# The High Energy X-ray Probe (HEX-P): Probing the physics of the X-ray corona
in active galactic nuclei

E. Kammoun 1,2,3,∗, A. M. Lohfink 4, M. Masterson 5, D. R. Wilkins 6, X. Zhao
7, M. Baloković 8,9, P. G. Boorman 10, R. M. T. Connors 11, P. Coppi 12, A. C.
Fabian 13, J. A. García 14,10, K. K. Madsen 14, N. Rodriguez Cavero 15, N.
Sridhar 10,16,17, D. Stern 18, J. Tomsick 19, T. Wevers 20, D. J. Walton 21,
S. Bianchi 3, J. Buchner 22, F. Civano 14, G. Lanzuisi 23, L. Mallick
24,25,10, G. Matt 3, A. Merloni 22, E. Nardini 2, J. M. Piotrowska 10, C.
Ricci 26,27, K.-W. Wong 28, A. Zoghbi 29,30,31, and the HEX-P Collaboration

###### Abstract

## 1

The hard X-ray emission in active galactic nuclei (AGN) and black hole X-ray
binaries is thought to be produced by a hot cloud of electrons referred to as
the corona. This emission, commonly described by a power law with a high-
energy cutoff, is suggestive of Comptonization by thermal electrons. While
several hypotheses have been proposed to explain the origin, geometry, and
composition of the corona, we still lack a clear understanding of this
fundamental component. NuSTAR has been playing a key role improving our
knowledge of X-ray coronæ thanks to its unprecedented sensitivity above 10
keV. However, these constraints are limited to bright, nearby sources. The
High Energy X-ray Probe (HEX-P) is a probe-class mission concept combining
high spatial resolution X-ray imaging and broad spectral coverage (0.2-80 keV)
with a sensitivity superior to current facilities. In this paper, we highlight
the major role that HEX-P will play in further advancing our insights of X-ray
coronæ notably in AGN. We demonstrate how HEX-P will measure key properties
and track the temporal evolution of coronæ in unobscured AGN. This will allow
us to determine their electron distribution and test the dominant emission
mechanisms. Furthermore, we show how HEX-P will accurately estimate the
coronal properties of obscured AGN in the local Universe, helping address
fundamental questions about AGN unification. In addition, HEX-P will
characterize coronæ in a large sample of luminous quasars at cosmological
redshifts for the first time and track the evolution of coronæ in transient
systems in real time. We also demonstrate how HEX-P will enable estimating the
coronal geometry using spectral-timing techniques. HEX-P will thus be
essential to understand the evolution and growth of black holes over a broad
range of mass, distance, and luminosity, and will help uncover the black
holes’ role in shaping the Universe.

##  2 Keywords:

accretion, X-ray astronomy, black holes, active galactic nuclei, quasars,
tidal disruption events

##  3 Introduction

Active galactic nuclei (AGN) are known to be prolific X-ray emitters. The hard
X-rays from AGN are thought to be generated through Compton scattering of
thermal UV-soft X-ray seed photons from the accretion disk by a hot electron
plasma located in the vicinity of the black hole, known as the corona (e.g.,
Vaiana and Rosner, 1978; Rybicki and Lightman, 1979; Haardt and Maraschi,
1991, 1993). The resulting X-ray continuum spectrum is usually approximated by
a power law of photon index $\Gamma$, with a high-energy cutoff
($E_{\mbox{\scriptsize cut}}$) at tens to hundreds of keV (e.g., Fabian et
al., 2017; Tortosa et al., 2018, 2023). Within the Comptonization framework,
$\Gamma$ and $E_{\mbox{\scriptsize cut}}$ can be mapped into the electron
temperature111The cutoff energy is commonly estimated to be $\sim 2-3$ times
the electron temperature (Petrucci et al., 2001a). ($kT_{\rm e}$) and the
coronal optical depth ($\tau$) plane (see, e.g., Titarchuk and Lyubarskij,
1995; Zdziarski et al., 1996; Beloborodov, 1999; Petrucci et al., 2001a, b;
Middei et al., 2019). The X-ray corona thus holds key information about the
physical processes occurring near the supermassive black hole (SMBH), offering
insights into the energetic phenomena associated with AGN. Understanding the
physics of the X-ray corona in AGN is crucial for unraveling the accretion
process, comprehending the origin of the emission from AGN, studying black
hole feedback and its impact on galaxy evolution, probing strong gravity
regimes, and investigating magnetic fields and particle acceleration.

###  3.1 The geometry of the X-ray corona

Constraining the geometry (e.g., size and location) of the X-ray corona offers
an opportunity to test models for the formation of the corona as each
mechanism predicts a distinct geometry. For example, the “two-phase” corona
model proposed by Haardt and Maraschi (1991) predicts an extended corona. In
contrast, within the framework of the failed jet model proposed by Ghisellini
et al. (2004), the corona is expected to be compact. To date, the most
promising constraints come from microlensing variability. The microlensing of
quasars by stellar components in the lensing galaxy results in a complex
magnification pattern (e.g., Wambsganss, 2006). The relative motion of the
lensed quasar, the galaxy and its stellar components, and the observer leads
to uncorrelated variability. The variability amplitude depends significantly
on the size of the emitting region, with a larger amplitude from a smaller
emitting region. Thus, the size of the source region can be estimated by
modelling the light curve of a lensed quasar (e.g., Chartas et al., 2002,
2009, 2016; Kochanek et al., 2007a, b; Mosquera et al., 2013). Reis and Miller
(2013) compiled measurements of the X-ray-emitting region in lensed quasars
and found the sizes range from $\sim$ one to tens of gravitational radii
($r_{\rm g}=GM_{\rm BH}/c^{2}$, where $G$ is the gravitational constant,
$M_{\rm BH}$ is the mass of the black hole, and $c$ the speed of light). We
highlight that current results are based on only a few strongly lensed
quasars.

X-ray spectral and spectral-timing properties of nearby bright Seyfert
galaxies, notably the detection of X-ray reverberation lags, led to the
similar conclusion that the X-ray source is compact and lies within a few
gravitational radii of the black hole (Fabian et al., 2009; De Marco et al.,
2013; Cackett et al., 2014; Emmanoulopoulos et al., 2014; Uttley et al., 2014;
Kara et al., 2016). Modelling the continuum optical-to-X-ray spectral energy
distribution (SED) of a few bright Seyfert galaxies also yields similar
estimates of the corona size (e.g., Petrucci et al., 2013; Done et al., 2013;
Porquet et al., 2019). This has been recently applied also to a bright quasar
at intermediate redshift (Kammoun et al., 2023). Further evidence of a small
physical size of the corona emerges from varying obscuration of the corona by
clouds in the broad line region (e.g., Risaliti et al., 2011; Sanfrutos et
al., 2013; Gallo et al., 2021). These results motivated the adoption of the
lamp-post geometry to describe the disk–corona system, where the X-ray corona
is assumed to be a point-like source located on the symmetry axis of the disk
(e.g., Matt et al., 1991; Martocchia and Matt, 1996; Miniutti and Fabian,
2004). Such a geometry could be physically realized by collisions and shocks
within an ejection flow or a failed jet (e.g., Henri and Pelletier, 1991;
Henri and Petrucci, 1997; Ghisellini et al., 2004), and has been assumed in
detailed models for the calculation of reflection spectra (e.g., Dauser et
al., 2013; García et al., 2014; Niedźwiecki et al., 2016; Ingram et al., 2019;
Mastroserio et al., 2021; Dovčiak et al., 2022). We note that recently several
works are considering a variety of more realistic 3D coronal geometries (e.g.,
Wilkins et al., 2016; Gonzalez et al., 2017; Zhang et al., 2019b).

Several works have adopted an approach similar to the one of (Soltan, 1982) to
constrain the radiative efficiency ($\eta$) of accreting SMBHs (e.g., Fabian
and Iwasawa, 1999; Yu and Tremaine, 2002; Marconi et al., 2004; Shankar et
al., 2004, 2009, 2020; Raimundo and Fabian, 2009; Raimundo et al., 2012;
Aversa et al., 2015; Zhang and Lu, 2017). This cosmology-independent approach
enables constraining the radiative efficiency by comparing the energy density
of quasar/AGN radiation with the local mean SMBH mass density. All these works
resulted in constraining $\eta$ to be larger than 0.1. More recently, Shankar
et al. (2020) favor $\eta\sim 0.12-0.2$ which is in line with spinning SMBH,
suggesting an average spin value larger than $\sim 0.5$. For these spin values
the half of the accretion disk emission is produced within the inner
$20\,r_{\rm g}$ (see e.g., Agol and Krolik, 2000; Dovčiak et al., 2022). The
inner edge of the accretion disk is thus required to extend close to the black
hole. Consequently, the bulk of the emission should be emitted from a compact
region, which is in agreement with the results obtained from X-ray
observations.

###  3.2 The heating and cooling mechanisms of the X-ray corona

While the heating mechanism of the corona is still uncertain, magnetic
reconnection has been suggested to play an important role (e.g., Galeev et
al., 1979; Di Matteo et al., 1997; Merloni and Fabian, 2001a, b; Sironi and
Beloborodov, 2020; Sridhar et al., 2021, 2023; Gupta et al., 2023). Within
this scenario, the magnetized coronæ could generate cyclo/synchrotron
radiation that can be observed in the radio/mm band (e.g., Laor and Behar,
2008; Inoue and Doi, 2014; Panessa et al., 2019). In fact, low levels of radio
emission has been almost ubiquitously detected in radio-quiet AGN. In many
cases, the emitting region is unresolved and associated with a compact, sub-
kpc nuclear region (see e.g., Smith et al., 2016; Panessa et al., 2019, and
references therein). Given the expected physical size of the X-ray emitting
region, the coronal mm-wave synchrotron emission is expected to be self-
absorbed, and it would therefore be more easily detectable in the mm than in
the radio. The size ($R$) of the self-absorbed synchrotron source decreases
with frequency following $R\propto\nu^{-7/4}$. Thus, the synchrotron emission
from an X-ray corona sized source would peak at $\sim 100\,\rm GHz$ (e.g.,
Raginski and Laor, 2016; Inoue and Doi, 2018). Several observational studies
hint towards a coronal origin of the mm-wave nuclear emission in non-jetted
AGN (e.g., Behar et al., 2015, 2018; Doi and Inoue, 2016; Inoue and Doi, 2018;
Kawamuro et al., 2022, 2023; Petrucci et al., 2023). In particular,
correlations between mm continuum emission and X-ray emission have been found
in various samples of AGN, with an average ratio between the $\sim
100-200\,\rm GHz$ and X-ray continuum of $\sim 10^{-5}-10^{-4}$ (Behar et al.,
2015, 2018; Kawamuro et al., 2022; Ricci et al., 2023). Interestingly, this
relation is consistent with what has been observed in coronally active stars
(Guedel and Benz, 1993), which are magnetically heated, similar to what is
expected for AGN coronæ, further supporting the proposed coronal origin for
the $100-200\,\rm GHz$ continuum emission, and in turn the idea that the
corona is magnetically heated.

A physically compact X-ray corona is expected to be radiatively compact,
meaning that the luminosity to radius ratio ($L/R_{\rm c}$) is large. The
compactness is usually described by the dimensionless parameter
$\ell=\frac{L}{R_{\rm c}}\frac{\sigma_{\rm T}}{m_{\rm e}c^{3}}$, where
$\sigma_{\rm T}$ is the Thompson scattering cross section and $m_{\rm e}$ is
the electron mass (Guilbert et al., 1983). For large values of $\ell$, the
number density of the high-energy photons can lead to electron-positron pair
production due to photon-photon collisions. In this case, feeding the corona
with more energy will lead to the production of more particles to share the
energy, which will limit the increase of the temperature of the corona and
prevent pair production from becoming a runaway process and exceed
annihilation. This can be assimilated to an $\ell$-dependent thermostat
(Svensson, 1984; Zdziarski and Lightman, 1985; Stern et al., 1995).
Observational constraints on the coronal parameters so far obtained are in
agreement with the hypothesis that pair production and annihilation may act as
an effective thermostat controlling the coronal temperature (Fabian et al.,
2015, 2017). It is common to assume that the coronal plasma is thermal.
However, several AGN show electron temperatures that are much lower than
expected for a purely thermal pair plasma (see e.g., Ursini et al., 2016;
Baloković et al., 2015; Reeves et al., 2021; Kara et al., 2017; Tortosa et
al., 2018; Bertola et al., 2022). These low temperatures could be indicative
of a hybrid plasma composed of a mixture of thermal and non-thermal particles
(Zdziarski et al., 1993; Ghisellini et al., 1993; Fabian et al., 2017). Only a
small fraction of non-thermal electrons with energy above 1 MeV would be
needed to result in runaway pair production. In this case, the cooled pairs
share the energy which reduces the temperature of the corona.

It is common to model the Comptonization spectrum using a phenomenological
power law with a high energy cutoff. Various physical models are also used and
all of them describe the current data equally well. These models differ in
their assumptions for the spectrum of the seed photons and the geometry of the
Comptonizing region. Various analytic Comptonization models such as CompTT
(Titarchuk, 1994) and CompPS (Poutanen and Svensson, 1996) are used, allowing
different simple geometries. Recently, more physically self-consistent
calculations such as EQPAIR (Coppi, 1999), BELM (Belmont et al., 2008), MoCa
(Tamborra et al., 2018), and MONK (Zhang et al., 2019b) include a variety of
geometries, hybrid coronæ, and polarization.

###  3.3 Observational constraints

The unprecedented sensitivity of NuSTAR (Harrison et al., 2013) above 10 keV
has been instrumental to study the hard X-ray spectra of AGN. NuSTAR enabled
high-energy cut-off measurements from single epoch observations for a
relatively large sample of local (Ballantyne et al., 2014; Matt et al., 2014;
Baloković et al., 2015, 2020; Ricci et al., 2017; Tortosa et al., 2018; Akylas
and Georgantopoulos, 2021; Kamraj et al., 2022; Tortosa et al., 2023) and
intermediate-redshift AGN for the first time (see e.g., Kammoun et al., 2017,
2023; Lanzuisi et al., 2019; Bertola et al., 2022; Marinucci et al., 2022b,
and references therein). It should be mentioned that previous studies (with
BeppoSAX, RXTE, INTEGRAL, etc.) allowed these measurements for a much smaller
sample of the brightest nearby AGN (e.g., Nicastro et al., 2000; Petrucci et
al., 2004; Malizia et al., 2014; Lubiński et al., 2016). NuSTAR observations
allowed us not only to constrain the coronal temperature by accurately
measuring values of $E_{\mbox{\scriptsize cut}}$, but also to probe its
variability (e.g., Ballantyne et al., 2014; Zoghbi et al., 2017) the origin of
which remains unclear.

The Swift/BAT AGN sample comprises all AGN detected by the all-sky $14-195$
keV BAT survey (Baumgartner et al., 2013; Oh et al., 2018). Ricci et al.
(2017) have estimated a median high-energy cutoff of local unobscured AGN
(i.e., $\log\left(N_{\rm H}/\rm cm^{-2}\right)<22$) of $E_{\rm cut}=210\pm
36\,\rm keV$. Ricci et al. (2018) subsequently found evidence for an inverse
correlation between $E_{\mbox{\scriptsize cut}}$ and Eddington ratio
($\lambda_{\rm Edd}$) where the median $E_{\mbox{\scriptsize cut}}$ increases
from $E_{\rm cut}=160\pm 41\,\rm keV$ to $E_{\rm cut}=370\pm 51\,\rm keV$
considering AGN with $\lambda_{\rm Edd}$ larger and smaller than 0.1,
respectively. This could be suggestive that the properties of the X-ray corona
evolve with the accretion rate. More recently, Kamraj et al. (2022) included
NuSTAR observations of this sample and estimated the mean electron temperature
to be $kT_{\rm e}=84\pm 9\,\rm keV$, which is consistent with the values
reported previously in the literature. However, the authors were not able to
recover the correlation between the electron temperature and any of the
accretion parameters. It is worth mentioning that differences in coronal
spectra may be expected between obscured and unobscured AGN, within the
framework of the orientation-based unified model of AGN (Antonucci, 1993; Urry
and Padovani, 1995), especially if the corona has a net velocity perpendicular
to the accretion disk (most likely related to jet formation; see e.g., Malzac
et al., 2001; Markoff et al., 2005). However, Baloković et al. (2020)
estimated a median $E_{\rm cut}=290\pm 20\,\rm keV$ for a sample of 130 local
obscured AGN, consistent within the current large uncertainties with the
values found by Ricci et al. (2017) for unobscured AGN.

More recently, the launch of the Imaging X-ray Polarimetry Explorer (IXPE;
Weisskopf et al., 2022) has enabled studies of X-ray polarization in AGN for
the first time (in the $2-8\,\rm keV$ range). X-ray polarization is a powerful
technique to explore the geometry of the X-ray emitting region (see e.g.,
Schnittman and Krolik, 2010; Beheshtipour et al., 2017; Tamborra et al., 2018;
Zhang et al., 2019b). So far, a total of four non-blazar AGN have been
observed (NGC 4151, MCG–05-23-16, the Circinus galaxy, and IC 4329A). Only
upper limits could be determined for MCG–05-23-16 (Marinucci et al., 2022b;
Tagliacozzo et al., 2023) and IC 4329A (Ingram et al., 2023), which are not
conclusive on the geometry of the X-ray source. The Circinus galaxy shows a
polarization of $28\pm 7$ per cent (at 68 per cent confidence level), with a
polarization angle of $18^{\circ}\pm 5^{\circ}$, roughly perpendicular to the
radio jet (Ursini et al., 2023). Given the large obscuration in this source,
this polarization is most likely due to reflection off cold material in the
torus (e.g., Ghisellini et al., 1994; Goosmann and Matt, 2011; Marin et al.,
2018; Ratheesh et al., 2021) and cannot be interpreted as the intrinsic
polarization of the X-ray source. As for NGC 4151, a polarization of $4.9\pm
1.1$ per cent with a polarization angle of $86^{\circ}\pm 7^{\circ}$ (at the
68 per cent confidence level) are reported by Gianolli et al. (2023). This is
consistent with a radially extended source of polarization that is
perpendicular to the radio jet, i.e., parallel to the accretion disk. Similar
results have also been observed in the black hole X-ray binary Cygnus X-1
(Krawczynski et al., 2022).

The High-Energy X-ray Probe (HEX-P; Madsen et al., 2023, in preparation) will
address many of the open questions related to the physics of the X-ray corona
in AGN. In this paper, we showcase how HEX-P will play a major role in
advancing our understanding of the physics of the X-ray coronæ in AGN. In
Section 4 we briefly present the properties of the mission. Characterisations
of the the X-ray coronæ in unobscured and obscured AGN with HEX-P are
discussed in Section 5 and Section 6, respectively. Section 7 shows how
studying the X-ray coronæ in high-redshift quasars will be possible with
HEX-P. Section 8 addresses the capabilities of HEX-P in studying the X-ray
coronæ of transient sources. Section 9 demonstrates how HEX-P will allow us to
better constrain the coronal geometry. Finally, we present a summary and a
brief discussion in Section 11.

##  4 Mission design

HEX-P is a probe-class mission concept that offers sensitive broad-band
coverage ($0.2-80\,\rm keV$) of the X-ray spectrum with exceptional spectral,
timing, and angular capabilities. It features two high-energy telescopes
(HETs) that focus hard X-rays, and one low-energy telescope (LET) that focuses
lower energy X-rays.

The LET consists of a segmented mirror assembly coated with Ir on
monocrystalline silicon that achieves a half power diameter of
$3.5^{\prime\prime}$, and a low-energy DEPFET detector, of the same type as
the Wide Field Imager (WFI; Meidinger et al., 2020) onboard Athena (Nandra et
al., 2013). It has $512\times 512$ pixels that cover a field of view of
$11.3^{\prime}\times 11.3^{\prime}$. It has an effective passband of $0.2-25$
keV, and a full frame readout time of 2 ms, which can be operated in a 128 and
64 channel window mode for higher count-rates to mitigate pile-up and faster
readout. Pile-up effects remain below an acceptable limit of $\sim 1\%$ for
fluxes up to $\sim 100$ mCrab in the smallest window configuration. Excising
the core of the PSF, a common practice in X-ray astronomy, will allow for
observations of brighter sources, with a typical loss of up to $\sim 60\%$ of
the total photon counts.

The HET consists of two co-aligned telescopes and detector modules. The optics
are made of Ni-electroformed full shell mirror substrates, leveraging the
heritage of XMM-Newton (Jansen et al., 2001), and coated with Pt/C and W/Si
multilayers for an effective passband of $2-80$ keV. The high-energy detectors
are of the same type as flown on NuSTAR (Harrison et al., 2013), and they
consist of 16 CZT sensors per focal plane, tiled $4\times 4$, for a total of
$128\times 128$ pixels spanning a slightly larger field of view than the LET,
$13.4^{\prime}\times 13.4^{\prime}$.

The broad X-ray band pass and superior sensitivity will provide a unique
opportunity to study the X-ray corona physics across a wide range of energies,
luminosity, and dynamical regimes.

###  4.1 Simulations

All the simulations presented here were produced with a set of response files
that represent the observatory performance based on current best estimates
(see Madsen et al., 2023, in preparation). The effective area is derived from
a ray-trace of the mirror design including obscuration by all known
structures. The detector responses are based on simulations performed by the
respective hardware groups, with an optical blocking filter for the LET and a
Be window and thermal insulation for the HET. The LET background was derived
from a GEANT4 simulation (Eraerds et al., 2021) of the WFI instrument, and the
HET background was derived from a GEANT4 simulation of the NuSTAR instrument;
both simulations assume HEX-P is orbiting at the first Earth-Sun Lagrangian
point (L1).

##  5 Physics of the corona in unobscured AGN

Unobscured AGN ($\log\left(N_{\rm H}/{\rm cm^{-2}}\right)<22$) with their
clean view of the central engine have provided most of the knowledge we have
today regarding AGN coronæ, as outlined in Section 3. The view of the whole
Comptonization spectrum, and of the X-ray reflection arising from the
interaction with the disk, is what makes X-ray observations of unobscured AGN
so valuable. The constraints provided on these sources are key as we move
towards refining our picture of AGN coronæ with HEX-P.

NuSTAR observations of unobscured AGN have provided many coronal temperature
measurements, which are consistent with the runaway pair production scenario
discussed above. Interestingly, in some AGN the coronal temperatures have been
found to be much lower than what would be expected for pair-regulated thermal
coronæ. One likely explanation for this is that the energy distribution of the
electrons in the corona is not entirely thermal but has a non-thermal
contribution. Such a hybrid corona would have a lower observed coronal
temperature as the non-thermal high energy electrons create pairs even when
the bulk of the electrons have lower temperatures.

The hybrid nature of the electron distribution can be detected directly from
the X-ray spectrum. Comptonization from thermal electrons results in a power-
law like X-ray continuum with a fairly abrupt cut-off at high energies.
Consequently, an addition of non-thermal electrons, following a power-law like
distribution beyond the thermal peak, leads to an increase in hard X-rays, as
there are now electrons with even higher energies. This additional hard X-ray
emission beyond the thermal cut-off is usually referred to as a “hard tail”.

Using these differences in the resulting Comptonization spectrum, i.e., the
X-ray continuum, some evidence has been found in black hole X-ray binaries
that the X-ray continuum is caused by hybrid Comptonization (e.g., Cadolle Bel
et al., 2006; Cangemi et al., 2021; Zdziarski et al., 2021). The presence of
hybrid Comptonization in AGN has been widely hypothesized but a direct
detection of the hard tail itself is beyond the capabilities of current and
proposed future instruments. HEX-P, however, will allow indirect confirmation
of the presence of hybrid Comptonization in AGN and determine the strength of
its contribution.

Figure 1: Left: Simulated LET (red) and HET (black) spectra of Ark 564,
assuming $\Gamma=2.48$, $kT_{\rm e}=13\,\rm keV$ and an exposure time of 90
ks. The figure also shows the background levels of NuSTAR (grey) and HEX-P-HET
(blue): the background of HEX-P is significantly lower than that of NuSTAR.
Right: The blue circles correspond to simulated observations of Ark 564 with
HEX-P at five different flux levels and five different coronal temperatures
from 15 to 10 keV (see text). For clarity, uncertainities in the the
compactness of Ark 564 are not plotted. The grey empty circles represent
typical estimates of $kT_{\rm e}$ from other AGN performed with NuSTAR.

There are two primary methods to indirectly detect hybrid Comptonization: one
relies on the reflection spectrum of the high energy continuum (García et al.,
2015, see also Section 11), and the other tests temperature regulation by
electron-positron pair production. The latter case can be made by studying the
evolution of a source with unusually low coronal temperature over a period of
time. If the source is variable, we can study the evolution of its compactness
and coronal temperature with time. If the coronal temperature is set by pair
production and the accretion geometry remains unchanged222This can be assessed
by studying the X-ray reflection spectrum. then as the source flux increases,
the coronal temperature will decrease. If, on the other hand, the coronal
temperature is not regulated by pairs then the temperature may not change or
change in a different fashion. If we see the expected trend of lower coronal
temperature with increased flux, and if the observed coronal temperatures are
much lower than expected for thermal Comptonization, that would be strongly
indicative of hybrid Comptonization. The findings can then also be compared to
theoretical predictions. In rare cases, this is already possible with today’s
instrumentation for some of the brightest X-ray binaries (see Buisson et al.,
2019, for an example), but has remained out of reach for AGN. For the
observations to yield meaningful constraints our measurements need to be more
accurate at low fluxes than is possible with current instruments. However,
HEX-P is able to obtain much tighter constraints on the coronal temperatures
due to its low background at high energies (Figure 1, left), enabling a high-
quality study of coronal temperature variations in AGN for the first time.

To demonstrate this, we explored the feasibility of a study with HEX-P that
targets the evolution of the coronal temperature in the bright, local narrow-
line Seyfert 1 galaxy Ark 564. Ark 564 is ideally suited for such a study as
it exhibits a low measured coronal temperature (Kara et al., 2017; Lewin et
al., 2022) as well as observed coronal temperature and flux variations (Barua
et al., 2020). Our HEX-P simulations of Ark 564 are based on the absorbed 2-10
keV fluxes observed with RXTE over a 4-year period ranging from 1999 to 2003
and our simulations are further assuming a scenario where the coronal size
remains unchanged with luminosity333Our final conclusions do not depend on
this assumption, as the coronal size can also be estimated from the spectral
modeling.. During this period the observed 2-10 keV flux varies between
$1.0-5.0\times 10^{-11}\rm\,erg\,s^{-1}\,cm^{-2}$ (Rivers et al., 2013). We
simulate spectra at five evenly spaced levels within the observed flux range
assuming a model that consists of a thermal Comptonization continuum and its
associated reflection spectrum, both of which are treated with the
relativistic reflection model relxilllpcP (Dauser et al., 2020). The second
lowest flux level of $2.0\times 10^{-11}\rm\,erg\,s^{-1}\,cm^{-2}$ corresponds
very well to the 2018 joint XMM-Newton+NuSTAR observations of the source,
which were studied in detail in Lewin et al. (2022). We therefore adopt the
spectral parameters from their detailed modeling for this flux level (Table 3
in their paper), in particular they have determined that $\Gamma=2.48$ and
$kT_{e}=14.7$ keV at this particular flux level. Then as we vary the flux
level, we not only adjust the normalization of our spectral model but also
evolve the photon index from 2.40 to 2.56 to mimic a softer-when-brighter
effect and the coronal temperature from 15.1 keV to 10.0 keV to have the
coronal evolve mildly with flux, the reflection parameters are kept unchanged.
To determine how much exposure time was needed at each flux level, we
simulated spectra with increasingly longer exposure times until we were able
to recover the coronal temperature with an accuracy of $\pm 2.5\,$keV or
better at 90% confidence. From the simulations, we found that a study of Ark
564 with HEX-P with a total exposure time of 445 ks targeting the source at
different flux levels will constrain the coronal temperature to $\leq 2.5$ keV
and could yield important data points in the compactness-temperature plane
(blue points, Figure 1, right). At higher flux levels it is easy for us to
exceed the accuracy requirement of 2.5 keV and the error bars are therefore
smaller. We note here that accuracy of the coronal temperature constraint for
a certain simulated exposure time is model-dependent, where more complex
models lead to less accurate constraints.

To place the findings from the Ark 564 study into the context of the wider
population of radio-quiet, unobscured AGN, ideally a few similar studies of
other sources would also be conducted. Suitable study targets display large
variability on timescales of weeks to months and possess a low coronal
temperature, several of such sources have already been identified. Together,
with the findings from one-time observation of unobscured AGN and high
redshift targets, this will highlight whether Ark 564 is special in any way or
whether its coronal variations can be considered representative for the
population as a whole.

The much lower background of HEX-P at higher energies and the therefore much
improved constraints on the spectral shape in this energy range with shorter
exposure times will likely not only lead to a better understanding of hybrid
Comptonization, but also to a more accurate spectral decomposition of
unobscured AGN. This aspect is studied in more detail in Piotrowska et al
2023.

##  6 Physics of the corona in obscured AGN

Figure 2: Simulated LET (red) and HET (black) spectra of an absorbed AGN at
$z=0.02$ (left) and $z=0.5$ (right), assuming $\log\left(N_{\rm H}/\rm
cm^{-2}\right)=22,23,24$ (top to bottom), $\Gamma=1.9$, $E_{\rm cut}=150\,\rm
keV$, and an exposure time of 100 ks. The grey dotted lines represent the
power law components, the grey dashed lines represent the neutral reflection
component, and the grey dash-dotted lines represent the thermal component.

Within the framework of the orientation-based unified model for AGN
(Antonucci, 1993), differences in coronal spectra between obscured
($\log\left(N_{\rm H}/{\rm cm^{-2}}\right)\geq 22$) and unobscured AGN could
be expected if the corona has significant velocity perpendicular to the
accretion disk, possibly related to jet formation (Beloborodov, 1999; Markoff
et al., 2005; Liu et al., 2014). Hard X-rays ($\geq 15\,\rm keV$) are
essential to study the intrinsic properties of obscured AGN, as the obscuring
material is optically thin to these photons. Constraints on coronæ in obscured
AGN remain extremely scarce in the literature despite the prevalence of
obscured AGN both in the local universe and, particularly, at high redshift
(50–75% of the population; Hickox and Alexander 2018). A few constraints based
on data from non-focusing hard X-ray instruments are available from, e.g.,
BeppoSAX (Dadina, 2007), Suzaku (Tazaki et al., 2011), INTEGRAL (de Rosa et
al., 2012), Swift/BAT (Ricci et al., 2018), or a combination thereof (Molina
et al., 2013). However, the cutoff of the power-law continuum at high
energies, $E_{\mbox{\scriptsize cut}}$, has been well constrained using NuSTAR
data for only about a dozen bright, obscured AGN (e.g., Baloković et al.,
2015; Fabian et al., 2017; Buisson et al., 2018; Tortosa et al., 2018; Ursini
et al., 2019), while larger samples representative of the obscured AGN
population provide ensemble-level constraints mostly based on short NuSTAR
observations with limited photon statistics (Baloković et al., 2020). In
addition to $E_{\mbox{\scriptsize cut}}$, we note that constraining the photon
index in obscured AGN is crucial to study the intrinsic AGN spectrum and
understand the accretion process.

Figure 3: $\Gamma-E_{\rm cut}$ contour plots ($3\sigma$ confidence level) for
$z=0.02$ and $0.5$ (blue and grey contours, respectively) assuming
$\log\left(N_{\rm H}/\rm cm^{-2}\right)=[22,23,24]$ (left to right), $E_{\rm
cut}=50,150,$ and $300\rm\,keV$ (top to bottom), and simulating a 100-ks HEX-P
observation. The black dotted contours correspond to the $3\sigma$ confidence
level constraints for a source at $z=0.02$ with 30-ks exposure by XMM-Newton
combined with a 100-ks exposure by NuSTAR (including both modules, FPMA and
FPMB). Horizontal and vertical lines indicate the input values of $\Gamma$ and
$E_{\mbox{\scriptsize cut}}$, respectively.

Thanks to its low background, and broad-band energy coverage, HEX-P will
provide better constraints on the properties of the X-ray corona in obscured
AGN. We performed simulations of various configurations of absorbed AGN to
test the capabilities of HEX-P in constraining coronal parameters, assuming
uniform exposure times of 100 ks (typical exposure for the planned HEX-P
surveys; see Civano et al., 2023). We used the borus02 model (Baloković et
al., 2018), which self-consistently computes the reprocessed emission from a
sphere with conical cutouts at both poles, approximating a torus with variable
covering factor. We considered a torus with a half-opening angle of
$45^{\circ}$ and an inclination of $60^{\circ}$. We also considered a photon
index of the incident power law of $\Gamma=[1.6,1.9,2.2]$, and a high-energy
cutoff $E_{\rm cut}=[50,150,300]\,\rm keV$. The incident luminosity was
estimated to match the average values observed by Ricci et al. (2017) for
$z=0.02$ and $z=0.5$, resulting in unabsorbed $2-10\,\rm keV$ luminosities in
the range $10^{42.9-45.5}\,\rm erg\,s^{-1}$. We considered three values for
the column densities $\log\left(N_{\rm H}/\rm cm^{-2}\right)=[22,23,24]$. For
simplicity, we assume a homogeneous torus with the line-of-sight column
density equal to the equatorial value. In addition, the soft X-ray spectrum in
obscured AGN is typically characterized by the emission from photoionized gas
in the narrow line region. For simplicity, we include a soft APEC component,
which is commonly used in the literature to model this emission in absorbed
AGN. For this component we assume a temperature of $0.9\,\rm keV$ and a
normalization that is equal to $10\%$ that of the power law, as typically seen
in local AGN (see e.g., Kammoun et al., 2020). The total model results
observed $2-10\,\rm keV$ fluxes in the range $2.5-8.5\times 10^{-12}\,\rm
erg\,s^{-1}\,cm^{-2}$. Figure 2 shows example HEX-P spectra for an
intermediate configuration for all the simulated $N_{\rm H}$ values, assuming
$\Gamma=1.9$, and $E_{\rm cut}=150\,\rm keV$, for $z=0.02$ and $z=0.5$ (left
and right panels, respectively). Figure 3 shows the $E_{\mbox{\scriptsize
cut}}$-vs-$\Gamma$ contours at the $3\sigma$ confidence level for all the
configurations considered. HEX-P accurately measures the photon index over a
broad range of parameters. The uncertainty on $\Gamma$ is larger for higher
column densities, since increasing amounts of absorption limits the access to
the power law portion of the continuum to increasingly hard X-rays. This
yields larger uncertainties on $E_{\mbox{\scriptsize cut}}$ as well. The
uncertainty on $E_{\mbox{\scriptsize cut}}$ also increases with increasing
$E_{\mbox{\scriptsize cut}}$; only lower limits could be estimated for $E_{\rm
cut}=300\,\rm keV$. We note that the observed flux decreases by a factor of
$\sim 3$ between $z=0.02$ and $z=0.5$, which lead to an increase in the
uncertainty on the parameters higher redshift. For comparison with current
facilities, we simulate a joint 30-ks XMM-Newton plus 100-ks NuSTAR
(accounting for both modules, FPMA and FPMB) observation, which are typical
exposures with these facilities. For all cases, we adopt $z=0.02$ as it
results in tighter constraints than the $z=0.5$ case. The $\Gamma-E_{\rm cut}$
constraints are shown as black dotted contours in Fig. 3, demonstrating the
improvements enabled by HEX-P. We note that, for the flux levels of these
sources and the exposure time, many of these sources will be easily detected
in the HEX-P surveys (see Civano et al., 2023).

We would like to note that our borus02 model is setup in coupled mode in which
the line-of-sight and global column densities scale together. The reflection
component will be higher at higher line-of-sight column densities, possibly
contributing to the increased uncertainties in the higher $N_{\rm H}$ regime
that we show in Figure 3. To definitively test this would require knowledge of
the dependence between global column density and a wide range of different
physical AGN parameters as well as a wide range of different model
prescriptions for the obscurer, which is outside the scope of the current
work. It is also worth mentioning that near and above the Compton-thick
regime, the measurement of the energy cut-off is degenerate with the assumed
obscurer model. Different obscurer geometries have different Compton-
scattering behaviour, altering the reprocessed spectrum up to 400 keV (e.g.,
Buchner et al., 2019, 2021). Thus the obscurer geometry and the energy cut-off
are degenerate. Merely assuming an obscurer geometry may bias the energy cut-
off measurement. Here, for simplicity, a fixed obscurer model (borus02) was
used for generating and modeling the data. The complex task of inferring the
true obscurer geometry and the true energy cut-off thus would have to be
achieved simultaneously, but this is beyond the scope of this paper. Boorman
et al. (in prep.) discusses HEX-P’s ability to infer torus geometries in
heavily obscured AGN in more detail.

##  7 Physics of the corona in intermediate-/high-redshift quasars

The high-energy cutoff in the X-ray spectrum (at tens to hundreds keV) can be
directly linked to the properties of the corona. Thanks to NuSTAR, whose hard
X-ray sensitivity is orders of magnitude better than previous missions, a
large sample of AGN were observed with unprecedented data quality at hard
X-rays. The majority of the observed sources are AGN with low luminosity
($L_{\rm 2-10\,keV}$ = 1042–1044 erg s-1; see e.g., Kamraj et al., 2018; Ricci
et al., 2018; Baloković et al., 2020), so they are in a region of the $kT_{\rm
e}-\ell$ plane where a broad range of $E_{\rm cut}$ is allowed within the
framework of the runaway pair production hypothesis. This can be seen in the
left panel of Figure 4 (we assume here $E_{\rm cut}$ = 2 $kT_{\rm e}$, $R_{\rm
c}\sim$10 $r_{\rm g}$, and coronal luminosity $L_{\rm c}$ = $L_{\rm
0.1-200\,keV}$ $\sim$4 $\times$ $L_{\rm 2-10\,keV}$). Furthermore, most of
these sources are in the nearby Universe ($z<$0.1, Figure 4, right).
Therefore, only lower limits to their cutoff energy were measured at 90%
confidence level due to the limited bandpass in the rest frame of local
sources ($<$80 keV), covered by NuSTAR.

Figure 4: High-energy cutoff of AGN as a function of 2–10 keV luminosity
(left) and redshift (right) by NuSTAR (light blue circles) and non-focusing
telescopes (gray squares), based on Bertola et al. (2022). The pair-production
forbidden regions assuming different coronal geometries are plotted in orange
(hemisphere) and yellow (slab). So far, $E_{\mbox{\scriptsize cut}}$ was only
measured in three quasars at $z>1$ (red stars). $E_{\mbox{\scriptsize cut}}$
is not constrained for most of the low-redshift and low-luminosity AGN.

On the contrary, high-redshift ($z\geq 1$) radio-quiet quasars (mostly with
luminosity $L_{\rm 2-10\,keV}>10^{45}\rm\,erg\,s^{-1}$) are ideal sources to
constrain high-energy cutoffs thanks to the limited allowed $E_{\rm cut}$
range on the $kT_{\rm e}-\ell$ plane (assuming $kT_{\rm e}$ is indeed limited
by runaway pair production; see Figure 4, left) and the cosmological
redshifting of $E_{\rm cut}$ to the NuSTAR observable band. Therefore, they
constitute a promising sample to constrain the AGN coronal properties at high
luminosity and test the runaway pair production theory. So far, only three
$z>$1 quasars were observed with NuSTAR and had their high-energy cutoffs
constrained (Lanzuisi et al., 2019; Bertola et al., 2022, and Figure 4,
right). All three quasars (with $L_{\rm 2-10\,keV}>3\,\times\,10^{45}\,\rm
erg\,s^{-1}$) were found below the critical line on the $kT_{\rm e}-\ell$
plane as per pair-production models expectations, though note the large
uncertainties for two of the sources which leave ambiguous results (Figure 4,
left). The well constrained cutoff energies of the three high-$z$ quasars
($\lambda_{\rm Edd}>0.1$) were 1.5–2.5 times lower than the 160 keV measured
by Ricci et al. (2018) for sources with high Eddington ratio. Therefore,
observing high-redshift quasars, where the coronal properties could be well
constrained, is crucial to study the possible evolution of the corona with the
accretion parameters such as the accretion rate and the SMBH mass.

Despite the recent progress in constraining the coronal properties of high-
redshift, high-luminosity quasars, the observations are extremely time-
consuming. A total of $\sim$375 ks NuSTAR and 150 ks XMM observations were
spent on the above three sources. The long exposures required in these
observations lead to a strong limitation in constraining AGN coronal
properties statistically.

Figure 5: Left, upper panel: 140 ks NuSTAR (blue) and 69 ks XMM-Newton (red)
spectra of 2MASS J1614346+470420 at $z=1.86$ (adapted from Lanzuisi et al.,
2019). Left, lower panel: 140 ks HEX-P simulation of the same quasar (HET in
black, LET in red). Right: contours of $E_{\mbox{\scriptsize cut}}$ and photon
index of the NuSTAR+XMM-Newton observations (red, green, and blue correspond
to 68%, 90%, and 99% confidence level, respectively) and HEX-P simulations
(dark yellow, yellow, and light yellow correspond to 68%, 90%, and 99%
confidence level, respectively).

HEX-P, with larger effective area, lower background, and broad band coverage,
is well suited to efficiently study the coronal properties of high-redshift,
high-luminosity quasars. We compare the 140 ks NuSTAR\+ 69 ks XMM-Newton
observation of 2MASS J1614346+470420 ($z=1.86$) from Lanzuisi et al. (2019) to
a simulated 140 ks HEX-P observation in Figure 5. Note that L1 orbit of HEX-P
implies that the HEX-P observation would take half the total observatory time
compared to the NuSTAR observation, since NuSTAR is in low-Earth orbit and
does not re-point during Earth occultations. The 90% confidence level
constraints on $E_{\mbox{\scriptsize cut}}$ are 106${}_{-40}^{+100}$ keV from
NuSTAR+XMM-Newton and 103${}_{-20}^{+30}$ keV from the simulated HEX-P
spectra, implying significantly improved constraints on $E_{\mbox{\scriptsize
cut}}$ by HEX-P in the same exposure time and half the clock time. The high-
energy cutoff and the photon index contours (at 68%, 90%, and 99% confidence
levels) derived from NuSTAR+XMM-Newton and simulated HEX-P spectra are plotted
in the right panel of Figure 5. The contours show that HEX-P can constrain
$E_{\mbox{\scriptsize cut}}$ at high confidence.

Figure 6: $E_{\rm cut}-\Gamma$ contour plots (68% and 90% confidence levels)
measured from the simulated spectra of the quasars with different redshifts
($0.65<z<4.5$), assuming an input 2–10 keV flux of 1 $\times$ 10-12 erg s-1
cm-2 and a HEX-P exposure of 50 ks. We simulated three set of high-energy
cutoff for each source, $E_{\rm cut}$ = 50, 100, and 200 keV (blue, green, and
yellow). The white crosses indicate the input values.The best-fit value of
each spectrum with its 90% confidence level uncertainty is indicated on the
upper right corner of each panel.

HEX-P will not only provide better constraints on AGN coronal properties but
it will also efficiently probe AGN coronæ in a large sample of high-redshift,
high-luminosity quasars. We demonstrate the constraining power of HEX-P for
coronal cutoff energy measurements of intermediate-/high-redshift, high-
luminosity quasars. We simulated nine HEX-P observations of quasars with 50 ks
exposures assuming an input 2–10 keV flux of $F_{\rm 2-10\,keV}$ = $10^{-12}$
erg s-1 cm-2. We note that cosmic X-ray background population synthesis models
predict $\sim$ 200 quasars at $z>$1 with such flux level (see e.g., Gilli et
al., 2007). The input redshifts range goes from $z$ = 0.65 to $z$ = 4.5, so
that their $2-10$ keV intrinsic luminosity range over $\log\left(L_{\rm
2-10\,keV}/\rm erg\,s^{-1}\right)=$ 45.25–47.25. We simulated source spectra
assuming a cutoff power law with a Galactic absorption of 1020 cm-2 and photon
index of $\Gamma$ = 1.90. We simulated three different high-energy cutoffs,
$E_{\rm cut}$ = 50, 100, and 200 keV, to explore the parameter space. Assuming
the same supermassive black hole mass, log($M_{\rm BH}$/$M_{\odot}$) = 9.2 and
a $2-10$ keV luminosity to bolometric luminosity conversion factor of 20
(e.g., Vasudevan and Fabian, 2007; Lusso et al., 2012), the Eddington ratios
of the sample range over $\lambda_{\rm Edd}$ = 0.2–20.

Figure 6 shows the $E_{\rm cut}-\Gamma$ contours measured from the simulated
spectra of the quasars with different redshifts and different high-energy
cutoffs. The best-fit high-energy cutoffs of each simulated spectrum with its
90% confidence level uncertainties are indicated on the upper right corner of
each panel. As expected, $E_{\rm cut}$ is better constrained at higher
redshift thanks to the cosmological redshifting, given the fact that we
conserve the flux ($F_{\rm 2-10\,keV}$ = $10^{-12}$ erg s-1 cm-2) of each
simulated source at different redshifts. Likewise, sources with $E_{\rm cut}$
= 50 keV are better constrained than higher $E_{\rm cut}$ sources at all
redshifts due to the band coverage of HEX-P. We found that it is difficult to
constrain the high-energy cutoff of intermediate redshifts ($z<$1.1) sources
with $E_{\rm cut}$ = 200 keV in a 50 ks exposure.

We tested the $E_{\rm cut}$ constrained by HEX-P assuming different photon
indices ($\Gamma$ = 1.60, 1.90, and 2.20) under a few redshifts ($z$ = 0.85,
1.40, 2.20, and 3.50) assuming the same flux as above. The results are
reported in Table 1. We found that harder photon index provides better
constraints on $E_{\rm cut}$ than softer photon index at all redshifts because
more high energy photons are obtained for harder photon indices and a given
$2-10\,\rm keV$ flux. The differences of the constraints on $E_{\rm cut}$
between softer and harder photon indices are much larger at lower redshifts
than at higher redshifts. Nevertheless, HEX-P could constrain the $E_{\rm
cut}$ in most circumstances even for the sources with quite soft photon index,
$\Gamma$ = 2.20.

Table 1: Constraints on the high-energy cutoff with different photon indices under different redshifts. $z$ |  $\Gamma$ = 1.6 |  $\Gamma$ = 1.9 |  $\Gamma$ = 2.2 |  $\Gamma$ = 1.6 |  $\Gamma$ = 1.9 |  $\Gamma$ = 2.2 |  $\Gamma$ = 1.6 |  $\Gamma$ = 1.9 |  $\Gamma$ = 2.2
---|---|---|---|---|---|---|---|---|---
|  | 50 keV |  |  | 100 keV |  |  | 200 keV |
0.85 |  51${}^{+12}_{-9}$ |  50${}^{+17}_{-10}$ |  51${}^{+42}_{-15}$ |  100${}^{+42}_{-24}$ |  100${}^{+78}_{-31}$ |  98${}^{+104}_{-34}$ |  205${}^{+237}_{-74}$ |  213${}^{+u}_{-96}$ |  213${}^{+u}_{-117}$
1.40 |  51${}^{+9}_{-7}$ |  52${}^{+14}_{-9}$ |  51${}^{+23}_{-12}$ |  102${}^{+30}_{-20}$ |  101${}^{+40}_{-20}$ |  96${}^{+70}_{-30}$ |  205${}^{+140}_{-60}$ |  196${}^{+240}_{-70}$ |  205${}^{+u}_{-90}$
2.20 |  50${}^{+6}_{-6}$ |  53${}^{+9}_{-7}$ |  50${}^{+11}_{-8}$ |  99${}^{+23}_{-16}$ |  100${}^{+30}_{-19}$ |  101${}^{+39}_{-23}$ |  197${}^{+80}_{-50}$ |  199${}^{+120}_{-60}$ |  190${}^{+175}_{-55}$
3.50 |  50${}^{+6}_{-5}$ |  50${}^{+6}_{-5}$ |  51${}^{+7}_{-6}$ |  99${}^{+20}_{-15}$ |  101${}^{+20}_{-15}$ |  101${}^{+23}_{-17}$ |  197${}^{+80}_{-45}$ |  197${}^{+80}_{-45}$ |  197${}^{+80}_{-45}$

eROSITA (Predehl et al., 2021) observations will provide an all-sky survey of
X-ray sources, which will be the deepest survey in the soft X-rays ever
performed. With the ongoing SDSS and DESI surveys, and future Euclid, 4MOST,
Subaru Prime Focus Spectrograph, and SPHEREx spectroscopic surveys, a large
number of high-redshift, high-luminosity, radio-quiet AGN are expected to be
discovered prior to the launch of HEX-P. The combination of these large
surveys should therefore provide a large sample of high-redshift quasars for
HEX-P coronal physics studies in the next decade.

##  8 Physics of the corona in transient sources: Changing-look/changing-state
AGN and Tidal Disruption Events

The growing number of time-domain surveys across the electromagnetic spectrum
are revealing extreme variability in the accretion process onto supermassive
black holes. These objects break from the standard stochastic variability seen
in steadily accreting AGN, and thus provide a unique window into the
formation, stability, and heating of the corona.

Changing-look AGN (CLAGN; also often referred to as changing-state AGN) show
rapid changes between optical spectral types through either the appearance or
disappearance of broad emission lines on timescales of months to years (for a
recent review, see Ricci and Trakhtenbrot, 2022). These sources challenge the
classic unified model for AGN, whereby the appearance of broad emission lines
is solely a function of viewing angle. Instead, the change between optical
spectral types is often coupled with an associated increase or decrease in the
observed optical and X-ray flux, suggesting that the accretion rate is another
key factor in AGN unification (e.g., Elitzur et al., 2014). The driving
mechanism behind these changing-look events is still under debate, with
theories including state transitions reminiscent of the behavior in black hole
binaries (Noda and Done, 2018; Ruan et al., 2019), radiation pressure disk
instabilities (Sniegowska et al., 2020), propagating cooling fronts linked to
changes in the magnetic torques in the innermost regions of the accretion flow
(Ross et al., 2018; Stern et al., 2018), and transient events like tidal
disruption events (TDEs; e.g., Merloni et al., 2015; Ricci et al., 2020).

Although CLAGN are still a relatively rare phenomenon with only about 100 such
systems known to date, they provide unique insights into the dynamic behavior
of the inner accretion flow and unveil physics that is impossible to probe in
standard steady-state AGN. One extreme example is evidenced by the recent
discovery of the disappearance and recreation of the X-ray corona in 1ES
1927+654. In late 2017, the source underwent one of the fastest changing-look
events to date, whereby broad emission lines were caught forming on timescales
of months (Trakhtenbrot et al., 2019). Shortly after the start of the
changing-look event, X-ray observations of 1ES 1927+654 revealed negligible
hard X-ray emission, indicating that the X-ray corona had vanished in this
source (Ricci et al., 2020, 2021). The corona began to reappear as the source
brightened in the X-ray, although it remained extremely soft compared to
standard AGN ($\Gamma\approx 3-3.5$; Ricci et al., 2021; Masterson et al.,
2022). Two competing theories have been suggested to explain the behavior in
this kind of system: a TDE occurring in an AGN (Ricci et al., 2020), or an
inversion in the magnetic flux polarity (Scepi et al., 2021; Laha et al.,
2022). Both theories can effectively cut off the energy supply to the corona,
leading to its destruction.

Figure 7: Top: X-ray light curve of 1ES 1927+654 from Masterson et al. (2022),
including 7 simultaneous XMM-Newton/NuSTAR observations and $\sim$ 500 NICER
observations. The pre-outburst X-ray luminosity is shown as an orange dashed
line, and the Eddington luminosity for a $10^{6}\,M_{\odot}$ black hole is
shown as a green dot-dashed line. The three evolutionary phases are indicated
with vertical black dotted lines. The three XMM-Newton/NuSTAR epochs simulated
with HEX-P are shown with grey boxes. Bottom: Fractional uncertainty on
$\Gamma$, $E_{\mathrm{cut}}$, and $kT_{\mathrm{bb}}$ for HEX-P simulations of
the changing-look AGN 1ES 1927+654 in three different evolutionary stages
during its 2018–2021 outburst. The left-most panel shows simulations during
the early super-soft phase (June 2018/Epoch 1), the middle panel shows
simulations during the peak X-ray luminosity phase (Nov. 2019/Epoch 4), and
the right panel shows simulations during the return to the post-outburst phase
(Jan. 2021/Epoch 7). For each evolutionary phase, we simulated 25 spectra for
6 different HEX-P exposure times based on the best-fit XMM-Newton/NuSTAR model
from Masterson et al. (2022). The blue dashed lines show the joint XMM-
Newton/NuSTAR constraints on $\Gamma$ and $E_{\mathrm{cut}}$ (or
$kT_{\mathrm{bb}}$ for the early super-soft state, in which the cutoff energy
cannot be constrained), which can be reached with HEX-P with much shorter
exposure times (10–25 ks). The spectral models and key coronal parameters used
to simulate the HEX-P spectra are given in Table 2.

As a unique opportunity to witness, for the first time, the formation of the
corona in an AGN, there have been extensive X-ray resources used to study this
enigmatic outburst of 1ES 1927+654. To date, NICER has observed the target for
more than 1 Ms, and there have been 8 simultaneous XMM-Newton/NuSTAR
monitoring periods, totalling more than 400 ks with each telescope. NuSTAR
observations have been crucial for disentangling the soft photon index and low
cutoff energy as the corona formed, which is not possible with either XMM-
Newton or NICER alone due to their limited bandpasses. However, both XMM-
Newton and NICER have been critical to unlocking information about the inner
accretion flow with their soft X-ray spectra, including providing crucial
information about rapid variability of the soft X-ray flux and the broad 1 keV
line, that was recently linked to reflection from the inner disk (Masterson et
al., 2022). To fully understand this system therefore requires a broad
bandpass, which is provided by a single observatory in HEX-P.

To assess the contributions HEX-P can make to our understanding of the corona
in CLAGN, we simulated the three different phases in the outburst of 1ES
1927+654 as would be seen with the LET and HET on HEX-P. For each phase, we
simulated HEX-P spectra using the best-fitting XMM-Newton/NuSTAR models from
Masterson et al. (2022) with six exposure times ranging from 10 to 150 ks. The
model used for each phase, along with the key coronal parameters used in the
simulations, are given in Table 2. Figure 7 shows the resulting constraints on
key coronal parameters $\Gamma$ and $E_{\mathrm{cut}}$ for each evolutionary
phase, as well as constraints on $kT_{\mathrm{bb}}$ of the dominant thermal
component in the early super-soft phase (in which constraints on
$E_{\mathrm{cut}}$ were not possible given the super-soft nature of the
source). We compare these constraints to the ones achieved by 50-ks joint XMM-
Newton/NuSTAR observations, which vary with the changing spectral shape and
flux of the source. We find that HEX-P can reach similar constraints as from
joint XMM-Newton/NuSTAR observations in a fraction of the exposure time. These
simulations show that HEX-P could probe the evolution of the cutoff energy and
photon index on roughly 10-25 ks timescales, meaning that either the source
could be visited relatively frequently to track the evolution over weeks-
months timescales, or that a single long stare could be broken into many 10-25
ks segments in which the evolution of $\Gamma$ and $E_{\mathrm{cut}}$ could be
studied. The X-ray flux of 1ES 1927+654 varied by an order of magnitude in
timescales as short as 8 hours while the corona was in the process of forming
(Ricci et al., 2020), which would allow HEX-P to provide unprecedented
constraints on the evolution of the temperature and compactness of a newly
forming corona for the first time. Since the discovery of 1ES 1927+654, other
sources with similarly dramatic X-ray variability have been discovered (e.g.,
a repeating TDE discovered by eROSITA, which shows a repetitive creation and
collapse of the X-ray corona; Liu et al. 2023).

Table 2: Spectral models and parameters used in simulations of 1ES 1927+654
with HEX-P

Phase | XSPEC Model | $\Gamma$ |  $E_{\mathrm{cut}}$ (keV) |  $kT_{\mathrm{bb}}$ (eV)
---|---|---|---|---
1 |  tbabs $\times$ ztbabs $\times$ (zbbody + zpower + gsmooth(xillverTDE)) | 3.4 |  –† | 88
2 |  tbabs $\times$ ztbabs $\times$ (zbbody + zcutoffpl + gsmooth(xillverTDE)) | 3.6 | 8.1 | 205
3 |  tbabs $\times$ ztbabs $\times$ (zcutoffpl + relxillCp) | 2.3 | 23.4 | –

†In this epoch, the source was too soft for proper determination of the cutoff
energy, and hence, we used a simple power law to model the spectrum. Note that
in the first joint XMM-Newton/NuSTAR observation of 1ES 1927+654, the source
was undetected above 3 keV with XMM-Newton and was not detected with NuSTAR.

Moreover, non-jetted TDEs present another opportunity to witness the formation
of the corona around SMBHs, and HEX-P will play a major role in improving our
understanding of various aspects of TDEs. Although detailed simulations go
beyond the scope of the current manuscript, we highlight some of these science
cases below. The majority of X-ray emitting, non-jetted TDEs show dominant
thermal emission, presumably from the accretion disk, but a handful of sources
have shown evidence for the formation of a corona in late-time ($\sim$ few
years after ignition) X-ray observations (e.g., Wevers et al., 2021; Yao et
al., 2022). As TDEs are believed to transition from super- to sub-Eddington
when the mass fallback rate drops as stellar debris is accreted, they are
another powerful probe of how the corona evolves with changes to the accretion
state. A correlation between the Eddington ratio and X-ray spectral state has
been already observed in a (small) sample study (Wevers, 2020), with a slope
that is remarkably similar to that seen in AGN samples (see e.g., Lusso and
Risaliti, 2016). HEX-P studies will enable constraints on the formation and
evolution of the corona while simultaneously constraining the high-energy
cutoff energy over a wide range of accretion rates in individual SMBHs. This
direct probe the disk-corona connection during different accretion states will
allow for a comparison to state transitions in black hole binaries (Remillard
and McClintock, 2006) and serve as a test of the scale-invariant nature of
accretion.

X-ray observations of TDEs can also provide independent estimates of SMBH
masses and spins, which are otherwise difficult to probe in dormant SMBHs.
Detailed X-ray spectral modelling of the thermal continuum in TDEs can provide
constraints on the mass and the spin of the central black hole (e.g., Wen et
al., 2020). With its wide energy coverage, HEX-P will be able to advance this
technique by mitigating the inherent uncertainties and bias introduced by the
presence of an X-ray corona. Furthermore, X-ray variability is also a powerful
probe of SMBH properties. In particular, the SMBH spin can be estimated
through the detection of X-ray quasi-periodic oscillations and/or disk
precession (e.g., Pasham et al., 2019), and the SMBH mass can be measured with
the break frequency of the power spectral distribution (PSD) (McHardy et al.,
2006). In TDEs, the PSD behaviour has been shown to change with Eddington
ratio (Saxton et al., 2012; Wevers et al., 2021), and hence can provide an
independent estimate of the SMBH mass in TDEs. Additionally, if TDEs form a
corona at late times when the accretion rate has dropped and the disk is
geometrically thin, then the Fe K$\alpha$ line can be used to constrain the
corona geometry and SMBH spin (e.g., Yao et al., 2022). Thanks to its broad
energy coverage, high throughput, low background, and L1 orbit permitting long
observations, HEX-P will be able to improve our understanding of TDEs using
all of these techniques. This will provide several independent mass and spin
measurements of quiescent black holes, which would add invaluable information
to constrain the demographics of quiescent low mass SMBHs ($<10^{7}M_{\odot}$)
that is not readily available through other means.

##  9 Coronal geometry

One of the key advances in recent years has been the ability to measure the
location, geometry and structure of the corona using X-ray reflection and
reverberation from the inner regions of the accretion disk. Determining the
structure of the corona in this way places important constraints on the
mechanism by which the corona is formed and energized by the accretion flow,
for example distinguishing between a corona that is formed in a jet or failed
jet, from a corona that is formed over the surface of the inner accretion
disk.

In addition to understanding the physics of the corona itself, measurements of
the location and geometry of the corona are important to validate assumptions
that underpin the use of X-ray reflection spectroscopy to measure fundamental
properties of the black hole, such as its spin (Fabian et al., 1989; Dauser et
al., 2013; Risaliti et al., 2013; Mallick et al., 2022). One of the
predominant techniques employed to measure black hole spin is based upon
identifying the innermost radius of the accretion disk (assumed to coincide
with the innermost stable circular orbit, ISCO) from the extremal redshift
detected in emission lines (namely the iron K line) within the reflection
spectrum (see Brenneman and Reynolds, 2006; Reynolds, 2021, and Piotrowska et
al., 2023). The intensity of the reflection and the line emission profile,
however, depends upon the geometry of the corona, which provides the primary
source of illumination. Typically, either a point source (or ‘lamppost’) is
assumed, or a phenomenological power law is used to model the emissivity
profile of the disk (i.e., the intensity of the reflected flux as a function
of radius). If the model corona or power law over-predicts reflection from the
inner disk, the corona is at a greater height or is more extended than assumed
in the model, and it is possible that the spin of the black hole is
underestimated (Fabian et al., 2014). First-hand measurements of the location
and geometry of the corona reduce this systematic uncertainty and the
degeneracy that arises between extended coronæ and low spins.

HEX-P will enable measurements of the location, geometry and structure of the
corona via both broadband X-ray spectroscopy and timing. The former is based
upon direct measurement of the emissivity profile of the disk from the
reflection observed in time-averaged X-ray spectra. The latter technique
utilizes the light travel time between the corona and disk as variations in
the luminosity of the primary X-ray emission reverberate off of the accretion
disk. In addition, further constrains on the geometry of the X-ray corona are
available from X-ray polarisation measurements, which can distinguish between
X-ray emission from a compact corona, a corona associated with a jet, or an
extended slab-like corona extending over the inner accretion disk (see e.g.,
Marinucci et al., 2022a; Gianolli et al., 2023; Tagliacozzo et al., 2023;
Ursini et al., 2023).

###  9.1 Measurements of the corona via X-ray spectroscopy

The emissivity profile of the disk is encoded in the profile of the
relativistically broadened emission lines. The emission line we observe is the
line emission integrated over the entire disk, and is comprised of photons
experiencing different Doppler shifts and gravitational redshifts, which vary
as a function of position on the disk. This means that it is possible to fit
the observed line profile as the sum of contributions from different radii on
the disk, and the relative contribution of the line model from each radius
provides a measurement of the emissivity profile (Wilkins and Fabian, 2011).
The measured profile can then be compared to theoretical predictions for the
illumination of the disk by coronæ with different geometries and at different
locations (Wilkins and Fabian, 2012; Dauser et al., 2013). Alternatively, a
geometry can be assumed for the corona (usually a point source, or lamppost),
and a model can be fit directly to full reflection spectrum to make a
measurement of the coronal parameters; namely the height of the corona for the
lamppost (Dauser et al., 2016). The emissivity profile of the disk is
sensitive to the coronal height in the case of a lamppost geometry, or to the
radial extent of the corona if it were extended over the surface of the disk
(Wilkins and Fabian, 2012).

A further measurement of the corona comes from the ratio of the reflected flux
to the flux of the continuum that is observed directly from the corona,
referred to as the reflection fraction (Wilkins and Gallo, 2015; Dauser et
al., 2016). In the case of an isotropically emitting point source above an
infinite accretion disk in a flat spacetime, we expect a reflection fraction
$R_{\rm f}=1$, since exactly half of the emission from the corona is emitted
downwards to illuminate the disk, and half is emitted upwards to escape to be
observed as the continuum. Light bending around the black hole causes a
greater fraction of the rays to be focused towards the inner accretion disk,
enhancing the reflection fraction relative to the continuum when the corona is
confined to a more compact region of space, closer to the black hole,
providing another probe of the compactness of the corona (Fabian et al., 2012;
Parker et al., 2014; Walton et al., 2021). On the other hand, the reflection
fraction can be reduced below unity if it is outflowing at a mildly
relativistic velocity (for example if the corona is part of a jet or failed
jet). In this case, the coronal emission is relativistically beamed away from
the accretion disk, enhancing the fraction of rays that are able to escape to
be observed as part of the continuum (Beloborodov, 1999; Gonzalez et al.,
2017). The broad bandpass of HEX-P is vital to obtain an accurate measurement
of the total reflection fraction. In addition, by combining measurements of
the reflection fraction with measurements of the relativistically broadened
iron K line and the associated reverberation time lags, HEX-P will be able to
obtain measurements of the location, the geometry, and the motion of the X-ray
emitting coronæ around black holes.

###  9.2 X-ray reverberation from the inner accretion disk

The measurement of X-ray reverberation time lags adds a further dimension to
the picture of the corona. The reflection and line emission from the accretion
disk responds to short-timescale changes in luminosity of the primary X-ray
emission from the corona. There is, however, a time delay between variations
in the continuum and the correlated variations in the reflection due to the
additional light travel time between the corona and disk (Fabian et al., 2009;
Uttley et al., 2014). In the sample of AGN in which reverberation has been
detected, the measured time delays are short and correspond to the light
crossing time across distances of between approximately one and ten
gravitational radii (this is approximately equal to the radial co-ordinate of
the event horizon of a maximally spinning black hole). Such short time delays
indicate that the corona must be compact and confined to a small region of
space close to the black hole and innermost accretion flow (e.g., De Marco et
al., 2013; Kara et al., 2016; Mallick et al., 2021). The reverberation time
scale is primarily sensitive to the scale height of the corona above the disk
(Wilkins and Fabian, 2013; Cackett et al., 2014). In recent years, there have
also been advances in simultaneously modelling the X-ray spectrum and time lag
measurements under the assumption of a point-like corona, using models such as
reltrans (Ingram et al., 2019). With simultaneous fitting, it is possible to
obtain simultaneous constraints on both the height of the corona and the mass
of the black hole (e.g., Mastroserio et al., 2020).

HEX-P measurements of X-ray reflection and reverberation were simulated for
nearby AGN, typical of those in which reverberation from the inner disk has
been detected (e.g., Kara et al., 2015, 2016; Mallick et al., 2021). These are
predominantly narrow line Seyfert 1 (NLS1) AGN (Gallo, 2006), which show
strong, relativistically broadened iron K lines from the inner accretion disk
and a high degree of short-timescale variability in their X-ray light curves
(see e.g., Gallo, 2018, for a review). The observations were simulated from a
full spectral-timing reverberation model, which begins with general
relativistic ray tracing calculations between a point-like corona, the
accretion disk, and the observer, using the CUDAKerr code of Wilkins and
Fabian (2012) and Wilkins et al. (2016). These ray tracing calculations self-
consistently predict the illumination of the disk by the corona, and the
energy shifts and time delays of the observed reflection and reverberation in
the form of the impulse response function (Reynolds and Begelman, 1997;
Cackett et al., 2014; Uttley et al., 2014). The response function is then
convolved with the xillver model for the reflection spectrum produced in the
rest frame of the material in the disk (García et al., 2013) to predict the
full spectral-timing response. The simulations generate the observed X-ray
spectrum (including the instrumental background), and the light curves that
would be observed in different energy bands, folded through the response
functions of the HEX-P telescopes and detectors, allowing measurements of the
spectrum and time lags to be conducted as they would be for real observations.

Figure 8: Left: Simulated 500 ks HEX-P observation of a typical nearby AGN,
showing X-ray reflection and reverberation from the inner accretion disk. The
top panel shows the simulated X-ray spectra obtained with the LET and HET. The
relativistically broadened iron K line (around 6.4 keV) and Compton hump
(around 25 kev) that are characteristic of reflection from the inner disk are
clearly visible. The bottom panel shows the measured time lag vs. energy
spectrum, which shows the relative time at which different energy bands
respond as variations in the continuum emission reverberate off the disk. The
energy bands dominated by reflection from the disk (the soft X-ray excess,
below 1 keV, the broad iron line, and the Compton hump) lag behind the bands
dominated by the primary continuum observed directly from the corona. This
time delay arises from the additional light travel time between the corona and
accretion disk, and is a sensitive probe of the location and geometry of the
corona. Right: Simulated iron K reverberation lag measurements as a function
of black hole mass for a simulated sample of AGN based upon the BASS sample.
For each simulated AGN, the mass, luminosity, and redshift are representative
of the real sample, and the remaining parameters of the black hole, disk, and
corona are drawn from random distributions. Each of the simulated AGN is
observed for 300 ks with HEX-P. Error bars on the black hole mass values
assume mass measurements can be made via optical reverberation, and correspond
to a representative uncertainty of 0.15 dex.

The left panel of Fig. 8 shows the simulated broadband X-ray spectrum and time
lag spectrum for a typical Seyfert-like AGN displaying X-ray reverberation for
a deep 500 ks observation with HEX-P. Reverberation is simulated around a
maximally spinning ($a=0.998$) black hole with a mass of $10^{7}\,M_{\odot}$
at a redshift of $z=0.025$. The observed flux is equivalent to accretion at
0.3 times the Eddington limit, and 5 per cent of the total bolometric
luminosity is produced by the corona. Parameters of the corona and accretion
disk are selected to be representative of the typical reflection spectra
observed in nearby AGN (e.g., Mallick et al., 2018; Jiang et al., 2018). The
X-ray spectrum shows the characteristic features of X-ray reflection from the
inner disk: a) the iron K line around 6.4 keV, with an extended redshifted
wing comprised of the line photons emitted from the innermost radii of the
accretion disk; b) the Compton hump around 20 keV; and c) a soft excess formed
as soft X-ray lines are blended together by relativistic broadening. The
ability to simultaneously measure the soft X-ray band, the broad iron K line,
and the Compton hump will allow HEX-P observations to unambiguously separate
reflection from the accretion disk from other spectral components (e.g.,
Walton et al., 2014).

The characteristic form of the reflection spectrum is also evident in the time
lag spectrum. The energy bands dominated by reprocessed emission from the
disk, namely the soft excess, the iron K line, and the Compton hump, are seen
to respond to variations at later times than the energy bands dominated by
continuum emission observed directly from the corona (1.5–3 keV and 8–10 keV).
The broad bandpass covered by the combination of the LET and HET on HEX-P,
combined with the available effective area, mean that HEX-P is sensitive to
reverberation time lags across the full range of the reflection spectrum. It
will be possible to measure the differential time lag between the redshifted
wing of the iron K line, which arises from the inner disk, closer to the black
hole and corona, and the 6.4 keV core of the iron K line from larger radii.
The difference in time lag between the inner and outer disk is a sensitive
probe of the structure of the corona (Wilkins et al., 2016). HEX-P will be
able to routinely measure the reverberation time delays associated with the
Compton hump, only hints of which have so far appeared in NuSTAR observations
(Kara et al., 2015; Zoghbi et al., 2017).

###  9.3 Simulation of HEX-P reverberation measurements

To demonstrate the sensitivity of HEX-P to X-ray reverberation and its
effectiveness in measuring the structure of the corona across the supermassive
black hole mass spectrum, iron K reverberation measurements were simulated for
a sample representative of the 25 brightest nearby AGN in the BASS catalog,
each observed for 300 ks with HEX-P. For each of these simulated AGN, the
mass, bolometric luminosity, redshift and absorption were drawn from the BASS
catalog (Koss et al., 2022). The remaining parameters of the black hole,
accretion disk and corona were drawn randomly from distributions representing
the reverberation sample (since detailed spectral analyses are not available
for all of the BASS AGN444The probability distributions from which the
parameter values are drawn were chosen to be consistent with the range of
parameter values from the sub-set of the BASS AGN sample for which
spectroscopic measurements are available, detailed at www.bass-survey.com and
in Ricci et al. (2017).). The reverberation measurement was simulated as
described above, predicting the time lag vs. Fourier frequency that would be
measured by HEX-P between the continuum-dominated 1.2–4 keV band and the 4–7
keV iron K band. We select the frequency bin with the highest signal-to-noise
(i.e., the ratio between the lag and the lag uncertainty) as the
representative iron K lag value.

The right panel of Figure 8 shows the simulated iron K reverberation lag
measurements as a function of black hole mass for the BASS AGN sample,
compared to the equivalent measurements made in a sample of nearby Seyfert
galaxies using XMM-Newton (Kara et al., 2016). HEX-P will significantly detect
reverberation in the iron K line and trace the relationship between the
reverberation time lag and the mass of the black hole. From these simulations,
we expect that HEX-P will be able to measure iron K lags in a sample of AGN
between redshifts of $z=0.005$ and $z=0.14$, with bolometric luminosities
between $3\times 10^{43}$ and $10^{46}$ erg s-1 (assuming 5 per cent of this
is emitted across the broad 0.1-100 keV X-ray band). The simulations show that
iron K lags will be measured to high precision, with a median uncertainty of
13 per cent across the sample from 300 ks observations. Longer observations of
around 500 ks will enable an additional reverberation measurement to be made
in the Compton hump to constrain the broadband time lag spectrum, to which
more detailed reverberation models can be fit (as demonstrated in the left
panel).555In Figure 8, the error bars on mass measurements assume that the
mass can be measured by optical reverberation. We assume a representative
uncertainty of 0.15 dex on the black hole mass, the median of those obtained
by reverberation measurements in the Black Hole Mass Database (Bentz and Katz,
2015).

Notably, HEX-P will also be able to detect reverberation around the most
massive black holes in the sample ($\gtrsim 10^{8}$ M⊙), where the
reverberation timescale is expected to be longer (if it scales simply with the
gravitational radius around the black hole). The ability to measure long-
timescale lags using common Fourier techniques (Uttley et al., 2014) is
limited by the duration of continuous light curves that can be obtained. The
highly elliptical orbit of XMM-Newton limits individual observations, and
hence individual light curve segments, to $\sim 125$ ks. At its L1 orbit,
HEX-P will be able to perform much longer continuous exposures (in principle
up to 12 days), greatly lengthening the timescales that can be probed.
Combined with the fact that the highest mass AGN are some of the brightest in
the BASS sample, iron K reverberation is expected to be detected at high
significance.

##  10 Black hole X-ray binary coronæ

This work has focused on coronæ in the vicinity of extragalactic SMBHs,
primarily observed through hard X-ray emission from AGN. However, Galactic
black hole X-ray binaries (BHXRBs) can be viewed broadly as scaled-down
versions of AGN, particularly from the point of view of the compact regions
close to the BH (Falcke et al., 2004; Körding et al., 2006; McHardy et al.,
2006; Walton et al., 2012). Many BHXRBs – those with low-mass companion stars–
are transient sources, undergoing outbursts lasting weeks to years, during
which they display a full repertoire of accretion states and regularly exceed
X-ray fluxes of 1 Crab (see, e.g., Remillard and McClintock 2006; Tetarenko et
al. 2016). Given the broad dynamical range one can probe via studies of
BHXRBs, much can be learned about the long-term behaviour and evolution of AGN
through observations possible on human timescales. In addition, the high
fluxes of outbursting BHXRBs allow for characterization of the broadband X-ray
continuum with exquisite detail, making them a primary target for HEX-P.

As briefly discussed in Section 5, today’s X-ray observatories already allow
us to probe the coronal temperatures of BHXRBs as they decrease with
increasing source flux (Buisson et al., 2019; Cangemi et al., 2021; Zdziarski
et al., 2021), and these measurements have contributed greatly to our
understanding of BH coronæ. However, degeneracies remain in the broadband
X-ray continuum emission and geometry of BHXRB coronæ, as well as the AGN
ones, and this has been a hindrance to reflection studies aimed at
characterizing the inner accretion flow and measuring BH spin. With its
broadband coverage, large collecting area, and high energy sensitivity, HEX-P
will: (i) break degeneracies in the geometry of the irradiating corona
intrinsic to reflection models; and (ii) measure deviations from purely
thermal Comptonization models, allowing explicit constraints on the non-
thermal fraction of BHXRB coronæ. The broadband spectral coverage of HEX-P in
the $0.2-80\,\rm keV$ range, combined with its high-energy sensitivity (owing
to low X-ray background) will be especially powerful in constraining the
weaker power law emission in BHXRB soft states, advancing coronal constraints
spanning orders of magnitude in luminosity and across the spectral states. We
refer the reader to Connors et al. (2023, in preparation) for a full
description of HEX-P simulations of BHXRB coronæ.

##  11 Discussion and conclusions

In this paper, we show the advances that HEX-P will make in understanding the
physics of the X-ray corona in different types of AGN, ranging from local
Seyferts to quasars at cosmological redshifts. The broad passband, large
effective area, low background, and the ability to have continuous long
exposures will enable addressing many open questions regarding the physics of
the X-ray corona in AGN. HEX-P will allow an accurate estimate of the
continuum emission spectrum of the corona by measuring the photon index and
the high-energy cutoff. Assuming a certain geometry, these parameters can then
be mapped into optical depth and electron temperature within the framework of
Comptonization (see e.g., Middei et al., 2019). We demonstrate how HEX-P will
improve our understanding of the electron distribution in AGN coronæ, measure
coronal properties of obscured sources, and extend these analyses to $z\geq
1$. Furthermore, we show that HEX-P will track variations of bright transient
events such as CLAGN and TDEs on timescales of a few hours. HEX-P will allow
us to study the properties of the X-ray corona and probe the innermost regions
of the accretion disk via reverberation mapping. This will be crucial for
understanding how the corona interacts with the accretion disk, studying disk
reflection and consequently measure the BH spin in AGN over a broad range of
mass and luminosity, and probing AGN feedback (see Piotrowska et al., 2023, in
preparation). Furthermore, HEX-P will allow a detailed study of obscuration in
AGN and identify the fraction of Compton-thick AGN in deep surveys (see
Boorman et al., 2023, Civano et al., 2023, in preparation). All this will help
achieving a more complete picture of how SMBHs grow and evolve across cosmic
time, and the role they play in galaxy evolution.

Simultaneous observations with LET and HET, covering the $0.2-80\,\rm keV$
range, will allow an accurate measurement of $E_{\mbox{\scriptsize cut}}$ up
to $\sim 1\,\rm MeV$ in bright sources where ionized disk reflection is
important. This has been discussed in detail by García et al. (2015) who
showed that the soft part of the reflection spectrum, rich in fluorescent
lines and atomic features, depends strongly on the properties of the corona,
illuminating the accretion disk. García et al. (2015) demonstrated this in the
case of joint NuSTAR and Suzaku spectra. Here, we revisit the analysis
presented by those authors for HEX-P. We consider a simple spectrum comprised
of neutral Galactic absorption (TBabs; Wilms et al., 2000) and a power law
plus ionized reflection (relxill; Dauser et al., 2013; García et al., 2014).
In XSPEC parlance, the model is written as: ${\tt Model=TBabs\times relxill}$.
We assume a Galactic absorption $N_{\rm H}=5\times 10^{20}\,\rm cm^{-2}$. As
for the reflection component, we assume a photon index $\Gamma=2$, a black
hole spin $a^{\ast}=0.998$, a disk ionization parameter $\log\left(\xi/\rm
erg\,cm\,s^{-1}\right)=1.5$, an inclination $i=45^{\circ}$, a disk emissivity
index $q=3$, and an iron abundance equal to the solar value. García et al.
(2015) used similar parameters except for the reflection fraction which they
assumed to be $R_{\rm f}=3$. This enhances the reflection features in the
spectrum. We choose a more conservative value of $R_{\rm f}=1$. We then
simulate 100-ks observations of HEX-P for $E_{\mbox{\scriptsize cut}}$ ranging
between 50 keV and 1 MeV for $2-10\,\rm keV$ flux levels of 1 mCrab, 10 mCrab,
100 mCrab, and 500 mCrab. These simulations are intended to represent a wide
range of Galactic and extragalactic accreting black holes. We fit the
simulated spectra using only the HET and by using both LET and HET
simultaneously. Similarly to García et al. (2015), we find that HEX-P will
allow us to accurately estimate $E_{\mbox{\scriptsize cut}}$ up to $\sim
1\,\rm MeV$ for bright sources (Fig. 9). In addition, we also find that the
inclusion of LET significantly reduces the uncertainty on
$E_{\mbox{\scriptsize cut}}$. We note that these results assume that the soft
X-ray excess is in totality caused by the ionized relativistic reflection from
the inner parts of the disk. In fact, the nature of this excess is still a
matter of debate as it has been suggested that it could also be caused by the
presence of a warm optically thick corona (e.g., Petrucci et al., 2018, 2020).
Studying the effect of such model on the properties of the hot corona is
beyond the scope of this paper.

Figure 9: Results of fitting HEX-P simulations assuming relativistic
reflection, by employing only HET (left) and by adding the soft X-rays LET+HET
(right). The grey, yellow, blue, and red areas correspond to the 90%
confidence level on estimating $E_{\mbox{\scriptsize cut}}$ for observed flux
levels of 1 mCrab, 10 mCrab, 100 mCrab, and 500 mCrab, respectively.

###  11.1 Synergies with other facilities

We have demonstrated the power of HEX-P as a stand-alone instrument to advance
our understanding of the physics of X-ray coronæ in accreting black holes.
However, simultaneous broadband X-ray and multi-wavelength coverage will
provide a more comprehensive physical picture of accretion onto these objects.
HEX-P will play an instrumental role in such studies thanks to its broad
passband and high energy sensitivity as demonstrated throughout this paper.
The ability to probe the emission from the X-ray corona for a variety of
sources with a wide range of mass, luminosity, distance, and variability
timescales will make HEX-P an essential tool to provide a comprehensive
understanding of the behavior and physics of AGN coronæ, which is the dominant
source of X-rays in the universe.

For example, the recent advancements made by IXPE, observing X-ray
polarization in AGN for the first time, required a full characterization of
the broadband X-ray spectrum. This has been ensured via simultaneous
observations with XMM-Newton, NuSTAR, and Chandra. HEX-P will provide
equivalent energy coverage to the combination of XMM-Newton/Chandra and
NuSTAR, with significant improvements to the high energy sensitivity. HEX-P
will be an essential complementary observatory to future X-ray polarization
missions such as the X-ray Polarization Probe (XPP; Krawczynski et al., 2019;
Jahoda et al., 2019) and the enhanced X-ray Timing and Polarimetry mission
(eXTP; Zhang et al., 2019a).

As mentioned earlier, a correlation between the hard X-ray emission and mm-
band continuum has been observed in various samples of AGN (see e.g., Behar et
al., 2018; Kawamuro et al., 2022; Ricci et al., 2023). HEX-P will accurately
measure the optical depth and temperature of the X-ray corona, as well as
determine the fraction of non-thermal electrons producing the X-ray emission
for a large sample of AGN. In coordination with ALMA and similar facilities,
these parameters can then be used to model the continuum emission in the mm-
band and probe the magnetic field powering the corona. HEX-P will also
complement observations with the Square Kilometer Array and the next
generation of cosmic microwave background experiments (e.g., CMB-S4; Carlstrom
et al., 2019; Abazajian et al., 2022).

Moreover, HEX-P will play a crucial part in connecting soft X-ray and soft
gamma-ray (i.e., up to a few MeV) observations of BHXRBs. In particular, a
primary science goal of the Compton Spectrometer and Imager (COSI; Tomsick et
al., 2019), scheduled for launch in 2027 as a NASA Small Explorer (SMEX)
mission, is to characterize the Galactic $511\,\rm keV$ positron annihilation
line and the gamma-ray polarization of compact objects. COSI will be able to
detect a broadened and redshifted pair line from pair-dominated coronæ. The
broadband X-ray coverage of HEX-P will help constrain the continuum coronal
emission, which will be important for modeling the higher energy emission and
polarization measurements provided by COSI. HEX-P will also play a major role
in constraining the X-ray emission from gamma-ray emitting AGN detected by
COSI (see Marcotulli et al., 2023, in preparation), which will probe the
connection between the accretion disk and jets (Ajello et al., 2019), shedding
yet more light on the nature of the X-ray corona.

Recent systematic studies of the optical night sky, such as All Sky Automated
Survey for Super Novæ (ASASSN; Shappee et al., 2014), the Asteroid
Terrestrial-impact Last Alert System (ATLAS; Tonry et al., 2018), and the
Zwicky Transient Facility (ZTF; Bellm, 2014; Graham et al., 2019), are
providing large numbers of Galactic and extragalactic transients, which are
often associated with accretion onto compact objects. The discovery rate of
nuclear transients with dramatic X-ray variability, similar to 1ES 1927+654,
is expected to increase significantly in the next decade with the continued
advance of large-scale, time-domain surveys in the optical (e.g., photometry
from the Vera Rubin Observatory, spectroscopy from SDSS-V) and UV (e.g.,
ULTRASAT; Sagiv et al., 2014). Therefore, HEX-P will be well-positioned to
make significant discoveries and contributions to the X-ray evolution and
nature of the corona in transient accretion systems (as discussed in Section
8).

Finally, HEX-P will play a major role in the era of the high-resolution X-ray
spectroscopy, with the development of microcalorimeters achieving a resolving
power ($E/\Delta E$) larger than 1000 at energies up to 7 keV. The Resolve
instrument on XRISM (Tashiro et al., 2018; XRISM Science Team, 2022) launched
in August 2023, and the X-ray Integral Field Unit (X-IFU) on Athena (Nandra et
al., 2013; Barret et al., 2023), expected to be launched in the late 2030s,
will revolutionize our view of accretion onto compact objects. These
instruments will probe the fine structure in absorption and emission from gas
around accreting black holes. However, these gaseous structures are dictated
by the hard ionizing X-rays, and the accurate measurements of such features
will require a proper estimate of the underlying X-ray continuum out to tens
of keV. This energy coverage can be provided by NuSTAR in the 2020s and HEX-P
will extend and improve this coverage in the decades to come. In particular,
HEX-P will be able to constrain the coronal emission and reflection spectrum
at hard X-rays. This will be crucial for determining the illumination profile
of the accretion disk by the X-ray corona, and accurately measuring BH spin
for a broad range of mass and luminosity (see also Piotrowska et al., 2023, in
preparation). Combining the resolving power of the future generation of
microcalorimeters to the broad energy coverage of HEX-P will allow us to
address important questions about black hole growth and evolution, and to
understand the role of these objects in shaping the Universe.

###  11.2 Conclusions

We have demonstrated that HEX-P will provide an unprecedented look at the
X-ray coronæ of accreting black holes with a main focus on AGN, thanks to its
high sensitivity and broad energy coverage. The main results of the paper can
be summarised as follows:

  * •

HEX-P will accurately estimate the coronal properties of unobscured AGN. In
particular, it will study evolution of coronal temperature in bright, variable
AGN. This will allow us to measure the non-thermal fraction of electrons in
AGN coronæ and test the runaway pair production hypothesis.

  * •

HEX-P will also accurately estimate the coronal properties of obscured AGN up
to $z\sim 0.5$. This will be a crucial test of the orientation-based unified
model of AGN.

  * •

HEX-P will efficiently constrain the coronal properties of high-redshift
quasars ($z>$1.40) even with high energy cutoff up to 200 keV. HEX-P will also
extend the estimates of the coronal properties of powerful quasars beyond
cosmic noon up to $z\simeq 4.5$. This will allow us to study the evolution of
the X-ray properties of AGN across cosmic times.

  * •

HEX-P will play a major role in studying transient events such as CLAGN and
TDEs. It will allow us to witness the destruction, formation, and evolution of
the X-ray corona in these extreme events.

  * •

Finally, the high sensitivity of HEX-P that extends to hard X-rays and the
possibility of long continuous observations of bright AGN will enable accurate
reverberation mapping of the innermost regions of the accretion disk. This
will allow us to estimate the location of the corona and its illuminating
profile over a broad mass range.

## Funding

EK acknowledges financial support from the Centre National d’Etudes Spatiales
(CNES). MB acknowledges support from the YCAA Prize Postdoctoral Fellowship.
LM is supported by the CITA National Fellowship. CR acknowledges support from
Fondecyt Regular grant 1230345 and ANID BASAL project FB210003. The work of DS
was carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with NASA. XZ acknowledges NASA support under
contract number 80NSSC22K0012.

## Author Contributions

EK is responsible for the creation of the manuscript, contributed to the
science cases and authored Sections 3,6, 7,11, and made Figures 2, 3 and 9. AL
contributed to the science case and authored Section 5, and made Figure 1. MM
contributed to the science case and authored Section 8, and made Figure 7. DW
contributed to the science case and authored Section 9, and made Figure 8. XZ
contributed to the science case and authored Section 7, and made Figures
4,5,6. MB contributed to the science case and co-authored Section 5. PGB
contributed to the science case and co-authored Section 5. RMTC contributed to
the science case and authored Section 10. JAG contributed to the science case
and authored Section 4. KKM contributed to the science case and authored
Section 4. DS contributed to the science case and authored Section 4. JB
contributed to the science case and co-authored Section 5. TW co-authored
Section 8. All authors contributed to the final editing of the manuscript.

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