Vol.0 (200x) No.0, 000–000

11institutetext: Department of Astronomy and Institute of Theoretical Physics
and Astrophysics, Xiamen University, Xiamen, Fujian 361005, China;
<EMAIL_ADDRESS>
22institutetext: Key Laboratory for Research in Galaxies and Cosmology,
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, China<EMAIL_ADDRESS>
33institutetext: SHAO-XMU Joint Center for Astrophysics, Xiamen, Fujian
361005, China

# The width-flux relation of the broad iron K$\alpha$ line during the state
transitions of the black hole X-ray binaries

Hang-Ying Shui 1133 Fu-Guo Xie 2233 Zhen Yan 2233 Ren-Yi Ma 1133

(Received 2009 month day; accepted 2009 month day)

###### Abstract

The observation of varying broad iron lines during the state transition of the
black hole X-ray binaries (BHXBs) have been accumulating.In this work, the
relation between the normalized intensity and the width of iron lines is
investigated, in order to understand better the variation of iron lines and
possibly its connection to state transition. Considering the uncertainties due
to ionization and illuminating X-rays, only the effects of geometry and
gravity are taken into account. Three scenarios were studied, i.e., the
continuous disk model, innermost annulus model, and the cloud model. As shown
by our calculations, at given iron width, the line flux of the cloud model is
smaller than that of the continuous disk model; while for the innermost
annulus model, the width is almost unrelated with the flux. The range of the
line strength depends on both the BH spin and the inclination of the disk. We
then apply to the observation of MAXI J1631-479 by NuSTAR during its decay
from the soft state to the intermediate state. We estimated the relative line
strength and width according to the spectral fitting results in Xu et al.
(2020), and then compared with our theoretical width-flux relation. It was
found that the cloud model was more favored. We further modeled the iron line
profiles, and found that the cloud model can explain both the line profile and
its variation with reasonable parameters.

###### keywords:

accretion, accretion disks – black hole physics – X-rays: binaries

##  1 Introduction

As well known, the outbursts of the BHXBs usually go through different
spectral states in sequence, i.e., from the quiescent state to the hard state
(HS), the intermediate state (IS), the soft state (SS), and then back inverse
(e.g. Remillard and McClintock 2006).

The theoretical models for the accretion flows in the HS and SS have been
widely accepted. For the SS, the X-ray spectrum is dominated by the soft
thermal component, the hard power-law component is weak with the photon index
larger than 2, and the accretion flow is believed to be in the mode of cold
standard accretion disk (SAD) that extends to the innermost stable circular
orbit (ISCO) (e.g. Shakura and Sunyaev 1973), being sandwiched by rare hot
corona (e.g. Liang and Price 1977). For the HS, the power-law component
dominates the spectrum, and the accretion flow transforms to the hot accretion
flow inside the truncation radius (e.g. Narayan and Yi 1994; Yuan and Narayan
2014).

However, the accretion mode of the IS, which describes how the accretion flow
transits between the SAD and the hot accretion flow, is still unclear.
Geometrically, there are three possible scenarios. A natural scenario is the
gradual inward extending or outward receding of the outer cold disk down
to/from the ISCO when the accretion rate increases/decreases, or the
continuous disk model, as first proposed by Esin et al. (1997).

Additionally, considering the evaporation and condensing between the rare hot
gas/corona and the dense cold disk, theoretical studies have shown that an
annulus of dense cold gas may form or vanish in the inner region of the hot
accretion flow during the state transition (Liu et al. 2007, 2011; Qiao and
Liu 2011). This kind of two-phase accretion flow is referred as the innermost
annulus model in our paper, since the annulus is not very far from the ISCO.

In fact, the geometry of the two-phase accretion flow could be even more
complex. Instead of one annulus, the cold-phase gas may also be in form of
clouds or clumps. Such inhomogeous accretion flow could relates to some
instabilities (Kuncic et al. 1996; Blaes and Socrates 2001, 2003; Gammie
1998). For example, if the advection heating could not balance the radiative
cooling, the thermal instability would be triggered, and the luminous hot
accretion flow would collapse non-linearly, leading to the formation of clumps
(Yuan 2001, 2003; Xie and Yuan 2012). The existence of such cloudy accretion
flow have been shown in some numerical simulations (Wu et al. 2016; Bu and Gan
2018; Sadowski et al. 2017; Liska et al. 2022).

However, from observational point of view, it is still difficult to identify
the process of state transition. On the one hand, the observations with high
spectral resolution during the state transition are still lacking. On the
other hand, different observations may have different results. In some
observations, a second thermal component is needed to fit the spectrum, which
seem to favor the cloud or the annulus model (e.g., Walton et al. 2017; Miller
et al. 2018; Tomsick et al. 2018). While for some other observations, the
truncation radius has already reached the ISCO before the hard-to-soft
transition starts to occur (e.g. Parker et al. 2015; Buisson et al. 2019; Dong
et al. 2022), which seems to disfavor all the three scenarios mentioned above.

Recently, the state transition of the black hole X-ray binary MAXI J1631-479
has been observed by NuSTAR , where the evolution of the broad iron line was
obtained (Xu et al. 2020). The authors fitted the spectra with the continuous
disk model, and found that the truncation radius evolves from $<1.9~{}r_{g}$
to $12\pm 1~{}r_{g}$. However, the authors found that the reflection rate in
SS is usually high relative to that in IS, and the iron line flux is not
correlated with the strength of the coronal emission in short intervals. They
suggested that the thermal emission may also contribute to the line emission,
however detailed calculation was not given.

The observed evolution of line width and strength enlightened us to study the
possible relation between the flux and width of the broad iron. For the
present spectral resolution, it is still difficult to identify the scenario of
state transition by the fitting of iron line profile, as the fluctuation of
the profile is still too large. The variation of the line flux and flux may
provide us the clues of state transition for present observations.

The reflection features are related with a series of factors, such as the
illuminating X-ray spectra and intensity, the ionization degree of the cold
gas, the geometry and movement of the cold gas, and the gravitational field.
Considering that the former two factors are still difficult to be constrained
from theory and they vary for different outbursts and different stages of one
burst, as the primary work, we just take into account the effects due to
geometry and gravity. Our model is very simple, but the result is quantitative
and robust.

In Sec.2, our model is introduced, where the geometry of the accretion flow
are described by a set of three parameters. For different model of state
transition, the parameters vary in different ways. In Sec.3, we investigate
how the line strength changes with the line width for different models. All
the line strength are normalized by the illuminating hard X-rays and the line
flux in the soft state. The favored scenario for the state transition of MAXI
J1631-479 are discussed in Sec.4, and the summary is given in Sec.5.

Figure 1: The geometry of the inhomogeneous accretion flow in our model. The
dark grey rectangles denote the cold gas, i.e. disk and cloud. The light grey
triangle denotes the hot accretion flow or corona. The dashed line shows the
geometry of SAD in the inner region, which extends to ISCO, $r_{\rm{ISCO}}$.
The parameters $r_{\rm{cl}}$, $\Delta r$ and $r_{\rm{tr}}$ represent the inner
edge and radial size of the clumps, and the radius of truncation,
respectively. Figure 2: An example of the emission line from a two-phase
accretion flow, with the parameters being $a_{*}=0.9$, $i=30^{\circ}$,
$r_{tr}=20~{}r_{\rm{ISCO}}$, $r_{cl}=5~{}r_{\rm{ISCO}}$, and $\Delta
r=4~{}r_{\rm{ISCO}}$. The red solid line shows the total emission line, while
the blue dot-dash and the grey dot-dot-dash lines show the components from the
cold cloud and the outer cold disk, respectively. For comparison, the emission
line of the same width from the continuous disk is also shown by the blue
dashed line.

##  2 The Model

###  2.1 Geometric structure and justification of the annulus-like clump

In our phenomenological study, the accretion flow is described with a set of
geometric parameters, the variation of which can describe the three scenarios
of state transition. The geometry of the model is shown in Fig. 1. Interested
readers are referred to (Yu et al. 2017, 2018) for more details. There are
three components in the model. The continuous standard accretion disk lies on
the outside of a truncation radius, $r_{\rm tr}$. Inside $r_{\rm tr}$, the
disk is replaced by hot gas, either hot accretion flow or corona. The annulus
locates at the mid-plane within the hot gas, representing the cold clumps.
Since the inner hot accretion flow is hot and therefore geometrically thick,
its X-ray emission can illuminate not only the cold clumps but also the outer
cold disk.

It is necessary to investigate the geometry of the clump. For the problem of
state transition, the clump is not far from the central BH, suffering strong
tidal force. The tidal force would stretch the clump in radial direction,
meanwhile the differential rotation of the accretion flow would stretch the
clump into arc form, making the azimuthal size, $\Delta\psi$, relatively
large.

For present observations, the integration time of spectral analysis is
typically tens to thousands of seconds, in contrast to the orbital timescale
of accreting gas of less than a second. In this situation, the clump looks
just like annulus to the telescopes except that, the “observed” annulus is
dimmer, or the iron line flux is lower. The difference of flux directly
depends on the intrinsic $\Delta\psi$ for given radial size and position.
Considering the arc form mentioned above, the difference should be within one
order of magnitude. We caution that this result does not hold for AGNs, where
the orbital timescale can be minutes to hours, being comparable to the
integration time in X-ray spectral analysis.

We additionally note that, there is a degeneracy between the azimuthal size
and the radial size. The line strength depends on the area of the cloud. It
means that the iron line from a clump of limited azimuthal size is similar to
that from a narrower ring at the same radial position. Although the line
profiles are a bit different, the difference is too small to be distinguished
for present observations. In other words, the radial size and azimuthal size
of the clump are degenerate.

Considering the above reasons, it is acceptable to simplify the clump as
annulus to study the width-flux relation. For simplicity, in our calculation
we only consider the case with one cloud. As will be shown later, our results
are not affected significantly by the number of clumps, because the width and
flux of the iron line mostly depend on the radial position of the innermost
cloud and the total area of the clumps, respectively. Therefore the clump can
be well described via two parameters, i.e., the radial position,
$r_{\rm{cl}}$, and width of the annulus, $\Delta r$. Different scenarios of
the state transition can all be described by the variation of these three
parameters. For the continuous disk model, the parameters can change in two
ways. One is that $r_{\rm{tr}}$ decreases or increases during the state
transition, meanwhile $\Delta r=0$ all the time. The other is that
$r_{\rm{cl}}$ decreases or increases during the state transition, meanwhile
$r_{\rm{tr}}=\Delta r+r_{\rm{cl}}$ all the time. For the innermost annulus
model, $r_{\rm{cl}}\sim r_{\rm{ISCO}}$ all the time, and $\Delta r$ and
$r_{\rm{tr}}$ change with time; And for the cloud model, the parameters are
all free, being limited only by the geometry.

###  2.2 Broad skewed iron line profile

The observed line profiles can be calculated with the ray tracing technique,
which has been well developed and widely used (e.g. Cunningham 1975; Rauch and
Blandford 1994; Fanton et al. 1997; García et al. 2014). We used the method
and code developed in Fanton et al. (1997). Briefly, if the centers of the BH
and its image are set as the coordinate origins of the disk plane and image
plane, respectively, for each grid on the image plane and a given redshift
factor, the photon trajectory is determined according to the theory of general
relativity (Carter 1968). It is then possible to trace the trajectory from the
image plane backwards to the origin of emission according to the variation of
polar angle. If the clumps is assumed to be in the equatorial plane of the BH,
the polar angle would change from $\pi/2$ to the inclination angle of
eyesight. Considering the axisymmetry of the geometric and the accretion flow,
without the need to find the azimuthal position, the radial position where the
trajectory intersect with the disk plane can be solved.

For the obtained radial position, the emissivity and movement can be
calculated according to the structure of the accretion flow. Following the
usual way of treatment, we assume the emissivity, or the intensity of
emission, follows a power-law distribution, i.e., $I\propto r{{}^{-p}}$
(Wilkins and Fabian 2012; Dauser et al. 2013), where index $p=3$ is fixed to a
typical value. Since the movement of the cloud is still difficult to
dynamically constrain, we assume it to be Keplerian just like the outer cold
disk. Meanwhile, the solid angle that the image grid subtends to the point of
photon emission can be calculated according to the photon geodesics.
Subsequently, the observed flux can be obtained with coordinate conversion to
the observer. Finally, scan the image plane, and then bin the observed flux by
redshift factor, we can obtain the theoretical line profile, as well as the
total flux of the iron line.

Here we want to note the assumption we followed, $I\propto r{{}^{-p}}$.
Firstly, the influences of the illuminating X-rays and the reflection rate are
not included, in other words, the illuminating X-rays and reflection rate are
normalized in our calculation. Since the geometry parameters of our model are
fixed in the soft state, we select the illuminating X-ray flux, the reflection
rate, and iron line flux in the soft state as normalization factor, and
investigate the relative variation of the iron lines. Secondly, we set the
outer boundary of the cold disk at 200 $r_{\rm{ISCO}}$ , which does not affect
the result obviously.

When the truncation radius is close to the ISCO, the temperature of the cold
gas would be as high as about 1 keV, so the broaden of the line on the disk
plane due to the thermal Doppler effect should also be taken into account. So
we assume the temperature of the cold gas, both the cloud and the outer cold
disk, to be that of SAD at the corresponding radius. The iron line is
broadened following Gaussian distribution,
$I(\nu)=\frac{1}{\Delta{\nu}\sqrt{\pi}}e^{-\frac{(\nu-\nu_{\rm
0})^{2}}{\Delta{\nu}^{2}}}$, where $\nu_{0}=6.4~{}{\rm{keV}}$ is the line
frequency in the rest frame,
$\Delta\nu=\nu_{0}\sqrt{\frac{2kT}{m_{\rm{e}}c^{2}}}$ is the broadening
produced by the Doppler effect, $k$ is the Boltzmann constant, $m_{\rm
e}c^{2}$ denotes the rest energy of the electron, and $T$ is the temperature
of the cold gas.

Figure 3: The possible width-flux relation during the state transition, where
$F_{0}$ is the line flux from SAD. As example the BH spin and the inclination
angle are taken as $a_{*}=0.96$, $i=30^{\circ}$. The blue dash-dot line
corresponds to the continuous disk model, or the upper limit of the line flux
for given width. The red solid line corresponds to the relation of the cloud
model, where the cloud contribute 10% of the total line flux. And the grey
dotted lines correspond to the cloud of all possible sizes at given radii,
among which the one of highest width corresponds to the innermost annulus
model.

Figure 4: The possible width-flux relation for different BH spin (left panel),
inclination (middle panel), and truncation radius (Right panel).

##  3 Results

In order to show the influence of the clouds on the width and flux of the
emission line, Fig.2 is plotted. The blue dashed and dash-dot lines show the
iron lines emit from the outer cold disks truncate at $5~{}r_{\rm ISCO}$ and
$20~{}r_{\rm ISCO}$, respectively. The grey dot-dash-dot line is the
contribution of a cloud at $5~{}r_{\rm ISCO}$. The iron line from the two-
phase accretion flow, in which $r_{\rm tr}=20~{}r_{\rm ISCO}$ and $r_{\rm
cl}=5~{}r_{\rm ISCO}$, is the sum of the dash-dot and dot-dash-dot lines. As
can be seen that the width of the line depends most on the inner boundary of
the cold gas. Therefore, it is a natural result that for the given line width,
the flux of the continuous disk is the highest, since the area of reflection
is largest.

For a given BH spin and inclination angle, as the geometry of the cold gas
evolves, the flux and width of the emission line vary correspondingly. Because
the geometric parameters change in different ways, the relation between the
flux and width differs for different scenarios. The range of possible
relations are shown in Fig.3. It should be noted that the width we used is
defined by 10% of the peak flux instead of half maximum, otherwise a large
fraction of the red wing would be ignored. Moreover, the line flux is
normalized by the iron line in SS that is contributed by the complete area of
SAD, as mentioned before. Compare to the absolute flux, the normalized flux
shows the relative variation.

Considering an annulus at $r_{cl}$, when $\Delta r$ increases, the total line
flux increases as the area of reflection becomes larger, meanwhile the line
width remain almost the same since the red wing of the line profile mostly
depend on the inner edge of the ring, where the gravitational redshift is
maximal. The influence of $\Delta r$ is shown by the grey dashed lines. The
first one from right represents the width-flux relation for the innermost
annulus scenario. The blue dash-dot line indicates the highest flux, which
correspond to the largest cloud that fill up all the area between $r_{\rm cl}$
and $r_{\rm tr}$. In other words, it represent the relation for the continuous
disk scenario. The red solid line shows the lowest line flux that corresponds
to the most narrow ring we set. Theoretically, the line flux from the annulus
could be as small as zero. But from the view point of observation, the flux
from the cloud cannot be too small, otherwise the contribution by the cloud
would be undetectable. Here we set the most narrow ring be about 10% of the
gap region between $r_{cl}$ and $r_{tr}$. Generally speaking, the width-flux
relation should be within the region between the blue dash-dot line and the
red solid line.

The influence of the BH spin on the range of the line flux is shown in the
left panel of Fig.4, in which the upper and lower limits are plotted with the
same type of line as shown in the caption. When the spin increases, the
maximal width becomes larger as the ISCO is closer to the BH and the
gravitational effects and the Doppler effect are more significant. It can be
found that for the cases with broader lines, the range of the flux becomes
larger with increasing $a_{*}$. This is because the broader lines are produced
by cold gas closer to the BH, which suffers more gravitational redshift.

Similarly, the influences of the inclination angle and the truncation radius
are shown in the middle and right panels of Fig.4, respectively. When the
inclination angle becomes larger, or the line of sight is closer to the
equatorial plane, the width of the line increase significantly due to the
Doppler effect. Therefore the width-flux relation is very sensitive to the
inclination. For larger truncation radius the contribution of the outer cold
disk is smaller, as a consequence the lower limit, i.e. 10% of the peak flux
becomes smaller, which means the cloud could be smaller. And so it is easy to
understand why the lower limit decreases with increasing truncation radius.

##  4 Application to MAXI J1631-479

Although our model is very simple, it provides us a quantitative and case
independent results. Based on the spectral fitting of the observational data,
the spectra and intensity of the illuminating X-rays, as well as the degree of
ionization, can be obtained. It is then possible to estimate the normalized
iron line flux by use of the relation between reflection and ionization level
(e.g. Ross et al. 1999).

###  4.1 Previous observational results of MAXI J1631-479

The black hole X-ray binary MAXI J1631-479 was discovered in 2018 by the All
Sky X-ray Image Monitor (MAXI, Kobayashi et al. 2018). And after that many
interesting works have been done on this source. Xu et al. (2020) carried out
a spectroscopic analysis based on the observation of the Nuclear Spectroscopic
Telescope Array (NuSTAR ) during its 2019 outburst. They captured two epochs
during the decay of the 2018-2019 outburst and clearly detected the strong
relativistic disk reflection features in both SS and IS. Rout et al. (2021)
performed a combined spectral and timing analysis by use of the observational
data by the Neutron Star Interior Composition Explorer (NICER), and Bu et al.
(2021) presented the broadband timing properties from the Insight-Hard X-ray
Modulation Telescope (Insight-HXMT) observations.

Figure 5: The modeling of the broad emission line from MAXI J1631-479 observed
by NuSTAR . The observational data are taken from Xu et al. (2020), with the
red and blue dots correspond to Observation IDs 90501301001 (Obs1 part I) and
80401316004 (Obs3), respectively. The data above 7 keV of all observations are
ignored. The blue solid line is the modeling with the cloud model. The long
dashed line are the modelling to the line in SS with the continuous disk
model. The short dashed line is the modeling to the line in IS with continuous
disk model, which failed to explain either the width or the strength without
additional variation of observational parameters.

The observations of NuSTAR boast of the broad passband (3-79 keV) and the high
spectral resolution. Being free from the pile-up effect, they are well suited
to study relativistic reflection and Fe K$\alpha$ emission line. So in this
work we concentrate on the result by Xu et al. (2020). The authors separated
the two epochs of observation into four stages (Obs 1 part I and II, Obs 2,
and Obs 3). The first two (obs1 part I and part II) are obtained from the
observation of ID 90501301001, and the spectra are dominated by thermal
component, corresponding to the SS. The rest two stages (Obs 2 and 3) are from
the observations of ID 80401316002 and 80401316004, and the spectra are
dominated by the power-law component, corresponding to the IS. After
subtracting the continuum, i.e., thermal and power-law components, they found
excess in the 5-7 keV, or the iron line profile as shown in Figure 3 of their
paper.

Considering the iron line profile is relatively constant over shorter
intervals, we select the Obs1 part I and Obs3 to represent the disk dominant
and power-law dominate states. Here we summarize the related results from Xu
et al. (2020). The spin of MAXI J1631-479 to be $a=0.96$ and the inclination
angle to be $i=30^{\circ}$. When the source evolves from part I of Obs1 to
Obs3, the line width varies from about 3 keV to 2 keV, the line flux decreases
from about $8\times 10^{-10}~{}$${\rm erg~{}s^{-1}cm^{-2}}$to $2\times
10^{-10}~{}$${\rm erg~{}s^{-1}cm^{-2}}$, the illuminating X-ray flux (10-79
keV) increases from about $10^{-9}~{}$${\rm erg~{}s^{-1}cm^{-2}}$to
$10^{-8}$${\rm erg~{}s^{-1}cm^{-2}}$, the illuminating X-ray flux (3-10 keV)
reduces from about $2.5\times 10^{-8}~{}$${\rm erg~{}s^{-1}cm^{-2}}$to
$2.3\times 10^{-8}$${\rm erg~{}s^{-1}cm^{-2}}$, the photon index hardens from
about 2.5 to 2.4, the ionization parameter, $\xi$, decreases from about 60000
${\rm erg~{}cm~{}s^{-1}}$to 3000 ${\rm erg~{}cm~{}s^{-1}}$, and the thermal
and power-law components of the continuum are all fitted.

###  4.2 Width-flux relationship in MAXI J1631-479

For the given spin and inclination angle, the theoretical line width could be
as high as about 5 keV for SAD when $a=0.96$ and $i=30^{\circ}$, while as the
fitting results of the observation is about 3.4 keV. The difference of 1.6 keV
is taken to be the error range of the line width for consistence. For obs3,
the theoretical line width can not be determined, as the geometry of the
accretion flow is unknown. So we also assume it to be 1.6 keV.

As the line flux is estimated with a Gaussian line in Xu et al. (2020), we
assume an error of 0.2 dex, which means the possible range is about 40%. So
the possible region of obs1 in the width-flux plane can be limited, the center
of which is marked with solid circle point.

The relative line flux in IS is estimated as folows. According to the fitted
continuum, the total illuminating X-rays in the range of 7-20 keV, which is
most important in ionizing electrons in the innermost shell of iron, increases
by about 3 times in IS. To estimate the variation of reflection rate for
different ionization level, we used the “relxill” model in XSPEC to obtain the
reflection spectra and then compare the fluxes in 6-7 keV for the illuminating
X-ray spectra ($\Gamma=2.4$) and ionization degree ($\xi=10000~{}and~{}3000$
${\rm erg~{}cm~{}s^{-1}}$) corresponding to SS and IS, respectively. We found
that for the same intensity of illuminating X-rays, the iron line flux
decreased by about 2 times in IS. To compare with our theoretical relation,
the line flux should be normalized to the same intensity of illuminating
X-rays and the same reflection rate. Therefore the ratio of the line flux of
IS to that of SS should be further divided by 3 and multiplied by 2, which is
finally about 0.17. Considering the 40% uncertainty on the line flux, the
possible range of normalized flux is 0.10$\sim$0.22, as shown by the solid
square point.

Then we can compare the observation results with the our theoretical results.
As shown in Fig. 3, the normalized line flux for the soft state roughly agrees
with the theoretical result, or the continuous disk model. While the data
point for the intermediate state located below the continuous disk model, but
can still be well explained with the cloud model.

###  4.3 Modelling of the iron line profiles

To further check the result, instead of just considering the width and
strength, we modeled the variation of the line profile by changing the
parameters of the cold gas reasonably, as shown by the curves in Fig.5. The
line profile in the soft state can be modeled with a cold disk that extends to
the boundary of 3 $r_{\rm ISCO}$. For the intermediate state, the truncation
of the cold disk occurs at about 20 $r_{\rm ISCO}$, meanwhile the cold gas
inside the truncation radius is in the range of 10 - 12 $r_{\rm ISCO}$. Our
result agrees with that of Xu et al. (2020), as $r_{\rm ISCO}$ is close to
gravitational radius for fast spinning BHs.

We also tried to model the variation of the line profiles with the continuous
disk model by just changing the truncation radius, we find it is difficult to
model the profile when take into account the variation of the illuminating
hard X-rays and ionization degree. As show by the short dashed line in Fig. 5
for example, this model failed to explain the line strength and width at the
same time.

Moreover, it is interesting to explore the impact of the azimuthal size of the
cloud. If the azimuthal size at a given radius for each cloud is less than
$30^{\circ}$, which is about one-tenth of $2\pi$, the radial range would be at
least ten times larger, considering the degeneracy of the azimuthal and radial
size. If there is just one cloud, it should extend from $10~{}r_{ISCO}$ to
$30~{}r_{ISCO}$, which is obviously unreasonable because the outer boundary of
the cloud is larger than the truncation radius. So we expect that there should
not be just one cloud. As the emissivity of the iron line is decreasing with
radius, we expect that the number of the cloud should be at least 3. However,
according to the evaporation model of the corona, where the evaporation rate
depends on the radius, it is still possible for a annulus to form during the
state transition.

##  5 Summary

In this work, we investigated the variation of the width and strength of the
broad iron lines during the state transition, and explored the possible
process of state transition. When compare with the combined observational data
of MAXI J1631-479 , the cloud model is favored according to the width-flux
relation we obtained. Moreover, the cloud model can explain not only the
spectra but also the relatively high reflection rate in SS. If the azimuthal
size of the cloud is small, our modeling parameters indicate that the number
of cloud should be more than 2.

In this work, the line flux and width in SS is selected as the normalization.
It is necessary to further check the variation of the iron during SS.
Moreover, the ionization level is assumed to be the same for the clouds and
outer cold disk, we would further improve the calculation in future works.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China under grant number U2038108, 12192220, 12192223 and 12133008, the
National SKA Program of China (No.2020SKA0110102). F.G.X. is additionally
supported in part by the Youth Innovation Promotion Association of CAS
(Y202064).

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