#  Insight-HXMT, NuSTAR and INTEGRAL Data Show Disk Truncation in the Hard
State of the Black-Hole X-Ray Binary MAXI J1820+070

Andrzej A. Zdziarski  Nicolaus Copernicus Astronomical Center, Polish Academy
of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland<EMAIL_ADDRESS>Bei You
School of Physics and Technology, Wuhan University, Wuhan 430072, China
Astronomical Center, Wuhan University, Wuhan 430072, China  Michał Szanecki
Faculty of Physics and Applied Informatics, Łódź University, Pomorska 149/153,
PL-90-236 Łódź, Poland  Xiao-Bo Li  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  Mingyu Ge  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

We study X-ray and soft gamma-ray spectra from the hard state of the accreting
black-hole binary MAXI J1820+070. We perform analysis of joint spectra from
HXMT, NuSTAR and INTEGRAL. We find an overall agreement between the spectra
from all three satellites. Satisfactory fits to the data require substantial
spectral complexity, with our models including two Comptonization regions and
their associated disk reflection, a disk blackbody and a narrow Fe K$\alpha$
line. Our fits confirm the presence of the truncation of the reflecting
optically-thick disk at least at $>$10 gravitational radii. However, we find
that the HXMT data alone cannot significantly constrain the disk inner radii.

##  1 Introduction

The knowledge of the geometry of accretion flows onto black holes (BHs) is
crucial for understanding the physics of accretion. While there is largely a
consensus that the standard disk model of geometrically-thin, optically-thick
accretion disk (Shakura & Sunyaev, 1973; Novikov & Thorne, 1973) extending
down to the innermost stable circular orbit (ISCO) applies to the soft
spectral state of accreting BH binaries, there is an ongoing controversy
regarding the geometry of accretion in the hard spectral state. The paradigm
dominant for a number of years that the inner disk is truncated and replaced
by a hot flow (e.g., Done et al. 2007) has been questioned in papers claiming
the disk extends down almost to the ISCO also in the hard state, see Bambi et
al. (2021) for a recent review.

Here, we address this question for MAXI J1820+070, a bright transient
accreting BH binary, whose outburst was discovered in 2018 (Tucker et al.,
2018; Kawamuro et al., 2018). The presence of a disk extending almost to the
ISCO in its hard state was advocated in Kara et al. (2019), Buisson et al.
(2019) and Wang et al. (2021). On the other hand, studies by Wang et al.
(2020), Axelsson & Veledina (2021), Dziełak et al. (2021), De Marco et al.
(2021), Marino et al. (2021), Zdziarski et al. (2021b), Zdziarski et al.
(2021c), hereafter Z21c, and Kawamura et al. (2022) found the disk to be
significantly truncated over various stages of the hard state.

You et al. (2021), hereafter Y21, analyzed broad energy spectra (2–200 keV) in
the hard state from Hard X-ray Modulation Telescope (Insight-HXMT; Zhang et
al. 2014) for MAXI J1820+070 in the 2018 outburst. It was found that the
reflection fraction, defined as the ratio of the coronal flux that illuminates
the disk to that emitted outside, showed an overall decrease with the
decreasing hardness of the X-ray spectrum. The latter decrease, in turn, was
found (Kara et al., 2019; De Marco et al., 2021) to be strongly correlated
with the decreasing time lag of soft X-rays with respect to hard ones observed
by the Neutron star Interior Composition ExploreR (NICER; Gendreau et al.
2016). Y21 interpreted this effect as indicating that the corona is
relativistically outflowing with the outflow velocity increasing with the
decreasing coronal scale height. However, the spectral fits of Insight-HXMT
alone cannot put constraint on the inner radius of the disc. Here, we follow
up on the studies of Y21 and Z21c, and combine the data from Insight-HXMT used
in Y21 with those from Nuclear Spectroscopic Telescope Array (NuSTAR; Harrison
et al. 2013), the Spectrometer on INTEGRAL (Roques et al. 2003; SPI), and the
INTEGRAL Soft Gamma-Ray Imager (Lebrun et al. 2003; ISGRI) used in Z21c. We
apply two different X-ray spectroscopy models to the joint data, with the main
goal of estimating the truncation radius of the disk. Also, we estimate other
parameters of the X-ray source.

The parameters of binary relevant to our study are as follows. The distance is
$D\approx 2.96\pm 0.33$ kpc based on a radio parallax (Atri et al., 2020), and
$D\leq 3.11\pm 0.06$ kpc based on the proper motion of the moving ejecta
during the hard-to-soft state transition (Wood et al., 2021). The inclinations
of the binary and the radio jet are $i_{\rm b}\approx 66\arcdeg$–$81\arcdeg$
(Torres et al., 2020), $i_{\rm j}\approx 64\arcdeg\pm 5\arcdeg$ (Wood et al.,
2021), respectively, and the BH mass is $M\approx(5.95\pm 0.22){\rm
M}_{\sun}/\sin^{3}i_{\rm b}$ (Torres et al., 2020).

##  2 Observations and data reduction

Table 1: Contemporaneous observations of MAXI J1820+070 with Insight-HXMT, NuSTAR and INTEGRAL in the hard state  Epoch | Insight-HXMT | Start time | Exp. LE, ME | SPI Start time | ISGRI Start time | Exp. SPI | NuSTAR | Start time | Exp. A
---|---|---|---|---|---|---|---|---|---
| Obs. ID | End time | Exp. HE | End time | End time | Exp. ISGRI | Obs. ID | End time | Exp. B
1 | P011466100402–3 | 58201.514 | 3688, 4049 | 58201.555 | 58201.544 | 13604 | 90401309008 | 58201.526 | 3046
|  | 58201.747 | 5599 | 58201.757 | 58201.757 | 8796 |  | 58201.766 | 3214
2 | P011466101202–4 | 58212.203 | 3800, 3859 | 58212.181 | 58212.181 | 28507 | 90401309012 | 58212.200 | 12334
|  | 58212.541 | 5402 | 58212.606 | 58212.393 | 8305 |  | 58213.177 | 12964

For our analysis, we have chosen epochs of the hard state of MAXI J1820+070
for which simultaneous data from all of Insight-HXMT, NuSTAR and INTEGRAL are
available. We have found only two such epochs. They are the same as epochs 1
and 2 in Z21c, which work analyses the NuSTAR and INTEGRAL data only. In the
current study, we add to them the corresponding Insight-HXMT spectral data.
The two epochs correspond to the beginning of a plateau phase on the count-
rate/hardness plot, see Fig. 1 of Zdziarski et al. (2021b). As shown in that
work, see their table 2, the bolometric luminosity during that part of the
plateau phase was almost constant, at $\approx$15% of the Eddington luminosity
(assuming $D=3$ kpc and $M=8{\rm M}_{\sun}$).

The INTEGRAL data (see Table 1) are from SPI and ISGRI, which jointly cover
the 23–650 keV range, and are the same as those in Z21c. The analysis of the
SPI data is described in Roques & Jourdain (2019). Data reduction and spectral
extraction for ISGRI was done using the osa v. 11.1 software (Courvoisier et
al., 2003). The SPI and ISGRI data include 0.5% and 1% systematic errors,
respectively.

The data from NuSTAR (3–79 keV) were reduced with heasoft v.6.25, the
NuSTARDAS pipeline v.1.8.0, and CALDB v.20200912 from the source region of a
$60^{\prime\prime}$ circle centered on the peak brightness. The data were
grouped to the signal-to-noise ratio $\geq$50 below 69 keV and no grouping has
been applied at higher energies. Following previous spectral studies of the
NuSTAR data, e.g., Buisson et al. (2019), we have applied no systematic errors
to the data. The data consist of those from two focal plane modules, A and B,
see Table 1.

The Insight-HXMT data are those extracted by Y21. The chosen data sets are
listed in Table 1. The low, medium and high energy detectors, LE, ME, and HE,
are used in the 2–10, 10–30, and 30–200 keV ranges, respectively. The
individual Insight-HXMT data overlapping with those of NuSTAR have been
merged, as described in Table 1. The data were grouped according to the
recommendation of the Insight-HXMT team. In order to directly study the
spectral calibration of the Insight-HXMT detectors, we have applied no
systematic errors to these data. See Li et al. (2020) for estimates of the
actual systematic errors.

Table 2: The results of spectral fitting the NuSTAR +INTEGRAL data alone in the 3–650 keV range  Component | Parameter | Epoch 1 | Epoch 2
---|---|---|---
ISM absorption |  $N_{\rm H}$ $[10^{21}]$ cm-2 | 1.4f
Joint constraints |  $i$ $[\arcdeg$] | $59^{+3}_{-0}$ | $67^{+4}_{-1}$
| $Z_{\rm Fe}\,[\sun]$ | $3.6^{+0.5}_{-1.7}$ | $1.7^{+0.2}_{-0.1}$
Disk | $kT_{\rm in}$ | $0.5^{+0.1}_{-0.1}$ | $0.4^{+0.1}_{-0.1}$
and a narrow | $N_{\tt diskbb}\,[10^{3}]$ | $7^{+19}_{-6}$ | $10^{+5}_{-5}$
6.40 keV line | $N_{{\rm Fe\,K}\alpha}\,[10^{-3}]$ | $4^{+1}_{-1}$ | $2^{+1}_{-1}$
Thermal | $y_{\rm th}$ | $0.62^{+0.01}_{-0.01}$ | $0.58^{+0.02}_{-0.01}$
Comptonization | $\Gamma_{\rm th}$ | $1.66^{+0.01}_{-0.02}$ | $1.72^{+0.04}_{-0.01}$
and reflection |  $kT_{\rm e,th}$ [keV] | $11^{+1}_{-1}$ | $11^{+1}_{-1}$
| $R_{\rm in}\,[R_{\rm g}]$ | $13^{+5}_{-4}$ | $17^{+3}_{-5}$
| ${\cal R}_{\rm th}$ | $1.5^{+0.2}_{-0.3}$ | $0.9^{+0.2}_{-0.4}$
| $\log_{10}\xi_{\rm th}$ | $4.3^{+0.1}_{-0.1}$ | $3.9^{+0.1}_{-0.1}$
| $N_{\rm th}$ | $1.51^{+1.86}_{-0.03}$ | $2.26^{+0.01}_{-0.04}$
Hybrid | $y_{\rm h}$ | $1.18^{+0.01}_{-0.01}$ | $0.93^{+0.05}_{-0.04}$
Comptonization | $\Gamma_{\rm h}$ | $1.26^{+0.01}_{-0.01}$ | $1.27^{+0.04}_{-0.03}$
and reflection |  $kT_{\rm e,h}$ [keV] | $23^{+1}_{-1}$ | $19^{+1}_{-1}$
| $\gamma_{\rm min}$ | $1.30^{+0.02}_{-0.05}$ | $1.09^{+0.29}_{-0.01}$
| $p$ | $2.70^{+0.37}_{-0.09}$ | $3.45^{+0.14}_{-0.15}$
| $\Delta R\,[R_{\rm g}]$ | $20^{+11}_{-6}$ | $35^{+13}_{-9}$
| ${\cal R}_{\rm h}$ | $0.74^{+0.25}_{-0.08}$ | $0.72^{+0.06}_{-0.03}$
| $\log_{10}\xi_{\rm h}$ | $2.3^{+0.2}_{-0.3}$ | $1.7^{+0.2}_{-1.7}$
| $N_{\rm h}$ | $0.47^{+0.30}_{-0.04}$ | $0.55^{+0.05}_{-0.11}$
| $\chi_{\nu}^{2}$ | 920/789 | 1634/1304
Table 3: The results of spectral fitting the NuSTAR +INTEGRAL data alone in the 3–150 keV energy range  Component | Parameter | Epoch 1 | Epoch 2
---|---|---|---
ISM absorption |  $N_{\rm H}$ $[10^{21}]$ cm-2 | 1.4f
Joint constraints |  $i$ $[\arcdeg$] | $59^{+3}_{-0}$ | $59^{+7}_{-0}$
| $Z_{\rm Fe}\,[\sun]$ | $1.5^{+0.9}_{-0.1}$ | $2.0^{+0.7}_{-0.3}$
Disk | $kT_{\rm in}$ | $0.4^{+0.2}_{-0.1}$ | $0.4^{+0.1}_{-0.1}$
and a narrow | $N_{\tt diskbb}\,[10^{3}]$ | $49^{+1400}_{-39}$ | $13^{+69}_{-9}$
6.40 keV line | $N_{{\rm Fe\,K}\alpha}\,[10^{-3}]$ | $4^{+1}_{-1}$ | $2^{+1}_{-1}$
Soft | $\Gamma_{\rm s}$ | $1.69^{+0.01}_{-0.03}$ | $1.71^{+0.01}_{-0.01}$
Comptonization |  $kT_{\rm e,s}$ [keV] | $28^{+34}_{-6}$ | $11^{+1}_{-1}$
and reflection | $R_{\rm in,s}\,[R_{\rm g}]$ | $12^{+4}_{-4}$ | $63^{+58}_{-26}$
| ${\cal R}_{\rm s}$ | $0.7^{+0.2}_{-0.2}$ | $0.7^{+0.5}_{-0.2}$
| $\log_{10}\xi_{\rm s}$ | $3.8^{+0.4}_{-0.1}$ | $4.0^{+0.1}_{-0.1}$
| $N_{\rm s}\;[\times 10^{-2}]$ | $7.7^{+0.1}_{-0.2}$ | $5.7^{+0.1}_{-0.1}$
Hard | $\Gamma_{\rm h}$ | $1.43^{+0.04}_{-0.02}$ | $1.20^{+0.04}_{-0}$
Comptonization |  $kT_{\rm e,h}$ [keV] | $43^{+19}_{-17}$ | $26^{+1}_{-1}$
and reflection | $R_{\rm in,h}\,[R_{\rm g}]$ | $39^{+13}_{-8}$ | $47^{+13}_{-10}$
| ${\cal R}_{\rm h}$ | $1.26^{+0.39}_{-0.04}$ | $0.9^{+0.1}_{-0.2}$
| $\log_{10}\xi_{\rm h}$ | $0^{+2.4}$ | $1.8^{+0.4}_{-1.8}$
| $N_{\rm h}\;[\times 10^{-2}]$ | $5^{+3}_{-2}$ | $0.2^{+0.1}_{-0.1}$
| $\chi_{\nu}^{2}$ | 905/772 | 1585/1289

##  3 Spectral Fits

Here, we further develop spectral models of Zdziarski et al. (2021b) and Z21c
fitted to X-ray and soft $\gamma$-ray data from MAXI J1820+070 in the hard
state. In Zdziarski et al. (2021b), we fitted four observations by NuSTAR,
which cover the 3–79 keV energy range. We have found that those spectral data
require the presence of two different, softer and harder, Comptonization
components, and two corresponding reflection components, one strongly
relativistically blurred and strongly ionized, and one weakly blurred and
weakly ionized. No disk blackbody component was included. The Comptonization
was modelled as by purely thermal electrons. The harder component dominated
the broad-band X-ray flux in all cases. Our proposed geometry (see fig. 4 in
Zdziarski et al. 2021b) was that of a hot flow within the disk inner
truncation radius, $R_{\rm in}$, with a large scale height responsible for the
harder Comptonization, and a coronal flow covering the disk between $R_{\rm
in}$ and a transition radius, $R_{\rm tr}$. The softer primary component
reflected from the disk underneath the corona, while the harder one reflected
from the radii $>\\!R_{\rm tr}$. The truncation radii were found to be in the
range of $R_{\rm in}\approx 10$–$10^{2}R_{\rm g}$, where $R_{\rm g}$ is the
gravitational radius. As pointed out in Zdziarski et al. (2021b), the harder
component may instead be outflowing and forming a slow sheath of the jet.

In Z21c, we studied two sets of simultaneous observations of MAXI J1820+070 in
the hard state by NuSTAR and INTEGRAL. The joint data covered the 3–650 keV
range. Our major finding was that Comptonization by purely thermal electrons
was not sufficient to account for the emission at $\gtrsim$100 keV, while the
high-energy spectrum formed a smooth continuation of that at lower energies.
The full spectra were then well fitted by the same model as above except that
the hard Comptonization was by hybrid electrons, i.e., Maxwellian with a high
energy tail. The electron temperatures were found rather low, $kT_{\rm
e}\approx 10$–30 keV. The truncation radius was found to be relatively high,
$R_{\rm in}\approx 20$–$30R_{\rm g}$. Also, the average spectrum from all
NuSTAR and INTEGRAL observations in the hard state was studied. It was similar
in shape to those of the two individual observations, but with a much better
statistics at high energies, which allowed a reliable calculation of the rate
of e± pairs. That rate was found to be quite high, but balanced by pair
annihilation, leading to the equilibrium relative pair density being low, in
agreement with the absence of a pair-annihilation spectral feature. This was
due to the rather large Thomson optical depth of the hard-Comptonization
cloud, $\tau_{\rm T}\approx 5$.

We perform spectral fits using xspec (Arnaud, 1996). The uncertainties given
below correspond to the 90% confidence range for a single parameter,
$\Delta\chi^{2}=+2.71$. In order to assure that some local minima are not
missed, we also scan the parameter space using the steppar command of xspec.

###  3.1 The NuSTAR and INTEGRAL data alone

We first re-fit the NuSTAR and INTEGRAL data alone. We initially fit the data
with the same hybrid-Comptonization model as in Z21c (where these data are
also denoted as epochs 1 and 2). The model includes two Comptonization regions
with two corresponding reflection regions, see above and section 3.1 of
Zdziarski et al. (2021b) for a detailed description. In particular, the BH is
assumed to be maximally rotating, with the dimensionless spin parameter of
0.998, in which case the ISCO radius is at $\approx\\!1.24R_{\rm g}$. That
model fits very well the high-energy data of the SPI and ISGRI up to several
hundred keV. In the fits, the two reflection zones have very different
characteristic ionization parameters, $\lg\xi\sim 3.5$ and $\lesssim 1.7$,
respectively (Zdziarski et al. 2021b; Z21c).

As in Z21c, we allow for a residual uncertainty of the overall slope of the
spectral calibration, by multiplying the model spectrum by
$KE^{-\Delta\Gamma}$, where we fix $K=1$ and $\Delta\Gamma=0$ for the module A
of NuSTAR. The fitted exponent $\Delta\Gamma$ was found in Z21c and here to be
relatively small, $\lesssim\\!0.01$ for NuSTAR module B and for the INTEGRAL
spectra. Also, we require the fitted inclination to be within the joint range
of those found for the binary and the jet, $59\arcdeg\leq i\leq 81$ (Wood et
al., 2021; Torres et al., 2020).

In the model, the soft Comptonization is purely thermal (with a non-thermal
electron tail possible, but not affecting the parameters, see Z21c), whose
incident spectrum and its relativistic reflection are modelled using reflkerr
(Niedźwiecki et al., 2019). The hard Comptonization is hybrid, by electrons
with a predominantly Maxwellian distribution with a power-law electron tail
above certain energy, modelled with reflkerr_bb (Z21c). In both routines, the
incident Comptonization spectrum is modelled using compps (Poutanen &
Svensson, 1996). The temperature of the seed photons is kept at 0.2 keV,
following the results of fitting NICER data by Wang et al. (2020). The current
version of reflkerr_bb has a small increase in the accuracy of the non-thermal
tail and an improvement of the mapping of the used reflection spectra of
xillver (García & Kallman, 2010; García et al., 2018) on the grid of
Comptonization spectra with respect to the version used in Z21c. Still, this
has only a minor effect on the fitted parameters and $\chi^{2}$. This model
yields $\chi_{\nu}^{2}\approx 954/792$ and $1762/1307$ (which are very similar
to those in Z21c) for epochs 1 and 2, respectively.

However, since the Insight-HXMT LE data (which we fit below) extend down to 2
keV, we include a disk blackbody component to the model. This accounts for the
observed spectral softening at low energies (cf. fig. 7 in Zdziarski et al.
2021b). We use the diskbb model (Mitsuda et al., 1984). This significantly
improves the fits with $\chi^{2}\approx 940/790$ and $1680/1305$ for epochs 1
and 2, respectively.

Also, we add an unblurred reflection to the above model. This is motivated by
the maximum radius for which reflection is calculated in either reflkerr or
relxill being equal to $10^{3}R_{\rm g}$. At this radius, there is still a
measurable broadening of the Fe K line. We first add a Gaussian Fe K$\alpha$
line. For epoch 1 and 2, it yields $\chi_{\nu}^{2}\approx 920/789$ and
$1634/1304$, respectively i.e., the fit improvements are significant. We find
the line is required to have the centroid energy of 6.40 keV and the width
$\ll\\!1$ keV (indicating its origin in a cold and neutral medium), which we
then fix in the fit. We have then added instead a remote, static, reflection
using the hreflectnth model (Niedźwiecki et al., 2019) with the same
parameters of the electron distribution as the hard hybrid Comptonization
(modelled by reflkerr_bb) but with the ionization parameter fixed at the
minimum value possible in this model of $\xi=1$ erg cm s-1 (i.e., for the
remote reflector being nearly neutral). This results in virtually no
difference of the fitted parameters and $\chi_{\nu}^{2}$ with respect to the
model with a Gaussian line for epoch 2. This is because only the Fe K$\alpha$
line contributes to the model significantly in hreflectnth. Thus, in order to
keep the model as simple as possible, we use a Gaussian component hereafter.
Our model is then

$\displaystyle{\tt plabs*tbabs(diskbb+reflkerr+reflkerr\\_bb}$
$\displaystyle\qquad{\tt+gauss)}.$ (1)

Furthermore, Y21 considered a variable irradiation index, $q$, corresponding
to the disk irradiating flux of $\propto R^{-q}$. In the models above, the
standard value (corresponding either to the disk dissipation, Shakura &
Sunyaev 1973 or irradiation by a central point source) of $q=3$ was used.
However, we find that for the present data $q$ is only weakly constrained to
$\geq$2.2 for epoch 1, and allowing it to be free does not improve the
$\chi^{2}$. For epoch 2, we obtain a similar $q\approx 3.0^{+1.5}_{-0.8}$ (and
no $\chi^{2}$ improvement). Thus, we hereafter keep the fixed $q=3$. Our
fitting results for this case are given in Table 2. Following Zdziarski et al.
(2021b) and Z21c, we have allowed the values of $i$ and $Z_{\rm Fe}$ to be
different for the two epochs. The main motivation here is to see the effect of
different data sets and models for the fitted values. In our view, X-ray
spectral models cannot by themselves be taken as fully reflecting the physical
reality. They provide only some approximations, which then can be tested by
combining with results from timing and other studies (see, e.g., De Marco et
al. 2021). We also note that while the Fe abundance has to remain constant,
some changes of the fitted inclinations are possible due to precession and
warping of various parts of the reflecting disk.

The results in Table 2 for epoch 2 are relatively similar to those obtained by
Z21c without including the disk blackbody and narrow Fe K line components. We
can see that the fitted truncation radius decreased, from $31^{+9}_{-5}R_{\rm
g}$ to $17^{+3}_{-5}R_{\rm g}$, and the Fe abundance slightly increased. On
the other hand, the present fit for epoch 1 gives $i\approx 59\arcdeg$, lower
than that in Z21c. As noted above, we constrained $i$ to $\geq\\!59\arcdeg$.
When this constraint is relaxed, the best fit $i$ still remains at
$59\arcdeg$. Then, the Fe abundance has significantly increased, to a rather
unlikely (at the best fit) value of $Z_{\rm Fe}\approx 3.6^{+0.5}_{-1.7}$.
This is apparently an artefact of the fitted model. As discussed by García et
al. (2018), the reflector density being kept at a fixed low value in the used
reflection model may lead to $Z_{\rm Fe}$ being in some cases artificially
high.

We have then studied alternative models. For the sake of simplicity, we
consider here only epoch 2, which has a much longer exposure than epoch 1, see
Table 1. Above, the reflection of the soft components came from an inner part
of the disk and was much more strongly ionized than that of the hard
components, which came from an outer disk part, see fig. 4 in Zdziarski et al.
(2021b). We have thus three alternative possibilities. The reflection of the
hard component could be still from an outer part but with a high ionization,
or it could be from an inner part with either low or high ionization. We have
found that all three possibilities correspond to local minima with the values
of $\chi^{2}$ significantly higher than our previous spectral solution with
the hard component reflecting from an outer part with low ionization
(representing thus the global minimum). The values of $\Delta\chi^{2}$ are
$+66$, $+33$ and $+21$, respectively. Summarizing our investigations above,
our chosen hybrid-Compton model to broad-band spectra is most likely among the
considered models based on the $\chi^{2}$ criterion.

On the other hand, the main purpose of this work is to study the Insight-HXMT
spectral data (alone and together with those from NuSTAR and INTEGRAL) for
this source, which useful energy range is (as we find below) $\lesssim$150
keV. Furthermore, we would like to be able to compare our above results,
obtained with the reflkerr family of models, with those of the relxillCp
models (García & Kallman, 2010; Dauser et al., 2016; García et al., 2018).
Since relxillCp does not have an option to include a non-thermal tail in the
electron distribution, we limit the energy range of the fitted data to
$\leq$150 keV, which allows us not to include the high-energy electron tail,
which was found to be necessary to fit the INTEGRAL data at higher energies.
Also, given the structure of relxillCp, we have to consider the variant of
that model with two Comptonization zones and the two corresponding reflection
regions overlapping rather than adjacent, i.e., with two independent inner
reflection radii, in spite of the objections to their physical reality
discussed above. As shown in Zdziarski et al. (2021b), see their table 2, this
still has a relatively little effect on the fitted parameters. Our model is
then

$\displaystyle{\tt plabs*tbabs(diskbb+relxillCp_{\rm s}+relxillCp_{\rm h}}$
$\displaystyle\qquad{\tt+gauss)},$ (2)

where the indices ‘s’ and ‘h’ correspond to the soft and hard thermal
Comptonization components, respectively. Note that the low-energy spectral
index, $\Gamma$, is constrained to $\geq$1.20 in relxillCp.

Figure 1: The unfolded spectra and data-to-model ratios of the epochs (a) 1
and (b) 2, fitted with the model of Equation (1), see Table 4 for the
parameters. The error bars show the data from NuSTAR A and B (magenta and
cyan, respectively), SPI (blue), ISGRI (red), Insight-HXMT LE (green), ME
(yellow) and HE (black). The spectra are normalized to the NuSTAR A one. The
softer (red long dashes) and harder (blue short dashes) Comptonization
components are for thermal and hybrid electrons, respectively. The
corresponding reflection components are shown by the magenta dot-dashes and
cyan dots, respectively. The disk blackbody and the narrow Fe K$\alpha$ lines
are shown by the black dot-dashed and yellow dashed lines, respectively. No
rebinning has been applied to the plots.

The results of the fitting with this model are given in Table 3. We see that
we still obtain fits where the disk is significantly truncated, and the
truncation radii for the two reflectors are larger for both epochs (except for
$R_{\rm in,s}$ for the reflection of the soft Comptonization component for
epoch 1, which is almost the same as the corresponding one for the reflkerr
model) than the inner truncation radii obtained with the reflkerr model (of
Equation 1), see Table 2. The differences in the parameters with respect to
the reflkerr models are due to the differences in the models; in particular,
relxillCp uses the purely thermal Comptonization model of Zdziarski et al.
(1996) rather than the hybrid one of Poutanen & Svensson (1996). Still, the
two sets of results are relatively similar.

We note that the increase of $R_{\rm in}$ from epochs 1 to 2 in Table 3 is
associated with a decrease of the spectral index of the dominant hard
component, $\Gamma_{\rm h}$. This is expected if the cooling of this component
is reduced due to fewer disk blackbody photons impinging on that component. On
the other hand, we see that the electron temperature decreases, contrary to
the above expectation. This shows that the energy balance is more complex,
most likely involving changing the optical depth and cooling by synchrotron
photons. We note that a similar lack of a correlation between $T_{\rm e}$ and
the spectral hardness is seen in Cyg X-1, see figures 4 and 6 in Ibragimov et
al. (2005). On the other hand, an analogous increase of $R_{\rm in}$ from
epochs 1 to 2 is much weaker in the model using reflkerr, see Table 2. This
demonstrates a significant model dependence of the detailed fitting results.

###  3.2 The NuSTAR, INTEGRAL and Insight-HXMT data

We then include the Insight-HXMT data in our spectral models. We fit the LE,
ME and HE data in the 2–10, 10–30 and either the full range of 30–200 keV when
using Equation (1) and 30–150 keV with Equation (2). We allow $K$ and
$\Delta\Gamma$ to be free for the LE and ME data; for HE, we found that
$\Delta\Gamma$ is compatible with null for epoch 1, and allowing it free does
not improve the fit. We thus keep it fixed at null, and allow only $K$ to be
free.

Our results are given in Tables 4 and 5 and Figure 1. We find that the
addition of the Insight-HXMT data only moderately changes our results. In
particular, the reflecting disk is substantially truncated for all of the fits
during both epochs. In all cases, we find $R_{\rm in}\gtrsim 10R_{\rm g}$.
However, we see that using the present model and including the Insight-HXMT
data results in the reflection components being quite strong for epoch 1, see
Figure 1(a), while no such issue appeared for the NuSTAR +INTEGRAL data alone
fitted in Z21c.

We have then tested again the effect of allowing the irradiation index, $q$,
to be free. While it gave no fit improvement at all in the case of the NuSTAR
+INTEGRAL data alone, we find it now only slightly, by a few, decrease the
values of $\chi^{2}$, while the fitted values are $q\gtrsim 2$. Still, those
models have similarly large values of the disk truncation radii, and show no
significant changes in the parameters. We thus have not investigated those
models further.

We clearly see apparently spurious high-energy tails in the HE data above
$\sim$100 keV. The presence of those upturns have resulted in hardenings of
the best-fit models, and in consequence the ISGRI and SPI data are below the
model at $\gtrsim$100 keV. Since the calibration of the INTEGRAL detectors is
rather well established, with mutual agreement of the spectra from its three
high-energy detectors (including the PICSiT), as shown for MAXI J1820+070 in
Z21c, the actual departure of the high-energy part of the HE spectrum
corresponds to ratio of the HE and INTEGRAL spectra, and it is higher than the
departures from the present best fits.

While we find also low value of $\Delta\Gamma$ for the fitted LE and HE
spectra, we find the rather high values of the best fit $\Delta\Gamma\approx
0.05$, 0.06 for the ME data for epochs 1 and 2, respectively111This problem
appears to be specific to the Insight-HXMT data for MAXI J1820+070; no such a
large $\Delta\Gamma$ is found when fitting the Insight-HXMT spectra of either
Crab or the BH X-ray binary MAXI J1535–571..

Table 4: The results of spectral fitting the NuSTAR +INTEGRAL +Insight-HXMT data in the 2–650 keV energy range  Component | Parameter | Epoch 1 | Epoch 2
---|---|---|---
ISM absorption |  $N_{\rm H}$ $[10^{21}]$ cm-2 | 1.4f
Joint constraints |  $i$ $[\arcdeg$] | $59^{+3}_{-0}$ | $60^{+1}_{-0}$
| $Z_{\rm Fe}\,[\sun]$ | $2.3^{+0.3}_{-0.1}$ | $2.1^{+0.1}_{-0.2}$
Disk | $kT_{\rm in}$ | $0.50^{+0.02}_{-0.01}$ | $0.50^{+0.02}_{-0.02}$
and a narrow | $N_{\tt diskbb}\,[10^{3}]$ | $3.3^{+0.6}_{-0.6}$ | $2.6^{+0.7}_{-0.4}$
6.40 keV line | $N_{{\rm Fe\,K}\alpha}\,[10^{-3}]$ | $3.1^{+0.1}_{-0.1}$ | $2.3^{+0.4}_{-0.3}$
Thermal | $y_{\rm th}$ | $0.71^{+0.01}_{-0.01}$ | $0.61^{+0.01}_{-0.01}$
Comptonization | $\Gamma_{\rm th}$ | $1.62^{+0.01}_{-0.01}$ | $1.71^{+0.01}_{-0.01}$
and reflection |  $kT_{\rm e,th}$ [keV] | $15.5^{+0.4}_{-0.5}$ | $12.4^{+0.3}_{-0.2}$
| $R_{\rm in}\,[R_{\rm g}]$ | $15^{+1}_{-4}$ | $15^{+1}_{-3}$
| ${\cal R}_{\rm th}$ | $1.40^{+0.05}_{-0.04}$ | $0.49^{+0.02}_{-0.05}$
| $\log_{10}\xi_{\rm th}$ | $4.20^{+0.03}_{-0.01}$ | $3.88^{+0.02}_{-0.04}$
| $N_{\rm th}$ | $1.62^{+0.50}_{-0.06}$ | $2.67^{+0.18}_{-0.07}$
Hybrid | $y_{\rm h}$ | $1.38^{+0.02}_{-0.05}$ | $1.03^{+0.03}_{-0.03}$
Comptonization | $\Gamma_{\rm h}$ | $1.19^{+0.02}_{-0.01}$ | $1.27^{+0.02}_{-0.02}$
and reflection |  $kT_{\rm e,h}$ [keV] | $27.5^{+0.3}_{-2.4}$ | $23.1^{+2.2}_{-0.4}$
| $\gamma_{\rm min}$ | $1.26^{+0.16}_{-0.02}$ | $1.15^{+0.25}_{-0.02}$
| $p$ | $3.66^{+0.10}_{-0.08}$ | $3.7^{+0.3}_{-0.1}$
| $\Delta R\,[R_{\rm g}]$ | $17^{+4}_{-6}$ | $27^{+5}_{-6}$
| ${\cal R}_{\rm h}$ | $1.29^{+0.01}_{-0.02}$ | $0.73^{+0.09}_{-0.02}$
| $\log_{10}\xi_{\rm h}$ | $0^{+1.7}$ | $0^{+1.2}$
| $N_{\rm h}$ | $0.35^{+0.03}_{-0.01}$ | $0.46^{+0.01}_{-0.01}$
LE/NuSTAR A | $\Delta\Gamma\,[10^{-2}]$ | $0.9^{+0.1}_{-0.6}$ | $1.0^{+0.5}_{-0.5}$
ME/NuSTAR A | $\Delta\Gamma\,[10^{-2}]$ | $5.0^{+0.2}_{-0.5}$ | $6.2^{+0.4}_{-0.4}$
| $\chi_{\nu}^{2}$ | 1684/1192 | 2210/1706
Table 5: The results of spectral fitting the NuSTAR +INTEGRAL +Insight-HXMT data in the 2–150 keV energy range  Component | Parameter | Epoch 1 | Epoch 2
---|---|---|---
ISM absorption |  $N_{\rm H}$ $[10^{21}]$ cm-2 | 1.4f
Joint constraints |  $i$ $[\arcdeg$] | $59^{+6}_{-0}$ | $62^{+3}_{-3}$
| $Z_{\rm Fe}\,[\sun]$ | $1.9^{+0.2}_{-0.1}$ | $1.8^{+0.2}_{-0.2}$
Disk | $kT_{\rm in}$ | $0.48^{+0.03}_{-0.03}$ | $0.48^{+0.02}_{-0.03}$
and a narrow | $N_{\tt diskbb}\,[10^{3}]$ | $4^{+2}_{-2}$ | $3^{+1}_{-1}$
6.40 keV line | $N_{{\rm Fe\,K}\alpha}\,[10^{-3}]$ | $3^{+1}_{-1}$ | $2.0^{+0.4}_{-0.4}$
Soft | $\Gamma_{\rm s}$ | $1.71^{+0.01}_{-0.01}$ | $1.71^{+0.01}_{-0.01}$
Comptonization |  $kT_{\rm e,s}$ [keV] | $14^{+1}_{-2}$ | $13^{+1}_{-1}$
and reflection | $R_{\rm in,s}\,[R_{\rm g}]$ | $13^{+4}_{-3}$ | $70^{+47}_{-30}$
| ${\cal R}_{\rm s}$ | $0.69^{+1.31}_{-0.04}$ | $0.7^{+0.5}_{-0.3}$
| $\log_{10}\xi_{\rm s}$ | $3.95^{+0.30}_{-0.04}$ | $3.9^{+0.1}_{-0.1}$
| $N_{\rm s}\,[10^{-2}]$ | $6.4^{+0.1}_{-0.1}$ | $6.3^{+0.5}_{-0.4}$
Hard | $\Gamma_{\rm h}$ | $1.31^{+0.03}_{-0.04}$ | $1.21^{+0.03}_{-0.01}$
Comptonization |  $kT_{\rm e,h}$ [keV] | $33^{+2}_{-2}$ | $29^{+1}_{-1}$
and reflection | $R_{\rm in,h}\,[R_{\rm g}]$ | $41^{+11}_{-9}$ | $54^{+15}_{-10}$
| ${\cal R}_{\rm h}$ | $1.0^{+0.4}_{-0.2}$ | $1.0^{+0.2}_{-0.5}$
| $\log_{10}\xi_{\rm h}$ | $0^{+2.3}$ | $0^{+2.0}$
| $N_{\rm h}\,[10^{-2}]$ | $6.6^{+0.6}_{-0.4}$ | $5.7^{+0.3}_{-0.1}$
LE/NuSTAR A | $\Delta\Gamma\,[10^{-2}]$ | $0.9^{+0.6}_{-0.6}$ | $1.0^{+0.5}_{-0.5}$
ME/NuSTAR A | $\Delta\Gamma\,[10^{-2}]$ | $5.0^{+0.6}_{-0.6}$ | $6.1^{+0.5}_{-0.5}$
| $\chi_{\nu}^{2}$ | 1561/1158 | 2139/1675

Then, following the recommendation of the Insight-HXMT team, we have tested
the effect of ignoring the ME data in the 21–24 keV range. However, we have
found that including those data has virtually no effect on the fits, and
results only in a modest increase of $\chi^{2}_{\nu}$.

Table 6: The results of spectral fitting the Insight-HXMT data only in the
2–150 keV energy range

Component | Parameter | Epoch 1 | Epoch 2
---|---|---|---
ISM absorption |  $N_{\rm H}$ $[10^{21}]$ cm-2 | 1.4f
Joint constraints |  $i$ $[\arcdeg$] | $63^{+4}_{-3}$ | $59^{+8}_{-0}$
| $Z_{\rm Fe}\,[\sun]$ | $1.6^{+0.3}_{-0.3}$ | $2.0^{+0.4}_{-0.4}$
Disk | $kT_{\rm in}$ | $0.46^{+0.02}_{-0.02}$ | $0.45^{+0.05}_{-0.05}$
and a narrow | $N_{\tt diskbb}\,[10^{3}]$ | $4.8^{+2.3}_{-0.8}$ | $4.4^{+3.2}_{-1.7}$
6.40 keV line | $N_{{\rm Fe\,K}\alpha}\,[10^{-3}]$ | $2.6^{+0.7}_{-0.9}$ | $1.7^{+0.8}_{-0.8}$
Soft | $\Gamma_{\rm s}$ | $1.73^{+0.03}_{-0.03}$ | $1.75^{+0.02}_{-0.02}$
Comptonization |  $kT_{\rm e,s}$ [keV] | $13^{+1}_{-1}$ | $13^{+1}_{-1}$
and reflection | $R_{\rm in,s}\,[R_{\rm g}]$ | $230^{+770}_{-175}$ | $1.4^{+7.4}_{-0.2}$
| ${\cal R}_{\rm s}$ | $0.5^{+0.1}_{-0.2}$ | $0.5^{+0.6}_{-0.1}$
| $\log_{10}\xi_{\rm s}$ | $4.0^{+0.1}_{-0.1}$ | $3.6^{+0.3}_{-0.1}$
| $N_{\rm s}\,[10^{-2}]$ | $7.1^{+0.6}_{-0.3}$ | $6.9^{+0.5}_{-1.3}$
Hard | $\Gamma_{\rm h}$ | $1.27^{+0.05}_{-0.07}$ | $1.27^{+0.07}_{-0.06}$
Comptonization |  $kT_{\rm e,h}$ [keV] | $32^{+1}_{-1}$ | $32^{+2}_{-2}$
and reflection | $R_{\rm in,h}\,[R_{\rm g}]$ | $33^{+22}_{-8}$ | $59^{+44}_{-19}$
| ${\cal R}_{\rm h}$ | $0.5^{+0.1}_{-0.1}$ | $0.9^{+0.1}_{-0.1}$
| $\log_{10}\xi_{\rm h}$ | $0^{+1.8}$ | $0^{+2.7}$
| $N_{\rm h}\,[10^{-2}]$ | $6.1^{+0.4}_{-0.7}$ | $6.1^{+0.06}_{-0.04}$
LE/HE | $\Delta\Gamma\,[10^{-2}]$ | $-1.4^{+1.2}_{-1.4}$ | $-0.7^{+1.3}_{-0.3}$
ME/HE | $\Delta\Gamma\,[10^{-2}]$ | $4.8^{+0.2}_{-1.0}$ | $6^{+0}_{-1}$
| $\chi_{\nu}^{2}$ | 552/370 | 451/370

.

###  3.3 Insight-HXMT data alone

We have then fit the Insight-HXMT data alone. For the sake of simplicity, we
fit here only the relxillCp model, Equation (2). Given the results of the
joint fitting, we assume $\Delta\Gamma=0$, $K=1$ for the HE data. However, we
have found that allowing free values of $\Delta\Gamma$ for the LE and ME leads
to rather large, and obviously spurious, results. We thus have constrained
their absolute values to be within the uncertainty ranges of the corresponding
joint fits, see Table 5. Our results are given in Table 6.

We can see that these data alone poorly constrain the inner radius. It is
found very large for epoch 1, and very small for epoch 2. This is opposite to
the behavior found for the fits with the NuSTAR +INTEGRAL data alone, Table 3,
and those including the Insight-HXMT data, Table 5. In those cases, the inner
radius for epoch 2 was found to be several times that for epoch 1, with both
showing significant disk truncation. This particular Insight-HXMT result is
clearly spurious. It appears that the Insight-HXMT data are, unfortunately,
not suitable to determine the truncation radii during the evolution of MAXI
J1820+070.

##  4 Discussion

We have found a good agreement between the overall shapes of the Insight-HXMT
data and those of NuSTAR and INTEGRAL. The two main outstanding issues are the
slope of the ME data requiring a relatively large correction to bring it into
an agreement with the NuSTAR data, and the presence of apparently spurious
high-energy tails in the HE data at $E\gtrsim 10^{2}$ keV. We note, however,
that Li et al. (2020) estimate the systematic error of the HE data at 150 keV
to be as high as 5%, whose inclusion would strongly reduce the statistical
significance of the tail.

Our major robust result is yet another confirmation of relatively large inner
truncation radii of the reflecting disk in MAXI J1820+070, $R_{\rm in}\gtrsim
10R_{\rm g}$, for both the NuSTAR data alone and those including Insight-HXMT.
On the other hand, the fits of a large set of Insight-HXMT data in Y21 assumed
that the disk extends down to the ISCO of a maximally rotating BH, i.e.,
$R_{\rm in}\approx 1.24R_{\rm g}$. This disk was irradiated by a corona above
it (see equation 1 in Y21) with a free irradiation index, with the best-fit
values of $q\sim 0$. Such low values of $q$ are possible only if either the
irradiation of innermost parts of the disk is strongly reduced with respect to
the standard coronal dissipation profile (Novikov & Thorne, 1973) or those
disk parts are strongly fragmented. Alternatively, such values of $q$ may
mimic the effect of the zero-stress boundary condition at the ISCO for a
slowly-spinning BH, where the peak of the dissipation profile is at $>10R_{\rm
g}$. In fact, the parameters $q$ and $R_{\rm in}$ are strongly correlated with
each other (Wilkins & Fabian, 2012), and often cannot be independently
determined.

In the case of the highly accurate NuSTAR data, we have found no fit
improvement at all when allowing a free $q$, and the full compatibility of the
models with the standard value of $q=3$. When the Insight-HXMT data are added,
we find slight fit improvements only, and $q\gtrsim 2$. This difference with
respect to the results of Y21 is due to (1) the inner disk assumed at the ISCO
in Y21 while estimated from the spectral fits in the present work, and (2)
taking into account the spectral complexity of this source in the present
work, with the presence of two Comptonization regions, the softer one
dominating at softer X-rays, and a harder one at harder X-rays. Such
stratification is also indicated by the presence of hard lags in this source,
i.e., harder X-rays delayed with respect to softer ones (De Marco et al.,
2021), and by spectral-timing modelling of Kawamura et al. (2022). Spectral
complexity of this source was also found by Buisson et al. (2019), who fitted
the NuSTAR spectra with reflection of two lampposts at different heights. The
necessity of spectral complexity in MAXI J1820+070 is discussed in detail by
Zdziarski et al. (2021b).

Furthermore, the widths of the Fe K complexes fitted for the Insight-HXMT data
alone were found significantly different from the fits to the combined NuSTAR
and Insight-HXMT data. As found in Section 3.3, the Insight-HXMT data alone
did show the reflecting disk at the ISCO for a maximally rotating BH for epoch
2, opposite to the case of either the NuSTAR or combined data sets, Sections
3.1 and 3.2. On the other hand, the truncation radius for epoch 1 was found to
be very large (Table 6). This comparison indicates that the Insight-HXMT data
for MAXI J1820+070 alone are not suitable for detailed X-ray spectroscopy of
the Fe K region. On the other hand, the Insight-HXMT data together with those
of NuSTAR do show moderate truncation radii for both epochs and using two
different models.

The inner disk temperatures we have found are $kT_{\rm in}\approx 0.4$–0.5 keV
in all cases, which is significantly higher than the values obtained by
fitting the NICER data, which are $\approx$0.2 keV (Wang et al., 2020). This
indicates the presence of a further softening of the apparent Comptonization
continuum toward lower energies, see figure 7 in Zdziarski et al. (2021b)
rather the presence of two separate disk blackbody components. Then, our usage
of the disk blackbody model should be considered to be a phenomenological
description only of inhomogeneous Comptonization/reflection spectra. This is
indicated by the apparently complex structure of the accretion flow in the
hard state, implied, e.g., by the results of the frequency-resolved spectral
fitting of NICER data by Axelsson & Veledina (2021) and Dziełak et al. (2021),
see also Kawamura et al. (2022).

We can also compare our spectral fitting results to some analytical estimates
of truncation radius. First, we can use the normalization of the disk
blackbody spectral component. This normalization was found at $N_{\tt
diskbb}\approx 3$–$4\times 10^{3}$ in Tables 4–6. The implied $R_{\rm in}$
depends on the color correction, $\kappa$, see, e.g., equation (8) in
Zdziarski et al. (2021a). This correction has been found to be within
$\kappa\approx 1.3$–1.7 (Davis et al., 2005). Then, for $D=3$ kpc, $M=7{\rm
M}_{\sun}$ and $i=65\arcdeg$, we find $R_{\rm in}/R_{\rm g}$ within
$\approx$6–13, which is somewhat below our fitted values for joint data.
However, we can also estimate the blackbody temperature resulting from
irradiation of the disk close to $R_{\rm in}$ by the Comptonization emission.
The irradiating Comptonization fluxes of the inner, soft, component for epochs
1 and 2 are $F_{\rm C}\approx 3$–$4\times 10^{-8}$ erg cm-2 s-1. Absorption of
a part $(1-a)$ of this flux, where $a$ is the back-scattering albedo, results
in a quasi-blackbody emission at a temperature above the effective one, see
Zdziarski & De Marco (2020). This irradiating temperature is estimated in
equation (9) of Zdziarski et al. (2021a). Typical values of $a$ for ionized
reflecting media were estimated to be within $a\approx 0.3$–0.7 (Zdziarski &
De Marco, 2020). We also neglect here the reduction of the observed
Comptonization flux with respect to that irradiating the disk due to
scattering in the hot medium, which is a conservative assumption. Then that
formula gives the color temperature due to irradiation alone as $kT_{\rm
in,irr}\approx 0.6$–0.9 keV, i.e., more than the fitted inner temperatures.
This shows that the assumption that the fitted diskbb corresponds to actual
disk blackbody is unphysical, in agreement with the finding of a lower inner
temperature in the NICER data (Wang et al., 2020). Finally, we estimate the
inner radius at which irradiation would result in a given inner temperature,
equation (6) in Zdziarski et al. (2021a). This gives relatively large values
of $R_{\rm in}\gtrsim 50R_{\rm g}$. Thus, our estimates favor strong
truncation.

As we noted above, Y21 interpreted their fits of Insight-HXMT data as showing
the presence of an outflowing corona. A strong argument for it was a decrease
of the fitted reflection fraction with the decreasing spectral hardness, which
was, in turn, correlated with the shortening time lags of soft X-rays with
respect to hard ones (measured by NICER). This discovery was interpreted as
due to the Doppler de-boosting of the disk irradiation by a corona whose
velocity decreases with the height. Note that the values of ${\cal R}\ll 1$
found by them were due to a bug in previous versions of relxill222At the time
of preparing the publication of Y21, relxill v. 1.4.0 was the latest available
version, with the bug fixed in v. 1.4.3, see http://www.sternwarte.uni-
erlangen.de/~dauser/research/relxill/. This version has been used in the
present work.. After that corrections, the fitted values of ${\cal R}$
increase by a factor of $\approx$3, but the trend of the decreasing reflection
fraction remains present (You & Zdziarski, in preparation).

Still, it is possible that the hard Comptonization component originates in an
outflowing corona forming an inner part of the jet (as noticed by Zdziarski et
al. 2021b). This was also suggested by Ma et al. (2021) based on the behavior
of X-ray quasi-periodic oscillations. Then, bulk-motion Comptonization may
provide another spectral component (e.g., Mondal & Chakrabarti 2021). However,
including this physical process is beyond the scope of our paper.

Finally, we comment on the relatively low values of the fitted electron
temperatures for fits with hybrid Comptonization, Tables 2 and 4. As discussed
in Z21c, this may be due either to the onset of pair production by photons
scattered by the non-thermal tail (Coppi, 1999; Fabian et al., 2017) or due to
energy balance in the flow emitting synchrotron photons Poutanen & Vurm
(2009); Malzac & Belmont (2009). Still, MAXI J1820+070 in the hard state
fitted by hybrid Comptonization shows lower values of $kT_{\rm e}$ than Cyg
X-1 fitted by the same spectral model (McConnell et al., 2002), which may be
due to the disk truncation being stronger in Cyg X-1 than in MAXI J1820+070,
which effect may in turn be related to a relatively low Eddington ratio in Cyg
X-1. Indeed, there is ample evidence for the truncation radius in the hard
state to decrease with the increasing luminosity, see, e.g., Plant et al.
(2015) and Basak & Zdziarski (2016) for such behavior in the BH transient
binary GX 339–4. We note, however, that since both of our chosen data sets
correspond to almost identical bolometric luminosity (and the spectral shape),
we cannot here study the evolution of the truncation radius with either the
luminosity of the accretion rate.

##  5 Conclusions

Our main results are as follows.

We have performed a thorough analysis of the simultaneous NuSTAR and INTEGRAL
observations of MAXI J1820+070 in its hard spectral state. We have extended
the original analysis in Z21c, and found that addition of a disk blackbody and
unblurred reflection significantly improves the fits. We have tested source
geometries alternative to those used in Z21c, but found they provide worse
fits. We have also fitted models using the relxill X-ray spectroscopy model,
and found it yields similar results to our original models (using reflkerr).

Then, we have included the simultaneous Insight-HXMT data, and repeated our
fits with reflkerr and relxill. We have found a good mutual agreement between
the NuSTAR, INTEGRAL and Insight-HXMT data, with the fits yielding similar
parameters to the previous ones.

In all of the above cases, we have found the reflecting disk to be truncated,
with $R_{\rm in}\gtrsim 10R_{\rm g}$. We have also tested possible departure
of the irradiation index, $q$, from the canonical value of 3. However, we have
found allowing it free does not improve the fits, and the best fit values
remain close to 3. This provides a confirmation for the assumed coronal
geometry, with the dissipation rate in the corona following that of the disk.
On the other hand, constraints from re-emission of the absorbed part of the
irradiation flux yield $R_{\rm in}\gg 10R_{\rm g}$.

We have then fitted the Insight-HXMT data alone. We have found that they
cannot constrain the inner disk radius. Our recommendation is that constraints
on $R_{\rm in}$ from other instruments are used.

## Acknowledgments

We thank the referee for valuable comments. We acknowledge support from the
Polish National Science Centre under the grants 2015/18/A/ST9/00746 and
2019/35/B/ST9/03944, and from the Natural Science Foundation of China
(U1931203 and 11903024). Our work also benefitted from discussions during Team
Meetings in the International Space Science Institute (Bern).

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#  Configuration entropy in the soft wall AdS/QCD model and the Wien law

Nelson R. F. Bragaa<EMAIL_ADDRESS>a UFRJ — Universidade Federal do Rio de
Janeiro, Instituto de Física,
Caixa Postal 68528, Rio de Janeiro, Brasil Octavio C. Junqueiraa
<EMAIL_ADDRESS>a UFRJ — Universidade Federal do Rio de Janeiro,
Instituto de Física,
Caixa Postal 68528, Rio de Janeiro, Brasil

###### Abstract

The soft wall AdS/QCD holographic model provides simple estimates for the
spectra of light mesons and glueballs satisfying linear Regge trajectories. It
is also an interesting tool to represent the confinement/deconfinement
transition of a gauge theory, that is pictured as a Hawking-Page transition
from a dual geometry with no horizon to a black hole space. A very interesting
tool to analyze stability of general physical systems that are localized in
space is the configuration (or complexity) entropy (CE). This quantity,
inspired in Shannon information entropy, is defined in terms of the energy
density of the system in momentum space. The purpose of this work is to use
the CE to investigate the stability of the soft wall background as a function
of the temperature. A nontrivial aspect is that the geometry is an anti-de
Sitter black hole, that has a singular energy density. In order to make it
possible to calculate the CE, we first propose a regularized form for the
black hole energy density. Then, calculating the CE, it is observed that its
behavior is consistently related to the black hole instability in anti-de
Sitter space. Another interesting result that emerges from this analysis is
that the regularized energy density shows a behavior similar to the Wien law,
satisfied by black body radiation. That means: the momentum $k_{max}$ where
the energy density is maximum, varies with the temperature $T$ obeying the
relation: $T/k_{max}=constant$ in the deconfined phase.

##  1 Introduction

It was proposed in [1, 2, 3] that the configuration entropy (CE) works as an
indicator of the stability of physical systems. The CE, also known as
complexity entropy, is motivated by information theory, in particular by the
Shannon infomation entropy [4]. Currently, one finds in the literature many
interesting examples where the CE plays the role of representing stability, as
for instance [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29]. By comparing the CE for different states of
the same system, it was observed, for diverse physical systems, that the
smaller is the CE, the more stable is the state.

The Shannon information entropy [4] for a discrete variable with probabilities
$p_{n}$ for the possible values that it can assume is defined as:

$S=-\sum_{n}\,p_{n}\log p_{n}\,.$ (1.1)

It is a measure of the information contained in the variable. The
configuration entropy is defined as a continuous version of (1.1), that for a
one-dimensional system reads

$S_{C}[f]=-\int dk\,f({k})\ln f({k})\,,$ (1.2)

The quantity $f(k)$ is called modal fraction and is usually defined in terms
of the energy density in momentum space, $\rho(k)$, namely,

$f(k)=\frac{|\rho(k)|^{2}}{|\rho(k)|^{2}_{max}}$ (1.3)

where $|\rho(k)|^{2}_{max}$ is the maximum value assumed by $|\rho(k)|^{2}$.
The modal fraction can alternatively be defined to be a normalized function,
replacing in the denominator of eq. (1.3) the square of the maximum value of
the energy density in momentum space by $\int|\rho(k)|^{2}dk$. Such a
definition would be more similar to the Shannon entropy, where the
probabilities are normalized, but lead to negative values for the CE. This
happens because, in contrast to (1.1) where the probabilities satisfy
$p_{n}\leq 1$, the same rule does not apply to the continuous case, that
involves densities.

There are many situations in physics where is important to understand what are
the conditions that affect the stability of a system. In this work we are
interested in the case of the holographic soft wall model[30]. This AdS/QCD
holographic model provides simple estimates for meson and glueball masses and
also for the confinement/deconfinement temperature [31, 32]. The model assumes
an approximate duality beween fields in an anti-de Sitter (AdS) space with a
scalar background and light mesons in a strongly coupled gauge theory. In the
finite temperature case, the geometry changes. For temperatures $T$ above a
critical value $T_{c}$ the geometry dual to the gauge theory is an AdS black
hole, while for $T<T_{c}$ it is just a thermal AdS space. The black hole phase
corresponds to the deconfined plasma phase of the gauge theory. Stability of
the black hole space corresponds, in this case, to stability of the plasma.
The confinement/deconfinement transition is represented in holography in terms
of a Hawking-Page (HP) transition [33], as pointed out by Witten [34] .

The main purpose of the present work is to analyse the stability of the soft
wall model using the configuration entropy approach. As we will show, the
energy density is singular, so it is necessary to develop a regularization
procedure in order to find a density that can be used to define the modal
fraction and then the CE. A remarkable property of this regularized density is
that it obeys a relation similar to the Wien law for black body radiation. The
maximum value of the energy density occurs in a momentum $k_{max}$ that is
proportional to the temperature. The organization is the following. In section
2 we present a review of the soft wall model at finite temperature and explain
how does the Hawking-Page transition shows up. Then in section 3 we develop a
regularized form for the energy density. Section 4 is devoted to discuss the
behaviour similar to the Wien law displayed by such density. Finally, in
section 5 we present the result for the CE of the soft wall as a function of
the temperature. A discussion about the results and some conclusions are
presented in section 6.

##  2 Soft wall AdS/QCD model and Hawking-Page transition

In the soft wall AdS/QCD model [30], the gravitational part of the action is
given by

$I=-\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{g}e^{-\Phi}\left(R+\frac{12}{L^{2}}\right)\;,$ (2.1)

where $\kappa$ is the gravitational coupling and $R$ the Ricci scalar. The
dilaton field $\Phi=cz^{2}$ plays the role of introducing, in a
phenomenological way, an infrared energy scale, namely $\sqrt{c}$, in the
model. This action corresponds to a negative cosmological constant
$\Lambda=-\frac{12}{L^{2}}$ and the solution considered in the soft wall model
has a constant negative curvature $R$. At zero temperature the geometry is an
anti-de Sitter space with radius $L$. At finite temperature there are two
soutions. One is the thermal AdS space and the other the AdS black hole space.
One assumes that the dilaton field does not affect the gravitational dynamics
of the theory. This can be interpreted as the assumption that the fluctuations
of $\phi$ are very small as compared to the gravitational ones.

The thermal AdS space, in the Euclidean signature with compact time direction,
is described by

$ds^{2}=\frac{L^{2}}{z^{2}}\left(dt^{2}+d\overrightarrow{x}^{2}+dz^{2}\right)\;,$
(2.2)

and the AdS with a black hole, by

$ds^{2}=\frac{L^{2}}{z^{2}}\left(f(z)dt^{2}+d\overrightarrow{x}^{2}+\frac{dz^{2}}{f(z)}\right)\;,$
(2.3)

with $f(z)=1-z^{4}/z_{h}^{4}$, being $z_{h}$ the position of the black hole
horizon. For the black hole, the time has a period $\beta$ and the temperature
is $T=1/\beta=1/(\pi z_{h})$, to avoid a conical singularity of the metric on
the horizon [33]. In the thermal AdS case, the periodicity of time is, in
principle, not constrained. However, requiring that the assymtotic limit of
the two geometries at $z=\epsilon$, with $\epsilon\to 0$, are the same, one
finds it out that the period of the thermal AdS is $\beta^{\prime}=\pi
z_{h}\sqrt{f(\epsilon)}$.

The AdS geometries (2.2) and (2.3) are both solutions of the Einstein action,
with curvature $R=-20/L^{2}$. So that, in both cases, one obtains the on-shell
action

$I_{\text{on-shell}}=\frac{4}{L^{2}\kappa^{2}}\int
d^{5}x\sqrt{g}\,e^{-\Phi}\;.$ (2.4)

It is worth to define action densities $\mathcal{E}(\epsilon)=I/V$, where $V$
is the trivial transverse tree-dimensional space. For the thermal AdS it reads

$\mathcal{E}_{AdS}(\epsilon)=\frac{4L^{3}}{\kappa^{2}}\int_{0}^{\beta^{\prime}}dt\int_{\epsilon}^{\infty}dz\,z^{-5}e^{-cz^{2}}\;,$
(2.5)

and for the black hole it is:

$\mathcal{E}_{BH}(\epsilon)=\frac{4L^{3}}{\kappa^{2}}\int_{0}^{\pi
z_{h}}dt\int_{\epsilon}^{\infty}dz\,z^{-5}e^{-cz^{2}}\;.$ (2.6)

The regulator $\epsilon$ is necessary due to the presence of ultraviolet
divergences in the integration over $z$. However, defining
$\bigtriangleup\mathcal{E}$ as the difference between the black hole AdS and
the thermal AdS action densities

$\bigtriangleup\mathcal{E}=\lim_{\epsilon\rightarrow
0}\left[\mathcal{E}_{BH}(\epsilon)-\mathcal{E}_{AdS}(\epsilon)\right]\;,$
(2.7)

the UV divergences are cancelled, and one obtains the following result

$\bigtriangleup\mathcal{E}=\frac{\pi
L^{3}}{\kappa^{2}z_{h}^{3}}\left[e^{-cz_{h}^{2}}\left(-1+cz_{h}^{2}\right)+\frac{1}{2}+c^{2}z_{h}^{4}\text{Ei}(-cz_{h}^{2})\right]\;,$
(2.8)

where $\text{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt$. We show in Figure
1 a plot of $\bigtriangleup\mathcal{E}$ as a function of the temperature. One
can see in this figure that for temperatures above a critical value $T=T_{c}$,
$\,\bigtriangleup\mathcal{E}$ is negative while for smaller values it is
positive. So that, there is a Hawking-Page transition[33]. When $T>T_{c}$ the
black hole has a smaller action, so it is stable. For lower temperatures, the
thermal AdS is the stable phase. The critical temperature corresponds to
$cz_{h}^{2}=0.419035$, or, equivalently to:

$T_{c}=0.491720\sqrt{c}\;.$ (2.9)

Figure 1: $\bigtriangleup\mathcal{E}$ versus $T$. Action density difference
between thermal and black hole AdS spaces, with $L^{3}/\kappa^{2}=c=1$.

If one fixes the parameter $c$ by the spectrum of the $\rho$-meson, one gets
[31] $\sqrt{c}=318\,\text{M}eV$. The corresponding deconfinement temperature
is $T_{c}=191\,\text{M}eV$. For an interesting alternative study of
deconfinement transition in holographic QCD using entanglement entropy see
[35].

##  3 Black hole regularized energy density

The mass of a black hole can be determined from the expression [33, 34]

$M=\frac{\partial I}{\partial\beta}\;.$ (3.1)

Considering the regularized difference between the black hole and AdS actions

$I=I_{BH}-I_{AdS}\;,$ (3.2)

one finds an energy per unit of transverse volume $V$

$E=M/V=\frac{\partial\bigtriangleup\mathcal{E}}{\partial\beta}\;,$ (3.3)

with $\bigtriangleup\mathcal{E}$ defined by eq. (2.8). The natural definition
of a black hole energy density $\rho_{BH}(z)$ in the relevant holographic $z$
coordinate would be one satisfying:

$\int_{0}^{\infty}\rho_{BH}(z)\,dz=E\;.$ (3.4)

In order to find such a density one can apply the derivative with respect to
$\beta$ on $\bigtriangleup\mathcal{E}$ but before integrating over coordinate
$z$. That means, considering $\bigtriangleup\mathcal{E}$ defined as the
difference between the expression in eq. (2.6) and the one in eq. (2.5) ,
taking into account the fact that the limits of integrations depend on
$\beta$.

It is convenient to introduce the coordinate $u=z/z_{h}$. Using equations
(2.5) and (2.6), one finds

$\displaystyle\bigtriangleup\mathcal{E}(u)=\frac{4L^{3}}{\kappa^{2}}\left[\beta\int_{\epsilon\pi/\beta}^{1}dz\,u^{-5}\left(\frac{\beta}{\pi}\right)^{-4}e^{-c(\frac{u\beta}{\pi})^{2}}-\left(\beta-\frac{\pi^{4}\epsilon^{4}}{2\beta^{3}}\right)\int_{\epsilon\pi/\beta}^{\infty}du\,u^{-5}\left(\frac{\beta}{\pi}\right)^{-4}e^{-c(\frac{u\beta}{\pi})^{2}}\right]\;.$
(3.5)

The derivative of (3.5) w.r.t. $\beta$ yields, after taking the $\epsilon\to
0$ limit,

$E\,=\,\lim_{\epsilon\to
0}\frac{\partial\bigtriangleup\mathcal{E}}{\partial\beta}=\frac{4L^{3}}{\kappa^{2}}\left[\frac{1}{2z_{h}^{4}}+\int_{1}^{\infty}du\left(\frac{3\pi^{4}}{u^{5}\beta^{4}}+\frac{2c\pi^{2}}{u^{3}\beta^{2}}\right)e^{-c\left(\frac{u\beta}{\pi}\right)^{2}}-\frac{7\epsilon^{4}}{2}\int_{\frac{\epsilon\pi}{\beta}}^{\infty}du\,\frac{\pi^{8}}{u^{5}\beta^{8}}e^{-c\left(\frac{u\beta}{\pi}\right)^{2}}\right]\;.$
(3.6)

The first (non-integrated) term of eq. (3.6) comes from the differentiation of
the integration limit with respect to $\beta$. It can be written in terms of
the last term, since one can easily show that

$\frac{1}{2z_{h}^{4}}=2\epsilon^{4}\int_{\epsilon\pi/\beta}^{\infty}du\frac{\pi^{8}}{u^{5}\beta^{8}}e^{-c(\frac{u\beta}{\pi})^{2}}\;.$
(3.7)

So, all the terms in the expression (3.6) for the energy E can be written as
integrals over the coordinate $z$ and one can associate the integrand with a
density. Actually, the result of the integral (3.7) does not depend on the
value of the upper limit of integration, as long as it is greater than the
lower limit. So , we will take it to be at $u=1$. Finally, returning to the
$z$-variable, one finds

$E=\lim_{\epsilon\to
0}\left(\int_{\epsilon}^{z_{h}}\rho_{1}^{BH}(z)\,dz+\int_{z_{h}}^{\infty}\rho_{2}^{BH}(z)\,dz\right)\;,$
(3.8)

where

$\displaystyle\rho^{BH}_{1}(z)$ $\displaystyle=$
$\displaystyle-\frac{4L^{3}}{\kappa^{2}}\frac{3\epsilon^{4}}{2z^{5}z_{h}^{4}}e^{-cz^{2}}\;,\quad\quad\quad\text{if}\quad\epsilon\leq
z\leq z_{h}\;,$ $\displaystyle\rho^{BH}_{2}(z)$ $\displaystyle=$
$\displaystyle\frac{4L^{3}}{\kappa^{2}}\left(\frac{3}{z^{5}}+\frac{2c}{z^{3}}\right)e^{-cz^{2}}\;,\quad\,\text{if}\quad
z>z_{h}\;.$ (3.9)

The expression (3) shows that the BH mass is concentrated in the ultraviolet
and infrared regions, i.e., in the extremes of $z$-direction. Such a
regularized result for the energy density makes it possible to calculate the
corresponding momentum space density, that is necessary in order to find the
CE.

##  4 Wien law in the soft wall model

The BH energy density, $\rho_{BH}(z)$, depends only on the Poincaré coordinate
$z$. The corresponding Fourier transform takes the form

$\displaystyle\widetilde{\rho}(k)=\frac{1}{2\pi}\lim_{\epsilon\to
0}\left(\int_{\epsilon}^{z_{h}}dz\,\rho^{BH}_{1}(z)e^{ikz}+\int_{z_{h}}^{\infty}dz\,\rho^{BH}_{2}(z)e^{ikz}\right)\;,$
(4.1)

with $\rho^{BH}_{1}(z)$ and $\rho^{BH}_{2}(z)$ defined by eq. (3). The
relevant quantity for the calculation of the modal fraction, and then the CE,
is the squared absolute value of the black hole energy density in Fourier
space, that can be written as:

$\displaystyle|\widetilde{\rho}(k)|^{2}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{2\pi}\lim_{\epsilon\to
0}\left(\int_{\epsilon}^{z_{h}}\rho^{BH}_{1}(z)\cos(kz)\,dz+\int_{z_{h}}^{\infty}\rho^{BH}_{2}(z)\cos(kz)\,dz\right)\right]^{2}$
(4.2) $\displaystyle+$ $\displaystyle\left[\frac{1}{2\pi}\lim_{\epsilon\to
0}\left(\int_{\epsilon}^{z_{h}}\rho^{BH}_{1}(z)\sin(kz)\,dz+\int_{z_{h}}^{\infty}\rho^{BH}_{2}(z)\sin(kz)\,dz\right)\right]^{2}\;.$

One can use numerical methods in order to compute these integrals, since they
do not present analytical solutions. The results must be independent of the UV
regulator $\epsilon$. As the numerical computation is performed with a finite
value of $\epsilon$, the limit $\epsilon\to 0$ is effectively obtained by
identifying the order of magnitude for which taking smaller values of
$\epsilon$ would not change the results, considering the degree of precision
in which one is working. We found it out that from $\epsilon\sim 10^{-16}$ to
smaller values there is no change in any of the results of this work, so this
value was used in our computations.

The functions $|\widetilde{\rho}(k)|^{2}$ present a global maximun
$|\widetilde{\rho}(k)|^{2}_{max}$, at a finite value of $k$, that we will name
as $k_{max}$. Analysing the variation of the value of $k_{max}$ with the
temperature, it is found that

$\frac{k_{max}}{T}=K,$ (4.3)

where $K$ is a constant. Using the normalization
$\frac{4L^{3}}{2\pi\kappa^{2}}=1$ and taking $c=1$, that corresponds to
working with the dimensionless temperature $T\equiv T/\sqrt{c}$, one finds
that $K=8.790\times 10^{4}$, with an error of $\pm 0.001\times 10^{4}$ or
$0.01\%$. In Figure 2-(A), we plot the absolute value of the BH energy density
at $T=0.65$, which shows a behaviour that is the same for all values of $T$.
Namely, $|\widetilde{\rho}(k)|^{2}$ goes to zero as $k\rightarrow 0$, for
$k>0$ it increases until it reaches the global maximum, and then oscillates,
tending to zero for large momentum.

Figure 2: (A) $|\widetilde{\rho}(k)|^{2}$ versus momentum at $T=0.65$; (B) the
same at $T1=0.55$ (blue), $T2=0.6$ (orange); and $T3=0.65$ (green).

As $T$ increases, $|\widetilde{\rho}(k)|^{2}_{max}$ and $k_{max}$ increase, as
shown in Figure 2-(B), where we plot $|\widetilde{\rho}(k)|^{2}$ at three
different temperatures, $T_{3}>T_{2}>T_{1}$.

This behaviour is represented in Figure 3, where we plot the points displayed
in Table 1.For the confined phase, corresponding to $T<T_{c}$, there is no
black hole, so there is no energy density. For $T>T_{c}$, the points represent
the relation described by eq. (4.3).

$T$ | $|\widetilde{\rho}_{BH}(k)|^{2}_{max}$ | $k_{max}$ | $T$ | $|\widetilde{\rho}_{BH}(k)|^{2}_{max}$ | $k_{max}$
---|---|---|---|---|---
$\,\,\,0.50\,\,\,$ | $\,\,\,1.2092\times 10^{24}\,\,\,$ | $\,\,\,43949\,\,\,$ | $\,\,\,1.00\,\,\,$ | $\,\,\,7.7388\times 10^{25}\,\,\,$ | $\,\,\,87900\,\,\,$
$\,\,\,0.55\,\,\,$ | $\,\,\,2.1422\times 10^{24}\,\,\,$ | $\,\,\,48345\,\,\,$ | $\,\,\,1.05\,\,\,$ | $\,\,\,1.0371\times 10^{26}\,\,\,$ | $\,\,\,92295\,\,\,$
$\,\,\,0.60\,\,\,$ | $\,\,\,3.6106\times 10^{24}\,\,\,$ | $52740$ | $\,\,\,1.10\,\,\,$ | $\,\,\,1.3710\times 10^{26}\,\,\,$ | $96683$
$\,\,\,0.65\,\,\,$ | $\,\,\,5.8365\times 10^{24}\,\,\,$ | $57135$ | $\,\,\,1.15\,\,\,$ | $\,\,\,1.7900\times 10^{26}\,\,\,$ | $101088$
$\,\,\,0.70\,\,\,$ | $\,\,\,9.1046\times 10^{24}\,\,\,$ | $61530$ | $\,\,\,1.20\,\,\,$ | $\,\,\,2.3108\times 10^{26}\,\,\,$ | $105474$
$\,\,\,0.75\,\,\,$ | $\,\,\,1.3773\times 10^{25}\,\,\,$ | $65924$ | $\,\,\,1.25\,\,\,$ | $\,\,\,2.9521\times 10^{26}\,\,\,$ | $109873$
$\,\,\,0.80\,\,\,$ | $\,\,\,2.0287\times 10^{25}\,\,\,$ | $70321$ | $\,\,\,1.30\,\,\,$ | $\,\,\,3.7152\times 10^{26}\,\,\,$ | $114275$
$\,\,\,0.85\,\,\,$ | $\,\,\,2.9187\times 10^{25}\,\,\,$ | $74712$ | $\,\,\,1.35\,\,\,$ | $\,\,\,4.6593\times 10^{26}\,\,\,$ | $118665$
$\,\,\,0.90\,\,\,$ | $\,\,\,4.1127\times 10^{25}\,\,\,$ | $79117$ | $\,\,\,1.40\,\,\,$ | $\,\,\,5.7955\times 10^{26}\,\,\,$ | $123066$
$\,\,\,0.95\,\,\,$ | $\,\,\,5.6887\times 10^{25}\,\,\,$ | $83505$ | $\,\,\,1.45\,\,\,$ | $\,\,\,7.1537\times 10^{26}\,\,\,$ | $127457$
Table 1: Maximum of BH density, $|\widetilde{\rho}(k)|^{2}_{max}$, with the
corresponding momentum $k_{max}$, at different temperatures. Figure 3:
$k_{max}$ versus $T$ for $T>T_{c}$.

The wavelength associated with the momentum $k_{max}$ is
$\lambda_{max}=2\pi/k_{max}$. Replacing this relation in (4.3), one gets the
analogous of Wien’s displacement law

$\lambda_{max}T=C\;,$ (4.4)

wherein $C=\frac{2\pi}{K}=7.148\times 10^{3}$. This equation shows that the BH
energy density will peak at different wavelength that are inversely
proportional to the temperature, analogously to the black-body radiation
spectrum.

##  5 Configuration entropy and stability

Now, let us study the stability of the soft wall model by computing the
configuration entropy (CE) following the prescriptions presented in the
previous sections. First we calculate the modal fraction (1.3), as defined in
the previous section, being $k_{max}$ the value of $k$ where
$|\widetilde{\rho}(k)|^{2}$ assumes the maximum value. The square of the
Fourier transform of the energy density is obtained from equation (4.2) using
the spatial density defined in eq. (3). Then the CE is calculated using
equation (1.2). This way, one can determine the CE as a function of the
temperature. Below $T_{c}=0.491720$, the stable phase is the thermal AdS. So,
there is no black hole and the CE is zero since, as expressed in (3.2), we are
considering the AdS as the “empty space” background. In the deconfined phase,
where the black holes are stable, the numerical computations of the CEs at the
same temperatures of Table 1, lead to the results displayed in Table 2 and
ploted in Figure 4.

$T$ | Black hole CE | $T$ | Black hole CE
---|---|---|---
$\,\,\,0.50\,\,\,$ | $327276$ | $\,\,\,1.00\,\,\,$ | $654774$
$\,\,\,0.55\,\,\,$ | $360026$ | $\,\,\,1.05\,\,\,$ | $687282$
$\,\,\,0.60\,\,\,$ | $392764$ | $\,\,\,1.10\,\,\,$ | $720018$
$\,\,\,0.65\,\,\,$ | $425406$ | $\,\,\,1.15\,\,\,$ | $752790$
$\,\,\,0.70\,\,\,$ | $458194$ | $\,\,\,1.20\,\,\,$ | $785476$
$\,\,\,0.75\,\,\,$ | $490936$ | $\,\,\,1.25\,\,\,$ | $818200$
$\,\,\,0.80\,\,\,$ | $523682$ | $\,\,\,1.30\,\,\,$ | $850874$
$\,\,\,0.85\,\,\,$ | $556308$ | $\,\,\,1.35\,\,\,$ | $883674$
$\,\,\,0.90\,\,\,$ | $589096$ | $\,\,\,1.40\,\,\,$ | $916450$
$\,\,\,0.95\,\,\,$ | $621846$ | $\,\,\,1.45\,\,\,$ | $949192$
Table 2: Black hole CE at different temperatures. Figure 4: CE versus T.

One can see in Figure 4 that the CE varies linearly with the temperature. In
particular, it is found that

$S_{C}=K_{c}T\;,$ (5.1)

where $K_{c}=(6.546\pm 0.001)\times 10^{5}$, meaning that black holes become
more unstable at higher temperatures. Such a result is consistent with the
expectation that black holes are subject to a process of evaporation by
Hawking radiation, and this effect should increase with the temperature.

Another outcome of this result for the CE is that there is also a linear
relation between the CE and $k_{max}$: $S_{C}\sim k_{max}$. The jump from
$S_{C}=0$ in the confined hadronic phase to $S_{C}\sim T$ in the plasma phase,
represents in a consistent way the release of the color degrees of freedom
that happens at $T=T_{c}$. From the correlation between the Wien law and the
CE, one concludes that states with smaller values of $k_{max}$ are more
stable.

##  6 Analysis of the results and conclusions

Studying the thermodynamics of the black hole geometry in the soft wall
AdS/QCD model, we obtained, in eq. (3), a regularized form for the energy
density in the holographic coordinate $z$. This regularized expression made it
possible the calculation of the square of the absolute value of energy density
in momentum space $|\widetilde{\rho}(k)|^{2}$, given by eq. (4.2). The results
obtained for the black hole phase display a behaviour with the form of the
Wien’s displacement law for black body radiation. Associating a wavelength
$\lambda$ with the inverse of $k$ one finds it out that the the value
$k_{max}$ where the maximum of $|\widetilde{\rho}(k)|^{2}$ occurs, corresponds
to a wavelength $\lambda_{max}$ that is inversely proportional to the
temperature, as shown in eq. (4.4).

We also found that the CE is proportional to $k_{max}$ and to the temperature.
This is a nontrivial issue. Recently it was show in Ref. [29] that the BH
configuration entropy in the hard wall holographic model is proportional to
$\log{T}$, showing a different behaviour with respect to the one found here
for the soft wall. It is known that the soft wall presents a better
holographic description of hadronic matter since the mass spectra obtained
with this model present linear Regge trajectories, as opposed to the hard wall
model. So, it is interesting to see that it also present a richer structure
from the point of view of the configuration entropy. The gap between the
thermal AdS and black hole phases is consistent with the transition from
hadronic to the plasma phase, or confinement/deconfinement transition. This
corresponds to the liberation of the color degrees of freedom at the Hawking-
Page critical temperature.

The result that the CE increases with the temperature, indicating that the
black hole becomes more unstable, is consistent with the expectation that
black holes undergo a process of evaporation that increases in intensity with
the temperature. The evaporation of anti-de Sitter black holes with planar
horizon was discussed in Ref. [36]. There, it was shown that the rate of
change in the mass is proportional to a positive power of the mass itself. So,
the rate of change of the mass increases with the temperature. The conclusion
that follows is that the CE captures consistently the dynamical instability of
the black hole phase, associated with the evaporation. In the gauge theory
side of the holographic duality, this instability could be associated with the
instability of the plasma, that has a finite lifetime, undergoing a process of
hadronization after this time. Another interesting point to be remarked is
that the vanishing of the CE for $T<T_{c}$ is a result consistent with the
interpretation that the instability captured by the CE is associated with the
black hole evaporation. In the confined phase there is no black hole, so there
is no evaporation.

Acknowledgments: The authors are supported by CNPq - Conselho Nacional de
Desenvolvimento Científico e Tecnológico. This work received also support from
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Finance Code 001.

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# GloSoFarID: Global multispectral dataset for Solar Farm IDentification in
satellite imagery

Zhiyuan Yang and Ryan Rad  Khoury College of Computer Science, Northeastern
University
Vancouver, Canada
{yang.zhiyu<EMAIL_ADDRESS>
###### Abstract

Solar Photovoltaic (PV) technology is increasingly recognized as a pivotal
solution in the global pursuit of clean and renewable energy. This technology
addresses the urgent need for sustainable energy alternatives by converting
solar power into electricity without greenhouse gas emissions. It not only
curtails global carbon emissions but also reduces reliance on finite, non-
renewable energy sources. In this context, monitoring solar panel farms
becomes essential for understanding and facilitating the worldwide shift
toward clean energy. This study contributes to this effort by developing the
first comprehensive global dataset of multispectral satellite imagery of solar
panel farms. This dataset is intended to form the basis for training robust
machine learning models, which can accurately map and analyze the expansion
and distribution of solar panel farms globally. The insights gained from this
endeavor will be instrumental in guiding informed decision-making for a
sustainable energy future. https://github.com/yzyly1992/GloSoFarID

###### Index Terms:

photovoltaic, solar panel farm, multispectral images, dataset, sustainability

##  I Introduction

Solar Photovoltaic (PV) technology has emerged as a key solution for meeting
the global demand for clean and renewable energy. By harnessing solar power
for electricity generation, solar PV plays a significant role in mitigating
climate change and reducing pollution. Its ability to replace traditional
power generation methods reduces global carbon emissions and lessens
dependence on finite resources, paving the way for a sustainable future [1].

A critical aspect of the global transition to clean energy is the precise
monitoring of solar panel farms, which are integral to renewable energy
infrastructure worldwide. Understanding the growth and distribution of these
farms is essential for assessing progress towards a sustainable energy
landscape [2]. Therefore, monitoring these farms is a pivotal step in
comprehensively understanding the global adoption of clean energy resources.

In line with this goal, our project aims to make a significant contribution by
creating a comprehensive dataset of multispectral satellite images of solar
panel farms. This dataset will be a foundational resource for developing
effective machine learning models for monitoring solar panel farms globally.
The focus on using these models highlights the importance of accurate and
advanced monitoring techniques in promoting a sustainable and cleaner energy
future worldwide. This dataset is a pioneering effort, representing the first
global compilation of solar panel farm data, and includes comprehensive
multispectral satellite information. Our dataset enhances existing resources
in three ways:

  * •

It employs mid-resolution data and includes rural areas globally, addressing
previous limitations in geographic coverage.

  * •

It provides multispectral satellite images with $13$ multispectral bands,
offering more comprehensive information that comes in an easy-to-use format
with a complementary how-to-use guide and script.

  * •

The dataset is updated with recent data from $2021$ to $2023$, doubling the
size of the existing data sources and ensuring current and relevant insights
into the global solar energy landscape.

Figure 1: Examples from our GloSoFarID dataset. Each row represents one sample
with all $13$ bands along with the ground truth mask where the gray color
indicates the solar farm area. See TABLE II for more details.

##  II Dataset Construction

To construct this dataset, we devised a methodical three-step process that
incorporates modern techniques. As illustrated in Fig. 2, our approach begins
with the creation of a robust initial training set, proceeds with the training
of state-of-the-art (SOTA) models, and concludes by processing the combined
outputs of the top-performing models to generate a novel dataset.

Figure 2: Dataset Construction Pipeline

###  II-A Gather Initial Training Dataset

Our first step involved aggregating data from diverse sources to assemble an
initial dataset. This dataset comprises three primary segments: Historical
Global Ground Truth from $2017$-$2018$ [3], Manually Annotated US Ground Truth
from $2022$-$2023$ by our team using Google Earth Engine (GEE), and Sentinel-2
Multispectral Satellite Imagery spanning $2017$-$2023$ [4]. The integration of
these datasets yielded a comprehensive initial training set of $10,230$
multispectral images, each with a resolution of $256\times 256$ pixels and
encompassing $13$ spectral bands. This rich and varied dataset lays the
groundwork for our subsequent analytical endeavors.

###  II-B Train SOTA Models

With the initial training set in place, we engaged several SOTA models,
renowned for their performance, for training. Utilizing a batch size of $16$
over $50$ epochs, our models demonstrated exceptional accuracy in identifying
solar panel farms from multispectral images. The top-performing model achieved
an outstanding $96.47\%$ Intersection over Union (IoU) and a $98.2\%$ F-score,
demonstrating its high accuracy in precisely segmenting solar panel areas
within the imagery.

###  II-C Generate New Dataset

Following the identification of the top three models based on predictive
accuracy, we employed them to predict and label data for the global dataset
covering $2021$-$2023$. The combination (logical OR) of predictions from these
models resulted in a unified dataset. To ensure the dataset’s quality and
reliability, we implemented a rigorous quality control process. This included
applying stringent filtering techniques to remove artifacts and isolated
clusters and systematically eliminating any erroneous or untrustworthy data
points. The resulting dataset, refined through these procedures and careful
manual inspection, represents the latest and most accurate collection of data.
Our comprehensive methodology ensures the delivery of a dataset distinguished
by its precision and reliability, suitable for a broad spectrum of
applications and analyses.

##  III Dataset Overview

This section presents a detailed analysis of the proposed dataset, including
comprehensive statistical information. It aims to give users a clear insight
into the dataset’s characteristics and strengths. Additionally, our Github
repository contains an easy-to-use guide for working with this dataset
regardless of the machine learning framework.

###  III-A Size, Region, and Time Span

The proposed dataset is comprised of a total of $13,703$ data samples,
spanning various global regions and capturing information from the years
$2021$ to $2023$. Fig. 3 displays the location and density of all data points
on a global map. Furthermore, Table I provides a detailed view of the
dataset’s composition and distribution

Figure 3: Global Distribution of Data Points. Sparse is represented in light orange, while dense is represented in dark orange. TABLE I: Dataset Composition Year & Region | Positive Sample | Negative Sample | Sum
---|---|---|---
2023 US∗ | 706 | 969 | 1675
2022 US∗ | 723 | 968 | 1691
2023 Global | 1774 | 1689 | 3463
2022 Global | 1803 | 1655 | 3458
2021 Global | 1787 | 1629 | 3416
Total | 6793 | 6910 | 13703
∗Samples from the US region have been manually annotated.

###  III-B Resolution and Dimension

This dataset maintains a spatial resolution of $10$ meters per pixel,
consistent with the resolution of Sentinel-2 satellite imagery. Each data
sample within the dataset is standardized to dimensions of $256\times 256$
pixels, and encompasses a rich spectral profile with $13$ bands [4]. Table II
summarizes the spectral information and Fig. 1 presents two samples
visualizing the $13$ bands along with the ground truth mask.

TABLE II: Dataset Bands Information Name | Wavelength | Description
---|---|---
B1 | 443.9nm (S2A) / 442.3nm (S2B) | Aerosols
B2 | 496.6nm (S2A) / 492.1nm (S2B) | Blue
B3 | 560nm (S2A) / 559nm (S2B) | Green
B4 | 664.5nm (S2A) / 665nm (S2B) | Red
B5 | 703.9nm (S2A) / 703.8nm (S2B) | Red Edge 1
B6 | 740.2nm (S2A) / 739.1nm (S2B) | Red Edge 2
B7 | 782.5nm (S2A) / 779.7nm (S2B) | Red Edge 3
B8 | 835.1nm (S2A) / 833nm (S2B) | NIR
B8A | 864.8nm (S2A) / 864nm (S2B) | Red Edge 4
B9 | 945nm (S2A) / 943.2nm (S2B) | Water vapor
B11 | 1613.7nm (S2A) / 1610.4nm (S2B) | SWIR 1
B12 | 2202.4nm (S2A) / 2185.7nm (S2B) | SWIR 2

###  III-C Compare to Existing Datasets

Numerous studies are dedicated to examining datasets featuring high-resolution
images of solar panels. SolarDK, a publicly available benchmark dataset,
focuses on the geographical area of Denmark [5]. This dataset encompasses
human-annotated information for both classification and segmentation tasks.
Furthermore, the UC Berkeley team developed another high-resolution dataset
utilizing Google Maps to collect satellite images from various states in the
U.S. [6]. Although these openly accessible datasets are valuable resources,
they lack diversity in covering non-urban regions and are restricted to
specific geographic locations and countries. To address these shortcomings,
our dataset overcomes these limitations by employing mid-resolution data and
encompassing rural areas across global regions on various continents.

In $2021$, L. Kruitwagen created a dataset that offers a global inventory of
photovoltaic solar energy generating units, presenting a comprehensive
overview of solar panel distribution worldwide [3]. This dataset involved an
extensive analysis, processing $550$ terabytes of imagery spanning
approximately $2$ months in real-time. While the outcome is an expansive and
insightful resource, it has certain limitations. It exclusively provides shape
files for solar panel areas, lacking crucial input source data. Moreover, it
captures information within an outdated timeframe, spanning from $2016$ to
$2018$. In response to these drawbacks, our proposed dataset aims to address
these issues comprehensively. It will include multispectral input satellite
images, offering a richer and more detailed dataset. Additionally, our dataset
will cover an up-to-date timeframe, ranging from $2021$ to $2023$, ensuring a
more current and relevant depiction of the global solar energy landscape.

##  IV Benchmark Models and Results

Running benchmark segmentation models on our dataset is essential to establish
a performance baseline and enable a fair comparative evaluation. This not only
enhances our understanding of the dataset but also provides a standardized
reference for the broader research community, fostering transparency,
consistency, and credibility in future studies. To analyze and benchmark the
dataset, we employed two widely recognized semantic segmentation models and
one simplified model, namely the Fully Convolutional Network (FCN), U-Net, and
Half-UNet depicted in Figure 4.

Figure 4: Architecture of FCN, Half-UNet, and U-Net TABLE III: Benchmark Model Results Model | Params | IoU (%) | F-score (%)
---|---|---|---
FCN[7] | 1,996,737 | 71.81 | 82.87
Half-UNet[8] | 224,609 | 71.48 | 82.17
U-Net[9] | 31,129,377 | 79.31 | 87.80

Figure 5: Qualitative Results of FCN, Half-UNet, and U-Net Models. The first
row displays the test samples in true color. Rows $2$-$4$ depict the
prediction masks for solar farm areas. In this representation, True Positive
is shown as white, True Negative as black, False Positive as red, and False
Negative as blue.

The FCN, introduced by Long et al. [7], is renowned for its ability to perform
end-to-end pixel-wise prediction through the extensive utilization of
convolutional layers, allowing it to maintain spatial information throughout
the network. This makes FCN particularly effective for semantic segmentation
tasks.On the other hand, the U-Net architecture, proposed by Ronneberger et
al. [9], features a U-shaped structure that incorporates both encoder and
decoder components. The encoder captures contextual information by
progressively downsampling the input, while the decoder facilitates precise
localization by upsampling and merging features from different resolutions.
The Half-UNet [8], a simplified model, offers a more lightweight alternative.
While retaining the essence of U-Net, Half-UNet reduces computational
complexity by utilizing only half of the U-shaped architecture. This
streamlining allows for faster processing and lower resource requirements,
making it an attractive choice for scenarios with limited computational
resources.

For the training of three models, a batch size of $16$ was selected to balance
computational efficiency and model convergence, and training was conducted
over $100$ epochs. To mitigate the risk of overfitting, we implemented early
stopping with a patience value set to $10$ based on validation loss. These
hyperparameter choices were made to strike a balance between model complexity
and training efficiency. The summary of the performance metrics and model size
are presented in TABLE III. Additionally, Fig. 5 illustrates the qualitative
results of the three models.

##  V Conclusion

This study presented a comprehensive global dataset of multispectral satellite
images of solar panel farms. Our dataset enhances existing resources in three
ways: by incorporating mid-resolution data from rural areas globally, using
multispectral satellite images for richer detail, and updating the timeframe
of data to $2021$-$2023$ for a current view of the global solar energy
landscape. The evaluation of this dataset using benchmark models has
established its robustness and set a baseline for future improvements. It has
proven effective in accurately identifying solar panel areas on a global
scale. This dataset is a significant asset for the solar energy research
community, potentially driving advancements in renewable energy development.

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