11institutetext: Università degli Studi di Palermo, Dipartimento di Fisica e
Chimica, via Archirafi 36 - 90123 Palermo, Italy 22institutetext: INAF/IASF
Palermo, via Ugo La Malfa 153, I-90146 - Palermo, Italy 33institutetext: IRAP,
Universitè de Toulouse, CNRS, UPS, CNES, Toulouse, France 44institutetext:
Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France 55institutetext:
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3
0HA, UK 66institutetext: Istituto Nazionale di Astrofisica, Osservatorio
Astronomico di Brera, via E. Bianchi 46, 23807 Merate (LC), Italy
77institutetext: CNRS, IRAP, 9 avenue du Colonel Roche, BP 44346, F-31028
Toulouse Cedex 4, France 88institutetext: Universitè de Toulouse, CNES, UPS-
OMP, F-31028 Toulouse, France

# Tracking the evolution of the accretion flow in MAXI J1820+070 during its
hard state with the JED-SAD model

A. Marino 112233 S. Barnier 44 P. O. Petrucci 44 M. Del Santo 22 J. Malzac 33
J. Ferreira 44 G. Marcel 55 A. Segreto 22 S. E. Motta 66 A. D’Aì 22 T. Di
Salvo 11 S. Guillot 7788 T. D. Russell 22<EMAIL_ADDRESS>
(Received XXX; accepted XXX)

###### Abstract

Context. X-ray binaries in outburst typically show two canonical X-ray
spectral states, i.e. hard and soft states, as well as different intermediate
states, in which the physical properties of the accretion flow are known to
change. However, the truncation of the optically thick disk and the geometry
of the optically thin accretion flow (corona) in hard state are still debated.
Recently, the JED-SAD paradigm has been proposed for black hole X-ray
binaries, aimed to address the topic of accretion and ejection, and their
interplay in these systems. According to this model, the accretion flow is
composed by an outer standard Shakura-Sunyaev disk (SAD) and an inner hot Jet
Emitting Disk (JED). The JED produces both the hard X-ray emission,
effectively playing the role of the hot corona, and the radio jets. The
disruption of the JED at the transition to the soft state coincides with the
quenching of the jet.

Aims. In this paper, we use the JED-SAD model to describe the evolution of the
accretion flow in the black hole transient MAXI J1820+070 during its hard and
hard-intermediate states. Contrarily to the previous applications of this
model, the Compton reflection component has been taken into account.

Methods. We use eight broadband X-rays spectra, including NuSTAR, NICER and
the Neil Gehrels Swift Observatory data, providing a total spectral coverage
of 0.8-190 keV. The data were directly fitted with the JED-SAD model. We
performed the procedure twice, considering two different values for the ISCO,
i.e. 4 RG (a∗=0.55) and 2 RG (a∗=0.95).

Results. Our results suggest that the optically thick disk (i.e. the SAD) does
not extend down to the ISCO in any of the considered epochs. In particular,
assuming $R_{\rm ISCO}$=4 RG, as the system evolves towards the
hard/intermediate state, we find an inner radius within a range of $\sim$60 RG
in the first observation down to $\sim$30 RG in the last one. The decrease of
the inner edge of the SAD is accompanied by an increase of the mass-accretion
rate. However, when we assume $R_{\rm ISCO}$=2 we find that the mass accretion
rate remains constant and the evolution of the accretion flow is driven by the
decrease of the sonic Mach number $m_{\rm S}$, which is unexpected. In all
hard-intermediate state observations, two reflection components, characterized
by different values of ionization, are required to adequately explain the
data. These components likely originate from different regions of the SAD.

Conclusions. The analysis performed provides a coherent physical evolution of
the accretion flow in the hard and hard-intermediate states and supports a
truncated disk scenario. We show that a flared outer disk could, in principle,
explain the double reflection component. The odd results obtained for $R_{\rm
ISCO}$=2 RG can also be considered as a further evidence that MAXI J1820+070
harbours a moderately spinning black hole, as suggested in other works.

###### Key Words.:

accretion – accretion disks – ISM: jets and outflows – X-rays: binaries –
X-rays, individuals: MAXI J1820+070

##  1 Introduction

A black hole X-ray binary (XRB) is a binary system composed of a stellar mass
BH accreting matter from a companion star (see Done et al. 2007, for a general
review). Almost all known black hole XRBs are transients (BHTs): they spend
most of the time in a quiescent state, at low X-ray luminosity (i.e., below
$\sim$ 1031 erg s-1), but can display sudden episodes of increased X-ray
activity called outbursts (with X-ray luminosity up to $\sim$ 1038-1039 erg
s-1 at the peak). When in outburst, these systems can be found in a variety of
accretion states characterized by different broadband X-ray spectral shape and
timing properties (see Remillard & McClintock 2006; Dunn et al. 2010; Belloni
& Motta 2016, for reviews). In particular, we distinguish between the hard
(HS) and soft (SS) spectral states. The HS is characterized by a cut-off
(around 100 keV when detected) power-law-like spectrum extending up to high
energies, and a cool (kT${}_{\rm disc}\sim 0.1-0.3\ {\rm keV}$) multi-color
disk blackbody. In the SS, spectra are dominated by a hotter multi-color disk
black-body component (kT${}_{\rm disc}\sim$0.8-1.0 keV), and sometimes an
additional steeper hard tail is detected as well. In all these states, a
further contribution to the overall spectrum is given by the reflection
component, a component thought to be due to the hot photons emitted by the
corona reprocessing on the accretion disk (see below). The study of the
reflection component, and in particular of the relativistically broadened Fe K
line at 6.4 keV, can be used as a powerful diagnostics tool to infer, e.g.,
the inner radius and ionization of the accretion disk and the inclination of
the system (see e.g. Reynolds & Nowak 2003, for a review).
The variety of spectral states in BHTs has traditionally been interpreted as
being due to different physical properties and geometry of the accretion flow
around the BH. In the HS, it is thought that the emission is due to thermal
Comptonisation by a hot, optically thin electron plasma (corona). The source
of soft seed photons is the (possibly truncated) optically thick accretion
disk (Shakura & Sunyaev 1973). Conversely, in the SS such a disk is expected
to extend down to the innermost stable circular orbit (ISCO, Bardeen et al.
1972) and only a marginal contribution from the corona is observed. However,
the consensus on the truncated disk model is not global and a number of
counter-arguments have been given as well (see, e.g. Zdziarski & De Marco
2020, for an extensive discussion on the topic). Among the critical issues,
spectral modelling of seemingly broad Fe K lines in BHTs indicated that the
optically thick disk was close or even coincident with the ISCO well within
the hard state (examples of these can be found in Miller et al. 2006; Tomsick
et al. 2008; García et al. 2015).
The controversies on the accretion flow geometry across spectral states is one
of the open problems regarding BHTs. When transients rise from quiescence,
their spectral evolution usually follows a standard path: the system moves
from quiescence to the hard state in a rise of luminosity of several orders of
magnitude, then evolves through the hard/intermediate (HIMS) and
soft/intermediate (SIMS) up to the soft state. Thereafter, the luminosity
decreases and the source exhibits the same transition in reverse but at a much
lower luminosity. The common behaviour of BHTs in making transitions from
hard-to-soft and from soft-to-hard states at different luminosity is called
hysteresis (Miyamoto et al. 1995; Zdziarski et al. 2004). The origin of such a
pattern is a matter of strong debate. It is also noteworthy that a relevant
fraction of BH binaries (about 40%, Tetarenko et al. 2016a) are ”hard-only” or
have displayed ”failed transition” outbursts (see, e.g. Hynes et al. 2000;
Brocksopp et al. 2004; Capitanio et al. 2009; Ferrigno et al. 2012; Del Santo
et al. 2016; Bassi et al. 2019; de Haas et al. 2021). BHTs are known to launch
mildly relativistic jets, which account for the emission from these systems
over radio and mid-IR frequencies. Jets are observed only in hard and hard
intermediate states, while they are ”quenched” around the transition to the
soft state (Fender et al. 1999; Corbel et al. 2000; Fender & Belloni 2004;
Russell et al. 2020) and a short-lived transient jet can be launched (e.g.,
Fender & Belloni 2004; Fender 2010; Rushton et al. 2017; Russell et al. 2019).
A third critical ingredient is represented by how the X-ray timing properties,
i.e. X-rays periodic and aperiodic variability, change according to the
accretion state of BHTs. In this context, one of the most intriguing
challenges consists in the interpretation of Quasi-Periodic Oscillations
(QPOs), i.e., peaks observed in the X-rays Fourier Power Density Spectra of
BHTs which are correlated with the spectral evolution of the systems (see
Ingram & Motta 2020; Motta et al. 2021, for recent reviews).

###  1.1 The JED-SAD paradigm

Much effort has been dedicated to interpreting the multi-wavelength spectral
behaviour and the timing properties of BH binaries in outburst. However,
taking into account the correlation between accretion and ejection has proven
to be difficult. A first attempt to describe globally such complex behavior
was addressed by Esin et al. (1997), who proposed a multi-flow configuration
for the disk. According to these authors, the accretion flow around BHs would
consist of an outer Shakura-Sunyaev accretion disk and an inner, less dense
and radiatively inefficient plasma phase, dubbed Advection Dominated Accretion
Flow (ADAF, Narayan & Yi 1995; Yuan 2001). However, the model suffered several
weaknesses, in particular it was not able to reproduce hard states at
luminosities comparable to the Eddington Limit (see the Introduction of Marcel
et al. 2018b, for an extensive discussion on the topic). In the following
decade, many updates were made to the model (see, e.g. Yuan 2001; Meier 2005;
Xie & Yuan 2012) to make it more suitable to describe the increasing number of
X-ray observations of BHTs. However, in none of these updated versions of the
model the correlation between accretion flow and jets has been addressed.
An attempt to connect accretion and ejection in a unified model has been
recently reported in a number of papers (Ferreira et al. 2006; Marcel et al.
2018b, a, 2019, 2020). This model connects the spectral evolution BHTs in
outbursts to changes in the multi-flow configuration of the accretion disk,
similarly to the approach of e.g., Esin et al. (1997). The main novelty of
this paradigm consists in its ability to also explain the appearance and
disappearance of the jet and their correlation with the accretion flow.
According to this model, the accretion disk is threaded with a large scale
vertical magnetic field $B_{\rm Z}(R)$. Recent numerical simulations have
shown that such magnetic fields become stronger near the compact object (e.g.,
Liska et al. 2018). As a consequence, the accretion flow is also expected to
become more magnetized as we approach the inner edge of the disk. In the
following we define the magnetization as
$\mu(R)=\displaystyle\frac{B^{2}_{Z}(R)}{\mu_{0}P_{\rm tot}}$, with $P_{\rm
tot}$ the total (radiation plus gas) pressure. In the outer regions of the
disk, $\mu\ll 1$ and particles are barely affected by the presence of the
vertical magnetic field. As a consequence, jet launching is inactive in this
region of the disk. Indeed, in order to launch magneto-centrifugally driven
jets (Blandford & Payne 1982), it has been shown that $\mu\gtrsim$0.1 must be
achieved (Ferreira & Pelletier 1995; Ferreira 1997; Jacquemin-Ide et al.
2019). For $\mu<10^{-3}$ or less, magnetic winds can be launched, possibly
carrying away a significant fraction of the mass. However, these winds exert a
negligible torque on the underlying disk (Zhu & Stone 2018; Jacquemin-Ide et
al. 2021). At these distances from the BH, particles are only subject to the
torque due to the internal turbulent viscosity, so that the disk can be well
described with the classic Shakura & Sunyaev (1973) model. The inner regions
are instead highly magnetized, with $\mu$ beyond the 0.1 threshold and
powerful jets can be launched. Jets carry away mass, energy and angular
momentum and exert then an additional torque (see, e.g. Ferreira & Pelletier
1993, 1995). Subsequently, accretion proceeds here at a supersonic velocity,
i.e. much higher than in the Shakura-Sunyaev accretion disk, and plasma in the
internal regions of the accretion flow result more rarefied. Summing up, the
accretion flow surrounding the BH has a two-flow configuration: it is composed
of a Shakura-Sunyaev disk in the outer regions of the flow, defined as
Standard Accretion Disk or SAD, and an inner, less dense and optically thin
Jet Emitting Disk (JED) (Ferreira et al. 2006). JED shares the same physical
properties of the hot corona, but additionally it also drives jets. Quite
intuitively, each accretion state could be obtained by mixing these
ingredients with different quantities, i.e. with a hybrid JED-SAD
configuration where the two realms extend over regions of different scale. In
this sense, such configuration is fundamentally determined by two main control
parameters: (1) the transition radius $R_{\rm J}$ between the JED and the SAD,
and (2) the inner mass accretion flow $\dot{M}_{\rm in}$ feeding the BH. A
detailed description of the model has been presented in Marcel et al. (2018b,
a) and we refer the reader to these papers for more details. This spectral
model has been successfully used to reproduce the X-ray spectral evolution of
the BHT GX 339$-$4 during four outbursts between 2001 and 2011 (Marcel et al.
2019, 2020). Moreover, using the $R_{\rm J}-\dot{M}_{\rm in}$ pairs that best
described the X-ray spectra in a hard state, these authors were also able to
reproduce the jet emission observed simultaneously in the radio band. Finally,
it has been recently shown that also some timing features could be explained
within the JED-SAD paradigm framework. Indeed, a direct proportionality has
been observed between the Keplerian frequency of the transition radius $R_{\rm
J}$ and the type-C QPO frequency in four different outbursts of GX 339$-$4
(Marcel et al. 2020). According to these results, this type of QPOs could
originate at the interface between two regions of different values of $\mu$,
i.e. being then strictly related to the existence of two different types of
accretion flow in the hard and hard/intermediate states of BHTs (Ferreira et
al. 2021, submitted).
The JED-SAD paradigm has successfully explained much of the observed accretion
and ejection phenomenology in GX 339$-$4\. However, the variety of different
behaviours observed in three decades of BHT studies (see, e.g. Dunn et al.
2010; Tetarenko et al. 2016b, for observational reviews) demands for other
tests of this model. In this paper, we report on the application of the JED-
SAD model to the BHT MAXI J1820+070 (ASASSN-18ey), hereafter MAXI J1820,
exploiting a broad X-ray data set including data from NuSTAR, the Neil Gehrels
Swift Observatory (hereafter Swift) and the Neutron Star Interior Composition
Explorer (NICER). This is the first time the JED-SAD model is applied directly
to the data through spectral fits and that Compton reflection process is taken
into account. It therefore represents a major test for the potentialities of
such paradigm.

###  1.2 MAXI J1820+070

MAXI J1820 is a BHT that was observed for the first time in the optical band
by the All-Sky Automated Search for SuperNovae ASSAS-SN (Shappee et al. 2014)
on 2018 March 3, and one week later by MAXI in X-rays (Kawamuro et al. 2018).
Detailed studies of the optical counterpart revealed that the system hosts a
stellar-mass black hole ($\sim$ 8.5 M⊙) accreting from a $\sim$ 0.4 M⊙
companion star (Torres et al. 2019, 2020). Since its discovery, the source
underwent a long (approximately one year) and bright outburst, becoming, at
its peak, the second brightest object in the X-ray sky. Due to its brightness,
the system was the object of an impressive multi-wavelength observing campaign
(see, e.g. Shidatsu et al. 2018; Paice et al. 2019; Hoang et al. 2019;
Trushkin et al. 2018; Bright et al. 2020; Tetarenko et al. 2021) and of a
large number of studies. The most recent and precise measure of the distance,
determined via radio parallax, is 3.0$\pm$0.3 kpc (Atri et al. 2020).
Furthermore, the system shows X-ray dips (Kajava et al. 2019) but not
eclipses, suggesting an inclination between 60∘ and 80∘. Other evidences of
the high inclination of the system are provided by optical spectroscopy
(Torres et al. 2019) and by the estimate of the inclination of the jet axis
(Atri et al. 2020; Wood et al. 2021), i.e. about 63∘. The orbital period of
the system has also been estimated being around 0.68 days (Patterson et al.
2018; Torres et al. 2020). In X-rays, the outburst was studied in detail in
hard state (see, e.g. Bharali et al. 2019; Buisson et al. 2019; Zdziarski et
al. 2021), soft state (see, e.g. Fabian et al. 2020) and in its final phase
(see, e.g. Xu et al. 2020). The truncation of the disk during the hard state
is one of the most controversial aspects of the system. On one hand, a number
of spectral-timing works pointed out that the disk reaches the ISCO already in
hard state, while a contracting lamppost corona is responsible for the hard
X-rays emission (Kara et al. 2019; Buisson et al. 2019; You et al. 2021; Wang
et al. 2021). On the other hand, a truncated disk scenario was also proposed
on the basis of further spectral-timing analyses, even on the same sets of
data (Zdziarski et al. 2021; De Marco et al. 2021; Axelsson & Veledina 2021).

##  2 Observations & Data reduction

In this paper we focus on the spectral analysis of the source in the high
luminosity hard state, i.e. from $\approx$ MJD 58190 until $\approx$ MJD
58300. With the aim of modelling broadband X-ray spectra, we included data
collected by Swift/XRT, NuSTAR, NICER and the Burst Alert Telescope (BAT)
onboard Swift. In order to obtain broad-band spectra from observations close
in time, we selected only XRT, NuSTAR and NICER observations which were quasi-
simultaneous, i.e. taken not more than one day apart from each other.
Furthermore, in order to avoid any discrepancy between BAT and NuSTAR due to
the spectral variability of the source, we extracted BAT spectra exactly over
the duration of each NuSTAR observation. These criteria led us to narrow down
the list of the available NuSTAR data to eight Epochs. We note that this set
of NuSTAR observations has been already analyzed and described by Buisson et
al. (2019). In the following, we will refer to these observations as Epochs
and we will label them with numbers from 1 to 8 in chronological order. It is
worth noticing that NICER observations cover five out of eight epochs and that
the BAT spectrum in Epoch 6 has been extracted in a longer time interval, due
to the low number of visits performed by Swift during the NuSTAR observation.
Details on the selected Epochs are reported in Table 1.

XRT
---
Epoch | ObsID | Start Time | Exposure
|  | (UTC) | (MJD) | ks
1 | 00010627001 | 2018-03-14 | 58191.9 | 1.0
2 | 00010627008 | 2018-03-19 | 58196.8 | 0.98
3 | 00010627013 | 2018-03-24 | 58201.1 | 0.98
4 | 00088657001 | 2018-04-04 | 58212.2 | 1.0
5 | 00010627038 | 2018-04-15 | 58223.3 | 1.7
6 | 00010627055 | 2018-05-04 | 58240.6 | 1.0
7 | 00088657004 | 2018-05-18 | 58255.9 | 1.9
8 | 00088657006 | 2018-06-28 | 58297.3 | 1.8
NuSTAR & BAT†
1 | 90401309002 | 2018-03-14 | 58191.9 | 11.8
2 | 90401309004 | 2018-03-21 | 58198.0 | 2.8
3 | 90401309008 | 2018-03-24 | 58201.5 | 3.0
4 | 90401309012 | 2018-04-04 | 58212.2 | 12.3
5 | 90401309013 | 2018-04-16 | 58225.0 | 1.8
6 | 90401309016 | 2018-05-03 | 58241.8 | 13.7
7 | 90401309019 | 2018-05-17 | 58255.6 | 9.4
8 | 90401309021 | 2018-06-28 | 58297.2 | 21.4
NICER
1 | 1200120103 | 2018-03-13 | 58191.0 | 10.7
3 | 1200120109 | 2018-03-24 | 58201.0 | 13.0
4 | 1200120120 | 2018-04-04 | 58212.0 | 6.5
5 | 1200120130 | 2018-04-16 | 58224.1 | 10.6
6 | 1200120143 | 2018-05-03 | 58241.2 | 4.0
Table 1: List of the XRT, NuSTAR and NICER observations of the source used in
this work. †: BAT survey data were extracted with the same exposure of each
NuSTAR observation, with the exception of Epoch 6, for which a longer ($\sim$
1 day) time interval was used to increase the statistics.

###  2.1 XRT

The 2018 outburst of the source was monitored regularly by the XRT telescope
on board Swift from MJD 58191 (2018, March 14) until MJD 58428 (2018, November
6), with a total of 75 observations performed in Window Timing (WT) mode. The
XRT data were first reprocessed with the task xrtpipeline, included in the
software package HEASOFT (v. 6.26.1). The source extraction procedure from the
cleaned event files was performed with ds9. Since all of the observations had
a high count-rate, i.e., always well above the limit for the pile-up
correction in WT (100 ct/s), we used an annulus region centered on the source
coordinates to extract spectra not affected by pile-up. The outer radius was
always chosen as $\sim$47”, while the inner radius of the annulus was selected
based on the registered count-rate in accordance with the Swift/XRT
guidelines111
https://heasarc.gsfc.nasa.gov/lheasoft/ftools/headas/xrtgrblc.html. In
particular, we used a $\sim$18” inner radius for Epoch 1, a $\sim$24” inner
radius for Epochs 2, 3, 5, 6 and a $\sim$28” inner radius for Epoch 4, 7 and
8. Each spectrum was re-binned with grppha in order to have 150 counts per
bin, allowing the use of the $\chi^{2}$ statistics.

###  2.2 NuSTAR

Data were reduced using the standard Nustardas task, incorporated in Heasoft
(v. 6.26.1). We extracted high scientific products (light curves and spectra)
using a circular area of 100” radius centered at R.A. = 18:20:22.0702, and
Dec. = +7:10:58.331, as source region. In order to take into account any
background non-uniformity on the detector, we extracted the background spectra
using four circles of $\sim$50” radii in different positions with negligible
contamination from the source. We then used Nuproducts to build spectra and
light curves. We used data from both the two hard X-ray imaging telescopes on
board NuSTAR, i.e. the focal plane mirror (FPM) A and B. The extracted spectra
were grouped using the optimal binning recipe by Kaastra & Bleeker (2016) in
order to have a grouping which reflects the spectral resolution of the
instrument in a given energy range and avoid any oversampling issue. We did
not sum the two spectra, but rather fitted them together by leaving a floating
cross-normalization constant as suggested by the NuSTAR team for bright
sources222On the FAQ page, issue 19:
https://heasarc.gsfc.nasa.gov/docs/nustar/nustar_faq.html.

###  2.3 BAT

Data from the BAT survey were also retrieved from the HEASARC public archive.
The downloaded data were processed using BAT-IMAGER software (Segreto et al.
2010). This code, dedicated to the processing of coded mask instrument data,
computes all-sky maps and, for each detected source, produces light curves and
spectra. Light curves in Crab units were extracted in three energy bands, i.e.
15-40 keV, 40-80 keV, 80-150 keV, with 1 day binning time (Fig. 1). Spectra
were extracted in the range 15-195 keV, with logarithmic binning (for a total
of 49 bins) and the official BAT spectral redistribution matrix was used.

###  2.4 NICER

NICER covered the 2018 outburst of MAXI J1820 with almost daily cadence,
resulting in hundreds of individual observations (ObsIDs
1200120101-1200120312). In this work, we analysed a sample of the NICER
observations which satisfied the requirement of being quasi-simultaneous, i.e.
taken within a day, with respect to the NuSTAR data. The data were reduced
using NICER-DAS 2019-05-21 v006. We selected Good Time Intervals using
NIMAKETIME and then applied them to the data via NIEXTRACT-EVENTS, selecting
events with PI channel between 25 and 1200 (0.25–12.0 keV). In order to fix
the distortions due to the NICER calibration uncertainties, we re-normalised
the spectra by using the residuals of a power-law fit to the Crab Nebula
(Ludlam et al. 2018). We used the public files nixtiaveonaxis20170601v002.arf
and nixtiref20170601v001.rmf as Ancillary Response File and Redistribution
Matrix File respectively, retrievable from the NICER website333See
https://heasarc.gsfc.nasa.gov/docs/nicer/proposals/nicer_tools.html.. As a
background spectrum we used the public background file nixtiback20190807.pi,
also available in the HEASARC archive.

##  3 Spectral analysis

XRT and BAT light curves and the related hardness ratios are shown in Fig. 1,
while in Fig. 2 the XRT Hardness Intensity Diagram (HID) of the whole 2018
outburst is shown. We used the (absorbed) 0.5-10 keV flux as an indicator of
the intensity of the source. In order to estimate the flux, we fitted each XRT
spectrum separately in the energy bands 0.5-2 keV and 2-10 keV with a simple
power-law (POWER in Xspec) model. We notice that a power-law model, although
not being an appropriate choice to describe adequately the spectrum in the
whole XRT energy range, approximate satisfactorily its shape in the two energy
bands considered. We associated an error equal to 10% of the estimated value
to each of these fluxes.
According to their hardness ratio and their timing properties (e.g., De Marco
et al. 2021), during Epoch 1 the system was in the canonical hard state, while
it remained in hard/intermediate state from Epochs 2 to Epoch 8. The latter
occurred during a short episode of re-hardening, as better highlighted in the
BAT light curve (see Fig. 1). In the following we report on the spectral
analysis of these Epochs.

Figure 1: From top to bottom, the Swift/XRT light curves and the related soft
X-ray HR, Swift/BAT light curves and the hard X-ray HR are shown. Energy
ranges are reported in the see legends. The NuSTAR observations are
highlighted with black dashed lines. Figure 2: XRT HID of the source during
the entire outburst. The epochs used in this work are highlighted by the
colors and different symbols (Sect. 3.2), i.e. circle for phase 1 (rise),
upper triangles for phase 2 (plateau) and lower triangles for phase 3
(decline).

The NuSTAR data usually range from 3-79 keV. However, a remarkable (and
unexpected) mismatch between FPMA and FPMB data is sometimes observed below 4
keV, likely due to a known instrumental issue (Madsen et al. 2020). This
mismatch was visible in our data. Therefore, we decided not to include data
below 4 keV, which are already covered by XRT. Similarly, also NICER data were
ignored below 4 keV, as they showed a soft excess, which could be related to a
further Comptonization or a quasi-thermal component, (see fig. 6, Zdziarski et
al. 2021), as well as several lines, both sharp instrumental features (Wang et
al. 2020) and broad physical lines. A proper investigation of these features
goes beyond the purposes of this manuscript and will be reported elsewhere.
Moreover, the BAT data showed an unexpected mismatch with NuSTAR below 30 keV,
possibly due to high systematics in this energy range for extremely bright
sources. We therefore ignored BAT data below 30 keV. So that, the final
broadband spectra covered the range 0.8-10 keV for XRT, 4-78 keV for NuSTAR,
4-10 keV for NICER and 30-190 keV for BAT.

Each spectrum was analyzed using the tbabs model in order to take into account
the effect of interstellar absorption, with the photoelectric cross sections
from Verner et al. (1996) and the element abundances from Wilms et al. (2000).
A constant component was also used to serve as cross-calibration constant. To
account for differences in the calibration of NuSTAR and NICER (see, also
Zdziarski et al. 2021) , we included a cross-calibration function which reads
$\textsc{const}\times{E^{\Delta\Gamma}}$, with $\Delta\Gamma$ the discrepancy
in photon index between the data sets (Ingram et al. 2017; Ursini et al.
2020). $\Delta\Gamma$ was left free to vary for NICER, while it was fixed to 0
for all the other instruments included in the broadband spectrum. Finally, as
each data-set was showing apparent systematics, we applied a 1% systematic
error to the analyzed spectra.

###  3.1 Basic ingredients of the JED-SAD spectral model

The eight X-ray observation were analyzed with the JED-SAD model by applying
it to the data in the form of a Xspec table. The key parameters in the model
are (but see Marcel et al. 2018a, , for a full description ):

  * •

The transition radius, $R_{\rm J}$, between the JED and the SAD, in units of
gravitational radii R${}_{\rm G}=GM/c^{2}$.

  * •

The mass accretion rate at the transition radius $R_{\rm J}$ or inner mass
accretion rate $\dot{M}_{\rm in}$, here expressed in units of Eddington mass
accretion rates $\dot{M}_{\rm Edd}=L_{\rm Edd}/c^{2}$.

  * •

The sonic Mach number $m_{\rm S}$ of the accretion flow, defined as the ratio
of the mass weighted accretion speed to the sound speed.

  * •

The ejection efficiency $p$444The ejection parameter is labeled $\xi$ in
Marcel et al. (2019). Here we use $p$ to avoid confusion with the ionization
parameter of the reflection model., which takes into account how effectively
the jet extracts angular momentum from the underlying JED, modifying then its
structure. Therefore in the JED, the accretion rate $\dot{M}(R)$ at any radius
$R$ scales with $R$ according to the formula $\dot{M}(R)=\dot{M}_{\rm
in}\left(\frac{R}{R_{\rm ISCO}}\right)^{p}$, with $p$ the ejection efficiency.

  * •

The magnetization $\mu$ (see Sect. 1.1).

  * •

The fraction $b$ of the accretion power Pacc which is released in the JED and
ultimately powers the jets, i.e. $b=2P_{\rm jet}/P_{\rm acc}$.

  * •

A geometrical dilution factor $\omega$, corresponding to the fraction of the
cold photons emitted by the SAD which cool down (through Inverse Compton) the
JED (we refer to Marcel et al. 2018a, and references therein for further
discussion on this topic). In the current version of the model, $\omega$
assumes values in the range 0-0.5.

  * •

The radius of the ISCO, $R_{\rm ISCO}$, which coincides with the innermost
radius of the JED555We note however that the JED-SAD calculations have been
performed in a Newtonian potential, meaning that the value obtained/chosen for
$R_{ISCO}$ needs to be taken with a grain of salt.

  * •

A normalization factor $K_{\rm JEDSAD}$, which determines how the luminosity
of the system scales with the distance $D_{\rm kpc}$ (in units of 1 kpc),
following the formula:

$K_{\rm JEDSAD}=\left(\frac{10}{D_{\rm kpc}}\right)^{2}$ (1)

As explored in detail by Marcel et al. (2018b), the JED emission in the
archetypal object GX 339–4 is best described by setting $\mu$ between 0.1 and
1.0, $b$=0.3, $\omega=0.2$ and $p$=0.01666This set of parameters, with the
exception of $\omega$ that was indeed found through spectral constraints, is
also consistent with a full MHD accretion-ejection solution with aspect ratio
$\epsilon=H/R\sim 0.1$, as expected for a hot JED (Ferreira 1997; Petrucci et
al. 2008). In the following we fix $\mu$ to 0.5 and adopt the same set of
values for $b$, $\omega$ and $p$ in the case of MAXI J1820+070. Contrarily to
the works on GX 339–4, we let $m_{\rm S}$ free to vary in order to investigate
its correlation with the high energy cut-off, which can be better constrained
in our work due to the NuSTAR and BAT coverage at hard X-rays. The value of
$R_{\rm ISCO}$ in MAXI J1820 is not precisely known. While it has been
reported that the ISCO in MAXI J1820+070 could be even lower than 2 RG (e.g.
Kara et al. 2019), it was recently found that the BH in the system is likely
slowly spinning (Fabian et al. 2020; Zhao et al. 2020; Guan et al. 2020), with
$a_{*}$ not higher than 0.5. We chose therefore to use an initial value of
$R_{\rm ISCO}$=4 RG (corresponding to a∗=0.55), approximately consistent with
the estimates reported in these works. The distance of the source was fixed at
3 kpc according to Atri et al. (2020). We will allow more freedom in $R_{\rm
ISCO}$ and $K_{\rm JEDSAD}$ and discuss the dependence of our results on the
choice of these parameters in Sect. 3.4-3.5. A summary of the parameters in
the model and the values adopted in this work is presented in Table 2.

Parameter |  | Description | Adopted
---|---|---|---
|  |  | Value(s)
$R_{\rm J}$ | (RG) | Transition radius | (free)
$\dot{M}_{\rm in}$ | $\left(\dot{M}_{\rm Edd}\right)$ | Inner mass-accretion | (free)
|  | rate |
$p$ |  | Ejection efficiency | 0.01a
$\mu$ |  | Magnetization | 0.5a
$b$ |  | Jet power | 0.3a
$\omega$ |  | Dilution factor | 0.2a
$m_{\rm S}$ |  | Sonic Mach number | (free)
$R_{\rm ISCO}$ | (RG) | ISCO | 4, 2
$D_{\rm kpc}$ | (kpc) | Distance in kpc | 3b
Table 2: Parameters used in the JED-SAD model. a: the parameter was set to the
same value found by Marcel et al. (2018b). b: in accordance with Atri et al.
(2020).

###  3.2 The two reflection models

A first analysis without any reflection component results in extremely poor
fits and strong residuals in the iron line and Compton hump regions. This was
expected because reflection is not taken into account in the JED-SAD model. We
therefore used a reflection table component, that we will label refl for
simplicity, based on the xillver reflection model (García et al. 2013). This
table has been produced by simulating spectra for different combinations of
$R_{\rm J}-\dot{M}_{\rm in}$ pairs, fitting them with a simple cut-off power-
law model and finally injecting the corresponding spectral index and high
energy cut-off in the xillver table. More details on this model will be given
in Barnier et al. (submitted). The parameters of this component are: the iron
abundance $A_{\rm Fe}$, the disk ionization parameter $\log\xi$ and a
normalization $K$. Since refl does not take into account the relativistic
blurring effects, we applied the convolution kernel kdblur, which smears the
reflection spectrum according to the original calculations by Laor (1991).
This model considers four additional parameters: the emissivity $\epsilon$,
the inner and outer radius of the reflecting disk $R_{\rm in}$ and $R_{\rm
out}$, the inclination $i$. In order to reduce the number of degrees of
freedom we set $\epsilon$ to the reasonable value of 3 (see, e.g. Dauser et
al. 2013; Xu et al. 2020) and $i$ to 70∘ (consistent with the estimated source
inclination, see e.g. Kajava et al. 2019). As the reflection component is
produced by the disk/SAD surface reprocessing the hard X-ray photons emitted
by the corona/JED, within the JED-SAD framework the inner radius of the
reflection $R_{\rm in}$ should coincide with the transition radius, $R_{\rm
J}$, between JED and SAD. In all spectral fits, $R_{\rm J}$ and $R_{\rm in}$
were therefore linked together.
The first model we applied, Model 1, reads:

${\rm\bf Model\ 1:}\textsc{tbabs}\times(\textsc{atable}(\textsc{jedsad})+\\\
+\textsc{kdblur}\times\textsc{atable}(\textsc{refl}))$ (2)

It is noteworthy that two reflection components were already suggested by Kara
et al. (2019) on the basis of reverberation time lags study with NICER data.
Thereafter a number of authors used a double-reflection model in the spectral
analysis of the same NuSTAR observations presented here (Buisson et al. 2019;
Chakraborty et al. 2020; Zdziarski et al. 2021). They all found both a low
ionisation reflection component and a highly ionised inner reflection
component, which contribute to the observed iron line profile providing
respectively a narrow core and a broad base. Therefore, we also tested a
second model, Model 2, where a second refl component has been added:

$\rm{\bf Model\ 2:}\textsc{tbabs}\times(\textsc{atable}(\textsc{jedsad})+\\\
+\textsc{kdblur}\times\textsc{atable}(\textsc{refl})+\textsc{kdblur}\times\textsc{atable}(\textsc{refl}))$
(3)

When Model 2 is applied, we use the subscripts ’1’ and ’2’ to refer to the
parameters of the inner and outer reflection components, respectively. The
purpose of this model is to reproduce a physical scenario where reflection
comes from two distinct regions, characterized by different ionisation
parameters and therefore presumably placed at different distances from the
illuminating corona (the JED here). Therefore we tied $R_{\rm in,1}$ to
$R_{\rm J}$ and $R_{\rm out,1}$ to $R_{\rm in,2}$, in order to place the two
reflection regions at separate but neighboring regions777This is only a
working assumption, as it cannot be excluded that regions may not be exactly
adjacent to each other., as shown in Figure 3. The outer radius of the remote
reflection is instead left free to vary. The normalisation and ionization
parameters were left free as well in both reflection components, so that we
have two different normalization and ionization values: $K_{1}$, $K_{2}$,
$\xi_{1}$ and $\xi_{2}$. All the other parameters, i.e. the inclination, the
iron abundance and the emissivity, were tied between the two reflection
components.

Figure 3: A sketch representing the proposed two-reflection geometry
underlying Model 2. From left to right: the black hole (in black), the inner
JED (violet) and the jets (lilac), the outer SAD (from red to yellow).

###  3.3 Spectral fits with $R_{\rm ISCO}$=4 RG

Based on the fit results described below, we can divide our observations in
three groups or ”phases”: phase 1 corresponds to Epoch 1, phase 2 includes
observations from Epoch 2 to Epoch 6 , phase 3 which corresponds only to
Epochs 7 and 8. The grouping of these Epochs is consistent with the XRT HID
(Fig. 2), where an analogous clustering of observations in three regions can
be observed. A similar subdivision was also proposed by De Marco et al.
(2021), with phases labeled as ”Rise”, ”Plateau” and ”Bright decline”,
respectively. In the following, we will treat these phases separately. In all
phases, we did not manage to constrain the values of $R_{\rm in,2}$ and
$R_{\rm out,2}$, as the fit was basically insensitive to variations of these
parameters. We therefore set $R_{\rm in,2}$ to 300 RG and $R_{\rm out,2}$ to
104 RG. We found values of $A_{\rm Fe}$ always between 2 and 3 times the solar
abundance888We caution the reader that the incompatibility between the best-
fit value of $A_{\rm Fe}=2.0-2.4$ found for Epoch 2 and the common value of
about 3 found for the other seven epochs, does not have a physical origin. For
consistency, we tried fixing $A_{\rm Fe}$ to a value of 3 also in Epoch 2, but
the fit, and particularly the iron line residuals, significantly worsened.
This may be due to systematics in NuSTAR between 8 and 11 keV which interfere
with a correct modelisation of the iron line region..
The final results are reported on Table 3, while corresponding best-fit models
and residuals are shown in Figure 4-5.

Spectral analysis in the case of $R_{\rm ISCO}$=4 RG |
---|---
Epochs |  | 1 | 2 | 3 | 4
NH |  $\times$1022 cm-2 | $0.560\pm 0.020$ | $0.190\pm 0.020$ | $0.192^{+0.014}_{-0.015}$ | $0.150\pm 0.020$
$R_{\rm J}$ | RG | $56.8^{+1.0}_{-1.1}$ | $35.9^{+5.0}_{-1.4}$ | $44.8^{+0.3}_{-0.5}$ | $43.9^{+1.1}_{-1.0}$
$\dot{M}_{\rm in}$ | $\dot{M}_{\rm Edd}$ | $0.838^{+0.008}_{-0.009}$ | $1.760\pm 0.080$ | $2.360^{+0.070}_{-0.050}$ | $2.120^{+0.060}_{-0.030}$
$m_{\rm S}$ |  | $1.301^{+0.017}_{-0.016}$ | $1.250^{+0.020}_{-0.050}$ | $1.243\pm 0.001$ | $1.250^{+0.020}_{-0.010}$
$A_{\rm Fe}$ |  | $2.9^{+0.3}_{-0.4}$ | $2.2^{+0.2}_{-0.2}$ | (3)
$R_{\rm in,1}$ | $R_{\rm G}$ | =$R_{\rm J}$
$R_{\rm in,2}$ | $R_{\rm G}$ |  | (300)
$R_{\rm out,1}$ | $R_{\rm G}$ | 104 | =$R_{\rm in,2}$
$R_{\rm out,2}$ | $R_{\rm G}$ |  | (104)
$i$ | ∘ | (70)
$\log{\xi_{1}}$ |  | $2.22^{+0.13}_{-0.23}$ | (3.50) | $3.71^{+0.02}_{-0.08}$ | $3.69^{+0.04}_{-0.13}$
$\log{\xi_{2}}$ |  |  | ¡1.70 | ¡1.65 | $2.07^{+0.22}_{-0.01}$
K1 | ($\times$10-6) | $42.0^{+17.0}_{-20.0}$ | $3.3^{+1.0}_{-0.9}$ | $3.6^{+0.4}_{-0.2}$ | $3.0^{+0.9}_{-0.4}$
K2 | ($\times$10-4) |  | $232.0^{+15.0}_{-22.0}$ | $0.6^{+0.1}_{-0.2}$ | $1.3\pm 0.5$
$\chi^{2}_{\nu}$ | (d.o.f.) | 1.01(746) | 1.03(755) | 1.00(1036) | 0.99(1029)
Epochs |  | 5 | 6 | 7 | 8
NH |  $\times$1022 cm-2 | $0.13\pm 0.01$ | $0.11\pm 0.01$ | $0.15\pm 0.02$ | $0.15\pm 0.02$
$R_{\rm J}$ | RG | $47.8^{+3.4}_{-2.1}$ | $45.8^{+1.3}_{-1.1}$ | $33.6^{+0.6}_{-0.5}$ | $33.3^{+1.2}_{-0.7}$
$\dot{M}_{\rm in}$ | $\dot{M}_{\rm Edd}$ | $1.56^{+0.07}_{-0.10}$ | $1.25^{+0.02}_{-0.04}$ | $1.00^{+0.05}_{-0.03}$ | $1.06^{+0.02}_{-0.04}$
mS |  | $1.25^{+0.02}_{-0.01}$ | (1.25) | ¿1.49 | ¿1.49
$A_{\rm Fe}$ |  | (3.0) | $3.4^{+0.3}_{-0.4}$
$R_{\rm in,1}$ | $R_{\rm G}$ | =$R_{\rm J}$
$R_{\rm in,2}$ | $R_{\rm G}$ | (300)
$R_{\rm out,1}$ | $R_{\rm G}$ | =$R_{\rm in,2}$
$R_{\rm out,2}$ | $R_{\rm G}$ | (104)
$i$ | ∘ | (70)
$\log{\xi_{1}}$ |  | $3.81^{+0.06}_{-0.03}$ | $3.80^{+0.05}_{-0.03}$ | $3.79^{+0.03}_{-0.02}$ | $3.48^{+0.13}_{-0.09}$
$\log{\xi_{2}}$ |  | $2.32^{+0.04}_{-0.19}$ | $3.04^{+0.07}_{-0.10}$ | $2.99^{+0.05}_{-0.12}$ | (2.00)
K1 | ($\times$10-6) | $3.5\pm 0.3$ | $3.9^{+0.7}_{-0.8}$ | $3.6\pm 0.4$ | $2.7\pm 0.6$
K2 | ($\times$10-5) | $0.70^{+0.50}_{-0.20}$ | $0.63^{+0.38}_{-0.10}$ | $0.61^{+0.18}_{-0.07}$ | 0.15${}^{+0.08}_{-0.12}$
$\chi^{2}_{\nu}$ | (d.o.f.) | 0.98(1146) | 1.01(991) | 1.12(794) | 1.15(748)
Table 3: Fit results for all epochs with Model 1 for epoch 1 and Model 2 for
the remaining epochs. Quoted errors reflect 90% confidence level. The
parameters which were kept frozen during the fits are reported between round
parentheses.  Figure 4: Energy spectra with the best-fit models, i.e. Model 1
for Epoch 1 and Model 2 for Epochs 2-4, and residuals. Data: XRT (blue), NICER
(green), NuSTAR (yellow-black) and BAT (red). Different linestyles were
adopted to distinguish between the different components, in particular: dot
for JED, dash-dot-dot-dot for SAD, dash-dot for inner refl, dash for the outer
refl. Figure 5: Energy spectra, best-fit model (Model 2) and residuals for
Epochs 5-8. Data: XRT (blue), NICER (green), NuSTAR (yellow-black) and BAT
(red). Different linestyles were adopted to distinguish between the different
components, in particular: dot for JED, dash-dot-dot-dot for SAD, dash-dot for
inner refl, dash for the outer refl.

#### Phase 1: The system in the hard state with one reflection component

Epoch 1 is satisfactorily fitted with both Model 1 and Model 2, with
$\chi^{2}_{\nu}$ (d.o.f.) of 1.01(749) and 0.90(748), respectively. In both
cases, the SAD is truncated far away from the BH, i.e. with $R_{\rm J}$ of
$\sim$60 RG and $\sim$ 150 RG for Model 1 and Model 2, respectively. However,
with Model 2, the value obtained for $\log{\xi}_{1}$ goes beyond the threshold
of 4, i.e. around 4.3. We consider such high value as unphysical for several
reasons. First of all, such a highly ionized medium at the edge of the disk is
hard to reconcile with it being truncated at 150 RG. Secondly, we note that
with the very high value found for $\log{\xi_{1}}$, the inner reflection
component has almost no significant fluorescence line and basically serves as
an additional continuum component. Finally, the spectral shape of the
reflection component computed with xillver for $\log{\xi}$¿4 is unreliable at
high energies999J. Garcia, private communication.. According to these
arguments, the presence of the second reflection component results most likely
spurious for Epoch 1. Using Model 1, $\log{\xi}$101010The subscript is removed
here since in Model 1 there is no need to distinguish between component ”1”
and component ”2”. is found in the range 2.0-2.4, which is instead more
plausible.

#### Phase 2. The outer reflection component arises

Epochs from 2 to 6 are poorly fitted by Model 1, with $\chi^{2}_{\nu}$ values
all above 1.25 and, most importantly, evident unmodelled structures between 6
and 10 keV in the residuals. Instead, the application of Model 2 results in
good fits and quite ”flat” residuals. The picture captured by the fits is
similar for the epochs in this group. The best-fit value of $R_{\rm J}$
decreases significantly with respect to the previous phase, i.e. attaining
values in the range 35-50 RG. Simultaneously, $\dot{M}_{\rm in}$ evolves
substantially, going from 1.8 $\dot{M}_{\rm Edd}$ (Epoch 2) to 2.4
$\dot{M}_{\rm Edd}$ (Epoch 3), and then decreases again to 1.3 in epoch 6.
$m_{\rm S}$ retains a well-determined value of about 1.25 for all the epochs
in this phase, with the only exception of epoch 6, where the parameter was
basically unconstrained and therefore fixed. In all epochs, ${\xi_{1}}$ is
higher than ${\xi_{2}}$ by at least an order of magnitude. Interestingly,
while $\xi_{1}$ is quite stable, an increasing trend can be observed in
$\log{\xi_{2}}$, which goes from an upper limit of 1.7 in epoch 2, to 3 in
epoch 6. The possibility of this trend being related to an evolution of the
inner radius of the outer reflection will be explored in the Discussion (see
Section 4).

#### Phase 3: The role of $m_{\rm S}$ in the re-hardening

Model 1 also gives an unacceptable fit for Epoch 7, but is instead compatible
with Epoch 8, with a $\chi^{2}_{\nu}$ (d.o.f.) of 1.15 (750). Model 2 gives a
good fit to Epoch 7 and an even better fit to Epoch 8, with a probability of
improvement by chance of $\sim$ 10-5 with respect to Model 1. We then conclude
that a double reflection scenario is also statistically preferred in phase 3.
As in the previous phase, $R_{\rm J}$ decreases again to about 33-34 RG, while
$\dot{M}_{\rm in}$ is constrained at $\sim$ 1 $\dot{M}_{\rm Edd}$. The most
apparent evolution with respect to the previous phase consists in a re-
hardening of the spectrum, as witnessed by the high energy cut-off of the BAT
data moving towards higher energies (see the BAT light curve, Fig. 1). Since
some level of degeneracy between $R_{\rm J}$ and $m_{\rm S}$ clearly exists
(see Fig. 6), such trend can be explained by an increase in both these
parameters. For example, Epoch 7 can be fitted keeping $m_{\rm S}$ fixed to
1.3, in consistency with the previous phase, and with $R_{\rm J}$ of 38-39
$R_{\rm G}$ or leaving $m_{\rm S}$ free, a case where $m_{\rm S}$ attains a
value of 1.5 and $R_{\rm J}$ decreases to 33-34 RG. Both fits are acceptable,
but keeping $m_{\rm S}$ fixed results in slightly unmodelled residuals in the
iron line region, contrarily to the case where $m_{\rm S}$ is free. We
therefore conclude that the spectral hardening in this phase is not reflected
in the values of $R_{\rm J}$, which actually suggests that the inner edge of
the disk keeps moving inwards as expected during the transition to the soft
state, but seems to be instead produced by an increase in $m_{\rm S}$,
producing a more rarefied and hotter accretion flow in the JED. Another
interesting evolution can be found in the values of $\xi_{1}$ and $\xi_{2}$:
while Epoch 7 has values compatible with Epoch 6 (and phase 2 in general), in
Epoch 8 both values $\log{\xi_{1}}$ and $\log{\xi_{2}}$ decrease
significantly, i.e. to values of $\sim$ 3.5 and 2.0 (which was fixed since it
was left unconstrained by the fit), respectively.
A summary on the evolution of the main parameters at play is presented in Fig.
7. We also show the evolution of the reflection components in Fig. 8.

Figure 6: JED solutions showing the impact of $m_{\rm S}$ and $R_{\rm J}$ on
the high energy cut-off. In the Left panel, we show different JED solutions
obtained keeping $R_{\rm J}$ to 40 RG for different values of $m_{\rm S}$. In
the Right panel, the different JED solutions are produced for different values
of $R_{\rm J}$, here expressed as $r_{\rm J}=R_{\rm J}/{\rm R}_{\rm G}$, and
$m_{\rm S}$ fixed to 1.5. The plot is used to display how increasing both
$m_{\rm S}$ and increasing $R_{\rm J}$ pushes the hard X-rays cut-off to
higher energies. Figure 7: Evolution of the best-fit parameters for $R_{\rm
J}$, $\dot{M}_{\rm in}$ and $m_{\rm S}$ over the eight analyzed Epochs. In all
panels, triangles and bold lines are used for fits performed with $R_{\rm
ISCO}$ at 4 RG. The reported errors are likely underestimated and an
additional, about 10%-20%, systematic error has to be considered to account
for the uncertainty on $\omega$ and $K_{\rm JEDSAD}$ (see Sect. 3.5).

Using the best-fit values found for $R_{\rm J}$, $\dot{M}_{\rm in}$ and
$m_{\rm S}$, it is possible to produce diagrams for the main physical
properties of the corresponding hybrid JED-SAD configuration. These diagrams
are presented in Fig. 9 for Epoch 1, 5 and 8, each of them chosen to represent
one of the three phases. In these plots, the accretion flow has been divided
into 31 radial zones, 30 for the JED, one for the SAD. The scale height $H$,
the Thomson optical depth $\tau_{\rm T}$, the electron temperature $kT_{\rm
e}$ and the emitted spectrum were calculated and plotted in Fig. 9 for each
one of these 31 regions (see Appendix C in Ursini et al. 2020, for further
details on how these plots were produced). The plots show that the increase in
$\dot{M}_{\rm in}$ in phase 2 corresponds to a geometrically thinner,
optically thicker and colder JED with respect to phase 1. The evolution of the
accretion flow is also apparent from the softening of the spectra emitted by
the single rings forming the JED. The re-hardening episode in phase 3 is
witnessed by the slight increase in the spectral high energy cut-off combined
with an increase in both $kT_{\rm e}$ and $\tau_{\rm T}$.

Figure 8: Best-fit models for inner and outer reflection in Epochs 2-7. Inner
reflection models have been multiplied by a factor 10 for clarity. Figure 9:
JED-SAD solutions for Epochs 1, 5 and 8, used to represent phases 1, 2 and 3,
respectively. In order to calculate these diagrams, we used the best-fit
parameters for $R_{\rm J}$, $\dot{M}_{\rm in}$ and $m_{\rm S}$ reported in
Table 3. In particular, for phase 1 we used $R_{\rm J}$=57 RG, $\dot{M}_{\rm
in}$=0.8 $\dot{M}_{\rm Edd}$ and $m_{\rm S}$=1.3, for phase 2 we used $R_{\rm
J}$=44 RG, $\dot{M}_{\rm in}$=1.7 $\dot{M}_{\rm Edd}$ and $m_{\rm S}$=1.3 and
for phase 3 the values adopted were $R_{\rm J}$=33 RG, $\dot{M}_{\rm in}$=1.1
$\dot{M}_{\rm Edd}$ and $m_{\rm S}$=1.5. In all panels, the radial
distribution is divided in 31 portions (annuli), 30 for the JED and one for
the SAD. In the top and middle panels, we present how the actual shape $z=H/R$
of the disk evolves with $R$. The distribution of the Thomson optical depth
($\tau_{\rm T}$, top-right color bar) and of the electron temperature
($kT_{\rm e}$, right color bar) is displayed with colors in the top and middle
panels, respectively. In the same plots, the event horizon of the BH is
represented by a half black circle. The total emitted spectrum is reported in
the bottom panels with a solid black line. The contribution from each of the
31 considered annuli is highlighted with a dotted line for the SAD region and
dashed lines for the JED regions. The sections of the plot corresponding to
the energy ranges that are not covered by XRT, NICER, NuSTAR and BAT data are
shown with grey background.

###  3.4 Spectral fits with $R_{\rm ISCO}$=2 RG

In Sect. 3.3, we kept the innermost radius $R_{\rm ISCO}$ frozen to 4 RG,
corresponding to a moderately spinning black hole, as apparently suggested by
recent spectral analysis (Fabian et al. 2020; Zhao et al. 2020; Guan et al.
2020). In this section, we replicated the previous fitting procedure after
setting $R_{\rm ISCO}$ to 2 RG, i.e. corresponding to a rapidly spinning BH
(a${}_{*}\sim$ 0.95).

In this new ensemble of fits, a subdivision in three phases can again be
individuated. Furthermore, a double reflection model is still necessary to fit
the Epochs, with the exception of Epoch 1, for which Model 2 results again in
oddly high values of $\log{\xi}_{1}$. However, a lower $R_{\rm ISCO}$
significantly affects the values obtained for $R_{\rm J}$, $\dot{M}_{\rm in}$
and $m_{\rm S}$. Also in this case, $R_{\rm J}$ decreases through the three
phases, but starts from almost 44 RG to then decrease to $\sim$20-30 RG and
finally to $\sim$15-20 RG in phase 2 and 3 respectively. The trend is
significant, since the discrepancy between these best-fit values is larger
than their errors. A different trend is observed in $\dot{M}_{\rm in}$, which
remained very stable throughout the eight Epochs analysed, with only a slight
increase in phase 3. The stability in $\dot{M}_{\rm in}$ is compensated by
$m_{\rm S}$ evolving blatantly, going from $m_{\rm S}\sim 1.1$ in phase 1 to
$m_{\rm S}\sim 0.75$ and finally rising again in phase 3 up to 1.5. If we
impose the same $m_{\rm S}$ for all the epochs, in conformity with the
analysis performed on GX 339–4 (Marcel et al. 2018b), the best fits found by
Xspec are strongly reflection-dominated. This happens because for $m_{\rm S}$
values beyond 1, the JED model is characterized by a rather high electron
temperature. The subsequently high energy cutoff is not compatible with the
spectral curvature well-traced by the BAT data in our spectra, which indicate
a cut-off at about 100 keV. The only way for the JED-SAD model to reproduce
such spectral curvature would be to decrease $R_{\rm J}$ and simultaneously
increase the temperature of the inner edge of the SAD. However, a hotter SAD
should also be visible in the XRT band, which does not show any sign of such
strong disk component. Since the JED-SAD is not suitable, the best solution to
describe the broadband spectrum consists in tuning up the inner (highly-
ionized) reflection component. Such reflection-dominated spectra are
challenging to explain, especially in the framework of the proposed JED-SAD
geometry. Similarly high reflection fractions have been explained in the past
with, e.g., a large covering fraction of the disk due to the presence of
clouds of colder matter (Malzac & Celotti 2002), with a compact corona close
to the BH emitting anisotropically due to light bending (Miniutti et al. 2003)
or due to a mildly-relativistic motion of the corona towards the disk
(Beloborodov 1999). Unfortunately all of these scenarios are hard to reconcile
with the proposed JED-SAD geometry. We conclude that with $R_{\rm ISCO}=2$ RG
the role played by $\dot{M}_{\rm in}$ in driving the evolution of the
accretion flow towards intermediate states is somehow replaced by an evolution
in $m_{\rm S}$.

###  3.5 Exploring the effect of the other JED-SAD parameters

In order to reduce the number of degrees of freedom, so far we kept some
parameters fixed in both Models 1 and 2. In the following we explore in more
detail their impact on the results obtained.
As shown by Marcel et al. (2018b), fig. 10, $p$ (which was labeled $\xi$ by
these authors) has little to no effect to the JED-SAD spectral shape.
Similarly, the choice of fixing $\mu$ to 0.5 should have a negligible impact,
especially since the high energy cut-off is basically insensitive to this
parameter (see fig. 6, Marcel et al. 2018b). On the contrary, the jet power
$b$ is expected to influence the JED shape at both hard and soft X-rays.
Broadly speaking, by increasing $b$ more power is funneled in the jets, making
the underlying JED rather cold, while with a lower $b$, the JED results
hotter. This effect would shift the position of the high energy cut-off and
therefore act as $m_{\rm S}$. However, $b$ is a function of the aspect ratio
$\epsilon$, which for a typical JED should be about 0.2-0.3. As shown by
Petrucci et al. (2010), Appendix A, we need $b$ values around 0.3-0.5 to
obtain such aspect ratios, as higher values would produce discs geometrically
too thin to be consistent with the expected JED geometry. We are therefore not
exploring the effect of varying $b$ any further.
Potential biases could be ascribed to the dilution factor $\omega$ and to the
parameter $K_{\rm JEDSAD}$, which is solely related to the distance. We recall
$\omega$ being a parameter regulating the amount of SAD photons penetrating
the JED and thereby cooling it. In the current model, it ranges between 0 (the
unlikely case where no photons from the SAD are intercepted by the JED) and
0.3 (when up to 30% photons can be intercepted). In order to check the effect
of $\omega$, we analyzed again Epoch 5 keeping $\omega$ fixed first to 0 and
then to 0.3. As expected, when we put $\omega$ to zero, the cooling effect
played by the SAD is replaced by lowering $R_{\rm J}$, which attains a value
of 27-31 RG, while also $\dot{M}_{\rm in}$ and $m_{\rm S}$ drop down to values
of about 1.1 $\dot{M}_{\rm Edd}$ and 1.0 respectively. Increasing $\omega$ to
0.3 only moderately impacts the outcomes of the fits, i.e. with $R_{\rm J}$
and $m_{\rm S}$ compatible with the results obtained with $\omega=0.2$ and a
15% increase in $\dot{M}_{\rm in}$. The distance of MAXI J1820 is well
constrained between 2.7 and 3.3 kpc (Atri et al. 2020). In the previous
sections, we froze $\omega$ to 0.2 and $K_{\rm JEDSAD}$ to 11.11,
corresponding to a distance of 3 kpc. We then computed again the best fits for
Epoch 5 freezing $K_{\rm JEDSAD}$ first to 13.7 (2.7 kpc), and then to 9.2
(3.3 kpc). Analogously to $\omega$, variations of $K_{\rm JEDSAD}$ do not
affect the overall physical scenario which was individuated in this phase
phase with $K_{\rm JEDSAD}=11.11$, but they result in a slight fluctuation of
the best-fit values of $\dot{M}_{\rm in}$, $R_{\rm J}$. For example, $R_{\rm
J}$ goes from $\sim$41 RG at $d$=2.7 kpc to $\sim$48 RG at $d$=3.3 kpc, i.e. a
range comparable with the estimated errors (about 2-5 RG). On the other hand,
$\dot{M}_{\rm in}$ oscillates between $\sim$1.58 and $\sim$1.70 $\dot{M}_{\rm
Edd}$ (for the lower and higher edge of the distance range respectively),
which is comparably higher than the range of variability associated to the
errors. These results point out that some caution must be taken when
considering the best-fit values obtained for $R_{\rm J}$ and $\dot{M}_{\rm
in}$ as face values.

##  4 Discussion

In the previous section we applied the physical model JED-SAD to fit the
XRT+NICER+NuSTAR+BAT observations of the source MAXI J1820+070 in hard and
hard/intermediate state. We included either one (Model 1) or two (Model 2)
reflection components. In the latest, more complex model, the two reflection
spectra originate from different but neighboring regions of the SAD disk. In
all the Epochs analyzed, with the exception of Epoch 1 (the only one in full
hard state), we found that only a 2-reflection solution gives acceptable fits.
In the following, we will discuss the main results obtained in this work and
their interpretation.
To fit our spectra we considered both a moderately ($R_{\rm ISCO}$=4 RG) and
rapidly ($R_{\rm ISCO}$=2 RG) spinning BH. We note that in the case of a
rapidly spinning BH, the evolution of the system is mainly driven by
variations in the sonic Mach number $m_{\rm S}$. This solution is different
with respect to the modelling of GX 339–4, in which $m_{\rm S}$ was fixed to
1.5 and the interplay between $R_{\rm J}$ and $\dot{M}_{\rm in}$ was
sufficient to describe the data (Marcel et al. 2019). We observed that this
trend is statistically significant, since imposing constant values of $m_{\rm
S}$ results in unphysical reflection-dominated scenario. A change in $m_{\rm
S}$ arises from a change in the torque due to the jet (through the toroidal
magnetic field). We see no clear reason for the jet torque to change on long
(¿ days) time scale, even if it is possible in principle. Although we discard
here this possibility, one should nevertheless keep it in mind and look for
traces of any correlation between these rapid accretion (increase in $m_{\rm
S}$) events and any jet signature.
On the contrary, in the fits performed keeping $R_{\rm ISCO}$ equal to 4 RG,
the $m_{\rm S}$ values are (almost) constant and the interplay between $R_{\rm
J}$ and $\dot{M}_{\rm in}$ plays again the main role. We therefore consider
the fits performed in the moderately spinning BH scenario more physically
reliable. This evidence is in line with the estimation of the spin value
obtained by e.g. Fabian et al. (2020) and Guan et al. (2020). Nevertheless,
even in the case $R_{\rm ISCO}$=4 RG, a (slight) increase in $m_{\rm S}$ is
observed going from phase 2 to phase 3, i.e. from 1.25 to 1.50. Imposing
$m_{\rm S}$ to 1.5 in Epoch 6, in continuity with Epoch 7, results in a
significantly worst fit where the reflection dominates over the direct JED
emission. While this is slightly different with respect to the results
obtained by Marcel et al. (2019, 2020), it is noteworthy that these authors
did not have data above 40 keV and, most importantly, they did not directly
perform spectral fits on the data. In future applications of the JED-SAD
model, the inclusion of high energy NuSTAR and especially BAT data might
therefore be necessary to highlight the role, if any, of $m_{\rm S}$ in
shaping the spectra of BHTs in hard state and, particularly, their high energy
cutoff.

###  4.1 The geometry of the accretion flow

The geometry of the accretion flow in the hard state of MAXI J1820 has been
inspiring a vivacious debate since its discovery. Several authors have
proposed that, when in hard state, the hot corona of MAXI J1820 contracts
while the accretion disk remains stable at the ISCO: this scenario was
proposed first by Kara et al. (2019) and recently supported by Wang et al.
(2021), based on NICER X-ray spectral and timing analysis. These results were
also corroborated by modeling the reflection component in NuSTAR spectra,
adopting two lamppost coronae models (Buisson et al. 2019; You et al. 2021).
However, Zdziarski et al. (2021) proposed that the truncated disk scenario can
still explain the hard and hard/intermediate state of MAXI J1820. According to
their spectral analysis performed with NuSTAR data on our Epochs 1,2,3 and 4,
the disk never extends down to the ISCO. A scenario where the disk is not only
truncated, but also the inner edge of the disk approaches the BH throughout
the hard state is also suggested by the spectral evolution of the quasi-
thermal component responsible for the thermal reverberation lags (De Marco et
al. 2021) and the evolution of the characteristic variability frequencies
observed in the iron line (Axelsson & Veledina 2021). It is noteworthy that a
trend for an approaching truncated inner disk is also qualitatively suggested
by Type-C QPOs (Buisson et al. 2019).
Regardless of the choice for $R_{\rm ISCO}$ and of the uncertainty in $K_{\rm
JEDSAD}$, all the fits performed by us show that the accretion disk in MAXI
J1820+070 is truncated in hard state and its inner edge moves inward during
the transition to the intermediate state. For $R_{\rm ISCO}$=4 RG, we found
$R_{\rm J}$ being about 50-60 RG in phase 1, 35-45 RG in phase 2 and 30-35 RG
in phase 3. A bit of caution is required in taking these measures as face
values, since the exact estimates of the truncation radius are instead
critically dependent on our choices for $R_{\rm ISCO}$ and $K_{\rm JEDSAD}$.
Our spectral analysis was performed on a data set which includes the data set
of Zdziarski et al. (2021) and reaches a similar conclusion on the geometry.
However, our work includes four more observations and provides a coherent
picture of the spectral evolution of the system, using a physical model. The
calculated transition radii $R_{\rm J}$ are consistent with the lower limits
on the truncation radius estimated by De Marco et al. (2021) by analysing the
quasi-thermal spectral component arising from the disk due to X-ray
irradiation from the corona (see, in particular, their fig. 9).
Finally, a multi-zone corona was invoked on the basis of both spectral
(Zdziarski et al. 2021) and temporal (Dziełak et al. 2021) behaviour of the
system in the same observations. We highlight that the JED model is also a
multi-zone model, as it takes into account the single contribution for each
”ring” of matter which composes the JED, each of them with variable
temperature and optical depth. Therefore, our approach is in line with other
works, where a single uniform plasma cloud was proven insufficient to
adequately describe the spectrum of Black Hole X-ray binaries (Nowak et al.
2011; Basak et al. 2017). Moreover a multi-zone model can naturally produce a
spectral shape which is not exactly a power law. In the energy range of the
iron line, this can have a direct impact on the broadness of the line and
could explain the absence of extreme red wings that would require an accretion
disk down to the ISCO.

###  4.2 On the nature of the second reflection component

In Kara et al. (2019) the line profile revealed by the NICER data was
evidently evolving across the hard and hard/intermediate state, with a
shrinking narrow core and a constant broad, blurred base. This has been
interpreted by invoking: (i) a disk reaching the ISCO in hard state, and (ii)
a contracting lamppost corona, which produces two reflection components, one
stable (the broad inner one) and one variable (the narrow outer one). In this
geometry, the top of this corona would mainly illuminate a distant region of
the disk, which produces the narrow line, contrarily to the bottom of the
corona, which would instead illuminate an inner region of the disk, where a
strongly blurred line arises. Using two reflection components was also
required in the spectral analyses by Buisson et al. 2019; Chakraborty et al.
2020; Zdziarski et al. 2021. In our modelling, the broad component in the iron
line is also stable and changes very slightly from Epoch 2 to 7 (see Fig. 8).
However, the disk does not reach the ISCO and the broadening of the line
profile is possibly due to the combination of two effects. Firstly, a
significant role is played by the underlying continuum. Indeed, previous
papers reporting a highly ionized disc reaching close to the ISCO in the hard
state consider a single Comptonization continuum, while a multi-zone
Comptonization, as considered in the JED-SAD model, is more physically
motivated. Furthermore, as a consequence of the high ionization parameter
estimated for the plasma in the inner disk ($\log{\xi_{1}}\sim 3.7-3.8$),
photons in the iron line tend to suffer multiple Compton scatterings, leading
to a further broadening of the line profile (Matt et al. 1996). In the JED-SAD
configuration the corona does not have a lamppost geometry and an explanation
analogous to Kara et al. (2019) can not hold. An alternative possibility could
be the inadequacy of our reflection models in dealing with real-life
reflection spectra. Indeed, we note that the reflection spectrum observed in
accretion disks results always from the continuum of reflection spectra
emitted at different radii. Using only one reflection component could just be
an oversimplified approximation in this case. Nevertheless, given the high
inclination of the system, some effects could also create an imbalance in the
contributions from the different parts of the disk, i.e. lowering the
contribution from the intermediate regions or increasing that from the outer
regions. We suggest two mechanisms potentially responsible for such imbalance,
sketched in Fig. 10.

Figure 10: Sketches representing two possible scenarios that might generate
the double reflection component used to model X-rays data of MAXI J1820+070 in
hard state. In particular, in panel a we show that a sudden jump in the scale
height of the disk might shadow part of the reflection coming from the
intermediate regions. In panel b, we show that the contribution from the on-
the-other-side-of-the-BH JED could illuminate particularly the outer regions
of the disk, leading to a boost in the outer reflection component.

One possibility could be that the disk, since viewed at such high inclination,
might create a ”shadow” zone where part of the reflection incoming between
$R_{\rm in,1}$ and $R_{\rm in,2}$ is obscured (see Fig. 10, panel a). As
highlighted by the Figure, a ”jump” in the scale height of the disk, located
sufficiently far away from its inner edge, could in principle generate such
self-shielding effect. A disk ”flared” in the outer region was also put
forward by Zdziarski et al. (2021) and Axelsson & Veledina (2021) in order to
explain the presence of the outer reflection component and the variability
associated to the iron line. However, it is not obvious whether a sudden
puffing up of the disk could happen or not in a classical Shakura-Sunyaev
disk, like the SAD model. A jump could arise from a transition in the
mechanism contributing mainly to the opacity in the disk. In the outer regions
of a Shakura-Sunyaev disk, the opacity is expected to be mainly due to free-
free absorption, while Thomson scattering dominates the opacity in the inner
regions of the disk (see equations (2.16) and (2.19) of Shakura & Sunyaev
1973). The geometrical thickness of the disk scales with the radius $R$
following a (slightly) different relation, i.e. it is proportional to
$R^{-21/20}$ in the inner, Thomson scattering dominated, regions and to
$R^{-9/8}$ in the outer regions. At the boundary between these two zones, i.e.
at a radius $R_{\rm jump}$, a jump is expected. We calculated $R_{\rm jump}$
according to our results for $\dot{M}_{\rm in}$ and assuming the same value
for $\alpha$ in both equations. We found $R_{\rm jump}$ between 3$\times$104
RG and 7$\times$104 RG, in all the epochs considered. In our fits, the
location of the outer reflector $R_{\rm in,2}$ was kept frozen to 300 RG. When
the parameter is thawed, it is completely unconstrained, as shown in Fig. 11).
Notwithstanding the uncertainty on $R_{\rm in,2}$, a boundary at beyond 104 RG
is likely located too far away to produce an extra-reflection component.
However, a crucial role might be played by irradiation from the inner regions
of the disk to the outer regions, an ingredient which is neglected in Shakura-
Sunyaev disks. Beyond some boundary radius, we expect that the upper layers of
the disk should be heated up by the impinging photons coming from the inner
regions, leading to evaporation and to a larger effective scale height.

Figure 11: Contour plots for the parameter $R_{\rm in,2}$ when left free and
not tied to $R_{\rm J}$.

Such effect would be efficient only where thermal winds are launched and
unfortunately this region is also located beyond 105-106 RG, i.e. where the
sound speed of the plasma overcomes the Keplerian speed (see, e.g.
Higginbottom et al. 2017). This argument would exclude that the optical winds
detected in MAXI J1820 in hard state (Muñoz-Darias et al. 2019; Sánchez-
Sierras & Muñoz-Darias 2020) could play the role of shadowing reflection from
intermediate regions. However, the presence of a large scale vertical magnetic
field make real-life accretion disks more layered and ”puffed up” (Zhu & Stone
2018; Jacquemin-Ide et al. 2021). It is therefore possible that the boundary
radius at which the disk is geometrically thickened by irradiation may be
located even closer and reasonably impacts the reflection spectrum observed
from the disk at high inclinations. In order to confirm this, irradiation must
be properly taken into account in models of magnetized disks around
BHs111111As done e.g. in Zhu et al. (2020) for Young Stellar Objects..
However, this is way beyond the scope of the present paper.

An alternative scenario is sketched in Figure 10, panel b. In principle, the
radiation from the JEDs on the two different sides of the BH with respect to
our line of sight could illuminate two differently located regions of the SAD,
originating the two reflection components. Indeed, photons coming from the
other-side-of-the-BH JED may not be able to impinge on the SAD before $R_{\rm
in,2}$, producing an excess in the reflection component coming from this
radius. The picture presented here is likely oversimplified, as it may require
a finely tuned light bending of the radiation coming from the other side of
the BH and it does not take into account any possible occultation from the
jet, expected to be optically thin.
In addition, it is unclear how the evolution, mainly in ionization, of the
outer reflection component fits within the proposed scenarios (see Fig. 8).
Such trend could be explained with an approaching of $R_{\rm in,2}$, which was
instead kept frozen to the assumed value of 300 $R_{\rm G}$. Indeed, as shown
in Fig. 11, $R_{\rm in,2}$ remains essentially unconstrained by the fit, so
that an approaching trend may be hidden here. If we consider the irradiation
from the inner disk as responsible of the puffing up of the outer disk and the
subsequent self-shielding effect, the fading in X-ray luminosity of the system
during Epochs 2 to 8 could be somehow correlated to the observed variability
in the outer reflection. However, this can not be confirmed without a proper
MHD modelling of magnetized disks that takes into account how irradiation
impacts the scale height of disks.

##  5 Conclusions

In this paper, we reported on the spectral study of the BH transient MAXI
J1820+070 in hard state exploiting the JED-SAD accretion-ejection paradigm.
This is the second object the model has been applied to and the first time
that Compton reflection is taken into account in detail. We investigated the
spectral behaviour of the system in 8 epochs, spanned over roughly 100 days,
during which the system was always in hard and hard/intermediate state. Due to
the uncertainty on the spin of the BH, we considered two possible values for
$R_{\rm ISCO}$, i.e. 4 RG (a∗=0.55) and 2 RG (a∗=0.95), respectively. In the
first case, only two parameters, $R_{\rm J}$ and $\dot{M}_{\rm in}$, drive the
largest changes in the spectral modelling: $R_{\rm J}$ decreases throughout
the period, while $\dot{M}_{\rm in}$ first increases, i.e. during the rise,
and then decreases, i.e. during the plateau and the decline (see De Marco et
al. 2021, for the definition of these phases). For a rapidly rotating BH,
$\dot{M}_{\rm in}$ remains quite constant, while a decrease in the sonic Mach
number $m_{\rm S}$ is observed and seems to drive the overall evolution of the
accretion flow. This scenario is unlikely and suggests that the BH in MAXI
J1820 is rather moderately spinning, as also found by Fabian et al. (2020) and
Guan et al. (2020). The geometry of the system consists in a truncated disk,
i.e. $R_{\rm J}$ never goes below 15 RG, with the inner radius decreasing
during the monitored period, in agreement with e.g. Zdziarski et al. (2021)
and De Marco et al. (2021). In order to successfully describe the spectra, two
reflection components have to be taken into account for all the epochs
considered (except for the first one): one component is highly ionized and
originating from the edge of the SAD, while the other is less ionized and
presumably originates from an outer region of the disk. While the inner
reflection component is stable, the outer reflection component evolves in a
way that is compatible with a scenario where the region responsible for such a
component is approaching the BH. A self-shielding effect, due to both the high
viewing angle and a flared outer disk, might offer a viable explanation for
such a phenomenon.
It is worth noticing that the exact $\dot{M}_{\rm in}$-$R_{\rm J}$ values
estimated here depend on the choice of $R_{\rm ISCO}$, the uncertainty on the
distance of the system and some simplifying assumptions on the dilution factor
$\omega$ and the jet power fraction $b$. However, we argued that these
assumption do not affect the main conclusions provided by this work, i.e. the
truncated disk geometry and the double reflection component, but only provide
additional uncertainty on the best-fit values presented here. Further
investigations are necessary to obtain more reliable and precise measurements
of the truncation radius and the inner mass-accretion rate. One possibility
would be to check if the obtained $R_{\rm J}$ and $\dot{M}_{\rm in}$ values
can be used to fit the radio observations of this system, as successfully done
by Marcel et al. (2019) for GX 339–4. Results discussing the jet behavior and
brightness using the JED-SAD model for MAXI J1820 will be presented in a
forthcoming publication.

###### Acknowledgements.

AM, MDS, AS, SM, ADA, TDS and TDR acknowledge a financial contribution from
the agreement ASI-INAF n.2017-14-H.0 and from the INAF mainstream grant (PI:
T. Belloni, A. De Rosa). AM and TDS acknowledge financial contribution from
the HERMES project financed by the Italian Space Agency (ASI) Agreement n.
2016/13 U.O. POP, SB and JF acknowledges financial support for the CNES space
national agency and the CNRS PNHE. We thank R. La Placa for fruitful
discussion.

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