# Hybrid Comptonization and Electron-Positron Pair Production in the Black-
Hole X-Ray Binary MAXI J1820+070

Andrzej A. Zdziarski  Nicolaus Copernicus Astronomical Center, Polish Academy
of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland<EMAIL_ADDRESS>Elisabeth Jourdain  CNRS; IRAP; 9 Av. colonel Roche, BP 44346, F-31028
Toulouse cedex 4, France  Université de Toulouse; UPS-OMP; IRAP; Toulouse,
France  Piotr Lubiński  Institute of Physics, University of Zielona Góra,
Licealna 9, PL-65-417 Zielona Góra, Poland  Michał Szanecki  Faculty of
Physics and Applied Informatics, Łódź University, Pomorska 149/153, PL-90-236
Łódź, Poland  Andrzej Niedźwiecki  Faculty of Physics and Applied Informatics,
Łódź University, Pomorska 149/153, PL-90-236 Łódź, Poland  Alexandra Veledina
Department of Physics and Astronomy, FI-20014 University of Turku, Finland
Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-10691 Stockholm, Sweden  Space Research Institute of
the Russian Academy of Sciences, Profsoyuznaya Str. 84/32, 117997 Moscow,
Russia  Juri Poutanen  Department of Physics and Astronomy, FI-20014
University of Turku, Finland  Space Research Institute of the Russian Academy
of Sciences, Profsoyuznaya Str. 84/32, 117997 Moscow, Russia  Nordita, KTH
Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23,
SE-10691 Stockholm, Sweden  Marta A. Dziełak  Nicolaus Copernicus Astronomical
Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland;
<EMAIL_ADDRESS>Jean-Pierre Roques  CNRS; IRAP; 9 Av. colonel Roche, BP 44346,
F-31028 Toulouse cedex 4, France  Université de Toulouse; UPS-OMP; IRAP;
Toulouse, France

###### Abstract

We study X-ray and soft gamma-ray spectra from the hard state of the accreting
black-hole binary MAXI J1820+070. We perform analysis of two joint spectra
from NuSTAR and INTEGRAL, covering the range of 3–650 keV, and of an average
joint spectrum over the rise of the hard state, covering the 3–2200 keV range.
The spectra are well modelled by Comptonization of soft seed photons. However,
the distributions of the scattering electrons are not purely thermal; we find
they have substantial high-energy tails, well modelled as power laws. The
photon tail in the average spectrum is detected well beyond the threshold for
electron-positron pair production, 511 keV. This allows us to calculate the
rate of the electron-positron pair production and put a lower limit on the
size of the source from pair equilibrium. At the fitted Thomson optical depth
of the Comptonizing plasma, the limit is about 4 gravitational radii. If we
adopt the sizes estimated by us from the reflection spectroscopy of $>$20
gravitational radii, the fractional pair abundance becomes much less than
unity. The low pair abundance is confirmed by the lack of both an annihilation
feature and of a pair absorption cutoff above 511 keV in the average spectrum.

††software: HEAsoft (v6.25; HEASARC 2014), NuSTARDAS (v.1.8.0), reflkerr
(Niedźwiecki et al., 2019), xillverCp (v. 1.4.3; García & Kallman 2010; García
et al. 2018), ireflect (Magdziarz & Zdziarski, 1995), osa (v. 11.1;
Courvoisier et al. 2003), xspec (Arnaud, 1996), compps (Poutanen & Svensson,
1996)

##  1 Introduction

Compton scattering of soft seed photons has for long been shown to explain
well hard X-ray spectra of the accreting black-hole (BH) X-ray binaries (XRBs)
in their hard spectral state. The electron distribution appears to be
predominantly thermal at mildly relativistic temperatures, $kT_{\rm e}$, see,
e.g., Done et al. (2007) and Zdziarski et al. (2020) and references in those
works. However, the form of the spectral high-energy cutoff in some sources
indicates the presence of a substantial high-energy non-thermal tail beyond a
Maxwellian electron distribution (e.g., McConnell et al. 2002; Wardziński et
al. 2002; Poutanen & Vurm 2009; Zdziarski et al. 2017; Walter & Xu 2017;
Cangemi et al. 2021). On the other hand, instead of being thermal, the
dominant Comptonization process could be scattering on the bulk motion of
fast-moving plasmoids generated in magnetic reconnection, in which case a
photon tail due to electron acceleration can appear as well (Beloborodov,
2017; Sironi & Beloborodov, 2020).

A related effect is e± pair production in photon-photon collisions. High-
energy tails in the spectra of sources explained by hybrid Comptonization
usually cross the threshold for this process, $m_{\rm e}c^{2}$, significantly
increasing the number of pair-producing photons with respect to the purely
thermal case. Pair equilibria (in which the rate of pair production is
balanced by that of pair annihilation) in hybrid plasmas were studied in
Zdziarski et al. (1993), Coppi (1999), Gierliński et al. (1999), McConnell et
al. (2002), Malzac & Belmont (2009) and Fabian et al. (2017).

Here we study these effects in MAXI J1820+070. It is a transient BH XRB, whose
outburst was discovered in 2018 (Tucker et al., 2018; Kawamuro et al., 2018).
The source is relatively nearby, at an accurately measured distance of
$d\approx 3.0\pm 0.3$ kpc (Atri et al., 2020). Also, the inclinations of both
the binary and its radio jet are well constrained, as $i_{\rm b}\approx
66\arcdeg$–$81\arcdeg$ (Torres et al., 2020), $i_{\rm j}\approx 63\arcdeg\pm
3\arcdeg$ (Atri et al., 2020), respectively. The BH mass is given by
$M\approx(5.95\pm 0.22){\rm M}_{\sun}/\sin^{3}i_{\rm b}$ (Torres et al.,
2020). The outburst was extensively monitored by a number of observatories, in
particular by Nuclear Spectroscopic Telescope Array (NuSTAR; Harrison et al.
2013), the Spectrometer on INTEGRAL (Roques et al. 2003; SPI), and the Imager
on Board the INTEGRAL Satellite (Ubertini et al. 2003; IBIS). Here, we study
spectra from contemporaneous observations by those instruments during the
initial hard state of the outburst. They provide high-quality broad-band
spectra in the range from 3 keV up to $\gtrsim$1 MeV.

##  2 Observations and data reduction

Table 1: Contemporaneous observations of MAXI J1820+070 with INTEGRAL and NuSTAR in the hard state  Epoch | INTEGRAL | SPI Start time | Exposure | IBIS Start time | Exp. ISGRI |  NuSTAR Obs. ID | Start time | Exposure A
---|---|---|---|---|---|---|---|---
| revolution | End time |  | End time | Exp. PICSiT |  | End time | Exposure B
1 | 1934 | 58201.555 | 13604 | 58201.544 | 8796 | 90401309008 | 58201.526 | 3046
|  | 58201.757 |  | 58201.757 |  |  | 58201.766 | 3214
2 | 1938 | 58212.181 | 28507 | 58212.181 | 8305 | 90401309012 | 58212.200 | 12334
|  | 58212.606 |  | 58212.393 |  |  | 58213.177 | 12964
A | 1931–1951 | 58193.453 | 1147756 | 58193.455 | 765345 | 90401309004–16 | 58198.036 | 50176
|  | 58246.894 |  | 58246.892 | 1156304 |  | 58242.643 | 52799

We use the INTEGRAL data from SPI and IBIS. The latter consists of the ISGRI
and PICSiT detectors. We use the SPI data as published in Roques & Jourdain
(2019). Data reduction and spectral extraction for ISGRI was done using the
osa v. 11.1 software (Courvoisier et al., 2003). The analysis of the PICSiT
data follows the method of Lubiński (2009).

We also use data from NuSTAR. They were reduced with heasoft v.6.25, the
NuSTARDAS pipeline v.1.8.0, and CALDB v.20200912. We set saamode=strict,
tentacle=yes and STATUS== b0000xxx00xxxx000. The source region is a
$60^{\prime\prime}$ circle centered on the peak brightness. We group the data
to signal-to-noise ratio $\geq$50, but only below 69 keV, so we can utilize
the 3–79 keV band to the maximum.

The chosen data sets are listed in Table 1. Epochs 1 and 2 consist of
overlapping INTEGRAL and NuSTAR observations. They correspond to the beginning
of a plateau phase on the count-rate/hardness plot, see fig. 2 in Buisson et
al. (2019). For them, the PICSiT data have low statistics and we do not use
them. Then, we analyze the average INTEGRAL spectra for the initial hard
state, over the span of 53 d, and the corresponding average NuSTAR spectra,
which set is denoted as A in Table 1. These INTEGRAL observations start in the
middle of the initial sharp rise phase of the outburst and continue through
the middle of the plateau phase. This data set includes the average PICSiT
spectrum.

We use the SPI spectra from 23 keV up to 650 keV for epochs 1, 2, and up to
2.2 MeV for the average spectrum, and include in them a 0.5% systematic error
(Roques & Jourdain 2019; added in quadrature). We add the same systematic
error to the average NuSTAR spectra, in order to avoid the broad-band fit to
be dominated by the very good statistics of the latter. However, following
previous papers on the NuSTAR data from MAXI J1820+070, e.g., Buisson et al.
(2019), we do not add a systematic error to the individual spectra. Given
apparent calibration inaccuracies in the ISGRI and PICSiT data, we add a 1%
systematic error to each of them. Also, the fluxes from the ISGRI at its
lowest energies are substantially below those of the other detectors, and thus
we use it in the 32–500 keV range only. We use the PICSiT data in the 0.24–2
MeV range.

##  3 Spectral Fits

###  3.1 The method

In our spectral analysis, we follow the treatment of Zdziarski et al. (2021b)
(Z21) with some modifications related to the presence of high-energy non-
thermal tails in the studied spectra. We use the X-ray fitting package xspec
(Arnaud, 1996). The fit uncertainties are for 90% confidence,
$\Delta\chi^{2}\approx 2.71$. Residual differences between the calibration of
the NuSTAR Focal Plane Modules A and B and the INTEGRAL detectors SPI, ISGRI
and PICSiT are accounted for by the model plabs, which multiplies the model
spectra by $KE^{-\Delta\Gamma}$. We set $K$ and $\Delta\Gamma$ fixed at 1 and
0, respectively, for the NuSTAR A module, and find them fitted to the module B
as 0.99 and $\approx+0.01$, respectively. For the INTEGRAL epochs 1, 2, we
find $\Delta\Gamma\approx+0.008$–0.015 and $K\approx 1.24$–1.28. We account
for the ISM absorption using tbabs (Wilms et al., 2000) using the elemental
abundances of Anders & Grevesse (1989), with the column density toward the
source of $N_{\rm H}=1.4\times 10^{21}$ cm-2 (e.g., Kajava et al. 2019). At
this low value, absorption of photons at $E\geq 3$ keV is weak, but still
noticeable.

We model spectra from Comptonization going directly to the observer and those
reflected from an accretion disk around a Kerr BH taking into account atomic
and relativistic effects. We use the reflkerr (Niedźwiecki et al., 2019) and
reflkerr_bb routines. The former assumes the Comptonization is on Maxwellian
electrons, using the compps code (Poutanen & Svensson, 1996). The reflkerr_bb
is a new routine111See users.camk.edu.pl/mitsza/reflkerr. It can also account
for blackbody-like disk emission due to its irradiation and viscous disk
dissipation, analogously to the lamppost version reflkerr_lpbb (Zdziarski et
al., 2021a). However, we do not use that feature in this work., which allows
for the presence of a power-law tail in the distribution, using a modified
version of compps222The standard version of compps allows for the presence of
a tail in the form of ${\rm d}N_{\rm e}/{\rm d}\gamma\propto\gamma^{-p}$ only,
which does not correspond to a physical acceleration for low values of
$\gamma$. On the other hand, both versions give accurate spectra for any
$\tau_{\rm T}$, with the number of calculated scatterings $\approx
50+4\tau_{\rm T}^{2}$ (at the expense of a long calculation time for large
values of $\tau_{\rm T}$).. The tail is parametrized by the Lorentz factor
($\gamma_{\rm min}$) at which the electron distribution switches from thermal
to a power law in the momentum with an index, $p$, ${\rm d}N_{\rm e}/{\rm
d}(\beta\gamma)\propto(\beta\gamma)^{-p}$ (where $\beta c$ is the electron
velocity), up to a maximum Lorentz factor, $\gamma_{\rm max}$, assumed to be
$10^{3}$.

The thermal electrons are parametrized by their temperature, $kT_{\rm e}$, and
the Compton parameter, $y\equiv 4\tau_{\rm T}kT_{\rm e}/m_{\rm e}c^{2}$, where
$\tau_{\rm T}$ is the Thomson optical depth of the plasma. The Comptonizing
cloud is assumed to be spherical and we set ${\tt geom}=0$ in compps, which
gives a fast method, which, however, overestimates $y$. We therefore use an
option of ${\tt geom}=-5$ (for a sinusoidal distribution of the seed photons
in a sphere, Sunyaev & Titarchuk 1980), which calculation is very slow but
accurate, in order to find the actual values of $y$, and thus of $\tau_{\rm
T}$. We also evaluate the average 2–10 keV power-law spectral index, $\Gamma$,
of a fitted Comptonization component (unabsorbed and without reflection). The
seed photons for Comptonization are assumed to have a blackbody distribution
with the temperature of $kT_{\rm bb}=0.2$ keV, following the results from the
Neutron star Interior Composition ExploreR (NICER; Gendreau et al. 2016) of
Wang et al. (2020).

In reflkerr, the reflected spectra in the rest frame are calculated using
xillverCp (v. 1.4.3; García & Kallman 2010; García et al. 2018) up to
$\approx$10 keV and ireflect (Magdziarz & Zdziarski, 1995) at higher energies,
see Niedźwiecki et al. (2019) for details. Both the direct and reflected
components are integrated over the surface of a Keplerian disk taking into
account the relativistic effects. Here, we assume the standard emissivity
profile of $\propto R^{-3}$, which is the same as the disk viscous dissipation
at $R\gg R_{\rm ISCO}$, where $R_{\rm ISCO}$ is the radius of the innermost
stable circular orbit. The reflection fraction, ${\cal R}$, is defined in
reflkerr as the ratio of the flux irradiating the disk to that emitted outside
in a local frame. We assume a rotating BH with the dimensionless spin of
$a_{*}=0.998$, for which $R_{\rm{ISCO}}\approx 1.237R_{\rm{g}}$, where $R_{\rm
g}\equiv GM/c^{2}$. However, the effect of this assumption is negligible at
$R\gg R_{\rm{ISCO}}$.

We then use the same spectral model as Z21, with two Comptonization
components, soft and hard, which reflect from two zones in the disk. As found
in Z21, and confirmed in this study for the individual observations, fits with
this model show that the hard component dominates both the total flux and the
emission at the peak of the $EF_{E}$ spectrum (where $F_{E}$ is the
differential energy flux) and beyond it. The reflection of this component is
found to be weakly blurred relativistically and from a weakly ionized medium.
The less luminous soft component dominates in soft X-rays, and its reflection
is both more blurred and from a strongly ionized medium. In the fits, we
determine the radial range of the soft reflection as between the inner disk
truncation radius, $R_{\rm in}>R_{\rm{ISCO}}$, up to a transition radius,
$R_{\rm tr}\equiv R_{\rm in}+\Delta R$. The hard reflection is from radii
$>R_{\rm tr}$. The fit parameters $R_{\rm in}$ and $\Delta R$ are determined
predominantly by the soft and hard reflection, respectively. Each of the two
reflection zones is characterized by its ionization parameter, $\xi\equiv 4\pi
F_{\rm irr}/n$, where $F_{\rm irr}$ is the irradiating flux and $n$ is the
density of the reflecting part of the disk.

Figure 1: The NuSTAR (magenta and cyan), SPI (blue) and ISGRI (red) unfolded
spectra of epochs (a) 1, (b) 2, fitted in the 3–80 keV range with the two-
component model with thermal electrons (black solid curves), but shown in the
entire ranges of the data. The spectra are normalized to the NuSTAR A module.
The fits significantly underpredict the spectra at $\gtrsim$100 keV. The
softer and harder Comptonization components are shown by the red long dashes
and blue short dashes, and the corresponding reflection components are shown
by the magenta dot-dashes and cyan dots, respectively.

However, the locations of the two Comptonizing plasmas (hard and soft) is not
determined by the fits. A possible geometry is shown in fig. 4 of Z21. There,
the soft component forms a corona covering the disk between $R_{\rm in}$ and
$R_{\rm tr}$, and its reflection is from the disk underneath. The hard
component forms either a hot flow or a slow jet sheath with a large scale
height at radii $<R_{\rm in}$. Its inner location is supported by its
dominance of the total luminosity. Since the disk between $R_{\rm in}$ and
$R_{\rm tr}$ is covered by the soft corona, the reflection of the hard
Comptonization component is from remote parts of the disk, at $R>R_{\rm tr}$.
While this geometry is not unique, our results below do not depend on it.

Table 2: The results of spectral fitting for our two-component coronal model (soft thermal and hard hybrid)  Component | Parameter | Epoch 1 | Epoch 2 | Average
---|---|---|---|---
ISM absorption |  $N_{\rm H}$ $[10^{21}]$ cm-2 | 1.4f
Joint constraints |  $i$ $[\arcdeg$] | $70^{+3}_{-3}$ | $66^{+1}_{-1}$ | $65^{+3}_{-1}$
| $Z_{\rm Fe}\,[\sun]$ | $1.5^{+0.4}_{-0.1}$ | $1.2^{+0.3}_{-0.1}$ | $1.1^{+0.1}_{-0.1}$
Thermal Comptonization | $y_{\rm th}$ | $0.54^{+0.01}_{-0.01}$ | $0.53^{+0.01}_{-0.02}$ | $0.68^{+0.01}_{-0.01}$
and reflection | $\Gamma_{\rm th}$ | $1.80^{+0.02}_{-0.03}$ | $1.87^{+0.02}_{-0.02}$ | $1.75^{+0.02}_{-0.01}$
|  $kT_{\rm e,th}$ [keV] | $12^{+1}_{-1}$ | $12^{+1}_{-1}$ | $20^{+1}_{-1}$
| $R_{\rm in}\,[R_{\rm g}]$ | $22^{+5}_{-5}$ | $31^{+9}_{-5}$ | $105^{+7}_{-67}$
| ${\cal R}_{\rm th}$ | $0.63^{+0.02}_{-0.13}$ | $0.42^{+0.01}_{-0.01}$ | $0.37^{+0.04}_{-0.05}$
| $\log_{10}\xi_{\rm th}$ | $3.61^{+0.05}_{-0.02}$ | $3.51^{+0.02}_{-0.03}$ | $3.51^{+0.02}_{-0.02}$
| $N_{\rm th}$ | $2.76^{+0.10}_{-0.38}$ | $2.90^{+0.01}_{-0.04}$ | $3.31^{+0.05}_{-0.17}$
Hybrid Comptonization | $y_{\rm h}$ | $0.95^{+0.05}_{-0.01}$ | $0.85^{+0.01}_{-0.01}$ | $1.04^{+0.01}_{-0.01}$
and reflection | $\Gamma_{\rm h}$ | $1.33^{+0.01}_{-0.03}$ | $1.43^{+0.01}_{-0.01}$ | $1.39^{+0.01}_{-0.01}$
|  $kT_{\rm e,h}$ [keV] | $21^{+2}_{-1}$ | $21^{+1}_{-1}$ | $33^{+5}_{-1}$
| $\gamma_{\rm min}$ | $1.24^{+0.06}_{-0.07}$ | $1.24^{+0.01}_{-0.01}$ | $1.31^{+0.08}_{-0.09}$
| $p$ | $3.06^{+0.54}_{-0.52}$ | $2.98^{+0.15}_{-0.01}$ | $3.55^{+0.20}_{-0.08}$
| $\Delta R\,[R_{\rm g}]$ | $850^{+60}_{-620}$ | $300^{+240}_{-80}$ | $<140$
| ${\cal R}_{\rm h}$ | $0.38^{+0.03}_{-0.01}$ | $0.31^{+0.01}_{-0.05}$ | $0.90^{+0.15}_{-0.05}$
| $\log_{10}\xi_{\rm h}$ | $2.47^{+0.07}_{-0.06}$ | $0^{+1.72}$ | $0^{+1.70}$
| $N_{\rm h}$ | $1.28^{+0.23}_{-0.02}$ | $1.09^{+0.01}_{-0.02}$ | $0.51^{+0.01}_{-0.02}$
| $\chi_{\nu}^{2}$ | 954/792 | 1759/1307 | 2275/2011

###  3.2 Fit results

Figure 2: The NuSTAR (magenta and cyan), SPI (blue), ISGRI (red) and PICSiT
(black; for A only) unfolded spectra and data-to-model ratios of the epochs
(a) 1, (b) 2, (c) A, fitted in the full ranges with the two-component coronal
model (black solid curves), see Table 2. The spectra are normalized to the
NuSTAR A one. The softer (red long dashes) and harder (blue short dashes)
Comptonization components are for thermal and hybrid electrons, respectively.
The corresponding reflection components are shown by the magenta dot-dashes
and cyan dots, respectively.

We first fit the NuSTAR data for epochs 1 and 2 assuming that the two
Comptonization zones have the same electron temperature. We obtain results
similar to those in table 2 in Z21, with $R_{\rm in}\gg R_{\rm ISCO}$. The
inclinations obtained from the fits are within $i\approx 60$–$70\arcdeg$,
which agrees with the binary and jet measurements (Section 1). As in Z21, we
do not impose the same inclination for each data set, in order to demonstrate
that each of them gives $i$ compatible with the observational constraints, as
well as to account for possible flaring and precession of the disk. We then
include the SPI and ISGRI data and fit them simultaneously with NuSTAR up to
80 keV. We find an excellent agreement of those data with NuSTAR. In the next
step, we include the data $>$80 keV, and plot the resulting $EF_{E}$ spectra
in Figure 1. We find that both the SPI and ISGRI data at $E\gtrsim 100$ keV
are significantly above the model fitted at 3–80 keV, while they are in a good
mutual agreement. In particular, the spectra measured by the SPI and ISGRI
extend up to 400 keV with significant non-zero fluxes, while the models fitted
in the 3–80 keV range predict only very weak such emission.

These results show that fitting the NuSTAR data with thermal Comptonization in
order to determine the rate of e± pair production and constrain its
equilibrium with the rate of pair annihilation in accreting BHs (Fabian et
al., 2015) may be unreliable. At least in MAXI J1820+070, the NuSTAR spectra
fitted by models with a thermal electron distribution do not provide reliable
predictions for the spectra around 511 keV, and thus for the pair production
rate. This conclusion agrees with that of Coppi (1999) and Fabian et al.
(2017), who found that allowing for a hybrid electron distribution reduces the
minimum electron temperature at which significant pair production can take
place.

We then fit the broad-band spectra for all three epochs with the thermal
Comptonization model, but allowing $kT_{\rm e}$ of the two Comptonization
clouds to be different. The resulting fits remain rather poor. In particular,
we obtain $\chi^{2}_{\nu}\approx 984/794$, 1835/1309, 2437/2013 for epochs 1,
2, A, respectively, with significant residuals at the highest energies. These
models also require very strong reflection from the harder component, e.g.,
${\cal R}_{\rm h}\approx 2.8$, 3 (constrained to $\leq$3) for epochs 2 and A,
respectively, which puts in question their physical reality.

We then allow for the presence of a non-thermal tail in the electron
distribution of the hard component, which adds two free parameters,
$\gamma_{\rm min}$ and $p$. We find that this strongly improves the fits, with
the spectra and data/model ratios shown in Figure 2 and the parameters given
in Table 2. With the new values of $\chi^{2}$, the probabilities for the
improvement to be by chance estimated by the F-test for epochs 1, 2, A equal
$\approx 5\times 10^{-6}$, $10^{-12}$, $10^{-30}$ respectively. We find
$\gamma_{\rm min}\approx 1.2$–1.4 and $p\approx 2.5$–3.7. The index for the
average spectrum is somewhat steeper than in the case of the individual
observations. This may be due to the INTEGRAL and NuSTAR observations not
covering exactly the same periods with different exposures and/or the high-
energy tail weaker than in the case of epochs 1–2 during some of the other
INTEGRAL observations (as observed; Roques & Jourdain 2019). Figure 3 shows
the electron distribution corresponding to the best fit to epoch A. The tail
contains 8% of the electrons and 40% of their energy.

We have added a high-energy tail to the hard component only since its
Comptonization dominates at high energies in modelling the spectra of epochs 1
and 2, which is also the case for all the observations studied in Z21. Still,
we have tested the effect of adding a tail also to the soft component, but
found that while its presence is allowed (and likely), it leads to no decrease
of $\chi^{2}$ for either epoch 1 or 2. For the average spectrum, we found a
modest decrease of $\Delta\chi^{2}\approx-9$ with almost no change to the
other fit parameters. The parameters of the tail to the soft component are
poorly constrained, and the model becomes considerably more complicated. Thus,
we have opted for not including that tail.

Figure 3: The electron distribution of hard Comptonization component of the
best fit to epoch A, with $kT_{\rm e}=33$ keV and a power-law with $p=3.55$
above $\beta\gamma$ corresponding to $\gamma_{\rm min}=1.31$. The dashed curve
gives the corresponding pure Maxwellian distribution.

Apart from the presence of the non-thermal tail, the results shown in Table 2
for epochs 1 and 2 are very similar to those for the NuSTAR observations 2 and
4 analyzed by Z21, with the Fe abundance close to solar and the disk being
substantially truncated. On the other hand, the relatively large value of
$R_{\rm in}$ for epoch A may be biased by the averaging, and we do not claim
it corresponds to the actual average inner radius.

Instead of adding a non-thermal tail to the electron distribution, Roques &
Jourdain (2019) were able to fit the SPI data ($\gtrsim$20 keV) alone by
thermal Comptonization, reflection and an e-folded power law. However, the
NuSTAR data (3–79 keV) for this source require two reflection components
(Buisson et al., 2019) as well as two incident continua with different shapes
(Z21), which we have assumed in our joint fitting. We can still add another
spectral component to our two-component thermal-Compton fit described above.
We have found that an added e-folded power law has a very low spectral index,
$\Gamma\sim 0$. In the absence of self-absorption (which is ruled out at high
energies), a spectrum as hard can be obtained only from thermal Comptonization
close to saturation, with a pronounced Wien peak. We have thus replaced the
e-folded power law by thermal Comptonization using the thcomp model (Zdziarski
et al., 2020), which is well suited for saturated Comptonization. For the
average spectrum, we have obtained a reasonable fit with $\Gamma\approx 1.2$
and $kT_{\rm e}\approx 68$ keV for that additional component. Still, this
model yields $\chi^{2}_{\nu}\approx 2324/2010$, which is significantly higher
than the model with a non-thermal tail (Table 2), with
$\Delta\chi^{2}\approx+49$, as well as the data are much above the model at
$E\gtrsim 0.5$ MeV. The two models are not nested, so the F-test cannot be
used. Instead, the Akaike information criterion (Akaike, 1978) can be used,
which yields the relative likelihood of the model with the additional
component compared to that with the tail of $\approx\\!8\times 10^{-12}$,
where we used it as given by eq. (1) in Dziełak et al. (2019). Thus, we see
that an addition of a high-energy tail to the electron distribution yields a
much more likely fit than that with an additional spectral component.

##  4 Pair production

We express the photon-photon pair production rate through the differential
photon density at its threshold, $m_{\rm e}c^{2}$, and express the photon
energies as $\epsilon\equiv E/(m_{\rm e}c^{2})$. We denote that density, ${\rm
d}n/{\rm d}\epsilon$, at $\epsilon=1$ as $n_{1}$. In order to estimate it, we
assume a simple model of a sphere with a radius, $R$, a uniform photon
density, and the average photon escape time of $\approx\\!(R/c)(1+g_{\rm
C}\tau_{\rm T}/3)$, where $\tau_{\rm T}\equiv Rn_{\rm e}\sigma_{\rm T}$,
$g_{\rm C}\equiv\sigma_{\rm C}/\sigma_{\rm T}$, $\sigma_{\rm T}$ and
$\sigma_{\rm C}$ are the Thomson and Compton cross sections, respectively,
$n_{\rm e}$ is the total density of electrons and positrons, and $g_{\rm
C}(\epsilon=1)\approx 0.43$. This yields

$n_{1}\approx F_{E}(511{\rm keV})\frac{3(1+g_{\rm C}\tau_{\rm
T}/3)d^{2}}{R^{2}c},$ (1)

where $F_{E}$ is in units of keV cm-2 s-1 keV-1. Given that the spectra in the
vicinity of 511 keV have approximate power-law shapes, see Figures 2, 4, we
assume ${\rm d}n/{\rm d}\epsilon=n_{1}\epsilon^{-\Gamma}$. Thus, the formalism
developed in this section is independent of the hybrid model fitted to the
broad-band spectra. We further approximate the photon field within the source
as isotropic, allowing us to use the absorption coefficient for photon-photon
pair production by Gould & Schréder (1967)333Corrected in Brown et al. (1973).
See also the corrected first three coefficients of the series expansion in
Zdziarski (1988); the fourth is in error.. The absorption coefficient for
photons crossing a power-law isotropic photon field without cutoffs is given
by eqs. (B3) and (B5) of Svensson (1987), which corresponds to the radial
optical depth of

$\tau_{\gamma\gamma}(\epsilon)\approx\frac{7\epsilon^{\Gamma-1}n_{1}\sigma_{\rm
T}R}{6\Gamma^{5/3}(1+\Gamma)}.$ (2)

In the absence of cutoffs, the total pair production rate is infinite. Here,
we integrate the absorption coefficient corresponding to Equation (2) over a
power law with a sharp high-energy cutoff at $\epsilon_{\rm c}\equiv E_{\rm
c}/(m_{\rm e}c^{2})>1$, and include a factor of 1/2 to account for double
counting of pair-producing photons. Note that this neglects the effect of pair
absorption, which we will consider later. This yields,

$\dot{n}_{\gamma\gamma}\approx\frac{7C\ln\epsilon_{\rm
c}}{6\Gamma^{5/3}(1+\Gamma)}n_{1}^{2}\sigma_{\rm T}c.$ (3)

It depends logarithmically on the unknown high-energy cutoff. Equation (3)
also requires that the power law extends at low energies to at least
$\epsilon=\epsilon_{\rm c}^{-1}$. The neglect of the cutoff in calculating the
absorption coefficient means that the rate calculated as above is
overestimated, which we account for by the factor $C<1$. At $\epsilon_{\rm
c}=4$ and $\Gamma=3$, 4, $C\approx 0.71$, 0.78, respectively. Given the lack
of an observed high-energy cutoff, the actual value of $\epsilon_{\rm c}$
remains unknown, but the average spectrum shown in Figure 4 implies
$\epsilon_{\rm c}\gtrsim 4$.

We assume the produced pairs lose quickly their energy in excess of the
thermal one in Compton and Coulomb scattering and thermalize. The annihilation
rate by thermal e± is

$(\dot{n}_{+})_{\rm ann}=\frac{3}{32}u(2-u)n_{\rm e}^{2}\sigma_{\rm T}cg_{\rm
A}(\theta),$ (4)

where $u\equiv 2n_{+}/n_{\rm e}$ is the pair abundance, $n_{+}$ is the
positron density, $\theta\equiv kT_{\rm e}/m_{\rm e}c^{2}$ and $g_{\rm A}$ is
the correction to the Born rate, which can be fitted as (Svensson, 1982)

$g_{\rm A}\approx\left[1+2\theta^{2}/\ln(2\eta_{\rm
E}\theta+1.3)\right]^{-1},$ (5)

where $\eta_{\rm E}\equiv\exp(-\gamma_{\rm E})\approx 0.5616$ and $\gamma_{\rm
E}$ is Euler’s constant. Here, we neglect the small effect of annihilation of
e± from the non-thermal tail of the fitted distribution, whose tail contains
only a small fraction of all e±.

We then consider escape of pairs, which is very likely to be present as
advection to both the BH and the jet/wind. We parametrize it by the ratio of
the light travel time across the source to the time scale of pair escape,
$\beta_{\rm esc}\equiv(R/c)/[n_{+}/(\dot{n}_{+})_{\rm esc}]<1$ (following eq.
18 of Zdziarski 1985). This gives

$(\dot{n}_{+})_{\rm esc}=\frac{1}{2\tau_{\rm T}}\beta_{\rm esc}un_{\rm
e}^{2}\sigma_{\rm T}c.$ (6)

The rates of annihilation and escape become equal for

$\beta_{\rm esc}=\frac{3}{16}(2-u)g_{\rm A}(\theta)\tau_{\rm T}$ (7)

(not assumed below). Pair equilibrium corresponds to
$\dot{n}_{\gamma\gamma}=(\dot{n}_{+})_{\rm ann}+(\dot{n}_{+})_{\rm esc}$.
(This neglects the photon-particle and particle-particle pair production,
which, as we have checked, are not important for this source.) Its solution
gives the radius of

$R_{\rm eq}\approx\frac{4(7C\ln\epsilon_{\rm c})^{\frac{1}{2}}\sigma_{\rm
T}d^{2}(1+0.14\tau_{\rm T})F_{E}(511{\rm
keV})}{\Gamma^{5/6}[u(\Gamma+1)]^{\frac{1}{2}}\tau_{\rm T}c\left[(2-u)g_{\rm
A}+\frac{16\beta_{\rm esc}}{3\tau_{\rm T}}\right]^{\frac{1}{2}}},$ (8)

which is $\mathrel{\vbox{ \offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}u^{-1/2}$. The minimum radius at which pair
equilibrium can be established, which we denote as $R_{\rm pair}$, corresponds
to the pair dominance, i.e., $u\rightarrow 1$. Also, for known $\tau_{\rm T}$
and radius, e.g., setting $R_{\rm eq}=R_{\rm in}$ of Table 2, we can solve for
$u$. For a given $u$, the optical depth to pair production can be calculated
using Equation (2),

$\tau_{\gamma\gamma}(\epsilon)\approx\frac{\epsilon^{\Gamma-1}\tau_{\rm
T}(7u)^{\frac{1}{2}}\left[(2-u)g_{\rm A}+\frac{16\beta_{\rm esc}}{3\tau_{\rm
T}}\right]^{\frac{1}{2}}}{8\Gamma^{5/6}(\Gamma+1)^{\frac{1}{2}}(C\ln\epsilon_{\rm
c})^{\frac{1}{2}}},$ (9)

which is maximized at $u=1$ (at which $R=R_{\rm pair}$).

Table 3: High-energy and e± pair parameters  Epoch | $\Gamma$ | $\frac{EF_{E}(511{\rm keV})}{{\rm keV\,cm}^{-2}\,{\rm s}^{-1}}$ | $\tau_{\rm T}$ | $\frac{R_{\rm pair}}{R_{\rm g}}$ | $\left.\tau_{\gamma\gamma}\right(u\\!=\\!1)$ | $u(R_{\rm in})$
---|---|---|---|---|---|---
1 | $3.60^{+0.26}_{-0.24}$ | $2.48_{-0.54}^{+0.62}$ | 5.7 | 3.8 | 0.34 | 0.017
2 | $3.70^{+0.22}_{-0.20}$ | $2.35_{-0.43}^{+0.48}$ | 5.1 | 3.7 | 0.29 | 0.008
A | $3.74^{+0.05}_{-0.04}$ | $2.14_{-0.09}^{+0.09}$ | 4.0 | 3.7 | 0.23 | 0.0007
Figure 4: The average data above 160 keV (normalized to the SPI) fitted with a
power law: SPI (blue), ISGRI (red), PICsIT (black). Including an e± pair
annihilation line in the fit yields its null normalization. We plot the
maximum line allowed by the data at $2\sigma$.

We then calculate the total observed flux of photons from pair annihilation as
$\dot{N}_{\rm ann}=2(\dot{n}_{+})_{\rm ann}(4\pi R^{3}/3)/[4\pi
d^{2}(1+0.14\tau_{\rm T})]$ (in units of cm-2 s-1), where the factor of 2
accounts for two photons produced in each annihilation event and the flux is
reduced by scattering (and we neglect secondary pair production by the
annihilation photons). Setting $R=R_{\rm eq}$, we find

$\dot{N}_{\rm ann}\approx\frac{(7Cu\ln\epsilon_{\rm
c})^{\frac{1}{2}}(2-u)\tau_{\rm T}F_{E}(511{\rm
keV})}{4\Gamma^{5/6}(\Gamma+1)^{\frac{1}{2}}\left[(2-u)g_{\rm
A}+\frac{16\beta_{\rm esc}}{3\tau_{\rm T}}\right]^{\frac{1}{2}}}.$ (10)

The above rate assumes that annihilation of pairs escaping the hot plasma is
negligible. In the limit of $kT_{\rm e}\ll m_{\rm e}c^{2}$ (which is satisfied
in MAXI J1820+070), the observed energy spectrum ($F_{E,{\rm ann}}$ in units
of keV cm-2 s-1 keV-1) from pair annihilation becomes (Svensson et al., 1996)

$F_{E,{\rm ann}}\approx\dot{N}_{\rm
ann}\frac{\epsilon^{5/2}}{(\pi\theta)^{1/2}}\exp\left[-\frac{(\epsilon-1)^{2}}{\epsilon\theta}\right].$
(11)

Here, we assumed that the annihilating e± are purely thermal. This is
approximately satisfied for this source; e.g., the steady-state electron
distribution for epoch A contains only 8% of non-thermal e±.

We now use the data from MAXI J1820+070 to determine the values of the
parameters related to pair production. We have fitted the spectra of the three
epochs at $E\gtrsim 160$ keV by power laws, and show the results in Table 3
and in Figure 4 for the average spectrum. Then we use equation (8) assuming
$\epsilon_{\rm c}=4$, $C=0.75$ and $\beta_{\rm esc}=0.3$, and use the values
of $\tau_{\rm T}$ obtained from the broad-band fits. The obtained radii for
the pair dominance are given in Table 3, expressed in units of the
gravitational radius for the assumed BH mass of $8{\rm M}_{\sun}$. We find
$R_{\rm pair}\approx 4R_{\rm g}$ and, from equation (9), the corresponding
$\tau_{\gamma\gamma}(\epsilon=1)\approx 0.3$. At such low radii, GR effects
would increase the photon energies and density, and lead to some increase of
$R_{\rm pair}$. Also, a possible flattening of the photon distribution at
highest energies, as fitted in Figure 2c, would also increase $R_{\rm pair}$
by a modest factor. On the other hand, attenuation of the power law at
$\epsilon\gtrsim 1$, if present, would decrease $R_{\rm pair}$.

The actual radii of the plasmas emitting in the vicinity of 511 keV remain
unknown. Adopting the hybrid Comptonization model, this emission is produced
by the hard spectral component, which reflection is only weakly
relativistically blurred, see Section 3. As discussed there, this argues for
its size being relatively large, $\sim\\!R_{\rm in}$ or more. The values of
$R_{\rm in}$ are larger by an order of magnitude than $R_{\rm pair}$. By
solving $R_{\rm eq}=R_{\rm in}$, we find $u\ll 1$, see Table 3. Thus, the hot
plasma within $R_{\rm in}$ appears not to be dominated by pairs. In this case,
$\tau_{\gamma\gamma}\ll 1$, and our neglect of pair absorption is justified.
The low pair density in this source is also corroborated by the lack of any
apparent cutoff above 511 keV in the observed spectrum.

A low pair density within the emitting plasma is further strongly supported by
results of a fit to the average spectrum including an annihilation feature. We
have found that the best-fit normalization of the feature is null. For that,
we assumed the fitted value of $kT_{\rm e,h}=33$ keV; however, the null best
fit value is obtained at any $kT_{\rm e}\lesssim 150$ keV. We show the
annihilation spectrum corresponding to the $2\sigma$ upper limit in Figure 4.
This limit corresponds to $\lesssim$0.18 of the normalization corresponding to
the pair-dominated plasma ($u=1$), which, in turn, corresponds to $u\lesssim
0.014$ and $R_{\rm eq}\gtrsim 24R_{\rm g}$, in agreement with the constraints
from the fits to epochs 1 and 2 assuming $R\sim R_{\rm in}$.

##  5 Discussion

In the hard state, we have obtained the disk inner truncation radii of $R_{\rm
in}\approx 20$–$30R_{\rm g}$ (from spectral fitting to the individual
observations), while the equilibrium radii corresponding of pair dominance
assuming the fitted Thomson optical depths are $R_{\rm pair}\sim 4R_{\rm g}$.
If the size of the plasma emitting around 511 keV is comparable to $R_{\rm
in}$ (or larger), as we argued above, its pair abundance is low. Still, the
obtained values of $R_{\rm pair}$ hint at the importance of pair production in
regulating the physical state of the flow. If pairs were just unimportant, the
minimum radius limited by pair equilibrium could have any value, but instead
we find $R_{\rm pair}\sim R_{\rm ISCO}$, suggesting that pair production
somehow knows the fundamental size scale of the system. The pairs produced by
the non-thermal photon tail may contribute to regulation of the temperature of
the thermal component of the electron distribution (Coppi, 1999; Fabian et
al., 2017).

Alternatively, the obtained low values of the electron temperature, $kT_{\rm
e}\sim 20$–30 keV, could be due to the energy balance in the flow itself.
Poutanen & Vurm (2009) and Malzac & Belmont (2009) suggested $T_{\rm e}$ can
be regulated by thermalization in a synchrotron boiler (Ghisellini et al.,
1988). Such regulation also takes place in advection-dominated accretion flows
(ADAFs), but the electron temperatures of such flows are much higher, e.g.,
Yuan & Narayan (2014). On the other hand, the existing ADAF models consider
only thermal electrons, while the presence of non-thermal tails can greatly
increase the synchrotron cooling (see below), which effect, could, in
principle, reconcile this model with the data. Some studies of extended
accretion flows with hybrid electrons were performed by Veledina et al.
(2013), but without including heating from ions.

An important diagnostic, strongly confirming the low pair dominance
independently of the pair equilibrium calculations, is the absence of an
annihilation feature in the observed spectrum. Purely thermal plasmas do not
show distinct pair annihilation lines even if dominated by pairs (Zdziarski,
1986; Stern et al., 1995; Svensson, 1996). However, the data for this source
rule out purely thermal plasmas, see Figure 1. On the other hand, hybrid
plasmas can emit X-ray spectra well reproduced by Comptonization on mostly
thermal electrons, but still exhibit relatively narrow annihilation features.
The absence of such a feature in the average spectrum therefore implies the
pair abundance to be very low.

The pair equilibrium formalism developed in Section 3 is independent of the
specific hybrid model fitted to the data. In particular, the pair production
rate is calculated for a general photon power-law spectrum, and is then
independent of the thermal-Compton parts of the observed spectra. Thus, this
rate remains approximately valid even if the high-energy tail is produced in a
spatially different region (e.g., some region of the jet) than that of the
thermal Comptonization, and it depends mostly on the characteristic size of
that region. However, the rates of pair annihilation and escape strongly
depend on the Thomson optical depth of the pair-producing plasma, which then
yields $R_{\rm pair}\mathrel{\vbox{
\offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}\tau_{\rm T}^{-1}$. We used the values of
$\tau_{\rm T}$ of the hard Comptonization region from our hybrid fits, which
then yields $R_{\rm pair}$ as given in Table 3. Alternative models with lower
$\tau_{\rm T}$ of the pair-producing region would then yield larger $R_{\rm
pair}$. Then, the pair abundance would depend on both that $R_{\rm pair}$ and
the size of that region. While we have been unable to find acceptable models
to the data with a separate region emitting the photon high-energy tail, we
cannot rule out their existence.

The best-fit steady-state power-law indices of the electron tail are in the
range of $p\approx 3.0$–3.6. The Compton (in the Thomson limit) and
synchrotron energy losses steepen the distribution of accelerated relativistic
electrons by unity, giving their index as $p_{\rm acc}=p-1$ (though we caution
that some of the emitting electrons in our case are only mildly relativistic).
Thus, the implied values of $p_{\rm acc}$ are in good agreement with those
from shock acceleration. In particular, indices $p_{\rm acc}\approx 2.2$–2.4
are found for collisionless shocks (Sironi et al., 2015). On the other hand,
magnetic reconnection can readily give particle spectra of similar indices
(Sironi et al., 2016; Ball et al., 2018; Sironi & Beloborodov, 2020).

The presence of high-energy tails in the electron distribution likely results
in the synchrotron emission of the electrons being greatly enhanced with
respect to the Maxwellian case (Wardziński & Zdziarski, 2001; Veledina et al.,
2011, 2013; Poutanen & Veledina, 2014). This is likely to result in the
synchrotron emission becoming an important source of the seed photons for
Comptonization, with a typical self-absorption turnover at $\lesssim$10 eV. We
have thus run spectral models in which the seed photon temperature was set to
10 eV, and found similar values of $\chi^{2}$ and minor differences in the
best-fit parameters with respect to our main calculations (with $kT_{\rm
bb}=200$ eV). Thus, either the disk blackbody and synchrotron emission can
provide the dominant seed photons, with the spectral fits at $>$3 keV not
allowing us to distinguish between these possibilities.

We note that the NuSTAR data were fitted by Buisson et al. (2019) with $R_{\rm
in}\approx 5R_{\rm g}$ and $i\approx 30$–$35\arcdeg$, both much lower than our
values. As discussed in Z21, the NuSTAR data for this source yield two
separate solutions, one low-$i$, low-$R_{\rm in}$, and the other high-$i$,
high-$R_{\rm in}$. However, the low-$R_{\rm in}$ one is ruled out by the
binary and jet measurements, showing $i\geq 60\arcdeg$ (Torres et al., 2020;
Atri et al., 2020), leaving the high-$R_{\rm in}$ solution as applicable to
this source.

##  6 Conclusions

We have studied two joint spectra from the 2018 hard-state of MAXI J1820+070
from NuSTAR and INTEGRAL. We have found an excellent agreement between the
NuSTAR and SPI spectra in the overlapping region of 23–79 keV, and an overall
agreement between NuSTAR and ISGRI. We have found the resulting 3–650 keV
spectra are well fitted by a two-component Comptonization model, with each
component having different spectral index and separate reflection regions.
However, the harder component, dominant at high energies, is required to be
emitted by hybrid electrons, whose distribution consists of a Maxwellian and a
high-energy tail. The presence of the tail is required at a very high
significance. A similar tail can be present also in the soft component;
however, its parameters are poorly constrained and its presence would only
weakly affect the fitting results due to the very low contribution of that
component at high energies.

Except for the presence of the tail, the parameters fitted to the spectra are
similar to those fitted by Z21 to the NuSTAR data of the observations taken
within the same state of the source (in particular those of their epochs 2 and
4). In particular, we find the disk is truncated at $\approx$20–$30R_{\rm g}$.
The electron temperatures are low, $\approx$20–30 keV for the harder
component, while the optical depths are relatively large, $\tau_{\rm T}\approx
4$–5.

We have also obtained an average hard-state spectrum of MAXI J1820+070, based
on the data spanning 53 d. This spectrum is quite similar to those of the two
individual observations, but extends to $\approx$2 MeV. It also includes the
PICSiT data, which are found to be in an excellent agreement with those of the
SPI.

The average spectrum crosses 511 keV, the threshold for e± pair production. We
have developed a formalism calculating the rates of pair production,
annihilation and escape in hybrid plasmas, and obtained formulae for the
equilibrium radius at a given pair abundance and the optical depth to pair
absorption. We have found that the minimum possible radius from the
equilibrium condition, for a pair-dominated plasma, is $\approx\\!4R_{\rm g}$.
This is, however, much less than our estimates of the size of the hot plasma,
$R\sim R_{\rm in}\sim 20$–$30R_{\rm g}$. At such radii, the equilibrium pair
abundance is $\ll$1\. This conclusion is confirmed by the absence of an
annihilation feature in the average spectrum. Even at its $2\sigma$ upper
limit, the implied size of the hot plasma is $\gtrsim 20R_{\rm g}$,
corresponding to a low pair abundance. Also, we have found no steepening of
the spectrum above 511 keV, which would occur due to pair absorption if the
plasma were pair dominated.

We consider it possible that pair production still acts as a thermostat,
limiting the electron temperature. On the other hand, the low temperature can
be due to an efficient cooling by synchrotron photons, which are copiously
emitted by hybrid plasmas.

## Acknowledgments

We thank the referee for valuable comments. This work is based on observations
with INTEGRAL, an ESA project with instruments and Science Data Center funded
by ESA member states (especially the PI countries: Denmark, France, Germany,
Italy, Switzerland, Spain; but also with a significant contribution from
Poland) and with the participation of Russia and the USA. The INTEGRAL SPI
project has been completed under the responsibility and leadership of CNES.
The SPI team is grateful to ASI, CEA, CNES, DLR, ESA, INTA, NASA, and OSTC for
their support. We acknowledge support from the International Space Science
Institute (Bern), and the Polish National Science Centre under the grants
2015/18/A/ST9/00746, 2019/35/B/ST9/03944 and 2014/13/B/ST9/00570. AV and JP
acknowledge the Academy of Finland grants 309308 and 333112, respectively.

.

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