# The Donor of the Black-Hole X-Ray Binary MAXI J1820+070

Joanna Mikołajewska Nicolaus Copernicus Astronomical Center, Polish Academy of
Sciences, Bartycka 18, PL-00-716 Warszawa, Poland  Andrzej A. Zdziarski
Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka
18, PL-00-716 Warszawa, Poland  Janusz Ziółkowski  Nicolaus Copernicus
Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716
Warszawa, Poland  Manuel A. P. Torres Instituto de Astrofísica de Canarias,
E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad
de La Laguna, E-38206 La Laguna, Tenerife, Spain  Jorge Casares Instituto de
Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain  Departamento de
Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain

###### Abstract

We estimate the parameters of the donor of the accreting black-hole binary
MAXI J1820+070. The measured values of the binary period, rotational and
radial velocities and constraints on the orbital inclination imply the donor
is a subgiant with the mass of $M_{2}\approx 0.49^{+0.10}_{-0.10}{\rm
M}_{\sun}$ and the radius of $R_{2}\approx 1.19^{+0.08}_{-0.08}{\rm
R}_{\sun}$. We re-analyze the previously obtained optical spectrum from the
Gran Telescopio Canarias, and found it yields a strict lower limit on the
effective temperature of $T>4200$ K. We compile optical and infrared fluxes
observed during the quiescence of this system. From the minima $r$ and
$i$-band fluxes found in Pan-STARSS1 Data Release 2 pre-discovery imaging and
for a distance of $D\approx 3$ kpc, reddening of $E(B$–$V)=0.23$ and
$R_{2}\approx{1.11R_{\odot}}$, we find $T\lesssim 4230$ K, very close to the
above lower limit. For a larger distance, the temperature can be higher, up to
about 4500 K (corresponding to a K5 spectral type, preferred by previous
studies) at $D=3.5$ kpc, allowed by the Gaia parallax. We perform evolutionary
calculations for the binary system and compare them to the observational
constraints. Our model fitting the above temperature and radius constraints at
$D\approx 3$ kpc has the mass of $0.4M_{\odot}$, $T\approx 4200$ K and solar
metallicity. Two alternative models require $D\gtrsim 3.3$–3.4 kpc at
$0.4M_{\odot}$, $T\approx 4500$ K and half solar metallicity, and
$0.5M_{\odot}$, $T\approx 4300$ K and solar metallicity. These models yield
mass transfer rates of $\sim\\!\\!10^{-10}M_{\odot}$/yr, compatible with those
based on the estimated accreted mass of $\approx\\!2\times 10^{25}$ g and the
time between the 2018 discovery and the 1934 historical outburst.

##  1 Introduction

The outburst of the transient accreting low-mass X-ray binary (LMXB) MAXI
J1820+070 was discovered on 2018-03-06 in the optical range with the $V$-band
magnitude of 14.88 (Tucker et al., 2018). It was detected in X-rays six days
later (on 2018-03-11; Kawamuro et al. 2018). Given the high observed
brightness and long duration of the outburst, it was the subject of a large
number of observing campaigns, whose results have led to numerous insights
into the nature of this source. Torres et al. (2019, 2020) have unambiguously
determined that the accretor is a black hole (BH). However, the donor
parameters have remained relatively loosely constrained (see Sections
2.1–2.3). More accurate determinations of the donor radius and temperature are
important for modelling the emission of the accretion flow in the quiescent
state of this system (Poutanen et al., 2022). Also, the metallicity of MAXI
J1820+070 is of importance for calculation of the inner radius of the
accretion disk during the outburst. On one hand, Buisson et al. (2019) found
the Fe abundance to be higher than solar by a factor of $\approx$3–9 and the
disk extending close to the innermost stable circular orbit from fitting
reflection model spectra to X-ray data in the luminous hard state. On the
other, fits by Zdziarski et al. (2021) obtained a fractional Fe abundance of
$\approx$1.1–1.6 and a truncated disk. However, Fe abundances from spectral
fits are subject to significant systematic errors related to the effects of
the assumed density of the reflector (García et al., 2018). Therefore, an
independent estimate of the abundances of the donor would be highly valuable
in constraining the results of X-ray spectroscopy.

In this work, we consider evolutionary models of the binary, with the goal of
constraining better the mass and radius of the stellar companion as well as
its spectral type and metallicity. In Section 2.1, we review available
constraints on the distance and binary parameters, in Section 2.2, we compile
available mid-infrared to ultraviolet photometric measurements, which
constrain the radius and temperature of the donor, in Section 2.3, we consider
its metallicity, in Section 2.4, we estimate the average mass accretion rate,
and in Section 3, we perform the evolutionary calculations. Section 4 provides
a summary of our results.

##  2 System parameters

Figure 1: The results of Monte Carlo simulations (sampled $10^{5}$ times each)
of the probability distribution for (top) the mass of the BH, (middle) the
donor mass, and (bottom) the donor radius. The vertical solid lines mark the
median, and the dashed lines show the 68% confidence, ranges.

###  2.1 The distance and binary parameters

MAXI J1820+070 is relatively nearby, with a distance of $D\approx 2.96\pm
0.33$ kpc measured based on a radio parallax (Atri et al., 2020). This agrees
well with the current Gaia EDR3 parallax measurement of
$D=2.81^{+0.70}_{-0.39}$ kpc (Bailer-Jones et al., 2021). Then, Wood et al.
(2021) determined $D\leq 3.11\pm 0.06$ kpc based on the proper motion of two
moving transient ejecta (Mirabel & Rodríguez, 1994) during the hard-to-soft
state transition, assuming the approaching and receding ejecta were identical.
The upper limit of $3.17$ kpc corresponds, however, to the minima of both
angular velocities and to the bulk Lorentz factor of
$\Gamma\rightarrow\infty$. The Lorentz factors of the transient ejecta
measured in X-ray binaries are generally low, at most a few (Miller-Jones et
al., 2006). In the present case, $D=3$ and 3.1 kpc correspond to rather large
$\Gamma\geq 3.4$ and 5.1, respectively. Therefore, the measurement by Wood et
al. (2021) appears to imply $D\lesssim 3$ kpc for realistic ejecta Lorentz
factors.

The binary orbital period is $P=0.68549\pm 0.00001$ d (Torres et al., 2019)
and the inclination of the binary is constrained at $3\sigma$ to $i\approx
66.2\arcdeg$–$80.8\arcdeg$ (Torres et al., 2020). While the latter is
consistent with the inclination of the jet, $i_{\rm j}\approx 64\pm 5\arcdeg$
(Atri et al., 2020; Wood et al., 2021), the jet and binary axes have been
estimated to be strongly misaligned (Poutanen et al., 2022), in which case the
jet inclination cannot be used to constrain the system masses.

Torres et al. (2020) established the rotational line broadening of the
companion of $v_{\rm rot}\sin i=84\pm 5$ km s-1, which in combination of the
radial velocity semi-amplitude ($K_{2}=417.7\pm 3.9$ km s-1) yields (assuming
cororation and Roche-lobe filling) the mass ratio of $q=M_{2}/M_{1}\approx
0.072\pm 0.012$, where $M_{1}$ and $M_{2}$ are the masses of the BH and its
companion star. These uncertainties are 1-$\sigma$. Torres et al. (2020),
following Marsh et al. (1994), obtained 2-$\sigma$ limits of $5.73<M_{1}/{\rm
M}_{\sun}<8.34$ and $0.28<M_{2}/{\rm M}_{\sun}<0.77$ by combining the
1-$\sigma$ uncertainties in the measured orbital parameters with the extreme
limits for the range of allowed inclinations. On the other hand, here we have
performed Monte Carlo simulations assuming Gaussian distributions of $v_{\rm
rot}\sin i$ and $K_{2}$ (which give $q$ via eq. 4 of Ziółkowski & Zdziarski
2017), $P$, and a flat distribution of $\cos i$ within the range given above.
This gives us probablilistic estimates of $M_{1}$ and $M_{2}$ and the donor
radius, $R_{2}$ [$=P(v_{\rm rot}\sin i)/(2\pi\sin i)$]. The results are shown
in Figure 1, and they imply $M_{2}=0.49^{+0.10}_{-0.10}{\rm M}_{\sun}$,
$R_{2}=1.19^{+0.08}_{-0.08}{\rm R}_{\sun}$ and $M_{1}=6.75^{+0.64}_{-0.46}{\rm
M}_{\sun}$, where we give the median and the 68% confidence ranges. Since the
radii of main-sequence stars can be roughly approximated by $R_{\rm MS}/{\rm
R}_{\sun}\approx(M/{\rm M}_{\sun})^{0.9}$, the above range of the donor radius
implies that the donor is a moderately evolved star (a subgiant).

The donor average spectrum from observations by the Gran Telescopio Canarias
(GTC) during the quiescence after the outburst was found to approximately
match those of K3–5 V main-sequence templates (Torres et al., 2020). The
effective temperatures of those stars are given by Pecaut & Mamajek (2013) as
4840–4450 K. We have, however, re-analyzed that spectrum and found a strict
limit of $T>4200$ K based on the lack of molecular bands (TiO and CN) expected
at $T\leq 4200$ K. This limit approximately corresponds to a K6 stellar type,
and we use it in our analyses below.

Table 1: The minimum observed fluxes for MAXI J1820+070 Source | Band |  $\langle\lambda\rangle\,[\mu$m] |  $F_{\nu,{\rm obs}}\,[\mu$Jy] |  $\langle F_{\nu,{\rm obs}}\rangle\,[\mu$Jy] | $10^{A_{\lambda}/2.5}$ | Dates
---|---|---|---|---|---|---
PS1 | $y$ | 0.963 | $265\pm 10$ | $290\pm 3$ | 1.33 | 2011-09-11, 2012-03-30
PS1 | $z$ | 0.868 | $238\pm 1$ | $256\pm 1$ | 1.40 | 2013-04-24
PS1 | $i$ | 0.755 | $141\pm 6$ | $220\pm 1$ | 1.55 | 2013-05-31
PS1 | $r$ | 0.622 | $89.8\pm 4.5$ | $197\pm 1$ | 1.77 | 2013-05-31, 2013-06-19
PS1 | $g$ | 0.487 | $54.9\pm 3.9$ | $117\pm 1$ | 2.15 | 2012-06-11, 2013-06-10
ZTF | $r$ | 0.622 | $139\pm 5$ |  | 1.77 | 2021-02-09
ZTF | $g$ | 0.487 | $60.4\pm 4.1$ |  | 2.15 | 2020-10-13
Table 2: Selected infrared and optical observations of MAXI J1820+070 in quiescent states Source | Band |  $\langle\lambda\rangle\,[\mu$m] |  $F_{0}$ [Jy] | $m_{\lambda}$ |  $F_{\nu,{\rm obs}}\,[\mu$Jy] | $10^{A_{\lambda}/2.5}$ | Date | Reference
---|---|---|---|---|---|---|---|---
WISE | $W2$ | 4.6 | 3631 | $17.95\pm 0.06$ | $239_{-13}^{+14}$ | 1.02 | 2010-01-14–2010-07-17 | Cutri et al. (2021)
WISE | $W1$ | 3.35 | 3631 | $17.59\pm 0.04$ | $334_{-11}^{+12}$ | 1.04 | 2010-01-14–2010-07-17 | Cutri et al. (2021)
2MASS | $K$ | 2.22 | 655 | $15.12\pm 0.12$ | $588_{-61}^{+69}$ | 1.08 | 1999-07-23 | Cutri et al. (2003)
2MASS | $H$ | 1.60 | 1150 | $15.44\pm 0.10$ | $764_{-69}^{+75}$ | 1.13 | 1999-07-23 | Cutri et al. (2003)
2MASS | $J$ | 1.26 | 1580 | $15.87\pm 0.07$ | $711_{-42}^{+45}$ | 1.20 | 1999-07-23 | Cutri et al. (2003)
GSC2.3.2 | $I$ | 0.812 | 2681 | $17.24\pm 0.43$ | $341_{-111}^{+166}$ | 1.46 | 1989–2000 | Lasker et al. (2008)
USNO-B1.0 | $R$ | 0.658 | 3247 | $16.76\pm 0.30$ | $642_{-155}^{+204}$ | 1.71 | 1949–1965 | Monet et al. (2003)
GSC2.3.2 | $R$ | 0.658 | 3247 | $18.05\pm 0.42$ | $196_{-63}^{+69}$ | 1.71 | 1987–1999 | Lasker et al. (2008)
GTC | $r^{\prime}$ | 0.623 | 3631 | $17.61\pm 0.25$ | $328_{-67}^{+75}$ | 1.77 | 2019-06-08–2019-08-07 | Torres et al. (2019, 2020)
USNO-B1.0 | $B$ | 0.442 | 4067 | $17.94\pm 0.30$ | $271_{-65}^{+86}$ | 2.37 | 1949–1965 | Monet et al. (2003)
GSC2.3.2 | $B$ | 0.442 | 4067 | $18.87\pm 0.41$ | $115_{-36}^{+53}$ | 2.37 | 1987–2000 | Lasker et al. (2008)
Table 3: The average UV–optical spectrum of MAXI J1820+070 observed on 2020-07-20–23  $\nu\,[10^{14}$ Hz] | $16.0\pm 2.7$ | $13.5\pm 1.5$ | $11.7\pm 1.6$ | $8.76\pm 0.99$ | $6.82\pm 0.34$ | $5.51\pm 0.33$ | $4.87\pm 0.45$ | $4.01\pm 0.32$ | $3.36\pm 0.14$
---|---|---|---|---|---|---|---|---|---
$F_{\nu,{\rm obs}}\,[\mu$Jy] | $9.99\pm 1.17$ | $14.0\pm 2.0$ | $31.2\pm 3.3$ | $59.5\pm 5.9$ | $87.9\pm 11.3$ | $148\pm 4$ | $238\pm 15$ | $303\pm 6$ | $415\pm 10$
$10^{A_{\lambda}/2.5}$ | 5.39 | 7.65 | 4.28 | 2.92 | 2.27 | 1.84 | 1.62 | 1.34 | 1.12

Figure 2: The extinction-corrected spectra of the binary during its quiescent
states assuming $E(B$–$V)=0.23$. The red and blue symbols give the minimum
fluxes measured by PS1 and ZTF, respectively; see Table 1. The black error
bars give the fluxes from the observations listed in Table 2. The green error
bars show the average spectrum obtained by Poutanen et al. (2022), see Table
3. The black solid curve show the blackbody with $R_{2}=1.11{\rm R}_{\sun}$,
$D=3$ kpc and $T=4230$ K, whose $i$-band flux is at the upper limit of the PS1
measurement, see Equation (3) below. The amplitude of the flux variability due
to ellipsoidal variability is $\lesssim\pm 10\%$ only.

###  2.2 Photometric properties

We have compiled available ultraviolet to mid-infrared measurements of the
binary during its quiescence from literature and public databases. These are
listed in Tables 1 to 3, which also provide references to the photometric
databases that were explored. In order to constrain the spectral distribution
of the donor star, the collected fluxes need to be corrected for interstellar
extinction. The total reddening through the Galaxy in the direction of the
source has been estimated by Schlafly & Finkbeiner (2011) as $E(B$–$V)=0.197$
and as $E(B$–$V)=0.23$ by Schlegel et al. (1998). Since the source is located
$\approx$0.5 kpc above the Galactic disk (see Section 2.3), the extinction
toward it is likely to equal the total one in the direction of the target. We
can also use the relationship between the extinction in the $V$-band ($A_{V}$)
and the H column density ($N_{\rm H}$) estimated from fitting X-ray data
(Güver & Özel, 2009), $N_{\rm H}=(2.21\pm 0.09)\times 10^{21}A_{V}\,{\rm
cm}^{-2}$. Here $A_{V}$ is given by $A_{V}=3.1E(B-V)$. The values of $N_{\rm
H}$ obtained from fitting XMM-Newton (Kajava et al., 2019), NICER (Bharali et
al., 2019) and the X-Ray Telescope (Burrows et al., 2000) onboard of Neil
Gehrels Swift (Wang et al., 2021) data for MAXI J1820+070 are
$1.4^{+0.3}_{-0.3}\times 10^{21}$, $1.6^{+0.3}_{-0.2}\times 10^{21}$,
$1.73^{+0.10}_{-0.07}\times 10^{21}$ cm-2, respectively. These values
correspond (including the uncertainty of the conversion) to the range of
$E(B$–$V)\approx 0.20$–0.26. To calculate the extinction at other wavelengths,
$A_{\lambda}$, we used $A_{V}$ and the fitting formulae of Cardelli et al.
(1989). Then, the observed magnitudes, $m_{\lambda}$, can be transformed into
the the observed, $F_{\nu,{\rm obs}}$, and unabsorbed, $F_{\nu}$, fluxes, as

$\displaystyle F_{\nu,{\rm obs}}$ $\displaystyle=F_{0}10^{-m_{\lambda}/2.5},$
(1) $\displaystyle F_{\nu}$ $\displaystyle=F_{\nu,{\rm
obs}}10^{A_{\lambda}/2.5},$ (2)

where $F_{0}$ is the flux corresponding to a zero magnitude.

Figure 3: Top: the extinction-corrected [assuming $E(B$–$V)=0.23$] light
curves of the binary from the ZTF. The red and blue symbols correspond to the
$r$ and $g$ filters, respectively. The two magenta arrows indicate the middle
times of the observations of Torres et al. (2019, 2020) and of Poutanen et al.
(2022). Bottom: the 15–50 keV daily-averaged count rate from the Neil Gehrels
Swift BAT. This rate is shown only for $>$0.004 cm-2 s-1, at which boundary a
typical fractional error is $\lesssim$0.3. We see that major reflares of the
source were present until around MJD 58900, then followed by weaker activity
in both optical and X-rays.

We first consider spectral data from Pan-
STARSS1111https://catalogs.mast.stsci.edu/panstarrs/ Data Release 2 (PS1 DR2),
which give measurements in the $y$, $z$, $i$, $r$ and $g$ bands for years
previous to the outburst. We have searched for the overall minimum fluxes in
each photometric band, using the psfFlux and psfFluxErr values in the
Detection Table provided in PS1 DR2. Those fluxes were obtained using an
automated pipeline; thus single detections at the faint end may be not real,
but arise from a systematic noise (Chambers et al., 2016), and be
indistinguishable from real detections. In order to avoid them, we have
applied a stringent selection criterion. Namely, we used only double
detections from subsequent exposures separated by $\lesssim$0.5 h and with the
fluxes differing by $<$25%. The results are given in Table 1 and plotted in
red in Figure 2 after correcting for the extinction assuming $E(B$–$V)=0.23$.
For comparison, Table 1 also gives the average fluxes from the DR2 stacked
images, which are significantly higher than the minimum fluxes in the $i$, $r$
and $g$ bands.

Next, we have obtained the $r$ and $g$ light curves observed by the Zwicky
Transient
Facility222https://irsa.ipac.caltech.edu/docs/program_interface/ztf_lightcurve_api.html
(ZTF; Bellm et al. 2019), covering the discovery outburst and subsequent decay
up to 2021 August. They are shown in Figure 3. We see that several mini-
outbursts or reflares occurred after the end of the main outburst event, with
the last major reflare in 2020 April. For comparison, we also show the hard
X-ray light
curve333https://swift.gsfc.nasa.gov/results/transients/weak/MAXIJ1820p070/
from the Burst Alert Facility (Barthelmy et al. 2005; BAT) onboard of Neil
Gehrels Swift. We see that the optical reflares have been accompanied by X-ray
activity. From the ZTF light curves, we have obtained the weakest fluxes
measured in each filter, which are given in Table 1 and are plotted in blue
symbols in Figure 2.

Then, we have compiled some other available optical and infrared measurements
of MAXI J1820+070 during quiescence. We list them in Table 2 and plot in
Figure 2 with black error bars. Table 2 also gives the values of $F_{0}$. Most
of the fluxes are higher than the minima from PS1. In particular, we analyzed
the observations made during the states near quiescence in the $r$ band (of
the SDSS photometric system) reported in Torres et al. (2019, 2020). After
calibrating the photometry of MAXI J1820+070 against the PSF mean magnitude of
a non-variable star in the field, the range of those magnitudes is
$r^{\prime}\approx 17.87\pm 0.02$–$17.35\pm 0.02$, and its midpoint is
$r^{\prime}\approx 17.6$. Finally, we plot the average optical and UV fluxes
obtained by Poutanen et al. (2022), see Table 3. The arrows in Figure 3
indicate the middle times of the observations of Torres et al. (2019, 2020)
and of Poutanen et al. (2022).

Figure 4: The region in the $D$–$R_{2}$ space allowed by the constraints on
$R_{2}$ from the binary parameters, on $D$ from the Gaia parallax, and on $T$
from the blackbody [for $E(B$–$V)=0.23$] and spectral limits, see Equations
(3–4). The horizontal blue solid and dashed lines correspond to the lower and
upper limit on $R_{2}$ (Section 2.1), respectively; the vertical black solid
line gives the limit of $D\leq 3.5$ kpc, and the diagonal red solid line
corresponds to the limiting $D/R_{2}$ from the PS1 $i$-band flux and $T>4200$
K at $A=1$. If an accretion disk still contributes to that emission, $A>1$,
and the red line moves in the direction of the red arrow becoming steeper. The
allowed region is within the lower triangle delineated by the three solid
lines. The limit on $T$ increases diagonally perpendicular to the red line
from 4200 K on it to 4540 K at the lower right corner of the allowed space.
Three evolutionary models of Section 3, B, B05 and C, satisfy these
constraints, and are shown by the black solid circles with arrows (moving
horizontally for $A>1$).

The PS1 DR2 minimum fluxes shown in Figure 2 constrain the stellar
contribution, which we approximate as a blackbody, $F_{\nu}^{*}=\pi
B_{\nu}(T)(R_{2}/D)^{2}$, where $B_{\nu}(T)$ is the blackbody intensity and
$T$ is the effective temperature. The most restrictive measurement is that in
the $i$ band, for which we take its upper limit (Table 1) and assume
$E(B$–$V)=0.23$. We have found that the blackbody relationship for this
monochromatic flux can be locally approximated as a power law of
$T\propto(D/R_{2})^{0.45}$. The constraint, together with the temperature
limit from our analysis of the average spectrum of Torres et al. (2019, 2020),
can be written as

$\displaystyle\frac{D/2.95\,{\rm kpc}}{R_{2}/1.11{\rm R}_{\sun}}$
$\displaystyle\gtrsim A\left(\frac{T}{4200\,{\rm K}}\right)^{2.22},$ (3)
$\displaystyle T$ $\displaystyle>4200\,{\rm K},$ (4)

where Equation (3) is scaled to the lower limit on the stellar radius, see
Section 2.1. Here the constant $A$ accounts for the accretion disk being
expected to be present at some level during the entire quiescence (see, e.g.,
the models of Dubus et al. 2001), in which case $A>1$. On the other hand, the
flux measurements may be affected by ellipsoidal variability, whose full
amplitude we estimate as $<$0.2 mag, i.e., less than $\pm 10\%$ with respect
the average flux. This estimate is based on the ellipsoidal variability of the
BH LMXB A0620–00, which has the mass ratio $q=0.067\pm 0.01$ (Marsh et al.,
1994) and $66\arcdeg.5<i<73\arcdeg.5$ (Haswell et al., 1993), consistent
within the uncertainties with those of MAXI J1820+070. The full amplitude of
the ellipsoidal variability of A0620–00 was measured by Marsh et al. (1994) as
0.13 mag. They also performed detailed simulations of this effect for a range
of the inclinations, and their table 4 gives the theoretical range of
0.15–0.18 mag at $i=70\arcdeg$. Given the relative weakness of this effect, we
neglect it. We also set $A=1$. The fiducial distance of 3 kpc and the minimum
$R_{2}$ correspond to $T\leq 4230$ K. We plot a blackbody spectrum at $T=4230$
K in the solid black curve in Figure 2. This spectrum gives a $\approx$26%
contribution [at $E(B$–$V)=0.23$] to the average flux in the $r$ band of
Torres et al. (2019, 2020). This is moderately higher than the stellar
contribution of 16–21% estimated by them.

Equations (3–4) for $R_{2}\gtrsim 1.11{\rm R}_{\sun}$ imply a $D\gtrsim 2.95$
kpc, in agreement with all the constraints on $D$ listed in Section 2.1. On
the other hand, these equations do not constrain $D$ from above. We take here
$D\lesssim 3.5$ kpc from the Gaia parallax as a tentative conservative upper
limit. While the limit of Wood et al. (2021) is stringent at face value, it
relies on the two ejecta being exactly symmetric, which may not be the case,
and 3.5 kpc is within the $2\sigma$ uncertainty of the radio parallax, see
Section 2.1. We plot the resulting joint constraints on $D$ and $R_{2}$ in
Figure 4. At $D=3.5$ kpc and $R_{2}=1.11{\rm R}_{\sun}$, we have the maximum
$T\approx 4540$ K, corresponding to a K5 stellar type. This satisfies the
original constraint of Torres et al. (2020). We note that the above
constraints favor low donor radii if $D\lesssim 3$ kpc. At $D\leq 3$ kpc and
$T=4200$ K, $R_{2}\leq 1.13{\rm R}_{\sun}$. Note that the above constraints
can still be relaxed if there are some systematic error on the minimum PS1
fluxes, present in spite of our strict selection procedure.

###  2.3 Metallicity

MAXI J1820+070 appears to be a relatively old system and thus may have come
from an environment with a subsolar metallicity. Its position in the Cartesian
coordinates ($X,\,Y,\,Z$) centered on the Sun and the velocities calculated
from the coordinated proper motion, ${\rm pmRA}=-3.093\pm 0.091$ mas/yr and
${\rm pmDec}=-6.283\pm 0.094$ mas/yr (Gaia EDR3, Gaia Collaboration et al.
2021), the systemic velocity of $\gamma=-22$ km/s (Torres et al., 2019) and
the adopted distance of $D=3$ kpc, are

$\displaystyle X=2393\,{\rm pc},\,Y=1730\,{\rm pc},\,Z=529\,{\rm pc},$ (5)
$\displaystyle U=40.85\,{\rm km/s},V=-93.40\,{\rm km/s},\,W=-4.19\,{\rm
km/s},$

where $U,\,V,\,W$ are velocities along the same axes, with $X,\,U$ positive
towards the Galactic center, $Y,\,V$ positive in the direction of the Sun
motion around the Galaxy, and $Z,\,W$ positive out of the plane of the Galaxy.
MAXI J1820+070 is thus located in the thick Galactic disk as indicated by its
distance from the Galactic plane, $Z=529$ kpc, and a high peculiar velocity,
$V_{\rm pec}=94$ km/s, moving almost parallel to the Galactic plane. The
Galactic $U,\,V,\,W$ velocities also locate it among the extended thick-disk
population objects in Toomre diagram (e.g. fig. 1 in Feltzing et al. 2003).
The Galactocentric distance of MAXI J1820+070 is 5.9 kpc, which locates it in
the inner Galactic disk, where the average value of the metallicity is
$\langle[{\rm Fe/H}]\rangle=-0.55\pm 0.17$ (e.g. Bensby et al. 2011). The
subsolar metallicity of the donor is further supported by relatively strong Ca
I and Ti I absorption lines compared to Fe I lines in the spectrum presented
in fig. 3 of (Torres et al., 2019), similar to other thick disk objects (e.g.
Feltzing et al. 2003).

We also note that high peculiar velocities of BH LMXBs are often attributed to
a BH natal kick velocity, see, e.g. Atri et al. (2019). That study included
MAXI J1820+070 and three other systems with no known (at that time) systemic
velocities and accurate distances, using instead a hypothetical kick velocity
distribution modelled by Monte Carlo simulations. Based on them, they
tentatively preferred the natal kick origin of the peculiar velocity in MAXI
J1820+070. Still, they also note that most LMXBs are relatively old systems,
and thus may have come from low metallicity environments like globular
clusters.

On the other hand, metallicities higher than solar were found in a number of
donor stars in LMXBs (see review in Casares et al. 2017). Still, this does not
appear to be universal. One example is the binary V934 Her/4U 1700+24
consisting of a red giant accreting on a neutron star. Detailed photospheric
abundance analysis for this system resulted in a slightly subsolar
metallicity, ${[\rm Fe/H}]\approx-0.5$ and $[\alpha/{\rm Fe}]=+0.27$ (where
$\alpha$ represents the average of the Mg, Si, and Ca elements), and the
$[\alpha/{\rm Fe}]$ versus [Fe/H] relation close to the mean one without any
notable peculiarity to this object (Hinkle et al., 2019). Similar
$\alpha$-element and Fe abundances were found in many symbiotic giants with
white dwarf companions (Gałan et al., 2016, 2017).

###  2.4 Average mass transfer rate

The bolometric flux of the source varied within only a factor of 2–3 during
most of the 2018 discovery outburst, i.e., $\approx$200 d, see, e.g., Shidatsu
et al. (2019), Fabian et al. (2020), namely within $\approx(0.5$–$1.5)\times
10^{-7}$ erg cm-2 s-1. In order to estimate the total fluence of that
outburst, we adopt the flux of $1.0\times 10^{-7}$ erg cm-2 s-1 during 200 d.
Assuming $D=3$ kpc and the accretion efficiency of $\epsilon=0.1$, we obtain
the accreted mass of $M_{\rm accr}\approx 2\times 10^{25}$ g. However, given
the uncertainties on the fluence, distance and accretion efficiency, the
uncertainty of this mass is probably by a factor of two.

After the 2018 discovery outburst, two historical outbursts in 1898 and 1934
were discovered by examining publicly available photographic plates (Kojiguchi
et al., 2019). Based on these data, the typical interval between outbursts of
this source was estimated by those authors as 40 yr. This implies that the
outburst within 1970–1980 was missed by observers of the period. However,
intervals between outbursts of BH LMXBs are often unequal, and it is also
possible that there was no outburst since 1934, giving the interval from the
previous outburst of 84 yr. If we assume that the mass accreted onto the BH in
2018 equals to that transferred from the donor during the interval since the
previous outburst (which theoretical assumption is clearly uncertain), the
average mass transfer rate is $-\dot{M}_{2}\approx 1.2\times 10^{-10}{\rm
M}_{\sun}$/yr, with an uncertainty, which we estimate to be by a factor of
two. Alternatively, the estimate of $-\dot{M}_{2}$ would increase to $\approx
2.6\times 10^{-10}{\rm M}_{\sun}$/yr if the actual interval between outbursts
was 40 yr.

##  3 The evolutionary status

We use the Warsaw stellar evolution code of Paczyński (1969, 1970) described
in Pamyatnykh et al. (1998) and Ziółkowski (2005). Our version of the code is
calibrated to yield the H mass fraction of $X=0.74$, the metallicity of
$Z=0.014$ (we use here the symbols $X$ and $Z$ in a different meaning than in
Section 2.3), and the mixing length parameter of 1.55 for the Sun at the solar
age. The method of the evolutionary calculations in this work follow those
used in Ziółkowski & Zdziarski (2017, 2018) and Zdziarski et al. (2019). We
refer the reader to Zdziarski et al. (2019) for details of the method.

Figure 5: The radii of partially stripped, moderately evolved stars with solar metallicity vs. the mass of the He core for total donor masses of $M_{2}=0.3$, 0.4, 0.5, 0.6, 0.7 and $0.8{\rm M}_{\sun}$. They are shown (from bottom to top) by the green, red, yellow, violet, blue and orange thick solid curves, respectively. The evolution proceeds from left to right at the constant $M_{2}$. The horizontal dashed lines with the corresponding colors (from bottom to top) show the radii of the Roche lobe around the donor for the above values of $M_{2}$ (almost independent of $M_{1}$). The crossings of the corresponding evolutionary tracks and horizontal lines determine the positions of the models.  Table 4: The results of the evolutionary calculations for the donor  Model | $\displaystyle{\frac{M_{2}}{{\rm M}_{\sun}}}$ | $\displaystyle{\frac{R_{2}}{{\rm R}_{\sun}}}$ | $\displaystyle{\frac{L_{2}}{{\rm L}_{\sun}}}$ | $T\,[{\rm K}]$ | $\displaystyle{\frac{M_{\rm c}}{{\rm M}_{\sun}}}$ | $\displaystyle{-\dot{M}_{2,10}}$
---|---|---|---|---|---|---
A | 0.3 | 1.012 | 0.278 | 4169 | 0.070 | 0.43
A2 | 0.3 | 1.012 | 0.219 | 3924 | 0.290 | 0.155
B05 | 0.4 | 1.114 | 0.452 | 4492 | 0.050 | 1.0
B | 0.4 | 1.114 | 0.347 | 4200 | 0.065 | 0.68
B2 | 0.4 | 1.114 | 0.273 | 3948 | 0.084 | 0.27
C05 | 0.5 | 1.200 | 0.587 | 4621 | 0.042 | 1.3
C | 0.5 | 1.200 | 0.437 | 4289 | 0.060 | 0.91
D | 0.6 | 1.275 | 0.560 | 4430 | 0.051 | 1.1
D2 | 0.6 | 1.275 | 0.414 | 4102 | 0.130 | 0.49
E | 0.7 | 1.343 | 0.749 | 4643 | 0.034 | 1.25
F | 0.8 | 1.404 | 1.019 | 4903 | 0.003 | 1.55

We have first followed the evolution in the core mass–stellar radius plane of
partially stripped stars for total masses of $M_{2}=0.3$, 0.4, 0.5, 0.6, 0.7
and $0.8{\rm M}_{\sun}$ and at the solar metallicity. The results are shown in
Figure 5 and Table 4, where these models are denoted by letters A, B, C, D, E
and F, respectively. In Figure 5, the evolution proceeds from left to right at
the constant $M_{2}$. The evolutionary engine lies in the hydrogen burning
shell, which adds newly synthesized helium to the core, increasing its mass.
The horizontal thin lines with the corresponding colors (from bottom to top)
show the radii of the Roche lobe around the donor for the above values of
$M_{2}$, and $M_{1}=7{\rm M}_{\sun}$. The crossings of the corresponding
evolutionary tracks and horizontal lines determine the positions of our donor
models. The relatively small core masses of those models confirm that the
donor is in an early subgiant phase, with the radius only a factor of about
two larger than a main-sequence star of the same mass (as noticed in Section
2.1). Given that the donor metallicity is relatively uncertain (Section 2.3),
we also calculated some models for a metallicity of half solar, $Z=0.007$ (B05
and C05), and twice solar, $Z=0.028$, (A2, B2 and D2), whose results are also
given in Table 4. The seen anticorrelation of the metallicity and temperature
in our evolutionary models is a well-known effect, resulting from an increase
of the opacities with the metallicity, present also in main sequence stars.

Figure 6: The rates of the evolutionary change of the donor radius (red curve)
and that of its Roche lobe (green curves) as functions of the donor mass loss
rate, $-\dot{M}_{2}$. The conservative solution corresponds to model B05, with
the averaged accretion rate on the BH of $\langle\dot{M}_{1}\rangle=1.0\times
10^{-10}{\rm M}_{\sun}$ yr-1. The results of our evolutionary model include
now the effect of mass outflow from the outer layers of the star. The green
curves show ${\rm d}\ln R_{\rm 2,L}/{\rm d}\ln M_{2}$ for $\alpha=0$, 1, 1.1,
and 1.2, from bottom to top. The values of $\beta$ along those curves are
given by $\langle\dot{M}_{1}\rangle/|\dot{M}_{2}|$, i.e., $\beta$ =1 at the
beginnings of the green curves. The intersection of the red curve and the top
green one gives a possible self-consistent and non-conservative model of MAXI
J1820+070, see Section 3.

The assumed BH mass, $M_{1}$, influences only one model parameter, namely the
rate of the mass transfer, $-\dot{M}_{2}$. However, this effect is rather
weak. For models A and F, changing $M_{1}$ to $3.17{\rm M}_{\sun}$ and
$11.1{\rm M}_{\sun}$, respectively (to keep the mass ratio equal 0.072), would
change $-\dot{M}_{2}$ from 4.3 to $4.4\times 10^{-11}{\rm M}_{\sun}$/yr, and
from 1.55 to $1.50\times 10^{-10}{\rm M}_{\sun}$/yr, respectively.

We compare now the models of Table 4 with the Monte Carlo constraints obtained
in Section 2.1 and other constraints obtained in Sections 2.2, 2.3 and 2.4. As
shown based on the fluxes measured before the outburst (Section 2.2), an
approximate maximal stellar blackbody contribution at $D=3$ kpc has $T=4200$ K
and $R_{2}=1.13{\rm R}_{\sun}$. This closely corresponds to our model B, which
support an approximate solar metallicity of the donor. If we still impose the
spectral type to be within K5–K3 (which requires $D\sim 3.5$ kpc), the
temperature has to be larger, which approximately corresponds to our model B05
with $T=4492$ K. This model has half-solar metallicity, which agrees with the
finding of Section 2.3 that MAXI J1820+070 is likely to belong to extended
Galactic thick disk population. On the other hand, it is in tension with the
X-ray spectral fitting of Zdziarski et al. (2021), which favor the metallicity
to be at least solar. Our model C, with $M_{2}=0.5{\rm M}_{\sun}$ and
$R_{2}=1.2{\rm R}_{\sun}$, also satisfies our constraints, yielding $T=4289$ K
and $D\gtrsim 3.3$ kpc. These three models are shown in the $R_{2}$–$D$
parameter space in Figure 4. We see that the constraints shown on Figure 4
allow the metallicity to be only slightly above solar. On the other hand, our
models with metallicity twice solar, A2, B2 and D2, have the temperatures of
3924 K–4102 K, which are below our strict lower limit of 4200 K.

The calculated mass-loss rate for model B05 is $-\dot{M}_{2}\approx 1.0\times
10^{-10}{\rm M}_{\sun}$/yr, which agrees well with that estimated in Section
2.4 of $1.2\times 10^{-10}{\rm M}_{\sun}$/yr with an uncertainty by a factor
of two. For model B, $-\dot{M}_{2}$ is only slightly lower.

The above agreement implies that the mass loss rate approximately equals the
accretion rate averaged over the interval since the previous outburst,
$\langle\dot{M}_{1}\rangle$, i.e., the mass transfer is conservative. However,
this assumption is uncertain, with outflows and not full exhaustion of the
disk during outburst being possible. Indeed, mass loss from a disk wind in
MAXI J1820+070 was found by Muñoz-Darias et al. (2019). This motivates us to
consider also models with non-conservative mass transfer. We note that taking
this effect into account does not change the model parameters of the donor. It
only increases the model values of $-\dot{M}_{2}$. We consider here non-
conservative mass transfer for model B05, following the approach of Ziółkowski
& Zdziarski (2018). Such transfer is described by two parameters, $\alpha$ and
$\beta$. As in Ziółkowski & Zdziarski (2018), we define $\alpha$ as the
specific angular momentum of the mass leaving the system in units of the
specific angular momentum of the donor measured from the center of mass
(Verbunt, 1993), with $\alpha\lesssim 1$ (as discussed, e.g., in Ziółkowski &
Zdziarski 2018). The parameter $\beta$ is defined as the fraction of the mass
lost by the donor that is accreted onto the accretor (Rappaport et al., 1982),
i.e., $-\dot{M}_{2}=\langle\dot{M}_{1}\rangle/\beta$.

We assume $\langle\dot{M}_{1}\rangle=1.0\times 10^{-10}{\rm M}_{\sun}$ yr-1.
The results are shown in Fig. 6. We compare here the rates of the Roche-lobe
change for $\alpha=0$, 1, 1.1 and 1.2 with the rate implied by our
evolutionary model as a function of
$\beta=\langle\dot{M}_{1}\rangle/|\dot{M}_{2}|$. Apart from the conservative
solution ($\beta=1$), a non-conservative solution is possible for
$\beta\approx 0.05$ and $\alpha=1.2$. A caveat for it is its relatively large
$\alpha$, requiring the outflow to occur far from the BH, around the disk
tidal radius (see fig. 1 of Ziółkowski & Zdziarski 2018). Thus, the simplest
and most likely solution remains the conservative case.

##  4 Summary

We have used the 1-$\sigma$ deviations for the binary parameters of MAXI
J1820+070 and a uniform distribution in $\cos i$ for the range of inclinations
given in Torres et al. (2020) to calculate the donor star mass and radius. We
obtain 68% confidence values of $M_{2}=0.49^{+0.10}_{-0.10}{\rm M}_{\sun}$ and
$R_{2}=1.19^{+0.08}_{-0.08}{\rm R}_{\sun}$ (as well as
$M_{1}=6.75^{+0.64}_{-0.46}{\rm M}_{\sun}$ for the BH mass).

We have compiled available optical and infrared measurements of MAXI J1820+070
in quiescence, which constrain the spectrum of the donor. The strongest
constraint is given by the minimum fluxes from PS1 DR2 in the $i$ and $r$
bands, which gives the maximum allowed blackbody temperature of $T<4230$ K at
the lower limit on the donor radius allowed by the binary parameters
($R_{2}=1.11{\rm R}_{\sun}$), $D\approx 3$ kpc and $E(B$–$V)=0.23$. This $T$
is at our lower limit of the temperature compatible with the average GTC
spectrum of $T>4200$ K, and it is less than those implied by the original
classification of the stellar type of the donor within the range of K3–5
(Torres et al., 2020). Requiring the latter can be reconciled with our
constraints if $D\sim 3.5$ kpc, yielding $T\lesssim 4540$ K, of a K5 stellar
type. The distance of $D=3.5$ kpc is allowed by the Gaia parallax, but it is
larger than those obtained from the radio parallax and the moving ejecta
method.

We then considered the source location in the Galaxy and peculiar velocity,
which indicate MAXI J1820+070 belongs to the extended thick-disk population.
That population is characterized by the metallicity less than solar. We also
estimated the fluence during the 2018 outburst. Using the time elapsed since
the previous detected outburst of the system, we evaluate the average mass
accretion rate (equal to the average transfer rate for conservative accretion)
of $-\dot{M}_{2}\sim 10^{-10}{\rm M}_{\sun}$/yr.

We finally considered a range of stellar evolutionary models that may
correspond to MAXI J1820+070. Our model B (see Table 4) approximates the above
constraints at the limiting $T=4200$ K, the donor mass of $M_{2}=0.4{\rm
M}_{\sun}$ and the solar metallicity (which approximately agrees with the
X-ray spectral fitting results of Zdziarski et al. 2021) and requires $D\geq
3$ kpc. An alternative at $D\gtrsim 3.3$ kpc is our model C with $T\approx
4300$ K. Another possibility at $D\gtrsim 3.3$ kpc is our model B05, with
$T\approx 4500$ K and the metallicity half solar, which, in turn, agrees with
the location of the source in the Galaxy. Thus, we cannot unambiguously
constrain the donor metallicity. Still, we find models with the metallicity
twice solar to be ruled out.

We also considered evolutionary models with non-conservative accretion.
However, we have found they require an unrealistically large specific angular
momentum of the mass leaving the system.

## Acknowledgments

We thank Juri Poutanen for providing us with their average optical-ultraviolet
spectrum of MAXI J1820+070, and the referee for valuable suggestions. We
acknowledge support from the Polish National Science Centre under the grants
2015/18/A/ST9/00746, 2019/35/B/ST9/03944 and 2017/27/B/ST9/01940. AAZ
acknowledges support from the International Space Science Institute (Bern).
MAPT and JCV have been supported by the Spanish MINECO under grant
AYA2017-83216-P and PID2020-120323GB-I00. MAPT acknowledge support via a Ramón
y Cajal Fellowship RYC-2015-17854.

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