11institutetext:  Nicolaus Copernicus Astronomical Center, The Polish Academy
of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Testing the presence of a dormant black hole inside HR 6819

A. Romagnolo A. Olejak A. Hypki G. Wiktorowicz K. Belczynski 11111111

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. HR 6819 was reported in Rivinius et al. (2020) to be a triple system
with a non-accreting black hole (BH). The inner binary system was defined as a
B3 III type star ($5-7{\rm\leavevmode\nobreak\ M}_{\odot}$ star estimated to
be at the end of its main sequence) and a dormant BH
($>4.2{\rm\leavevmode\nobreak\ M}_{\odot}$). The period of the inner binary
was estimated to be $\sim 40$ days with an eccentricity in the range
$0.02-0.04$. As the inner binary is not resolved, the third component may
actually be just spatially coinciding with the inner binary.

Aims. In this study we test whether the system’s inner binary can be
reconstructed using the isolated binary evolution in the Galactic field or
through the dynamical evolution within globular star clusters. Our goal is to
understand the formation of the HR 6819 inner binary.

Methods. To simulate the inner binary evolution we assumed that the influence
of the third body on the inner binary is negligible. We created synthetic
populations of BH-MS binaries for the Galactic disk and the Galactic globular
clusters to compare to the reported parameters of HR 6819 inner binary. We
have adopted very optimistic input physics, in terms of common envelope
evolution and BH formation, for the formation of binaries like the reported
inner HR 6819 binary.

Results. Despite our optimistic assumptions we cannot form systems like the
inner HR 6819 binary in globular clusters. Even with our extreme assumptions,
the formation of HR 6819 inner binary in the Galactic field population is not
expected.

Conclusions. We argue that HR 6819 does not host a dormant BH in support of
the conclusions reached by El-Badry & Quataert (2021); Bodensteiner, J. et al.
(2020) and by Safarzadeh et al. (2020).

###### Key Words.:

Binaries - Stars: black holes - Stars: individual: HR 6819

##  1 Introduction

Black holes (BH) in binary systems are predicted to be around 10$\%$ of the
whole Milky Way BH population, and most of them are BH-BH systems (Olejak et
al., 2020a). BHs with non-compact companions can be detected through
interactions with them, but they are possibly a very small fraction of the
total Galactic population. So far only $\sim$20 Galactic BH systems have been
detected (Casares, 2007; Casares & Jonker,
2014)111https://stellarcollapse.org/sites/default/files/table.pdf. Most of
them are close X-ray binary systems transferring matter through an accretion
disk. For recent reviews on stellar-origin BHs see Bambi (2018). BHs may also
be detected as gravitational wave signal emitters (Abbott et al., 2016) and in
non-interacting binaries using astrometric and spectroscopic observations of
their companion stars. Single BHs can also be potentially detected through
microlensing (e.g. Wyrzykowski et al., 2016; Wiktorowicz et al., 2019), self-
lensing (Wiktorowicz et al., 2021) or while accreting from interstellar medium
(Tsuna et al., 2018). Theoretically Hawking radiation (Hawking, 1974) may
possibly allow for a detection of thermal radiation from the immediate
vicinity of a BH.

HR 6819 is reported to be a triple system harbouring a BH candidate (Rivinius
et al., 2020). The BH candidate mass was estimated on the basis of radial
motions of the visible star in the inner binary. The spectral type of the
companion star has been classified as B3 III. Using single-star evolutionary
tracks (Ekström et al., 2012) and observational data (Hohle et al., 2010), the
most likely range for the star was estimated to be between 5 and
$7{\rm\leavevmode\nobreak\ M}_{\odot}$ and the star was estimated to be at the
end of its main sequence. The lower limit for the BH candidate mass was set at
$4.2{\rm\leavevmode\nobreak\ M}_{\odot}$. The star was reported to be roughly
1 ${\rm\leavevmode\nobreak\ M}_{\odot}$ more massive than its unseen
companion, while the inner binary is supposed to have a period of $40.333\pm
0.004$ and an eccentricity of $0.03\pm 0.01$ . The outer orbit with the Be
star (with a mass of $\sim 6{\rm\leavevmode\nobreak\ M}_{\odot}$) is instead
unconstrained. On top of that, the outer star does not show any significant
radial velocity variation and may actually be just spatially coinciding with
the inner binary. Neither the FEROS spectrometer (Rivinius et al., 2020), nor
Gaia have a sufficient resolution to resolve the full triple system. The age
of the whole triple system was estimated to be between 15 and 75 Myr.

Other possible interpretations of the observational data cannot be excluded.
For instance, spectral features of HR 6819 may be also explained but a close
binary in a 4-day period, composed of two A0 stars of $\sim$ 2.3 M⊙ each with
no BH (Mazeh & Faigler, 2020). In an independent analysis of the system
spectrum El-Badry & Quataert (2021) also proposed the absence of a dormant BH.
In their interpratation HR 6819 would be composed of two optical components:
the B3 star and its Be stellar companion. Also Bodensteiner, J. et al. (2020)
proposed a similar solution. According to their study HR 6819 could be a
binary hosting a stripped B-type star and a rapidly-rotating Be companion.
Post-interaction lower-mass stars (exposed hot cores) can also mimic higher-
mass isolated stars. The temperature of B-type subdwarf can be higher than
$20,000$ K despite having mass $\lesssim 1{\rm\leavevmode\nobreak\ M}_{\odot}$
(e.g. Heber, 2009).

Also Safarzadeh et al. (2020) argued on the unlikelihood of a dormant BH
inside the HR 6819 inner binary. Their work was based on three different
considerations. (i) The small probability of having such a triple system
configuration in terms of mass and spectral type in the Galactic population of
stars. (ii) The extremely narrow range of orbital separations (both for the
inner and the outer orbit) for the system to be dynamically stable. (iii) The
very narrow set of BH kicks and ejected mass at the BH formation that fits the
observational constraints in terms of the orbital parameters for the inner
binary. Additionally, they showed that a chance alignment between a B III star
and a Be star could be a potential explanation for the observational data.

Stevance et al. (2021) pointed out, in their letter regarding the potential
existence of a dormant BH in NGC 1850, that the study of binary evolution in
stellar populations can be beneficial in order to test the strength of any
claim regarding the presence of an unseen BH in binary systems. In our study
we therefore try to reproduce the originally reported orbital parameters of
the HR 6819 within the framework of isolated binary evolution and dynamical
formation in globular clusters. In both cases we test very optimistic models
that can maximise the likelihood of reconstructing the HR 6819 inner binary
reported parameters (the optimal physics for these scenarios is described in
Sec. 3.1). In this way we will either set a best case (despite being
unorthodox in terms of input physics) scenario if we find any evolutionary
solution for the reported parameters, or we will add additional support for
the absence of a dormant BH in HR 6819 in case of no evolutionary solution. We
assumed that HR 6819 is a triple system with the third component either
unbound to the inner binary or the outer orbit so wide that it does not affect
its evolution. We consequently adopted the estimates from Rivinius et al.
(2020) as reference parameters of HR 6819 inner binary.

##  2 Method

###  2.1 StarTrack

In our study we used the StarTrack222https://startrackworks.camk.edu.pl
population synthesis code (Belczynski et al., 2002, 2008), which allows the
user to simulate the isolated binary evolution of the system with a wide
variety of initial conditions and physical parameters. The most up to date
description of StarTrack’s standard physical parameters and approaches, as
well as the adopted model for star formation rates and the metallicity
distribution of the Universe, can be found in Belczynski et al. (2020), with
two updates in §2 of Olejak et al. (2020b). We used a rapid SN engine model
(Belczynski et al., 2012; Fryer et al., 2012).

We introduced a significant modification compared to the StarTrack standard
assumptions, which is that all BHs (regardless of their mass) are formed via
direct collapse without asymmetric mass ejection in SN explosions. In such
scenario, BHs do not receive natal kicks except for potential Blaauw kicks
(Blaauw, 1961).

In our simulations we tested several scenarios with different values of
metallicity Z (see Table 3), neutrino mass loss during BH collapse $F_{\rm
loss}$ (in terms of percentage of total mass loss), efficiency of orbital
energy loss for CE ejections $\alpha$, the initial system semi-major axis a0,
and the stellar masses of progenitors at ZAMS, respectively MZAMS,a and
MZAMS,b.

Table 1: Tested parameter space for the isolated binary evolution of HR 6819.

Parameter | Min | Max
---|---|---
Z | 0.01 | 0.02
$F_{\rm loss}$ | 1% | 20%
$\alpha$ | 0.5 | 5
MZAMS,a [M⊙] | 10 | 150
MZAMS,b [M⊙] | 0.08 | 150
a0 [R⊙] | 2000 | 10000
333Here Z-metallicity, $F_{\rm loss}$-neutrino mass loss during BH formation,
$\alpha$-efficiency of orbital energy loss for CE ejection, $M_{\rm ZAMS,a/b}$
\- masses at ZAMS of respectively the BH progenitor star and its stellar
companion, and $a_{\rm{0}}$ \- initial semi-major axis.

###  2.2 MOCCA

To investigate a possible dynamical interactions influence on the formation of
HR 6819 within Galactic globular star clusters we used the numerical
simulations performed with the mocca code444http://moccacode.net. This is
currently one of the most advanced codes to perform full stellar and dynamical
evolution of real size star clusters (Giersz et al., 2014; Hypki & Giersz,
2013; Giersz, 1998). mocca is already used in a wide number of projects to
investigate the evolution of compact binaries e.g. BHs binaries (Hong et al.,
2020). Moreover, it is able to follow N-body codes closely and it provides
almost as much information about stars and binary stars (Wang et al., 2016).
The strong dynamical interactions are performed with the Fewbody code (Fregeau
et al., 2004).

In order to increase the reliability of the results from the possible
dynamical channel, the mocca code was fully integrated with the startrack
code. Now, mocca follows all the stellar and binary evolution processes with
startrack as it was done with SSE/BSE codes in previous mocca studies. In
total $120$ numerical simulations of globular clusters were performed. In each
simulation a cluster begun its evolution with $6\times 10^{5}$ initial objects
(binaries and single stars), a $95\%$ binary fraction was assumed, metallicity
of $Z=0.0105$ was adopted, with $60$ pc tidal radius and $2$ pc half-mass
radius. Each globular cluster was evolved.

All globular cluster models were evolved with the same input stellar/binary
physics (see Sec. 4) that increases possibility of HR 6819 inner binary
formation (see Sec. 3.1). However, each model was a different realization of
the initial parameters for stars and binaries within their adopted initial
distributions (star masses, binary orbits, initial positions and velocities
within a given cluster). This was done to ensure that we will not miss any
potential HR 6819 formation channel due to poor statistics.

##  3 Isolated binary evolution

In order to reconstruct the HR 6819 inner binary we have to recover some
scenarios that, similarly to Rivinius et al. (2020), gave a BH-MS binary with
component masses $>4.2{\rm\leavevmode\nobreak\ M}_{\odot}$ for the BH and
$5-7{\rm\leavevmode\nobreak\ M}_{\odot}$ for B star (BH inner binary orbit
companion). Additionally the two objects are supposed to differ in mass by
$\sim 1{\rm\leavevmode\nobreak\ M}_{\odot}$ (BH being less massive). Finally,
the observational constraints require an orbital period of $40.3$ days and an
eccentricity of $\sim 0.03$. From now on we will refer to the BH progenitor as
the primary star and its B companion as the secondary star.

###  3.1 Simulation

In isolated binary evolution calculations we can make an educated guess on how
we need to push evolutionary parameters to maximize the chance of formation of
a binary similar to reported as the inner binary of HR 6819. Our guess is
based on several hundreds of trials of running evolution of massive binaries.
We find that BH formation, CE evolution, metallicity of binary are factors
important in the formation of BH-MS binaries with parameters resembling
reported architecture of HR 6819. We allow for some extreme changes in several
parameters but which are still not excluded on theoretical grounds.
Metallicity is allowed to change in the range $Z=0.01-0.02$ allowed for the
Galactic disk, neutrino mass loss during BH formation is allowed to change in
the range $F_{\rm loss}=1-20\%$, and the efficiency of conversion of orbital
energy into the ejection of the stellar envelope during the CE evolution in
the range $\alpha=0.5-5$. After just a few experiments it becomes clear that
we have to assume no BH natal kicks and no mass loss during BH formation. More
experiments lead to the preferred values of $Z=0.0105$, $F_{\rm loss}=7\%$,
and $\alpha=5$. Such high CE efficiency requires extra sources of energy
besides orbital in ejection of the envelope, but can not be excluded (Fragos
et al., 2019; Santoliquido et al., 2020).

We evolve $N=10\times 10^{6}$ massive binaries, in which the primary mass (the
initially more massive binary component) is taken from the IMF 3-part broken
power law within a mass range of $0.08-150{\rm\leavevmode\nobreak\
M}_{\odot}$. The secondary component mass is instead taken from a flat
distribution of mass ratio (secondary to primary) within the mass range
$0.08-150{\rm\leavevmode\nobreak\ M}_{\odot}$. The orbital period is taken
from Mink & Belczynski (2015).

###  3.2 Finding systems resembling HR 6819

We can now select BH-MS binaries that resemble reported inner binary in HR
6819. We compute the total difference, which is a number measuring how a given
BH-MS binary generated in our simulation differs from HR 6819, by taking into
account the desired set of parameters (BH mass $4.2-6{\rm\leavevmode\nobreak\
M}_{\odot}$, B star companion $5-7{\rm\leavevmode\nobreak\ M}_{\odot}$,
orbital period $30-50$ days555We set an arbitrary confidence range of $10$
days around the observed value, even though it doesn’t correspond to the
actual margin of error ($0.004$ days) reported by Rivinius et al. (2020).,
eccentricity $0.02-0.04$). If the total difference is equal to $1.0$ it means
that a binary has the parameters within the requested ranges. If the value is
larger than $1.0$ it means that some of the parameters are getting outside the
required constraints. The difference for any of the four parameters (BH mass,
companion mass, period and eccentricity) is measured as a normalized
difference to the middle of the expected center of a range. These normalized
differences for the secondary star mass, the BH mass, the binary period and
the eccentricity are then respectively calculated in the following way:

$Comp_{\rm diff}(x)=\begin{cases}ABS(M_{\rm comp}-5.0)/6.0+1.0,&\text{if
}M_{\rm comp}<5\\\ ABS(M_{\rm comp}-7.0)/6.0+1.0,&\text{if }M_{\rm comp}>7\\\
1.0,&\text{otherwise}\end{cases}$ (1)

$BH_{\rm diff}(x)=\begin{cases}ABS(M_{\rm BH}-4.2)/5.1+1.0,&\text{if }M_{\rm
BH}<4.2\\\ ABS(M_{\rm BH}-6.0)/5.1+1.0,&\text{if }M_{\rm BH}>6\\\
1.0,&\text{otherwise}\end{cases}$ (2)

$Per_{\rm diff}(x)=\begin{cases}ABS(M_{\rm Per}-30)/40+1.0,&\text{if }M_{\rm
Per}<30\\\ ABS(M_{\rm Per}-50)/40+1.0,&\text{if }M_{\rm Per}>50\\\
1.0,&\text{otherwise}\end{cases}$ (3)

$Ecc_{\rm diff}(x)=\begin{cases}ABS(M_{\rm Ecc}-0.02)/0.03+1.0,&\text{if
}M_{\rm Ecc}<0.02\\\ ABS(M_{\rm Ecc}-0.04)/0.03+1.0,&\text{if }M_{\rm
Ecc}>0.04\\\ 1.0,&\text{otherwise.}\end{cases}$ (4)

The final total difference is a combination of the four differences:

$Total_{\rm diff}=BH_{\rm diff}\times Comp_{\rm diff}\times Per_{\rm
diff}\times Ecc_{\rm diff}.$ (5)

In our simulations we find no binary ($Total_{\rm diff}\sim 1$) resembling the
reported parameters of HR 6819 inner binary even with our rather large ranges
on some parameters (e.g., orbital period).

###  3.3 Calibration to Milky Way

In this step we extrapolate our results to correspond to the entire Milky Way
disc (isolated binary evolution). The disc mass was estimated by Licquia &
Newman (2015) to be 5.17 $\pm$ 1.11 $\times$ 10${}^{10}M_{\odot}$. We
therefore adopt a star formation rate (SFR) of $\sim 5{\rm\leavevmode\nobreak\
M}_{\odot}$ yr-1 and assume that it was constant in the disc over the last
$10$ Gyr. We need to scale up (multiply) our results up the factor:

$f_{\rm scale,bi}=\frac{M_{\rm disc}}{M_{\rm sim}}$ (6)

where $M_{\rm disc}=5\times 10^{10}$ is the total stellar mass of the Milky
Way disc and $M_{\rm sim}\sim 4.2\times 10^{9}$ is the stellar mass
corresponding to our simulation in case of 100% binarity, while $M_{\rm
sim}\sim 6.9\times 10^{9}$ in case of 50% binarity, i.e. 2 out of 3 stars in
binary systems.

The above calibration results in a total number of BH-MS binaries that were
formed in the Galactic disc over $10$ Gyr: $29,482,201$ for 100% binarity and
$17,997,947$ for 50% binarity.

Now we can distribute all BH-MS binaries uniformly over $10$ Gyr of disc
evolution and knowing (from population synthesis calculation) each system BH-
MS lifetime estimate the current number of BH-MS binaries present in Galactic
disk to be: $427,464$ for 100% binarity and $249,134$ for 50% binarity. None
of the two binarity cases shows any BH-MS system matching the reported HR 6819
configuration. Figure 1 illustrates the current Galactic population of BH-MS
binaries for 100% binarity and their differences from the reported HR 6819
binary parameters. As a reference for potential studies to test the
reliability of our results, we also show in figure 2 the distribution of BH
masses (top figure), MS companion masses (middle figure) and orbital periods
(bottom figure) for BH-MS binaries in the Galactic disc for 100% binarity,

Figure 1: Current predicted population of Galactic disc BH-MS binaries and
their difference (in terms of orbital parameters) for 100% binarity from the
reported parameters of HR 6819 inner binary Rivinius et al. (2020). No BH-MS
binary out of $427,931$ binaries resembles the HR 6819 inner binary to the
point of having a total difference = 1.

(a)

(b)

(c)

Figure 2: Current distribution in BH-MS Galactic disc of BH masses (above), MS
companion masses (middle) and orbital periods (below) for 100% binarity.

###  3.4 Statistical significance of the formation of HR 6819 inner binary

Although we can find an evolutionary solution for the formation of the
reported HR 6819 inner binary here we show that it is highly unlikely that
such a binary would be expected in the current population of observed Galactic
BH binaries. As pointed out in Section 3.3 we predicted up to $427,464\approx
4\times 10^{5}$ BH-MS binaries currently present in the Galactic disc, with
none resembling the reported HR 6819 inner binary. Additionally, our estimate
is done for very optimistic evolutionary assumptions on isolated binary
evolution. We conclude that finding a binary similar to the reported inner HR
6819 binary in the current sample of Galactic BH systems is not expected.

Despite the fact that we do not consider likely that such a system would form
in the Galactic disk, in the Appendix (see Sec. A) we describe for
completeness how such a system could have possibly formed and evolved in the
framework of the isolated binary evolution.

##  4 Globular cluster evolution

After performing all $120$ GC simulations we find the formation of $89,817$
BH-MS binaries. We find no BH-MS binary resembling the reported inner HR 6819
binary. The best candidate has a BH mass of $4.5{\rm\leavevmode\nobreak\
M}_{\odot}$, a companion MS star with mass of $6.7{\rm\leavevmode\nobreak\
M}_{\odot}$, an orbital period of $27.5$ days ($a=120{\rm\leavevmode\nobreak\
R}_{\odot}$), and an eccentricity of $e=0.014$ ($Total_{diff}\sim 2$). The GC
estimate ends here. We conclude that Galactic GCs do not form binaries
resembling the reported inner binary of HR 6819. However, for consistency with
the binary evolution estimate, we proceed with an estimate of GC current
population of BH-MS binaries.

We calibrate/extrapolate our results to represent entire population of the
Milky Way clusters. The current total mass of the Milky Way GCs is based on
GCs mass estimates from Baumgardt (2017)
666https://people.smp.uq.edu.au/HolgerBaumgardt/globular/ and equal to
$3.7\times 10^{7}{\rm\leavevmode\nobreak\ M}_{\odot}$. Mass loss of the star
clusters through the Hubble time depends on many factors (e.g., Meiron et al.
(2021)). For the sake of simplicity we assume that half of the star mass of
clusters was lost in their evolution. Thus, we arrive at the initial mass for
Milky Way GCs to be $7.4\times 10^{7}{\rm\leavevmode\nobreak\ M}_{\odot}$. On
the other hand the stellar mass included in our $120$ GC simulation was
$8.1\times 10^{7}{\rm\leavevmode\nobreak\ M}_{\odot}$. Therefore the
multiplication factor to our results is

$f_{\rm scale,gc}=\frac{7.4\times 10^{7}{\rm\leavevmode\nobreak\
M}_{\odot}}{8.1\times 10^{7}{\rm\leavevmode\nobreak\ M}_{\odot}}=0.9$ (7)

This gives us $8.2\times 10^{4}$ BH-MS binaries formed by GCs in Milky Way.
However, only a fraction of them will survive till present time, as GCs are
old and most BH-MS stars that has formed long ago are already gone or are not
BH-MS binaries anymore.

Star formation in the Galactic globular clusters was taking place at large
redshifts $z>2$ (Katz & Ricotti, 2013). We, conservatively (in context of
survival of BH-MS binaries with rather massive MS stars), assume that all
stars in Galactic GCs have formed at $z=2$, so $t\sim 10$ Gyr ago. Since we
know BH-MS lifetimes from our GC simulations, we can remove BH-MS binaries
that do not survive till the present time.

Figure 3:  Current predicted population of Galactic globular cluster BH-MS
binaries and their difference (in terms of orbital parameters) from the
reported parameters of HR 6819 inner binary Rivinius et al. (2020).

In Figure 3 we show the current population of Galactic globular cluster BH-MS
binaries. We see that there is no system resembling the reported HR 6819 inner
binary ($Total_{diff}=1$) and that there are only a very few systems with low
total difference in orbital parameters from HR 6819. The main reason why it is
hard to find dynamical candidates for HR 6819 system in globular clusters
stems from the fact that possible candidates (like the one with
$Total_{diff}\sim 2$) may be found only in the very beginning of the star
cluster evolution. Stars as massive as the companion to a BH
($>5{\rm\leavevmode\nobreak\ M}_{\odot}$) live only for very short time
($\lesssim 150$ Myr) and can not contribute to the current globular cluster
(old) stellar populations. Even if a GC produced of HR 6819 like system then
such a binary would be gone long time ago. Thus, dynamical origin for
potential HR 6819 is highly unlikely.

##  5 Conclusion

For the two major Galactic BH-MS formation channels we adopted evolutionary
physics that increases the possibility of formation of a system resembling the
reported parameters of the HR 6819 inner binary by Rivinius et al. (2020). The
evolution of stars in Galactic globular clusters does not allow for any
avenues for creating such binaries. Isolated evolution of binaries in Galactic
disc also is a highly unlikely site for formation of such binary.

We conclude that on the evolutionary grounds HR 6819 does not host a dormant
BH. In support of our conclusion, the absence of the dormant BH was explained
by very different system orbital architecture than proposed by Rivinius et al.
(2020). Based on the same observational data used by Rivinius et al. (2020),
other groups (El-Badry & Quataert, 2021; Bodensteiner, J. et al., 2020;
Safarzadeh et al., 2020) concluded as well that HR 6819 does not host a
dormant BH.

###### Acknowledgements.

AR, AO, AH and KB acknowledge support from the Polish National Science Center
(NCN) grant Maestro (2018/30/A/ST9/00050). AH was also supported partially by
Polish National Science Center grant 2016/23/B/ST9/02732.

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##  Appendix A Example of formation of HR 6819 inner binary in the framework
of the isolated binary evolution

Here we present an example of one evolutionary scenario in the isolated binary
evolution framework (Fig. 4) that we find that leads to the formation of a
binary that matches the observational constraints from Rivinius et al. (2020).
Initially the system consists at ZAMS of two stars of
$24.0{\rm\leavevmode\nobreak\ M}_{\odot}$ and $7{\rm\leavevmode\nobreak\
M}_{\odot}$. All input physics corresponds to our population synthesis

Figure 4:  Formation of a system resembling the reported HR 6819 inner binary
in the isolated binary evolution framework. It starts with two massive stars
at metallicity $Z=0.0105$. While the companion star remains in its main
sequence, the more massive star initiates a CE phase after $\sim 8.2$ Myr and
shortly after it collapses into a BH.

At the beginning the two stars are on a very wide, circular orbit (period
$\sim 2.6\times 10^{3}$ days, $e=0.0$). At $8$ Myr the primary evolved off its
main sequence (MS), expanded and evolved into a core helium burning giant. It
initiates a common envelope phase, after which the orbital period is reduced
to about $30$ days and the giant star loses its envelope. We then assume a
direct collapse of the primary star into a BH. Due to the neutrinos emission,
the eccentricity of the system slightly increases to $e=0.035$. We note the
formation of BH-MS binary.

Evolution of orbital period and eccentricty of this binary are shown in Figure
5. During the whole BH-MS phase (between $\sim 10-50$ Myr since the system
formation), orbital period is $\sim 36$ d, which is close to the reported
value of $40.3$ d. The eccentricity during BH-MS phase $e=0.035$ is in the
observed [0.02; 0.04] range.

Figure 5:  Evolution of the binary period (blue) and eccentricity (purple) of
the binary that forms reported HR 6819 inner like system in the isolated
binary evolution framework. The CE phase (red) takes place at $\sim 8.2$ Myr
and was initiated by the primary star, that collapses shortly thereafter into
a BH. BH-MS system (similar to HR 6819) phase is noted for $\sim 40$ Myr
($\sim 10-50$ Myr since ZAMS). We also show evolution of this binary beyond
BH-MS phase with two stable RLOF events (at $\sim 49$ Myr and $\sim 61$ Myr
since ZAMS) initated by expanding secondary.

Evolution of mass of both components of this binary is shown in Figure 6.
During BH-MS phase BH mass is $6.1{\rm\leavevmode\nobreak\ M}_{\odot}$ and its
MS companion mass is $7.0{\rm\leavevmode\nobreak\ M}_{\odot}$. Both masses are
within observational estimates of Rivinius et al. (2020). During the BH-MS
phase the difference in mass between the BH and its companion is $\Delta
M=0.9{\rm\leavevmode\nobreak\ M}_{\odot}$ and it is consistent with the
estimated value $\Delta M\sim 1{\rm\leavevmode\nobreak\ M}_{\odot}$ for the HR
6819.

Figure 6:  Evolution of mass the two components of the binary that forms
reported HR 6819 inner like system in the isolated binary evolution framework.
The primary (green line) loses most of its mass during the CE phase (red line)
and collapses into a BH at $8.6$ Myr after ZAMS. The secondary star (blue
line), remains in its MS with almost constant mass for $50$ Myr. Then it loses
most of its mass during two stable RLOF events (grey lines).

In Figure 7 we show the temporal evolution of the radii of the two components
for this binary in comparison with their Roche lobe radii. We note that during
BH-MS stage (time between $8.6$ and $49.8$ Myr) there is no RLOF episode and
the wind mass loss from the secondary star
($\dot{M}<10^{-7}{\rm\leavevmode\nobreak\ M}_{\odot}{\rm\leavevmode\nobreak\
yr}^{-1}$) is too weak to produce any significant X-ray emission at the
orbital separation found for this system (a=$\sim 108{\rm\leavevmode\nobreak\
R}_{\odot}$) during this stage. The BH in our example is thus dormant and in
agreement with the observational data.

Figure 7:  Radius evolution of the two components of the binary that forms
reported HR 6819 inner like system in the isolated binary evolution framework.
After the formation of BH, the secondary is well within its Roche lobe during
its entire MS lifetime ($<50$ Myr).

In Figures 5, 6, and 7 we also include prediction for the future evolution of
this binary system. During the whole BH-MS phase parameters of this binary
remain almost unchanged with the only exception of the radius of the secondary
star, which increases from $\sim 3{\rm\leavevmode\nobreak\ R}_{\odot}$ to
$\sim 6{\rm\leavevmode\nobreak\ R}_{\odot}$ as it evolves through its main
sequence. Secondary leaves MS at $\sim 50$ Myr and begins a rapid expansion
that eventually leads Roche lobe overflow. The system then enters stable mass
transfer phase which will last for $\sim 0.1$ Myr. After secondary loses its
H-rich envelope (BH does not accrete any significant amout of mass as
accretion proceeds on fast thermal-timescale), it becomes a naked He star with
mass of $1.37{\rm\leavevmode\nobreak\ M}_{\odot}$. RLOF and associated mass
and angular momentum loss from the binary leads to significant orbital
expansion to orbital period of $\sim 10^{3}$ days ($a=105$ AU — change this to
solar radius unit!) and the orbit gets circularized. Low mass helium star then
undergoes expansion during shell He-burning and it initiates second stable
RLOF at $\sim 61$ Myr since ZAMS. RLOF again expands the orbit to orbital
period of $1279$ days ($\sim$ 962 AU). After borh RLOF episodes the secondary
star lost more than 80$\%$ of its mass and it cools of to become a carbon-
oxygen white dwarf (WD). We note formation of wide (coalescence time much
larger than Hubble time) BH-WD binary.