpass, the
effects of the wind at low frequencies are more difficult to constrain.

Notably, the PSD of observation 701 remains quite smooth also after filtering
out wind-absorbed segments. We verified that this is due to residual wind
absorption. Indeed, when choosing a tighter threshold of hard and soft colours
for the selection of the NWA dataset (Sect. 2.1), the double-hump shape
appears (Appendix B). This different selection also allows for additional low-
frequency variability due to the wind in the Total dataset to be clearly
observed in this observation as well. The need for a different threshold
suggests that the average properties of the wind may be different between
consecutive passages at superior conjunction (as also indicated by the
different extension of the colour-colour track towards low values of soft
colours and the spread around the main track222We note that in general,
spectral hardening and/or softening of the source may also produce such
changes in the colour-colour tracks (e.g. see figure 2 of Grinberg et al.
2020). However, our best-fit models do not show significant difference in
spectral slope between these two observations, Table 2., during observations
201 and 701, Fig. 2). Nevertheless, a tighter threshold for the selection of
NWA events in observation 701 severely reduces the amount of usable exposure
(thus decreasing the signal-to-noise of spectral-timing measurements). For
this reason, and for consistency with the other observations, we decided to
use the same selection criteria as for the other observations (Sect. 2.1).

Finally, it is worth noting that the effects of the wind on the PSD of the
source are not limited to the soft (and more absorbed) energy bands, but can
be observed up to the highest sampled energies (i.e. $E=3$–10 keV), suggesting
rather high column densities.

Figure 3: PSD for the Total (blue), the NWA (green) datasets of each
observation (from top to bottom) in the soft (0.5–1.5 keV), intermediate
(1.5–3 keV) and hard (3–10 keV) energy bands (from left to right). For
observation 201 we also show the PSD obtained from selection of the wind-
dominated (characterized by a hard colour $\leqslant$ 0.8, Sect. 4.3) “DIPS”
dataset (in magenta).

##  4 Cross spectral analysis

Cross-spectral analysis informs us about the amount of correlated variability
and the causal relationship between different spectral components (e.g. Nowak
et al., 1999; Uttley et al., 2014). It is well-known that in the hard state of
BHXRBs the primary Comptonisation component and the disc vary in a linearly
correlated way, showing high levels of coherence on a broad range of
timescales (e.g. Wilkinson & Uttley, 2009). In addition, the disc is observed
to lead the variations of the Comptonisation component on the long timescales
($>1$ s, Uttley et al. 2011), while, for some sources, it is observed to
respond to hard X-ray variability on the shortest sampled timescales ($<1$ s)
(Uttley et al., 2011; De Marco et al., 2015; Kara et al., 2019; De Marco et
al., 2021). The former behaviour is usually interpreted in terms of inward
propagation of mass accretion rate fluctuations in a spectrally inhomogeneous
medium (Lyubarskii, 1997; Kotov et al., 2001; Arévalo & Uttley, 2006; Ingram &
van der Klis, 2013; Mushtukov et al., 2018; Bollimpalli et al., 2020), while
the latter is ascribed to the thermal response of the disc to variable hard
X-ray irradiation (thermal reverberation, e.g. Uttley et al. 2011; De Marco et
al. 2015; De Marco & Ponti 2016; De Marco et al. 2017; Kara et al. 2019; Wang
et al. 2020; De Marco et al. 2021). In this section, we analyse the X-ray
cross-spectral timing properties of Cyg X-1 in order to study the correlations
and causal relationship between the disc and the primary Comptonisation
components, and estimate the effects of the wind on such properties.

We first identified the energy bands where the disc and the Comptonisation
component dominate by fitting the time-averaged spectra of the source. In
order to minimise the effects of the wind on the primary X-ray continuum, the
spectra were extracted considering only the NWA dataset. The time-averaged
spectra of all observations were fit jointly. We discarded data below 0.7 keV
in order to avoid distortions due to electronic noise in Timing mode
(calibration file: XMM-CCF-REL-265, Guainazzi, Haberl and Saxton
2010)333https://www.cosmos.esa.int/web/xmm-newton/ccf-release-notes.

The model used for the fit is TBnew $\times$ [diskbb + nthComp + relxillCp] in
Xspec. It includes a multicolour disc component, diskbb, (Mitsuda et al.,
1984), and the soft excess, well modelled by a soft Comptonisation component,
nthComp (Zdziarski et al., 1996; Życki et al., 1999). Even though difficult to
physically interpret, this component is necessary to obtain a reasonable fit
(e.g. Zdziarski & Gierliński, 2004; Basak et al., 2017). The relxillCp (Dauser
et al., 2014; García et al., 2015) component has been used to model the hard
Comptonisation component and its associated reflection spectrum from the inner
disc. Among the different spectra, we tied the inclination and the column
density of the ISM to be the same for each observation, whereas for every
single spectrum, we tied the seed photon temperature to the inner disc
temperature and we fixed the high-energy cut-off of the hard Comptonisation
component at 100 keV (Basak et al., 2017). The column density of the ISM was
left free to vary in order to account for possible residual wind absorption.
Indeed, we found $\mathrm{N_{H}}\sim 0.9\,\times 10^{22}\,\mathrm{cm^{-2}}$,
slightly higher than the value measured in Galactic neutral atomic Hydrogen
column density surveys (i.e. $0.7\,\times 10^{22}\,\mathrm{cm^{-2}}$, HI4PI
Collaboration et al. 2016). We observe significant residuals at energies
between 1.5 and 2.5 keV, i.e. around the absorption edges of the detector,
likely associated with incorrect calibration. Indeed, after ignoring this
energy range the fit significantly improves, with a $\chi^{2}/dof=11582/8268$.
We found an inner disc temperature in the range $kT_{in}\sim$ 0.17–0.2 keV and
a spectral index for the hard Comptonisation component $\Gamma_{H}\sim$
1.3–1.4 (see Table 2).

In Fig. 4 we show the spectrum, the best-fit model and their ratio for each
observation.

Figure 4: The best-fit models from the joint fit of the NWA time-averaged
spectra of all the observations (observation 601 is divided in two due to a
gap in the data). The best-fit model (Tbnew $\times$ [diskbb + nthComp +
relxillCp]) is overplotted in black (solid curve) for each observation. For
clarity, the single components of the model are reported only for observation
201: triple-dot-dashes correspond to the disc blackbody (diskbb), dots to the
soft excess (nthComp), dashes to the hard Comptonisation component (relxillCp
Comptonised part only) and dot-dashes to its reflection with the disc
(relxillCp reflection part only). There are large instrumental residuals in
the energy range 1.5–2.5 keV, so this range is excluded from the fit. The
bottom panels show the ratios of the data to the best-fit model for each
spectrum separately.

We see that the disc dominates below $\sim$ 1 keV and that the 2–10 keV energy
band is dominated by the hard X-ray primary emission. Thus we decided to use
the energy bands 0.3–1 keV and 2–10 keV for the computation of cross spectra.
We extracted light curves in these two energy bands, using the same sampling
parameters (time bin and segments length) as used for the computation of the
PSDs (Sect. 3). For each light curve segment, we computed the cross-spectrum,
then we averaged over the different cross spectra. Following Ingram (2019), we
rebinned the cross spectra in order to have a minimum number of 500 points per
frequency bin to obtain good S/N, so to avoid the complexities of defining the
Poisson noise contribution. This procedure was repeated separately for the NWA
and Total datasets of each observation. We used the average cross spectra to
compute time lags and coherence as detailed in Sects. 4.1 and 4.2.

###  4.1 Coherence as a function of frequency

The coherence measures the degree of linear correlation between two light
curves as a function of Fourier frequency (e.g. Vaughan & Nowak, 1997; Nowak
et al., 1999; Uttley et al., 2014). Fig. 5 shows the intrinsic coherence
(computed using equation 8 of Vaughan & Nowak 1997) for the Total and the NWA
datasets, between the 0.3–1 keV and 2–10 keV light curves.

Above $\sim$ 10 Hz the intrinsic coherence has large uncertainties because of
a significant contribution from uncorrelated Poisson noise. Therefore, we
decided to limit our analysis to the frequencies less affected by Poisson
noise (i.e. 0.1–10 Hz).

We find that the intrinsic coherence of the Total dataset changes
significantly among the different observations (ranging between $\sim$
0.6–0.9), and we ascribe this behaviour to the presence of the wind. Indeed,
for this dataset, the lowest values of intrinsic coherence are registered
during the observations characterized by stronger absorption (observation 201
and 701). On the other hand, the intrinsic coherence of the NWA dataset is
high ($\sim$ 0.95) and consistent with being constant among observations, as
commonly observed (e.g. Nowak et al., 1999; Pottschmidt et al., 2003; Grinberg
et al., 2014). In other words, when considering the “bare” emission from the
X-ray source, there are no indications of significant changes of coherence
between variability in the two energy bands.

Figure 5: Intrinsic coherence between the 0.3–1 keV and the 2–10 keV energy
bands for the Total (blue) and NWA (green) datasets. For observation 201, the
DIPS dataset (hard colour $\leqslant$ 0.8) is shown in magenta

.

###  4.2 Lag-frequency spectra

We then computed the frequency-dependent time lags between the same soft and
hard energy bands used to compute the intrinsic coherence in Sect. 4.1. The
time lags were estimated as $\tau(\nu)=\phi(\nu)/2\pi\nu$, where $\phi(\nu)$
is the phase lag of the average (over the different light curve segments of a
single observation) cross-spectrum. Results are shown in Fig. 6, for both the
Total (blue) and NWA (green) datasets of each observation.

We observe that the amplitude of the lag is positive at all frequencies,
indicating that rapid variability in the 2–10 keV energy band is delayed with
respect to the 0.3–1 keV band. Interestingly, there is no evidence of a
thermal reverberation lag in the 1–10 Hz range (which would manifest as a
negative lag representing a delay of the disc-dominated, 0.3–1 keV band with
respect to the Comptonisation-dominated 2–10 keV band). While hard lags are
commonly observed in BHXRBs (e.g. Nowak et al., 1999; Uttley et al., 2011; De
Marco et al., 2015), we note that the wind affects their amplitude at low
frequencies. Indeed, during observations 201 and (to a lesser degree) 701, the
Total datasets display slightly shorter low frequency hard lags than during
the other observations. We further investigated this issue in Sect. 4.3 for
the most absorbed observation. After correcting for the effects of wind-
absorption (NWA dataset in Fig. 6), the amplitude of the low frequency hard
lags in observations 201 and 701 increases, becoming consistent with the lags
measured in the other observations444The shorter low frequency hard lags for
observation 701 could be related to residual wind absorption as mentioned in
Sect. 3.

Figure 6: Lag-frequency spectra between the 2–10 keV and the 0.3–1 keV energy
band for Total (blue) and NWA (green) datasets. In magenta, the DIPS dataset
(with a hard colour $\leqslant$ 0.8) for observation 201.

###  4.3 The X-ray spectral-timing properties of the stellar wind

Given the observed differences between the Total and the NWA datasets, we
investigated in more detail the spectral-timing signatures of the X-ray
variability induced by the stellar wind. We did so by analysing time intervals
dominated by wind absorption, thus corresponding to the deeper phases of each
X-ray dipping event (Fig. 1). To this aim we focused on observation 201 (which
covers the first passage at superior conjunction), as this displays the
largest number of absorption dips in the light curve and the densest track in
the colour-colour diagram of the source (Figs. 1 and 2). We first studied the
dependence of the measured fractional rms on the amount of absorbing material
along the line of sight. We selected 8 different regions of soft and hard
colours which, for a given covering factor (Appendix A), would correspond to
increasing values (from A to H in Fig. 7) of the column density
$\mathrm{N_{H,w}}$. Using the same procedure as in Sect. 3, we computed the
average PSD of the data within each selected regions. The PSDs are extracted
in the soft, intermediate and hard energy bands used in Sect. 2.1, and are
displayed in Fig. 8. At both low and high frequencies we can observe
significant differences as a function of the amount of intervening gas. In
particular, we register a gradual increase of low frequency variability and a
decrease of high frequency variability power. Above a certain value of
$\mathrm{N_{H,w}}$ (regions D to H of the colour-colour diagram), in the soft
and in the intermediate bands (Fig. 8, upper and middle panel) the variability
power starts to be suppressed also at lower frequencies. In other words, at
the bottom of the dip, soft band variability is significantly damped on all
sampled timescales, while this suppression appears to occur only above an
increasingly higher frequency as the energy increases.

In order to better visualise these trends, we integrated the PSDs over the
0.16–0.6 Hz and the 2–5 Hz frequency intervals. The resulting fractional rms
in each frequency range is plotted separately (see Fig. 9) for each energy
band as a function of the selected region of the colour-colour diagram.

The 0.16–0.6 Hz fractional rms initially increases up to $\sim$ 23–25 percent.
Then it starts decreasing, reaching values of $\sim$ 10–17 percent. This
change in the overall trend occurs in regions of the colour-colour diagram
characterised by higher values of $\mathrm{N_{H,w}}$. In the softest band the
trend reversal occurs earlier in the colour-colour diagram track (i.e. at
higher values of the hard colour) than seen in higher energy bands. At soft
energies, the most affected by absorption, we also register the highest/lowest
values of maximum/minimum fractional rms. While the hardest energy band does
not show evidence of a drop of 0.16–0.6 Hz fractional rms, an increasing trend
can still be clearly observed, suggesting absorption affects also this region
of the spectrum.

On the other hand, the 2–5 Hz fractional rms steadily decreases (from $\sim$
15 to $\sim$ 12 percent) in all bands, eventually reaching a plateau (between
regions D and H).

The increase of fractional rms, observed at low frequencies in some regions of
the colour-colour diagram and at different levels for different energy bands,
can be explained in terms of variations of the column density of the absorbing
gas crossing the line of sight to the X-ray source. However, the suppression
of variability that occurs in the most absorbed phases/energy bands, and at
the highest sampled frequencies requires the presence of additional scattering
components (see discussion in Sect. 5).

Finally, we measured again the coherence and the lags between the 0.3–1 keV
and the 2–10 keV energy bands (as in Sects. 4.1 and 4.2), this time using a
dataset which selects only intervals strongly affected by the wind (i.e. with
a hard colour $\leqslant$ 0.8). This dataset (hereafter referred to as “DIPS”)
roughly corresponds to the regions D to H in Fig. 7. The results are
overplotted in Figs. 5 and 6. A comparison with the Total and NWA datasets
clearly demonstrates that wind absorption is the main cause of the lower
coherence and shorter hard lags measured in the Total dataset during each
passage at superior conjunction. Indeed, the X-ray variability associated with
the stellar wind is characterised by a low intrinsic coherence ($\lesssim$
0.5, and dropping to zero above 2 Hz) and a negative (soft) lag of a few tens
of msec.

Figure 7: The colour-colour diagram of observation 201 showing the regions
selected for the study of the PSD as a function of the amount of absorption.
From A to H the different regions correspond (for a constant covering factor)
to an increasing value of $\mathrm{N_{H,w}}$.

Figure 8: PSDs for the selected regions in Fig. 7 in the soft (0.5–1.5 keV,
upper panel), intermediate (1.5–3 keV, middle panel) and hard (3–10 keV, lower
panel) bands. The triangles represent upper limits (at $3\sigma$ confidence
level). Figure 9: Fractional rms as a function of increasing wind absorption
(from region A to region H).

##  5 Discussion

Detailed studies of the stellar wind in Cyg X-1 have been mostly focused on
characterizing the way the wind influences the spectral properties of the
system, and generally limited to $E\gtrsim 1$ keV (e.g. Grinberg et al., 2015;
Hirsch et al., 2019). On the other hand, studies of the X-ray timing
properties of Cyg X-1 have been mostly focused on the X-ray source, neglecting
the effects of the wind (e.g. Pottschmidt et al., 2003; Axelsson & Done, 2018;
Mahmoud & Done, 2018a, b).

In this paper we found that the presence of a stellar wind significantly
affects the X-ray variability power of the source, as well as the amount of
coherence and the amplitude of the lags between variable emission in the soft
(0.3–1 keV) and hard (2–10 keV) energy bands. The observed behaviour changes
significantly as a function of the orbital phase, due to the orbital
modulation of wind absorption.

###  5.1 Effects of the wind on the X-ray variability power

We found that the stellar wind influences the observed X-ray variability power
of the source in a complex way, that depends on both energy and timescale
(Sects. 3 and 4.3). These results are discussed below considering the low and
high frequency behaviour separately.

####  5.1.1 X-ray variability at low frequencies ($\lesssim$ 1 Hz)

The variable X-ray absorption associated with the stellar wind has the effect
of increasing the long timescale/low frequency fractional rms of the source
(Figs. 3, 8, and red markers in Fig. 9). This wind-induced, excess X-ray
variability is observed in all the energy bands considered, including the
highest energies (3–10 keV), in line with spectral studies that showed that
the stellar wind can modify the broad band continuum up to very high energies
($\sim$ 10 keV, e.g. Grinberg et al., 2015).

By considering a clumpy wind model, Grinberg et al. (2015) investigated the
variability induced by clumps of variable size and porosity (a measure of the
mean free path among clumps), and constant mass, as they cross our line of
sight. The main inferred effect of these transits is an increase of the
observed variability power due to the variations of the $\mathrm{N_{H,w}}$ of
the intervening structured wind. According to this study, excess X-ray
variability would be observed only when averaging over timescales sufficiently
short so as to catch the passage of a single/small number of clumps (at lower
frequencies/longer timescales wind-induced variability would result washed out
as a consequence of averaging over a large number of transits). Although this
model may represent a simplification of the real conditions, it clearly shows
how variations of the $\mathrm{N_{H,w}}$ of the wind can produce an increase
of variability power as observed in the data analysed here (Figs. 3, 8, and
9). This would imply that the timescales sampled in our study ($\sim$ 1–10 s)
are short enough to catch the passage of single clumps. Grinberg et al. (2015)
also note that below/above a certain timescale/temporal frequency the wind
does not contribute to increase the level of X-ray variability further. This
timescale roughly corresponds to the time needed for a typical clump to
transit the line of sight, and thus can be used (knowing the wind velocity) to
estimate the typical size of the clumps. We can obtain an estimate of the
typical size of the clumps responsible for the observed excess variability in
Cyg X-1 by considering the maximum frequency, $\nu_{\mathrm{max}}$, at which
the variability power in the Total PSD is observed to be greater or equal to
the variability power in the NWA PSD. Assuming a wind terminal velocity of
$v_{\infty}=2400\ {\rm km\ s^{-1}}$ (Grinberg et al., 2015), the resulting
radial size of the clumps would be of the order of $l\sim$ 0.5–1.5 $\times
10^{-4}R_{\ast}$ (we note that the larger values of $\sim$ 0.02–0.2 $R_{\ast}$
previously found by Hirsch et al. 2019 are due to the larger minimum timescale
sampled in their analysis, i.e. 25.5 s). This range accounts for the different
values of $\nu_{\mathrm{max}}$ observed in the different observations. We note
however, that while this represents an order of magnitude estimate, the clumps
likely have a broad distribution of sizes. In particular, smaller clumps can
give a significant contribution to the observed high frequency X-ray
variability.

When considering the most absorbed stages (which correspond to lower values of
hard colour in the colour-colour diagram, Fig. 7, and for a given covering
factor, to higher $\mathrm{N_{H,w}}$, Fig. A), we found that the low-frequency
X-ray variability power in softer bands starts to be significantly suppressed
(Fig. 9). Large scale reprocessing/fluorescent scattering of soft band photons
in the stellar wind is a plausible mechanism to explain the decrease of low
frequency variability power during the dips. Indeed, light travel time
differences across this extended medium strongly dilute the intrinsic flux
variability of the X-ray source. This constant or slowly variable extended
emission component from the wind would dominate the soft X-ray band when the
direct continuum is highly absorbed, resulting in a net decrease of fractional
rms at these energies. We note that a similar effect might be produced by
scattering off dust layers in the ISM (known to produce a soft spectrum due to
the energy dependence of the dust scattering cross-section, Predehl & Schmitt
1995; Xiang et al. 2011; Ling et al. 2009). However, this seems less plausible
given the factor of $\sim$ 2 decrease in fractional rms observed during the
dips in the softest energy band (Fig. 9), which implies quite a large
scattering fraction.

####  5.1.2 X-ray variability at high frequencies ($\gtrsim$ 1 Hz)

The observations during passage at superior conjunction (observations 201 and
701) are characterized by a strong suppression of the high frequency
variability hump, a characteristic feature in the power spectrum of Cyg X-1.
This can be seen by comparing the Total (as well as the DIPS for observation
201) datasets with the NWA datasets (Figs. 3 and 13). Such a suppression is
observed at all energies, and according to Fig. 9, it starts occurring at
relatively low levels of wind absorption. Nonetheless, this behaviour cannot
be ascribed to absorption in the stellar wind, since variable absorption would
result in an increase of fractional rms, while constant absorption would
equally affect the variable and constant flux components, thus it would not
produce any change in the fractional rms.

Here, we consider the possibility that the suppression of the high frequency
variability power is due to scattering, thus smearing out the fastest
variability, and infer the optical depth of the scattering medium. The
timescales affected by scattering will be those shorter than the light
crossing time of the scattering medium, thus corresponding to the highest
temporal frequencies in the power spectrum. The temporal frequency where this
dampening becomes significant is expected to decrease as the optical depth
$\tau_{s}$ increases (e.g. Zdziarski et al., 2010).

Such a behaviour has been previously observed in a number of sources (e.g.
Belloni et al., 1991; Berger & van der Klis, 1994). Interestingly, in the case
of the BHXRB Cyg X-3 the analysis of the PSD required the presence of a gas
optically thicker than the wind (Zdziarski et al., 2010). Using equation 5 of
Zdziarski et al. (2010) we estimate that the drop observed in Cyg X-1 (at
$\nu\sim$ 5–10 Hz) would require $\tau_{s}\sim$ 0.5–1 for a scattering region
of the size of $R\sim$ 3$\times 10^{9}$ cm, as inferred for Cyg X-3. The need
for such a large $\tau$ is also confirmed by the fact that the suppression of
high frequency variability in the PSD of observation 201 is still observed at
very high energies (see Fig. 10, reporting the PSD of the Total and NWA
datasets of observation 201, extracted in the $E=$ 6–10 keV energy band).
These results support the hypothesis of the presence of an optically thick gas
intercepting the line-of-sight to the X-ray source during superior
conjunction. However, previous analyses suggest $\tau_{w}\ll 1$ for the
stellar wind in Cyg X-1 (Zdziarski, 2012), remarkably lower than required from
our estimates. Therefore, as previously seen in Cyg X-3, a gas optically
thicker than its stellar wind is needed to explain the drop of high frequency
variability seen in Cyg X-1. This optically thicker gas may be associated with
an accretion bulge, possibly formed by collision of the stellar wind with the
edge of the disc (e.g. Poutanen et al., 2008; Zdziarski et al., 2010). To
further test this hypothesis, we extracted PSDs of observation 201 from two
adjacent time intervals selected near superior conjunction, one characterized
by low absorption and the other dominated by strong dips (as shown in the
inset of Fig. 11). The high frequency humps are dampened in both PSDs (Fig.
11, see the comparison with the NWA dataset of observation 501, which shows
the shape of the PSD, far from superior conjunction), i.e. both inside and
outside the strongest absorption dips, thus excluding denser wind clumps as
the origin of the variability drop, and supporting the hypothesis of the
presence of a scattering bulge.

Figure 10: PSD of observation 201 for the Total (blue) and NWA (green)
datasets in the 6–10 keV energy band.

Finally, we note that the high frequency variability power is not completely
suppressed. Indeed, studying observation 201 in more detail, we observe that
the decrease of high frequency fractional rms is followed by a plateau during
the most absorbed stages (Fig. 9). This high frequency variability may be due
to intrinsic source variability, from photons that reach the observer without
being scattered, or to residual contribution from variable absorption. For
example, considering the clumpy wind scenario discussed above, wind clumps
producing excess variability at low frequencies as they cross our line of
sight, will likely have a distribution of sizes. This implies that smaller
clumps can exist which would contribute to the high frequency X-ray
variability of the source.

Figure 11: PSDs extracted from two adjacent time intervals (see inset) during
observation 201 near superior conjunction, respectively characterized by low
(‘NOdip”, purple) and strong (‘dip”, orange) absorption. The PSDs are
extracted in the 0.3–10 keV energy band. Overplotted for comparison is the NWA
PSD of observation 501 near inferior conjunction (dashed line), which shows
the intrinsic shape of the PSD of Cyg X-1 (i.e. free from absorption and
possible scattering effects at superior conjunction).

###  5.2 Effects of the wind on X-ray coherence and lags

We found that another important effect associated with the presence of the
stellar wind is a decrease of intrinsic coherence between the soft (0.3–1 keV)
and the hard (2–10 keV) energy bands (Fig. 5) on all time scales. On the other
hand, in agreement with previous studies, the intrinsic coherence of the
source is close to 1 when time intervals free from wind absorption are
selected (NWA in Fig. 5). In addition, we showed that the amplitude of the
hard X-ray lags, commonly observed in Cyg X-1 and a characteristic feature of
BHXRBs (e.g. Uttley et al., 2011; De Marco et al., 2015), results reduced when
the source is at superior conjunction. This effect is particularly strong
during observation 201 (Fig. 6), possibly due to denser clustering of
absorption events (Fig. 1).

The observed loss of coherence requires the emergence of one (or more) non-
linear variability components during the most absorbed phases of the orbit,
and it is clearly a direct or indirect consequence of the presence of the
stellar wind. Indeed, when considering only the most absorbed stages of
observation 201 (DIPS in Fig. 5, upper left panel) low coherence is observed
at all frequencies (the average low frequency coherence is $\sim$ 0.5, a
factor of $\sim$ 2 lower than in the NWA dataset). Given the presence of the
wind absorbing medium, the most plausible way to explain the observed loss of
coherence between highly absorbed soft bands and primary continuum-dominated
hard bands is the non-linearity of absorption variability. This may be due
either to motions of clouds that are optically thicker in soft than in hard
bands, or to the non-linear response of the absorbing gas to variations of
hard X-ray irradiation. We note that a similar drop of coherence ascribable to
intervening absorption structures was reported in the Seyfert 1 galaxy NGC
3783 De Marco et al. (2020). Simulations showed that changes in the photo-
ionisation state of the wind, as a consequence of variability of the
irradiating X-ray flux, could explain the observed decrease of coherence
during obscured states in that source (see also the recent results from
Juráňová et al. 2022).

Interestingly, the DIPS dataset of observation 201, reveals the presence of an
additional, low frequency ($\lesssim$ 0.2 Hz) soft (negative) lag component
(Fig. 6, upper left panel), which has the net effect of reducing the amplitude
of the observed low frequency hard X-ray lags when the wind-absorbed data are
not filtered out (Total in Fig. 6, upper left panel). This lag is likely the
result of large scale scattering off the wind, becoming dominant when the
direct continuum is blocked due to strong line-of-sight absorption. Another
possibility is that the lag is due to recombination delays (Silva et al.,
2016), thus scaling inversely with the density of the absorbing gas (Nicastro
et al., 1999; Behar et al., 2003; Krongold et al., 2007; Kaastra et al.,
2012). However, a detailed modelisation with appropriate spectral-timing
models and a higher energy-resolution study of the low frequency soft lag
associated with the DIPS dataset would be needed to confirm this hypothesis.

##  6 Conclusions

In this paper, we studied how the stellar wind influences the observed X-ray
spectral-timing properties of Cyg X-1, focusing on the short timescales
($\lesssim$ 10 s), which map the innermost regions of the accretion flow. To
this aim we made use of the XMM-Newton data from the CHOCBOX monitoring
campaign, which allowed us to extend our analysis down to soft X-ray energies
($E\gtrsim 0.3$), where wind absorption is stronger. The monitoring covers two
consecutive passages at superior conjunction (i.e. $\phi_{\mathrm{orb}}=0$).
This phase of the orbit is characterized by intense clustering of strong
absorption dips caused by the stellar wind, Fig. 1 (e.g. Bałucińska-Church et
al., 2000; Feng & Cui, 2002; Ibragimov et al., 2005; Wilms et al., 2006;
Poutanen et al., 2008; Boroson & Vrtilek, 2010; Grinberg et al., 2015).

Our main findings can be summarised as follows.

  * •

The most absorbed orbital phases are characterized by an increase of the
fractional variability at low frequencies ($\lesssim$ 1 Hz), and a suppression
at high frequencies ($\gtrsim$ 1 Hz). As a consequence, the double-hump PSD
shape typical of the hard state of Cyg X-1 results smoothed out when the
source is at superior conjunction. In the least absorbed orbital phases (away
from superior conjunction) only the increase of low frequency fractional rms
is observed, while the high-frequency PSD hump is not affected.

  * •

We ascribe the increase of low frequency fractional variability power due to
variations of the column density of the intervening clouds (e.g. as a
consequence of relative motions of the wind clumps). The observed timescales
correspond to an average radial size for the clumps of $l\sim$ 0.5–1.5$\times
10^{-4}R_{\ast}$ (assuming a wind terminal speed of $2400\ {\mathrm{km}\
\mathrm{s}^{-1}}$).

  * •

The suppression of high frequency variability power in the observations at
superior conjunction requires the presence of an optically thicker (than the
gas in the wind) medium. We associate this component with an accretion bulge,
possibly formed by collision of the stellar wind with the edge of the disc.
The presence of this component was independently proposed in the past to
explain some timing properties of Cyg X-1 and Cyg X-3 (Poutanen et al., 2008;
Zdziarski et al., 2010).

  * •

The most absorbed stages of the orbit are characterized by a decrease of
intrinsic coherence between soft (absorption-dominated) and hard (primary
continuum-dominated) bands. We ascribe the loss of coherence to non-linear
variability of the absorbing gas. In addition, the observation most affected
by absorption dips shows the emergence of a long, low frequency soft lag,
which contributes to reduce the amplitude of the hard X-ray lags intrinsic to
the broad band continuum. A plausible explanation for this lag is large scale
reprocessing or recombination within the wind.

## Acknowledgements

This work is based on observations obtained with XMM-Newton, an ESA science
mission instrument, and contributions directly funded by ESA Member States and
NASA. The authors thank Ileyk El Mellah for providing insightful comments. The
research leading to these results has received funding from the European
Union’s Horizon 2020 Programme under the AHEAD2020 project (grant agreement n.
871158). EVL was partially supported by the Polish National Science Center
under the grant No. 2021/41/B/ST9/04110. BDM acknowledges support from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No. 798726 and Ramón y Cajal Fellowship
RYC2018-025950-I. AAZ and AR acknowledge support from the Polish National
Science Center under the grants No. 2015/18/A/ST9/00746 and
2019/35/B/ST9/03944; and 2021/41/B/ST9/04110, respectively. TMB acknowledges
financial contribution from the agreement ASI-INAF n.2017-14-H.0 and from
PRIN-INAF 2019 N.15. SM acknowledges financial support from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program HotMilk (grant agreement No. 865637).

##  7 DATA AVAILABILITY

The data are publicly available from ESA’s XMM-Newton Science Archive and
NASA’s HEASARC archive.

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##  Appendix A Modeling the colour-colour diagram of observation 201

We modeled the colour-colour diagram of each observation of Cyg X-1 with a
simple absorption model. As an example, here we show the results obtained for
observation 201. We used Xspec to model the continuum using a partially
absorbed power law with spectral index 1.7 (tbpcf $\times$ powerlaw). We fixed
the covering fraction at $0.9$ and let the wind column density
$\mathrm{N_{H,w}}$ vary (from $0.1\times 10^{22}\,\mathrm{cm^{-2}}$ to
$3.5\times 10^{23}\,\mathrm{cm^{-2}}$). This allowed us to build the colour-
colour diagram track show in Fig. 12 and overplotted to the data. The model
also includes ISM absorption (TBnew) with Galactic $\mathrm{N_{H}}=0.7\times
10^{22}\,\mathrm{cm^{-2}}$ (Basak et al., 2017; HI4PI Collaboration et al.,
2016) where abundances from Wilms et al. (2000) were used.

For such a model, the threshold of hard and soft colours used in Sect. 2.1 to
filter out data characterised by strong wind absorption, corresponds, for a
covering factor of 0.9, to a wind column density of $\mathrm{N_{H,w}}\geqslant
1.08\times 10^{22}\mathrm{cm^{-2}}$.

Figure 12: Colour-colour diagram of observation 201 with the simulated model
(TBnew $\times$ tbpcf $\times$ powerlaw in Xspec) for a partial covering
factor of 0.9 (red line), overplotted on the data. Each red point on the model
curve corresponds to a different value of $\mathrm{N_{H,w}}$ (from $0.1\times
10^{22}\,\mathrm{cm^{-2}}$, first point on the top, to $3.5\times
10^{23}\,\mathrm{cm^{-2}}$, last point on the bottom, arbitrarily spaced). The
green points correspond to the selected NWA dataset.

##  Appendix B Selection of the NWA dataset for observation 701

We verified that a stricter (from that used in Sect. 2.1) selection of hard
and soft colours allows recovering the double-hump shape of the PSD of
observation 701 (corresponding to the second passage at superior conjunction).
For this we chose hard colours $\geqslant$ 1.05 and soft colours $\geqslant$
1\. Results are shown in Fig. 13.

Figure 13: PSD for Total (blue) and NWA (green) light curves in the soft
(0.5–1.5 keV), intermediate (1.5–3 keV) and hard (3–10 keV) band for
observation 701 after selection of datasets characterised by hard colour
$\geqslant$ 1.05 and soft colour $\geqslant$ 1.

##  Appendix C Spectral fitting parameters

We report (Table 2) the broad band continuum best-fit parameters for the model
Tbnew $\times$ [diskbb + nthComp + relxillCp] fitted jointly to the spectrum
of each XMM-Newton observation of Cyg X-1 (as described in Sect. 4).

Table 2: Best-fit parameters for each XMM-Newton observation of Cyg X-1. The $kT_{in}$ and $kT_{e}$ parameters are, respectively, the inner temperature of the disc and the electron temperature of the soft Comptonisation component. The $\Gamma_{H}$ and $\Gamma_{S}$ parameters are, respectively, the spectral index of the hard and soft Comptonisation components. Errors are indicated with a confidence level of 90%. Unconstrained parameters (indicated with $f$) were kept fixed at the best-fit value. Component |  Parameter |  201 |  501 |  $601_{\mathrm{A}}$ |  $601_{\mathrm{B}}$ |  701
---|---|---|---|---|---|---
diskbb |  $kT_{in}$ (keV) |  $0.18_{-0.01}^{+0.01}$ |  $0.19_{-0.01}^{+0.01}$ |  $0.17_{-0.01}^{+0.01}$ |  $0.17_{-0.01}^{+0.01}$ |  $0.20_{-0.01}^{+0.01}$
nthComp |  $\Gamma_{S}$ |  $2.02_{-0.04}^{+0.04}$ |  $1.68_{-0.02}^{+0.08}$ |  $2.23_{-0.03}^{+0.01}$ |  $2.51_{-0.02}^{+0.09}$ |  $1.35_{-0.04}^{+0.08}$
|  $kT_{e}$ (keV) |  $0.80_{-0.04}^{+0.02}$ |  $1^{f}$ |  $1^{f}$ |  $1^{f}$ |  $<0.77$
relxillCp |  $\Gamma_{H}$ |  $1.38_{-0.02}^{+0.02}$ |  $1.32_{-0.02}^{+0.04}$ |  $1.35_{-0.02}^{+0.02}$ |  $1.42_{-0.05}^{+0.04}$ |  $1.38_{-0.01}^{+0.02}$