# How To Spin Black Holes Up In High-Mass X-ray Binaries And Not Merge In The
Attempt.

Enrique Moreno Méndez1
1Facultad de Ciencias, Universidad Nacional Autónoma de México, A. P. 70-543
04510, CDMX, México<EMAIL_ADDRESS>0000-0002-5411-9352

(September 2021)

###### Abstract

Astrophysical black holes (BHs) can be fully described by their mass and spin.
However, producing rapidly spinning ones is extremely difficult as the stars
that produce them lose most of their angular momentum before the BH is formed.
Binaries where the progenitor is paired with a low-mass star in a tight orbit
can produce rapidly spinning BHs (through tides), whereas those with massive
companions cannot (as they do not fit in such an orbit). A few rapidly-
spinning black holes (BHs) have been observed paired with very massive
companion stars, defying stellar-formation paradigm. Models which reduce the
stellar-core–envelope interaction (and winds) do not match observations nor
theory well; I show they also miss explaining the energetics. BH spins cannot
be produced during stellar collapse; using orbital spin through explosion-
fallback material does not match the observations; spinning BHs up through
accepted mass-transfer channels takes longer than their lifetimes, it is
usually discarded. I show that fast mass-transfer mechanisms, predicted to
merge the BH and star, successfully spin the BHs up and show a mechanism to
avoid said mergers and main dangers of the alternatives while naturally
explaining the observations. The implications are potentially paradigm-
shifting and far-reaching in the high-energy BH-astrophysics context.

###### keywords:

Hypercritical accretion – Common-envelope evolution – Mass transfer – Black-
hole spin – X-ray binaries

##  1 Introduction: The HMXB evolution context.

Cyg X-1, LMC X-1, M33 X-7, and IC10 X-1 are four high-mass X-ray binaries
(HMXBs 111Please refer to Table 1 for a list of acronyms and symbols used
throughout this article.; i.e., a class of BH binary, BHB, where a BH is
paired with a massive companion star, $M_{\star}>10\mbox{M${}_{\odot}$}$, with
ongoing mass transfer; see Tables 2 and 3 for a list of their physical
parameters); each hosting a Kerr BH (KBH; a BH with large spin or Kerr
parameter, $a_{\star}\gtrsim 0.8$)222The Kerr parameter is defined as
$a_{\star}\equiv cJ_{BH}/GM_{BH}^{2}$, where $J_{BH}$ is the angular momentum
of the BH, and $M_{BH}$ its mass; $|a_{\star}|<1$ for any KBH; $a_{\star}=0$
represents a (non-spinning) Schwarschild BH (SBH).. HMXBs are intermediate
stages of systems which produce a number of high-energy transient events.
These include Hypernovae/Gamma-Ray Bursts (HNe/GRBs; see Appendix A), short
GRBs (if at least a NS is present and $q$, the mass ratio, is not far from 1),
and binary BHs (BBHs) which, when they merge, are sources of the energetic
gravitational-waves events (GWs) detected by LIGO-Virgo. They are also
candidates for ultraluminous X-ray sources (ULXs; Appendices E, J). Their
formation-channel mechanism has, so far, eluded being understood.

There are stellar evolution models which try to explain the origin of large-
spin BHs in HMXBs by reducing the Taylor-Spruit dynamo (TS; Spruit, 2002) and
stellar winds from their evolution (e.g., Qin et al., 2019, ; but see Fuller &
Ma (2019) for the latest strong caveats against reducing TS; Appendices F,K)
and, as I will show in the next section, have a contradictory account of the
formation of their KBH. Moreno Méndez & Cantiello (2016) have shown that
imparting such spins at birth is not possible with SN kicks nor standing-
accretion-shock instabilities (SASIs). Meanwhile Batta et al. (2017); Schrøder
et al. (2018) have shown that angular-momentum-imbued fallback from CC SN
could spin the BHs up to $a_{\star}\lesssim 0.8$ (again, with caveats from
Fuller & Ma, 2019, see Appendix B where it is also shown that their parameters
are anticorrelated to the pertaining observables); however this would require
fine tunning just to accommodate the more modest spins among these HMXBs.

Full name | Acronym | Full name | Acroynm
---|---|---|---
Black Hole | BH | Kerr parameter (BH spin) | $a_{\star}$
Schwarzschild Black Hole ($a_{\star}\sim 0$) | SBH | Kerr Black Hole($a_{\star}\gtrsim 0.8$) | KBH
High-Mass X-ray Binary | HMXB | Intermediate-Mass X-ray Binary | IMXB
Supernova(s) | SN(e) | Low-Mass X-ray Binary | LMXB
Core Collapse | CC | Neutron Star | NS
Hypernova(s) | HN(e) | Shock Wave | SW
Gamma-Ray Burst | GRB | Standing Acretion Shock | SAS
Hypercritical Accretion | HCA | Standing Acretion Shock Instabilty | SASI
Super-Eddington Accretion | SEA | Common Envelope | CE
White Dwarf | WD | Grazing Envelope | GE
Mass Transfer | MT | Gravitational Wave | GW
Taylor-Spruit | TS | Black-Hole Binary (1 BH-1 star) | BHB
Blandford-Znajek | BZ | Binary BH (2 BHs) | BBH
Blandford-McKee | BM | Ultraluminous X-ray Source | ULX
Roche Lobe | RL | Roche-Lobe Overflow | RLOF
Laser Interferometer GW Observatory | LIGO | General Relativity | GR
Proto-Neutron Star | PNS | Quantum Mechanics | QM
Kelvin-Helmholtz | KH | Main Sequence | MS
Compact Object (BH or NS) | CO | Chemically-Homogeneous Rotationally Mixed | CHRM
Wolf-Rayet | WR | Bondi-Hoyle-Lyttleton | BHL
Table 1: Table of main acronyms and symbols used throughout the text.

Mechanisms were presented in Moreno Méndez et al. (2008); Moreno Méndez (2011)
which involve transferring mass from the stars and onto the BHs to spin them
up but in systems which are clearly unstable under the current mass-transfer
paradigm Klencki et al. (2021); Soberman et al. (1997) and would produce
stellar mergers rather than the observed HMXBs. In this article I show that,
by employing a couple of simple ideas based on forever foregone and overlooked
aspects of hypercritical accretion (HCA; see section 3; Moreno Méndez et al.,
2017; López-Cámara et al., 2019, 2020) and/or Super-Eddington accretion (SEA;
section 3), it is possible to stop the mass-transfer episode in a common-
envelope (CE; section 4), and/or grazing-envelope phase (GE; section 4; Soker,
2015; Shiber et al., 2017; Shiber & Soker, 2018; Moreno Méndez et al., 2017;
López-Cámara et al., 2019, 2020) and allow the formulation of simple
evolutionary tracks for the presently observed HMXB systems (Appendices G, H,
Table 4). Henceforth, removing the need to invoke a large stellar spin, along
with the risk of a HN/GRB explosion during the BH-formation process which
would likely disrupt such BHBs, and prevent the formation of these HMXBs.

Discussions on the implications of these results for ultraluminous X-ray
sources (ULX; by examining the likelihood of a long-lasting or a pulsed GE
phase in HMXBs systems; Appendices E,J), HN/GRB central engines (Appendix A),
and the implications for the formation of LIGO gravitational-wave (GW;
Appendix D) sources from binary BHs (BBHs), as well as the implications for
future stellar evolution models are included ; this, along with a general
description of a stellar evolution scenario (Appendix D and Figure 1), in the
context of our results, which outlines the formation channel which naturally
explains the observables in the six known BH HMXBs with emphasis on the four
with KBHs measurements and predictions for the other two. A set of possible
tests for our model can be found in Appendix G.

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
---|---|---|---|---|---|---|---
HMXB |  $M_{\star}$ [M⊙] |  $M_{BH}$ [M⊙] | $q$ | P orb [days] | $a_{\star}$ |  $E_{rot}$ [B] | Refs.
M33 X-7 | $70.0\pm 6.9$ | $15.65\pm 1.45$ | 10$\rightarrow$5 | 3.45 | $0.84\pm 0.05$ | 3,400 | i, ii, iii
Cyg X-1 | $40^{+2}_{-2}$ | $21.2\pm 2.2$ | 5$\rightarrow$2 | 5.60 | $>0.9985(3\sigma)$ | 10,000 | iv, v, vi
LMC X-1 | $31.79\pm 3.48$ | $10.91\pm 1.41$ | 7$\rightarrow$3 | 3.90917 | $0.92^{+0.05}_{-0.07}$ | 3,000 | vii, viii
IC 10 X-1 | $17-35$ | $10-15(?)$ |  $2\rightarrow 1$(?) | 1.45175 | $\gtrsim 0.8$ | $>1,300$ | ix, x, xi
NGC 300 X-1 |  $26$(?) |  $17\pm 4$(?) | $?\rightarrow 1.5$ | 1.366 |  $>0.8$? |  $>1,500$? | xii, xiii
Cyg X-3 |  $(8-15)$(?) |  $\lesssim(5-10)$(?) | $?\rightarrow 1.5$ | 0.20 |  $\lesssim 1$? |  $\sim 5,000$? | xiv
Table 2: Relevant properties of HMXB systems with large $a_{\star}$ BH: (1)
HMXB name. (2) Mass of stellar companion. (3) BH mass; (4) Mass ratio (given
by $q\equiv M_{\star}/M_{BH}$) necessary change for Conservative mass transfer
onto the BH to acheive the observed $a_{\star}$ by HCA). (5) Shows the
currently observed orbital period. (6) Lists the current measured BH spin
$a_{\star}$. (7) Current rotational energy, $E_{rot}$, with units in Bethes,
estimated from the currently observed BH spins and masses. Finally, (8) has
the references for the observational data. Data with “(?)” takes best
estimates in the literature. Data with “?” are best guesstimates in this paper
from available data and extrapolating from binary evolution. The references
are as follows: i, ii, iii are Liu et al. (2008); Orosz et al. (2007); Liu et
al. (2010); iv, v, vi are Zhao et al. (2021); Mirabel & Rodrigues (2003);
Miller-Jones et al. (2021); Kushwaha et al. (2021); vii and viii are Orosz et
al. (2009); Gou et al. (2009); ix, x, xi are Laycock et al. (2015, 2015);
Steiner et al. (2016); xii and xiii are Crowther et al. (2010); Binder et al.
(2021); and xiv is Koljonen & Maccarone (2017) respectively.

##  2 The BH spin-energy conundrum.

Let us first analyze a couple of revealing and relevant questions for HMXB
systems, where KBHs lurk, in order to elucidate the main issue behind the
evolutionary channel leading to their formation and why, the natal KBH
hypothesis, used in most of the literature related to these HMXBs, on top of
the convoluted evolutionary channels, has a fundamental flaw at its core :

  * •

A very massive star is about to collapse into a non-rotating, or Scwarschild
BH (SBH). Will it produce a SN explosion or will it collapse directly?
Unfortunately, the answer is not staright forward. Many large scale properties
of the collapsing star must be considered. But, furthermore, SN simulations,
necessary to answer such a question, are particularly intrincate computations,
as they require many microscocpic, mesoscopic and macroscopic minutiae to be
very well implemented, as well as GR and QM effects to be correctly quantified
(see, e.g., Janka, 2012). Even though the star posseses enough energy to blow
itself apart (about $10^{2}$ times the energy necessary to unbind all the
stellar envelope) in a SN explosion following its core collapse (CC) –while
its stellar core collapses and forms a compact object (CO; i.e., either a
neutron star, NS, or a BH) it releases its binding energy in a fraction of a
second– the answer is not trivial because about 99% of this energy will be
emitted as neutrinos, and most of these, may leave the star without ever
interacting with it (see Bethe et al., 1979, for details). Furthermore, the
collapsed core (which becomes partially neutrino-opaque) forms a hot proto-NS
(PNS), which radiates away great amounts of energy as it cools down and could
help the explosion succeed, however, the shockwave (SW) that may give rise to
the SN is in the worst place in the universe for its survival. The SW
(supersonic by nature) stalls at a few $10^{7}$ cm in a supersonic inflow of
the outer layers of the Fe-core of the star (photodisintegration of this Fe
into neutrons and protons removes about 1 B per $0.1\mbox{M${}_{\odot}$}$ that
crosses the SW3331 Bethe $\equiv 1$ B $\equiv 1$ foe $\equiv 10^{51}$ erg),
forming a standing accretion shock (SAS). If the SW can be revived, the SN
will proceed and the PNS will cool down into a NS or a low-mass SBH. However,
if no mechanism is found to channel part of the neutrino energy into the SAS,
the latter may subside and the SN will fail, allowing, thus, for the full
core, as well as most of the stellar envelope to collapse into a more massive
SBH (a fraction of the mass is lost through neutrinos, allowing for the
release of the gravitational binding energy); in a sense, we have the
unstopable force vs the unmovable wall problem to solve and find out what kind
of stellar SBH has formed, massive or not.

  * •

Now, let us compare to a collapsing a star which is allowed to have every
element, every trick in the SN-explosion toolkit available –or, even one up–
in the hypernova/gamma-ray-burst (HN/GRB) central-engine toolkit to build up
its chances of exploding (i.e., a Collapsar; MacFadyen & Woosley, 1999); let
us as well allow it access to an energy reservoir about a hundred times larger
than that for typical CCSNe, and which is readily available (as kinetic
enregy) even before CC. Would that star explode or keep that energy stored?
Such a question is much easier to answer! The star would produce as glorious
an explosion as allowed and radiate a substantial amount of its available
energy outright! Right?

Well, unfortunately, this natal-BH-spin/energy argument has not been brought
up in the context of the natal spin of KBHs in HMXBs until now. So we will
address it in some depth next.

Ever since the spin parameters of BHs in HMXBs started to be estimated by
different methods, and they were found to be extremely large, $a_{\star}\equiv
Jc/GM^{2}>0.8$ (Table 2, column (6)), a long list of papers have claimed that
these BH spins have to be natal (from the first for M33 X-7, Liu et al., 2008,
to the last estimate for Cyg X-1, Zhao et al. 2021)444This should include,
basically, every article with a BH spin measurement (for HMXBs) as well as
evolutionary model to explain them., i.e., all the BHs in HMXBs were born
KBHs, along with the immense ammount of stored energy this implies. Now, the
key point against this line of argument is this: Other than the large
$a_{\star}$ observed in these KBHs, all available dynamical evidence points to
the contrary: very low energy events formed these BHBs! E.g., all these
systems have massive stellar BHs, so they involve either heavy fallback of a
mostly failed SN or direct collapse (Cyg X-3 could be the exception here;
there is evidence it could be an IMXB; where its small orbital separation
suggests it could have had access to a central engine and still survived the
BH formation; Appendix A), they are still bound to their companion stars
(which is evidence of a low energy BH formation event, even while considering
the flagrant observational bias); eccentricities are low (and tidal
circularizations are of order Myrs, Zahn, 1977, 1975, 1989; Zahn & Bouchet,
1989); have low peculiar velocities (e.g., Cyg X-1 and its low velocity with
respect to the Cyg OB3 cluster, Mirabel & Rodrigues, 2003, ; Table 3 (column
12), which implies a $\Delta M_{SN}\lesssim 1\mbox{M${}_{\odot}$}$, and given
its 2 kpc distance to OB3, a BH age of $(7\pm 2)$ Myrs555These estimates might
need to be updated after the latest corrections to the distance by Miller-
Jones et al. (2021).); estimates of the energy of the SN (e.g. from SN
remnants) that gave rise to such a KBH are, unfortunately, not available; no
evidence of large jets (or BH spin orientation) to orbit missalignment (see
discussion on Cyg X-1 at the end of next section), nor strong precession (from
SN kicks; instead, alignmet would be expected if the spin is a product of mass
transfer onto the BH); there is no evidence of SN or, better yet, HN pollution
on the companions, although this is difficult to corroborate (the material
will dilute into the star with time); regardless, these absences can help
constraint the minimum age of the BHs. Kicks and mass-loss during a CCSN
would, typically, have increased their previous orbital separations, so it may
also be suspicious that they did not previously synchronize their spin with
their orbital period; if they had done so, they would have slowed down to
spins below $a_{\star}\lesssim 0.2$ (Moreno Méndez et al., 2008; Moreno
Méndez, 2011). Tables 2 and 3 show the available data.

(9) | (10) | (11) | (12) | (13) | (14) | (15) | (16)
---|---|---|---|---|---|---|---
Sp.T. |  $L_{X}$ [$10^{38}$ erg] | $e$ |  $V_{pec}$ [km s-1] |  $E_{bubble}$ [erg] | SNR? | $J-S$ | Refs.
O(7-8)III | 0.13-2.49 | 0.0185(77) | – | – | – | $\leq 3^{o}$ | xv, xvi, xvii, xviii
O9.7Iab | $0.05$ | 0.0189 | $9\pm 2$ | $7\times 10^{48}$ | Bubble | $(10-20)^{o}$ | xix, xx, xxi, xxii
O7III | 2.3 | 0.0256(66)? | – | – | – | – | xxiii, xxiv, xxv
WR | $0.7$ | – | – | – | – | – | xxvi
WR | $2.6(1)$ | – | – | – | – | – | xxvii, xviii
WN(4-7) | 0.5 | – | $\geq 9$ | – | Yes | ($>15^{o}$)? | xxix, xxx
Table 3: Spectral types and relevant dynamical properties of HMXB systems
(same HMXB order as in (1), Table 1) with large $a_{\star}$ BH: (1) HMXB name.
(9) Spectral type (Sp. T.) of stellar companion. (10) X-ray luminosity of the
BHB; (11) Eccentricity of the binary. (12) Peculiar velocity vs cluster. (13)
Energy deposited into bubble by jets. (14) Does it have an observable SNR?
(15) Orbital to spin (jet) inclination. (16) References for the observational
data. The references are as follows: xv, xvi, xvii, xviii are Liu et al.
(2008); Orosz et al. (2007); Valsecchi et al. (2010); Liu et al. (2010); xix,
xx, xxi, xxii are Zhao et al. (2021); Kushwaha et al. (2021); Gallo et al.
(2005); Casares & Jonker (2014); xxiii, xixv, xxv are Orosz et al. (2009); Gou
et al. (2009); Casares & Jonker (2014); xxvi is Barnard et al. (2008); xxvii
and xxviii are Barnard et al. (2008); Binder et al. (2015); xxix and xxx are
Sakari & Miller-Jones (2009); van Kerkwijk et al. (1996), respectively.

In order to estimate the available energy available to form a BH of mass
$M_{BH}$ and spin $a_{\star}$ we can use (e.g., Moreno Méndez et al. (2011)):

$\displaystyle E_{KBH,rot}$ $\displaystyle=$ $\displaystyle
f(a_{\star})M_{BH}c^{2},\;\;\;\;{\rm where}$ $\displaystyle f(a_{\star})$
$\displaystyle=$ $\displaystyle
1-\sqrt{\frac{1}{2}\left(1+\sqrt{1-a_{\star}^{2}}\right)}$ (1)

to obtain the order of magnitude of available kinetic energies during and
after CC if these BH spins were natal, as claimed (estimates listed in Table
2, column (7)). Having KBHs with a natal spin of such magnitude and such large
BH masses, $M_{BH}>10\mbox{M${}_{\odot}$}$ for all for well measured HMXBs
(Table 2, columns (3) and (6)), implies that they had between $10^{54}$ and
$10^{55}$ ergs available during CC! Even though, the available energy to a KBH
central engine will be a factor of 3 to 4 below the rotational energies
listed, all the kinetic energy is fully available to the star and its collapse
dynamics before the BH is formed 666This is also a lower limit, considering
for efficiencies and an actual energetic explosion which still left such a KBH
behind., so accretion disks and extremely strong magnetic fields will be
anything but absent. Thus, the energy to blow themselves appart, by forming a
centrifugally supported accretion disk, and building up to $10^{15}$-G
magnetic fields encompasing some $V\sim(10^{7}{\rm cm})^{3}$ wide volume is
present (they may be one or two orders of magnitude less intense as well as
one order of magnitude less extended, but its enlightening to look at the
comparatively low energy cost to produce them, for some $10^{51}$ erg, when
compared to the energy budget proposed by such an evolutionary model); the
formation of such KBH should have produced some spectacularly luminous events,
a large amount of mass loss, and disrupted the BHBs, or at least, left very
strong imprints in them which should be visible today, only, none of them did!
(see Appendix H for further discussion on this topic).

In principle, one could attack the energy argument by saying that these KBH
have so much energy that they rapidly dismantle the central engine and stop
the energy drain, keeping the KBH almost intact (the Goldielocks scenario in
Moreno Méndez et al., 2011; Brown et al., 2007, 2008; Moreno Méndez, 2014),
but this alternative can easily be discarded, for this scenario, given that
the (kinetic) energy is present before CC and can find different paths to be
used to reduce the natal mass and spin of the BH before it is even formed.
Appendix A, in the supplementary sections, addresses isues on the formation of
KBH central engines of HN/GRBs (where the BH spin must be natal) and its
relation to X-ray binaries (XBs). In general, it is more likely HNe/GRBs come
from IMXB, rather than HMXBs (for the formation of the first BH; the formation
of the second BH could produce HNe/GRBs in either case).

One model which cannot be dismissed by the energy argument is that the spin
may be acquired by caputring orbital angular momentum through SN-fallback
material (Batta et al., 2017; Schrøder et al., 2018), nonetheless, as shown in
Appendix B, it has a number of important caveats to overcome.

Fortunately, there is an alternative explanation: HCA during CE or GE
evolution after the formation of the BH. Nonetheless, it involves accepting
that a couple of dogmas have to be thrown out the window; I will show in the
following section how these may be easily overcome. I will now provide a
channel that prevents the merger during this stage, and naturally explains
other properties observed in these HMXBs.

##  3 On the Eddington limit, BHL, HCA/SEA.

When CE evolution is studied in a binary stellar system, the usual set up for
studying the removal of the CE and inspiral of the orbiting stars uses an
energy-conservation prescription (e.g., Klencki et al. (2021); Soberman et al.
(1997); Shore et al. (1994)): Gravitational binding energy of the CE vs the
energy difference between initial and final orbits. Usually mediated by the
efficiency by which the energy is converted ($\alpha$) and a radial density
profile of the stellar material in the CE ($\lambda$), which can be treated as
a product ($\alpha\lambda$), and binary systems which have undergone a CE
phase may be used to constrain/calibrate it (see Lee et al., 2002, for XR
binaries, where they find $\alpha\lambda\sim 0.2$).

Nonetheless, all HMXBs in our study, Cyg X-1, LMC X-1, M33 X-7, and IC 10 X-1
have mass ratio between companion star and BH ($q_{\rm system}\equiv
M_{\star}/M_{\rm BH}$) well above 1 (Table 1, column (4)), hence RLOF, CE/GE
mass transfer will be driven by at least one runaway process i.e., once it
begins, it cannot be stopped until a merger occurs, unless $q$ can be brought
below $q\lesssim 1$ beforehand. The actual state of these binaries, then
should not be attainable via a RLOF or CE where the donor star transfers mass
onto the BH, instead; they should be either in an ongoing CE stage or they
should have merged if RLOF has ever ensued after BH formation. In principle
this argument should discard the HCA during CE/GE scenario for BH spining in
these binaries from Moreno Méndez et al. (2008); Moreno Méndez (2011), but see
section 5.

Unfortunately, most of the research in this field invokes the Eddington limit
(for short reviews on the topic, see e.g., Brown & Weingartner (1994);
Chevalier (1989)) for Bondi accretion onto the BHs (i.e., spherically
symmetric; Bondi (1952)) which can be obtained from:

$\displaystyle\dot{M}_{\rm Edd}$ $\displaystyle\simeq$
$\displaystyle\frac{L_{\rm Edd}}{c^{2}}\simeq
10^{-15}M_{10}~{}\mbox{M${}_{\odot}$}~{}{\rm s}^{-1},\;{\rm and}$
$\displaystyle L_{\rm Edd}$ $\displaystyle\simeq$ $\displaystyle
10^{39}M_{10}~{}{\rm erg~{}s}^{-1},$ (2)

($M_{10}$ stands for a unitless $10-\mbox{M${}_{\odot}$}$ accretor) to argue
that, given the masses of the companions in these systems (Table 1, (2)),
their ages constrained to $\tau_{\rm HMXB}\lesssim 10$ Myr (even after
accounting for stellar rejuvenation of the companion stars from mass transfer
from the BH-progenitor stars Braun & Langer (1995)), the requirement of
transfering $5~{}\mbox{M${}_{\odot}$}$ to $15~{}\mbox{M${}_{\odot}$}$ onto the
BHs (to spin them up to $a_{\star}>0.8$ from initial low values) at the
Eddington rate is not attainable (Qin et al. (2019); Fuller & Ma (2019); Liu
et al. (2008); Zhao et al. (2021); McClintock et al. (2014); Liu et al.
(2010); Gou et al. (2009); Steiner et al. (2016); Valsecchi et al. (2010) and
a large etc.) as more than 10 Myrs are necessary.

However, these systems, either during RLOF mass transfer (or wRLOF; Mohamed &
Podsiadlowski (2007); El Mellah et al. (2018, 2019)), or during CE mass
transfer, are under conditions which are extremely different from Bondi
accretion. In any case, Bondi-Hoyle-Lyttleton Bondi & Hoyle (1944); Hoyle &
Lyttleton (1941); Edgar (2004) accretion is a much better approximation to
what is occurring, and as such, it should be considered in such estimates.
Still, BHL falls short from considering the density, velocity, and temperature
gradients, as well as the Coriolis and Centripetal/Centrifugal forces present
during the CE stage, even after jets are considered; numerical simulations
have been performed in order to account for these in Moreno Méndez et al.
(2017); López-Cámara et al. (2019, 2020) (where even hints of accretion-disk-
like flows are observed) and can be used to constraint the near BHL accretion
rates for BH accretion during CE (finding typical values of
$\dot{M}\simeq\eta\dot{M}_{\rm BHL}$ with $\eta\sim 0.1$). Along with the
substantial amount of transferable material, these conspire against the
Eddington limit and allow for mass-transfer rates which not only are super
Eddington777$\dot{M}_{\rm Edd}<\dot{M}\lesssim 3,000\dot{M}_{\rm Edd}$; mostly
due to the geometry of the flows: equatorial accretion, poloidal radiation or
jets, thus avoiding the main Bondi accretion bottleneck., but even
hypercritical (where cooling is mostly done through neutrino transport Brown &
Weingartner (1994); Chevalier (1989)). In fact, accretion rates have been
estimated numerically as well in the simple BHL mode for the newtonian case
see, e.g., Ruffert (1994); for the GR, or higher resolution of the accreting
BH case, convergence is quite clear (see Lora-Clavijo et al., 2015, this
result includes density gradients).

So, we have now shown that the Eddington luminosity is nowhere near an
impediment to accrete the necessary mass to bring a BH from $a_{\star}$ from
close to 0 all the way to 1 within a small fraction of the lifetime,
$\tau_{\rm HMXB}$, of these binaries; i.e., doubling the initial mass of the
BH (see Fig 6 in Brown et al. (2000), as suggested in Moreno Méndez et al.
(2008); Moreno Méndez (2011)) within a much shorter period of time as shown in
Moreno Méndez et al. (2017); López-Cámara et al. (2019, 2020); Moreno Méndez
et al. (2022) where a CO within the CE of a RG or RSG accretes at about
$\eta\dot{M}_{\rm BHL}$:

$\displaystyle\dot{M}_{\rm BHL}\equiv\epsilon\frac{4\pi\rho
G^{2}M^{2}}{(v^{2}+c_{s}^{2})^{3/2}}$ $\displaystyle\simeq$ $\displaystyle
10^{-5}\ \epsilon_{0.1}\ \rho_{-6}\ M_{10}^{2}\ v_{7}^{-3}\ {\rm
M}_{\odot}\;{\rm s}^{-1}$ (3) $\displaystyle\simeq$ $\displaystyle
10^{10}\dot{M}_{Edd}.$

Thus, a BH of $5~{}\mbox{M${}_{\odot}$}$ may double its initial mass within a
month and a $10~{}\mbox{M${}_{\odot}$}$ BH in a few days if the mass is
available (i.e., within its RL); even at accretion rates well below those of
BHL, e.g., for SEA $\dot{M}_{\rm SEA}\lesssim 10^{3-4}\dot{M}_{\rm Edd}$, the
necessary mass transfer and accretion can easily be acheived within a Myr.

##  4 On CE separation by ablation and by accretion.

From section 3, we see that the elephant in the room is not whether a BH can
accrete enough mass (which it shows can be done with HCA/SEA), if available
from a massive companion, but rather, how to stop the BH from merging with and
swallowing its companion star given that the mass transfer will be unstable
until the BH has more mass than the donor (see Klencki et al. (2021); Soberman
et al. (1997); Shore et al. (1994) for a more in depth analysis), which is not
close to the situation observed in most of these HMXBs.

Happily, the issue can be resolved by considering two tools available to
HCA/SEA due to the BHL accretion rate. The first one, a usually (and
correctly) disregarded issue in most CE binaries studies but which is highly
relevant here; it was suggested for a GE of a NS and a RSG in Papish et al.
(2015), further studied (NS/BH with RSG) in Moreno Méndez et al. (2017);
López-Cámara et al. (2019, 2020); here I propose it works in much more bound
stellar environments as well, as detailed next. An accreting CO, either a NS
or a BH, has a very deep potential well, unlike most stars, and thus, the
binding energy released by such accretion is of order

$E_{\rm B}=\epsilon M_{\rm acc}c^{2}$ (4)

with typical $\epsilon\sim 0.06$ to $\sim 0.1$ for NSs and SBHs
($a_{\star}\sim 0$) and as large as $\sim 0.4$ for extreme KBHs
($a_{\star}\lesssim 1$). This translates to a binding energy ($E_{\rm B}$)
release of order $10^{54}$ erg to $7\times 10^{54}$ erg when the accreted mass
is of order $M_{\rm acc}\sim 10\mbox{M${}_{\odot}$}$. But these are energies
comparable to those available to the central engine of a GRB/HN event (of
course, the luminosities are much smaller, being determined by the accretion
rate). Even if 99% of the energy is emmitted as neutrinos this is more than
enough energy for jets (Moreno Méndez et al., 2017; López-Cámara et al., 2019,
2020; Soker, 2015; Shiber et al., 2017; Shiber & Soker, 2018; Papish et al.,
2015) to remove a $M_{\rm CE}\gtrsim 40~{}\mbox{M${}_{\odot}$}$ CE (where I
take the stellar radius to be $\sim 4\mbox{R${}_{\odot}$}$; these are extreme
values, but they make the point that they work even for M33 X-7, the most
massive HMXB in our sample) with a binding energy of $E_{\rm B,CE}\lesssim
10^{51}$ erg. This can be achieved (in as little as a few days to as long as a
few thousand years) with BHL SEA/HCA (i.e., accounting for 1$\%$ neutrino- and
10$\%$ BHL-efficiency, for a total $0.1\%$ efficiency). In escence, this means
that, a 10-M⊙ BH which doubles its mass by BHL accretion will release enough
energy to unbind the CE/GE in which it participates. Hence, the mechanism
proposed in Moreno Méndez et al. (2008); Moreno Méndez (2011) can explain the
formation mechanism for the HMXBs with high-spin BHs without incurring into an
early merger; they may be born SBHs and HCA until the BH masses and spins have
reached their current values! This, of course, has the observational advantage
that the natal BH masses can be large ($M_{\rm BH}\gtrsim
10~{}\mbox{M${}_{\odot}$}$, allowing the binary to survive the formation of
the BH), and the SNe were, at best, low energy, if not failed, thus in
agreement with the large BH masses observed in the HMXBs. The substantial
amount of mass loss through this channel may prevent the orbital separation to
decrease. It is important to notice that this mechanism is self-regulated, if
too much energy is produced by, e.g., jets, accretion rates drop and so does
the jet luminosity, thus allowing the accretion to increase again López-Cámara
et al. (2019), so until, enough accretion and released energy have not been
acheived, the CE will not be lifted.

We may call such a mechanism CE-separation by ablation, and note that, under
the right conditions, it may lead to an extended period of a GE Soker (2015);
Shiber et al. (2017); Shiber & Soker (2018) or pulsed GE (which we may define
as when the mass transfer is intermitent due to KH; i.e., if the stellar
envelope is convective Klencki et al. (2021); Soberman et al. (1997); Shore et
al. (1994)).

As mentioned above Papish et al. (2015), under the assumption that accretion
disks form and launch jets, for systems involving WD, MS, and AGB stars, jets
can remove a CE; here, we have borrowed the model for HMXBs and use the larger
energy deposition of HCA/SAE onto the BHs, plus we note that jets can do the
job as shown in Moreno Méndez et al. (2017); López-Cámara et al. (2019, 2020)
(for a CO-RG CE; and for a MS in our previous estimate), especially if an
accretion disk can be formed, which is likely in a GE, but it is not always
clear within a CE (see López-Cámara et al., 2022), however, any photon or
kinetic outflows which may deposit the large amount of energy by the release
of the gravitational binding energy may do the job well enough Aguayo-Ortiz et
al. (2019, 2021); thus, to allow for larger photon luminosities, accretion
rates close to the HCA/SEA boundary would probably be favored.

For SEA (given its nature, a mostly geometrical effect) to occur it is
important to have something similar to BHL accretion, such that an accretion
disk forms (or a configuration that allows equatorial inflows and polar
outflows); such conditions naturally arise during RLOF, wRLOF (Mohamed &
Podsiadlowski (2007); El Mellah et al. (2018, 2019)888This is a very relevant
mode in these HMXBs (El Mellah et al., 2018, 2019), as it is close to current
observed status in the six HMXBs we discuss in this paper; see the current
x-ray luminosities in column (10) of table 3, they are consistent with wRLOF
MT at, e.g., $\sim 10^{-5}\mbox{M${}_{\odot}$}$ yr${}^{-1}\sim 10^{20}$ g s-1,
for an accretion efficiency of $\sim 10\%$, and $1\%$ in photons, this gives
$L\sim 10^{38}erg$ s-1., GEs, and/or CEs and that is why these stages in
binary stellar evolution should be more carefully considered. BHL facilitates
equatorial accretion (mostly driven by angular momentum transfer) and polar
outflows, e.g., photon radiation (jets). For HCA, independently of the
geometric components999E.g., CCSNe are HCA by nature, but the accretion is
spherically-symmetrtic, i.e. Bondi, however, due to the extreme conditions of
the infalling material, the Eddington limit is that for neutrinos (instead of
photons), which as mentioned earlier, help support the SAS until the Fe core
has fully collapsed and the SW may be revived., neutrino processes alleviate
the radiation transport of the binding energy due to an increase in accretion
rate boyond the limit of SEA. So, the issue of how much energy is radiated by
photons and how much by neutrinos arises (CEs are transparent to neutrinos, so
they cannot help remove the CE). At the HCA/SEA limit we have $\sim 100\%$
photons, whereas at CC we have 99% neutrinos, so we can always pick an
efficiency between 1 and 100% which will introduce an error bar of a couple of
orders of magnitude in our energy budget estimates. For 10% of BHL HCA onto a
10-M⊙ BH (eq.3, section 3), this amounts to a (photon) jet luminosity range of
$L_{\rm jet}\lesssim 4\cdot 10^{(47-49)}$ erg s-1. Which means, that in a
matter of a few hours to a few days, it could remove the CE of a massive star
such as those in HMXBs such as M33 X-7, LMC X-1, or Cyg X-1. A CE lasting up
to a few $10^{3}$ years could be removed with a jet of $L_{\rm jet}\gtrsim
10^{(40-41)}$ erg s-1, below that, a merger stage could be initiated. Of
course, this scenario should leave around a large amount of energized ablated
material that could be currently observable.

An immediate second tool, so simple and elegant it was hidden in plain sight,
now appears: What if all of the CE-available material is accreted onto the BH?
After removal of the binding energy and angular momentum by neutrinos, which
do not lift away the CE material, this option is quite likely for the BHL
accretion regime (and, obviously, overlooked and never considered in the
Eddington-limited-accretion-rate regime); it was even part of the CC process
during BH formation! But wait, is that not a merger by definition? Notice that
this is what occurs during a $q$-runaway, on a dynamical timescale. True, but
as long as $\dot{M}_{\rm BHL}>\dot{M}_{\rm MT}$ everything is solved when
$q\rightarrow 1$, as the $q$-runaway will stop. The $\alpha\lambda$ CE is no
longer an issue either as all material is depleted by accretion onto the BH as
well. It also solves part of the problem when the jets are not luminous enough
to lift the CE and only the KH-runaway is active. In the HCA regime (B or
BHL), all matter that can get rid of its excess angular momentum and kinetic
energy is up for grabs by the BH as long as it crosses the L1 (Lagrange 1)
point, the accretion rate is a no-issue (at about $\dot{M}_{\rm BHL}\simeq
1\mbox{M${}_{\odot}$}$ day-1 for a 10-M⊙ BH and a $\tau_{\rm KH}\gtrsim
10^{4}$ yrs for the stars in HMXBs). As shown in Klencki et al. (2021);
Soberman et al. (1997); Shore et al. (1994), if the mass ratio reaches either
$q\lesssim 1$ values (which would be unlikely before CC SN, or substantial
accretion and ablation), or if the mass transfer reaches a radiative layer in
the stellar envelope, the runaway RLOF or GE/CE MT process should stop (until
the stellar structure reaches a new thermal equilibrium configuration in a
$\tau_{\rm KH}$ timescale for HMXBs in the former case), breaking the unstable
MT and the CE phase appart; this may lead to a GE, or to a KH-mediated,
pulsed-GE stage.

One must be careful to remember that, even if the CE is radiative, but $q>1$,
conservative MT keeps shrinking the orbit in a dynamical runaway; so unless MT
is not conservative, e.g., if ablation helps to partially lift the CE, the
merger is not prevented until $q\sim 1$; and if it does, in a radiative layer,
even an underway merger should stop whenever the accretion rate surpases the
RLOF-MT rate (as the $q$-runaway process is gone, and along with it, the
$\alpha\lambda$ induced orbital drag/friction)! In the convective envelope
case, the second runaway will be activated on top of the $q$ runaway, and
material dumped onto the BH will accrete away at nearly BHL rate in a short CE
phase, which will, nonetheless, substantially alter the binary mass, spin, and
orbital parameters (shrinking the orbit for $q\gtrsim 1$ and conservative MT),
as soon as a radiative layer is found, or if jets are active, the merging
process stops. We may call this scenario CE-separation by accretion.

It is suggestive to note that for the HMXBs with WRs in our sample, even if
the masses are not well constrained, the $q$ values do not necessarily stray
away from $q\sim 1$. So, in principle they may have already evolved past such
a merging CE/GE phase, where the primary MS underwent MT RLOF, attempted a
merger, but then $q\rightarrow 1$, probably driven with CE-separation by
accretion, which allowed for nearly conservative MT, and thus, the orbital
period decreased from a few days to under 1.5 days, the H envelope was
accreted by the BH (spining it up), and then the, already underway, so-called
merger failed, providing us with further observational hints of this scenario.
In a sense, this, change of paridgm in how we approach accretion during CE MT
begs the question: Could you actually merge such systems with a BH when HCA is
such a powerful available accretion tool? It seems that, once $q\sim 1$, the
answer is no.

##  5 Recipee for preventing mergers during CE in HMXBs.

Four mechanisms conspire to cause a merger during a RLOF or CE/GE mass
transfer phase in HMXBs when the mass ratio of the star to the BH is $q\equiv
M_{\star}/M_{BH}>1$:

  * •

The $q$-runaway process: When the donor is more massive than the accretor,
$q>1$, conservative MT shrinks the orbital separation dynamically with swift
and catastrophic consequences (the star no longer fits in its RL).

  * •

The Kelvin-Helmholtz (KH) runaway process but only for convective envelope (or
incumbent layers of) stars or CEs. This is due to the layer being
adiabatically dominated, the Mass-Radio relationship becomes inverted and the
star expands when it losses mass Klencki et al. (2021); Soberman et al.
(1997); Shore et al. (1994).

  * •

Once the CE sets in, the $\alpha\lambda$ mechanism converts the orbit in
vacuum into an orbit in a medium, and thus, friction or drag causes the
orbital separation to decrease.

So, breaking links in this catastrophic chain of events can help stop mergers
or at least help slow them down until another link breaks apart; of particular
importance is bringing $q\lesssim 1$ which works on a dynamical timescale and
precipitates the $\alpha\lambda$ mechanism.

Two important mechanisms are available when HCA/SEA appears in the scene which
help break links of the merger chain of events:

  * •

Separation by ablation (section 3 and 4): The extreme amount of gravitational
binding energy released by the massive accretion process onto a CO can be used
to drive jets and winds (Appendix C,H) or other outflows (e.g., cocoons) which
help remove the CE and, in some cases, keep the stellar envelope within its RL
(see, e.g., Soker, 2015; Shiber et al., 2017; Shiber & Soker, 2018; Moreno
Méndez et al., 2017; López-Cámara et al., 2019, 2020).

  * •

Separation by accretion (Appendices C,D): This is due to the large HCA/SEA
rate during BHL accretion. All RLOF MT material can be accreted by the CO as
fast as it is transferred due to the BHL HCA/SEA nature of the accretion flows
(regulated by how fast the excess angular momentum, AM, can be lost). Thus,
this process can regulate the KH runaway, but not the $q$ runaway Nonetheless,
this HCA/SEA accretion mechanism can rapidly (a matter of days; see eq. 3 in
section 3) and fully stop a merger in process by bringing $q\rightarrow 1$
and, by full CE (accretion) removal, stop the $\alpha\lambda$ mechanism.

---
Figure 1: General scheme of the evolutionary channels with unstable CE stoped
by ablation or accretion with HCA/SEA for HMXBs with KBHs: (1) Top-left panel:
ZAMS masses $M_{1}>M_{2}$, $a_{i}\gtrsim(10^{2-3}\mbox{R${}_{\odot}$}$. (2)
Top-right panel: $M_{1}$ RLOFs, goes into a CE and transfers mass onto
$M_{2}$, orbital separtion shrinks, $a_{2}<a_{i}$ to a few
$10^{1}\mbox{R${}_{\odot}$}$; $q$ is inverted ending the first CE. (3) Middle-
left panel: $M_{1}$ evolves into pre-CC star (orbit too wide for
synchronization, else, too wide to produce large $a_{\star}>0.2$ of pre-CC
star), $a_{2}\sim a_{3}$. (4) Middle-right panel: CC BH formation, mass loss
to explosion makes orbit expand $a_{4}>a_{3}$, SBH born with $a_{\star}>0.2$.
(5) Bottom panel: CE/GE/pulsed GE phase substantially increases BH mass
$M_{BH,5}\simeq(1.5-2.5)M_{BH}$ to bring $a_{\star}$ to the 0.8 to 1 range;
this mass-transfer phase is regulated/stabilized by ablation, HCA/SEA full
envelope accretion, or both; ULX or pulsed ULX phase is likely in process. A
combination of masses and Cases (A, B, or C) can accomodate for the different
HMXBs in Table 1.

Finally, we can present a stellar evolution model which may connect the six
observed HMXBs we present in this paper. Particularly, explaining the
difference between the O-star and the WR systems. We will begin at the fourth
panel of Figure 1. A SBH just formed in a HMXB via a dark CC SN or direct
collapse which produced a massive BH with little binary orbital perturbation.
The O-star companion has recently (in panel 2) been recycled by the CE MT
which deposited the H envelope of the primary. wRLOF ensues for a few Myrs (in
the more massive HMXBs) to tens of Myrs (in the case of the lower mass
systems), thus the mass and AM accretion onto the SBH begins. As the O-star
components evolve, they begin filling their RLs, stable MT ensues due to the
$q>1$ and KH runaways. However, these are rapidly stopped by CE-separation by
ablation. Short episodes of RLOF MT and pulsed-GE evolution follow (thus brief
ULX-like stages may be present). Nonetheless, these are probably resposible
for most of the MT as well as the AM transfer which has produced the spins
currently observed in the KBHs in these HMXBs, particularly in the O-star
systems; this stage, represented in the fifth panel of Fig.1, is the current
state of M33 X-7, Cyg X-1 and LMC X-1.

In the case of the WR systems, IC 10 X-1, NGC 300 X-1 and, perhaps Cyg X-3,
they may have evolved beyond. The O-star left the MS and attempted to enter
the RG, only to quickly fill both RLs (its own and that of its BH companion).
Hence, a merger scenario set in, too much MT too fast; if the BH was buried
too deep into the CE, it may happen that no accretion disk nor jets are
produced, thus setting up the stage for CE-separation by accretion. The
consequence being substantial conservative MT onto the BH through HCA, further
decrasing the orbital separation, and bringing $q\rightarrow 1$ by enhancing
the mass of the BH and removing the H envelope from the O star; thus, causing
the merger to fail. Therefore, producing the WR systems we observe today, with
large BH masses, small orbital periods, and $q\simeq 1$. Ready and prime for
their debut as HN/GRB progenitors and, if they survive the fireworks, GW
sources when the two KBHs merge. It is noteworthy that HN/GRB explosions will
likely disalign the spins of the KBHs with respect to each other as well as to
the orbital AM (which this evolutionary channel would have previously
aligned).

What factors may influence whether CE-separation by ablation or by accretion
are the main channel at one point or another? It is likely that SEA has larger
photon luminosity and accretion-disk winds than HCA (which has a larger
overall luminosity, but neutrino dominated), so it will probably help the
ablation scenario. Thus, SEA needs accretion disk formation in order to
produce jets, whereas HCA does not. Hence, situations where the orbital
separation to stellar radius is large (small RL filling factor by the star)
and the RLOF occurs rather fast ($\tau_{\rm sync}>\tau_{\rm\star,expansion}$)
the stellar envelope will present itself as a large-velocity wind to the BH,
along with velocity and density gradients, Coriolis, etc., thus allowing for
accretion disk formation. So, important factors to consider are the number of
runaway processes active during RLOF, $q$, KH, $\alpha\lambda$. The faster the
RLOF occurs the less likely tidal synchronization will be present. This would
then favour CE separation by ablation. Instead, if the star has tidally
synchronized, then accretion while still assymetric, will have less AM to form
an accretion disk, so jets are less likely (but winds or other polar outflows
may still be present; section 4), thus going into HCA rather than SEA. Hence,
CE separation by accretion will be more likely.

The final relevant timescale in this process is, of course, the viscous
timescale required to fully accrete an accretion disk (if and when it forms)
into the BH (in a sense, it allows us to estimate how rapidly the
disk/accretor/infalling matter can radiate away all the binding energy and
angular momentum necessary to fall into the BH). Given the extended disks that
may form in HMXBs (from $R_{disk}\sim 10^{12}~{}{\rm cm}\sim r_{RL}$, i.e., if
the radius of the disk is that of the RL of the BH, and down to $10^{6}$ cm at
the ISCO101010Located between the ms and the mb, or marginally stable and
marginally bound orbits for SBH or KBH which are located between $1M\leq
R_{isco}\leq 9M$.) this is a relevant issue. Using eq. 3 in section 3, we see
that at $\dot{M}_{\rm BHL}\simeq 10^{-5}\mbox{M${}_{\odot}$}$ s-1 and assuming
that $10~{}\%$ of the energy is radiated (binding energy) away (e.g., by
jets), then $L_{\rm BHL}\simeq 10^{48}$ erg s-1. If we assume these objects
are observed as extreme ULXs, and these are obesreved up to $10^{42-44}$ erg
s-1, we could extrapolate a maximum accretion rate between of
$10^{-4}\dot{M}_{\rm BHL}\gtrsim\dot{M}_{\rm acc}\gtrsim 10^{-6}\dot{M}_{\rm
BHL}$ (or $10^{6}\dot{M}_{\rm Edd}\gtrsim\dot{M}_{\rm acc}\gtrsim
10^{4}\dot{M}_{\rm Edd}$) regulated by accretion-disk viscosity. To avoid
further deviating from our objective, we can simply state that the presence of
objects like ULXs to CCSNe (including all high-energy transients in between)
account for the presence of both SAE and HCA in nature.

One last caveat: McClintock et al. (2014) mention that a spin-orbit
missalignment larger than $\theta_{\rm so}\gtrsim 10^{o}$ could modify the
continuum-fitting $a_{\star}$ meassurements; Cyg X-1 has a $\sim 150$ day
modulation which has been associated (Ibragimov et al., 2007) to the
precession of the jets (and, likely, BH-spin and inner-disk angular momentum
vectors) with an orbital inclination of $(10-20)^{o}$. But it is unreasonable
to expect a correction from $a_{\star}\gtrsim 0.99$ values to
$a_{\star}\lesssim 0.2$ (rather than a small decrease in total spin; see
concerns expressed in Fuller & Ma, 2019). But this raises the, highly
unlikely, alternative that all HMXBs could have tilted spin-orbit alignments
and could be biased towards higher values of $a_{\star}$. However, there are
five out of five (Cyg X-1 among them) spins which are consistent with Fe
K$\alpha$ line meassurements; both these methods seem to be consistent with
each other, thus strengthening the case for the large $a_{\star}$ measured
values. If confirmed, the spin-orbit tilt in Cyg X-1 can help constraint the
SN kick perpendicular to the pre-CC orbital plane and, thus, the explosion
energy.

##  6 Conclusions

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
---|---|---|---|---|---|---|---|---|---|---|---
Models | Bound | Merge | $M_{BH}$ | $M_{comp}$ | $P$ | e | $a_{\star}$ | $E_{rot}$ | SO align | Bubble | Polluted
1 | ✓ | ✗ | ✓ | ✓ | ✓? | ✓ | ? | – | – | ? | ✓
2 | ✓ | ✗ | – | – | – | – | ✗ | ✗ | – | – | –
3 | ✓ | ✗? | ✓? | ✓? | ? | ✗ | ✓? | ✓ | ✓ | ✓ | ✓✓
4 | ✓ | ✓ | ✓? | ✓? | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ | –
5 | ? | ✗ | ✗ | ✓ | ✗ | ✗ | ✓? | ✗ | ✗ | ✓? | ✓
6 | ✓ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗
Obs. | ✓ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗
Table 4: Summary of best estimates of different evolution model outcomes
compared to observations in HMXBs (last row): Model 1, [2]; Model 2, SASI [4];
Model 3, [5,6]; Model 4, [7,8] under the assumption of a merger from CE; Model
5, [7,8]+TW (TW stands for conclusions from this work, i.e., adding the energy
necessary to explain a natal $a_{\star}=0$ vs a $a_{\star}\gtrsim 0.8$); Model
6, [2]+TW, Final row is for the obervables in the six HMXBs considered in this
work. (see Appendix C for more information). ✓ stands for an observationally
confirmed or positive parameter in the model; ✗ for a negative result, i.e.,
not confirmed by observations; $-$ appears when the model does not account for
the parameter; ✓✓ when it is expected to be large, and a question mark is
placed next to an extrapolated parameter or guesstimate, or alone if no
extrapolation is well suited or it is in doubt. The parameters answer the next
list of questions/parameters: (1) Model, (2) will the system remain bound
after a HN/GRB using the energy required to produce the predicted BH spin?,
(3) will the system merge in a CE before producing a HMXB?, (4) Is the
expected BH mass in agreement with those in the HMXB sample?, (5) Is the
expected companion mass in agreement with those in the HMXB sample?, (6) Is
the expected orbital period in agreement with the observed ones?, (7) How
about the expected eccentricity?, (8) Is the expected BH spin observed?, (9)
Is $E_{rot}$, the available energy at CC, consistent with observations?, (10)
Is the expected Spin-Orbit alignment consistent with observations?, (11) Are
they likely to form a accetion/jet powerwed bubble after BH formation?, (12)
Is the companion heavily polluted by a SN or HN/GRB?

In this article we have shown that the formation of natal KBHs in HMXBs is not
energetically consistent with the observables in these BHBs (App.A). As an
aside, the conclusions in Moreno Méndez et al. (2008); Moreno Méndez (2011)
about the impossibilty of producing large natal spins in massive stellar BHs
hold well to the observational evidence (see discussions in Apps.A,D); the
stellar evolution models proposed there are now on solid ground with our novel
MT model. Even after almost 5,000 stellar models in Qin et al. (2019) the
natal spins are produced under assumptions which observations do not favour,
as the results in Fuller & Ma (2019) and other sources have shown (App.F). The
alternative of failed SN fallback is still possible, but not observationally
favored; it also has trouble explaining the fastest spinning KBHs (Appendix
B).

I have shown that MT with HCA/SEA in CE/GE/pulsed-GE phases (KH regulated),
after the formation of a SBH in HMXBs, can be stabilized and mergers after
RLOF are fully preventable through the CE-separation by ablation, CE-
separation by accretion, or by a combination of both; the mechanisms described
in sections 2 and 5. The current population of three O-star and three WR-HMXBs
is suggestive of pre- and post-H-stripping during the attempted merger phases
through RLOF-MT going into CE/GE stages (as in section 4). This mechanism
naturally explains large BH masses, large companion masses, large BH spins,
current spin-orbit alignments, low eccentricities, low peculiar velocities,
low companion pollution, observed X-ray luminosities (through wRLOF with
possible intermitent periods of ULX; Appendices E,G,H,J), as well as the
identified energy bubbles (which could be a SNR, or, more likely, powerwed by
SEA, low HCA periods, or mass ejected by CE/GE phases); we have discussed most
of these parameters, currently observed in some of the HMXBs. Orbital periods,
efective temperatures, may be easily fitted as well (see Table 4 along with
its explanation in Appendix E). Appendix G has a set of possible tests for
some of the hypotheses and results put forward in this article.

I have discussed possible outlines for the evolution giving rise to these
HMXBs utilizing the new tools at the disposal of HCA or SEA (more binary
stellar evolution details can be found in Appendices G,H).

I have also provided highlights of the consequences of our scenarios for
observational consequences for HMXB evolution, ULX sources, the production of
HN/GRB central engines (see Apps.A,E) as well as the production of LIGO
sources, which, even accounting for possible HN/GRB events, may be aided at
remaining bound by a possible BH-mass distribution bifurcation due to the
HCA/SEA phase as well as highly-bound orbits.

The current BHBs in HMXBs, if capable of surviving the formation of the second
BH (which in some of these HMXB systems there is a large chance of producing a
HN/GRB event), have a reasonable chance of becoming a BBH which will merge
during a future GW transient event (Apprndix H).

CE/GE evolution is an extremely-important and outcome-defining phase of
binary- and multiple-stellar systems evolution which is still poorly
understood. How these phases come to an end, in general, is one of its
unsloved mechanisms. Here I have shown mechanisms which solve the merging
problem for systems involving COs which are capable of HCA/SAE at nearly BHL
accretion rates during MT phases. Thus, the implications for binary-evolution
and population-synthesis numerical codes, and the results they produce are
wide and important. Predictions on rates for HNe/GRBs, short GRB/Kilonovae,
ULXs, HLXs, as well as GW mergers, rely on these methods and thus, are
affected by the results I have presented here.

## Acknowledgements

Useful discussions with F. De Colle, D. López Cámara, A. Vigna-Gómez, and T.
Maccarone have benefited this work. The author has made substantial use of
arXiv and ADS services where all data and materials are available as indicated
by the references in this work.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## Appendix A: On HN/GRB central engines

Producing KBHs with natal spins is necessary to explain HN/GRB central
engines, the leading mechanisms to do this rely on resupplying angular
momentum through a stellar-binary channel (e.g., tides); in such cases, it is
neceassary for the pre-BH binary to have a substantially less-massive (as
compared to the BH progenitor star) companion so that the orbital separation
may be small enough to spin the pre-collapsing star up to large AM values.
Moreno Méndez et al. (2011); Brown et al. (2007, 2008); Moreno Méndez (2014)
estimate the natal $a_{\star}$ of the BHs and energies that should have been
available for a number of XBs in or near the Milky Way at the time of CC. The
IMXB LMC X-3 (Brown et al., 2008), with a modest-mass companion star, $M_{\rm
comp}\sim 5\mbox{M${}_{\odot}$}$, was the likeliest GRB/HN progenitor, as its
estimated natal spin of $a_{\rm\star,LMCX-3}\sim 0.45$, sets its energy budget
at around $60B$ (leaving behind the observed BH with spin of $a_{\star}\sim
0.25$), enough for a GRB/HN event, but unlikely to rapidly destroy the central
engine before the energy can be used (Goldilocks scenario Brown et al.
(2008)). In escence, given that the secondaries do no fit within short-enough
orbits to spin the BH progenitors up, HMXBs should have much larger BH masses
but much smaller BH spins ($a_{\star}\lesssim 0.05$, perhaps up to 0.2) and
rotational energies. So, not all BHs have to be born as SBHs, HN/GRB require
some degree of KBH, but more likely from IMXBs, not for HMXBs. Of course, the
descendant BBHs of IMXBs may be less likely to be detectable in a merger than
those of HMXBs given the lower masses involved (and given the likelyhood of
hosting a HN/GRB and the lower mass of their components, they are more likely
to be disrupted at BH formation; however, they may well be more common and fit
into more bound orbits, which could help compensate).

It is interesting to note that all three WR-BH HMXBs (IC 10 X-1, NGC 300 X-1
and Cyg X-3; Table 2) are already within the orbital separation (or obital
period below two days) where the tidal-synchronization timescale is short
(thousands of years or less) and can impart a substantial amount of angular
momentum onto the WR star before CC (see table 1). Thus, it should be expected
that these three systems produce a HN/GRB type of explosion (probably
producing very eccentric orbits as opposed to breaking up given that the
orbits are well bound and the WR are stripped, thus mass loss might not be
significant), and, likely (accounting for masses and orbital binding
energies), a future merger of two KBHs (if the BBHs survive) observable by a
GW observatory.

## Appendix B: On the fallback model to explain the BH spins.

The mechanism of spinning up BHs by SN fallback acretion of Batta et al.
(2017); Schrøder et al. (2018) seems capable of explaining BH spins in HMXBs
of $a_{\star}\lesssim 0.8$. It may still have problems with those around
$a_{\star}\sim 1$. Some issues with this model explaining the HMXBs we are
dealing with in this article are the following:;

  * •

Fuller & Ma (2019) point out that if the exploding star was H-stripped, there
is little or no fallback. These HMXBs have, most likely, gone through a CE
phase where the primary star transferred most of its H-envelope onto the
companion, thus the collapsing stars were in such a situation. However, the
argument of Fuller & Ma (2019) does not hold, as the mechanism is independent
of the composition; i.e., the relevant lever arm for the torque is not from
the H-envelope extension, but from the orbital separation, and if the He layer
can be lifted to a radius beyond the orbital separation (the amount of
material lifted may be relevant to the amount of specific AM required to be
transported between orbit and spin, though), then that should be enough for
the model in Batta et al. (2017); Schrøder et al. (2018) to have merit.

  * •

More relevant may be the issue that the lifted-envelope material, having low
velocity can be rapidly accreted by the companion (similar to wRLOF Mohamed &
Podsiadlowski (2007); El Mellah et al. (2018, 2019), where most of the
material is capured by the companion). If the orbital velocity is similar or
larger than the velocity of the lifted envelope it will be mostly swept and
accreted by the larger-cross-section and more-massive companion, thus,
effectively preventing accretion of mass and angular momentum by the BH.

  * •

Perhaps the most important problem with the Batta et al. (2017); Schrøder et
al. (2018) model however, is that the prediction of the most relevant
parameters are inverted with respect to HMXB distribution of these: From fig.3
in Batta et al. (2017), as well as columns (5) and (6) in Table 1 of this work
we can see that the relationship between BH spins to orbital separations is
inverted; larger spins should be observed for shorter orbital periods. This is
not to say that, perhaps, a different set of parameters or configurations
could resolve the issue or not, but that the most immediate predictions of the
model do not fit the data; hence, at the very least, fine tunning seems to be
necessary from the Batta et al. (2017); Schrøder et al. (2018) model to fix
this issue.

  * •

In the simulations presented, none explain the largest BH spins in the
observed HMXBs.

## Appendix C: Implications for and from ULXs.

If the HMXB systems in tables 2 & 3 formed their BH about $10^{6}$ years ago,
underwent a quick CE stopped by ablation or accretion as we propose, and
continued transfering mass through RLOF but mediated by a KH timescale (i.e.,
an intermitent or pulsed GE phase) at a rate that allows the current systems
to transfer and accrete, i.e., some $10\mbox{M${}_{\odot}$}$ per Myr period,
that gives an accretion rate of about $\dot{M}\sim
10^{-5}\mbox{M${}_{\odot}$}$ yr $\sim 10^{21}$ g s-1, which produces an
average luminosity of $L\simeq 10^{42}$ erg s-1 for the last Myr. This is
right where ultraluminous X-ray sources (ULXs) luminosities lie. Such an
accretion rate is borderline between the HCA and SEA regimes (within $10^{3}$
and $10^{4}$ Eddington). In a sense, ULXs may be our view of these HCA GE HMXB
systems as they transform their SBH into KBHs readying them for a future GW
merger with large BHs spins. As the HCA/SEA-fed BHs grow larger masses and
spins, their jets luminosity must increase as well; thus, as these systems
evolve and keep accreting from their companion stars, they may likely become
brighter ULX sources with time (unless the excess goes mostly through the
neutrino channel). Bubbles/cocoons detected in these systems could help put
constraints on photon vs neutrino fluxes produced by the HCA/SEA on these
systems, as well as if a continuous ULX stage of GE occurs or a, KH-timescale,
intermitent ULX through pulsed GE dominates.

The HMXBs we include in this work, actually, show luminosities closer to
$10^{38}$ erg s-1 (Table 2, (10)). Most likely (and from direct observation of
the current status of these HMXBs), accretion is wind fed most of the time,
and, every $\tau_{\rm KH}\gtrsim 10^{4}$ years, an episode of mass transfer
may occur, giving rise to a limited time period of HCA feeding frenezy and,
perhaps, intermitent ULX activity.

## Appendix D: On Taylor-Spruit dial downs.

The Taylor-Spruit (TS) dynamo (Spruit, 2002) is an important mechanism which
regulates the angular transport inside stars. As such, it has been calibrated
in stellar-modeling codes by using the observation of NSs, WDs and stellar
(Sun) spins (Heger et al., 2005; Suijs et al., 2008; Eggenberger et al., 2005,
respectively). Fuller & Ma (2019) discuss that the TS dynamo has in general
been underused in stellar models if the population of low mass stars is to be
explained. Using this calibration, but for massive stars, they find that most
single (or non-torqued up, e.g., by tides or mass transfer) BHs should be born
with low spins. However, by taking the Eddington-limit narrative, they also
conclude that BHs in HMXBs are born with large spins (without providing a
mechanism to acheive this).

Qin et al. (2019) assume in their models that, contrary to the results in
Fuller & Ma (2019); Heger et al. (2005); Suijs et al. (2008); Eggenberger et
al. (2005) (note that Fuller & Ma (2019) was released after Qin et al.
(2019)), the TS dynamo can be dialed down in their massive-stellar models
(winds are also set low compared to the typical values for the metallicities
of the systems they describe, however, clumping can justify this assumption)
in order to try to explain the observed large spins in the HMXB BHs; they
produce nearly 5,000 models (including some with CHRM models), but do not
fully reproduce the obesrvations (missing spins, orbital periods, etc.); they
do not take into account the energy available in the BH progenitor. In Moreno
Méndez (2014) a similar assumption was taken (dialing down TS and/or the
poloidal magnetic field which connects the evolved stellar core with the
stellar envelope) in the context stellar models for the progenitors of HN/GRB
Moreno Méndez et al. (2011); Brown et al. (2007, 2008); nonetheless, this is
required only for late and short stages of stellar evolution, after Case C
mass transfer; something which is explicitly allowed by, and hence consistent
with, the results in Fuller & Ma (2019).

## Appendix E: On obtaining the results presented in Table 3.

Conclusions reached by papers Qin et al. (2019) (row 1), Moreno Méndez &
Cantiello (2016) (row 2), Batta et al. (2017); Schrøder et al. (2018) (row 3),
and Moreno Méndez et al. (2008); Moreno Méndez (2011) (row 4), and if missing,
extrapolated from analytical energy estimates. Not all these papers are aimed
at reproducing these HMXBs but only to explain their large spins (in fact
Moreno Méndez & Cantiello (2016) aims to show that natal kicks, through a
directed SASI, cannot produce such large spins), so they are not expected to
match the observations. Estimates derived from the conclusions of this paper,
and applied to the conclusions of Qin et al. (2019) are presnted in row 5; the
conclusions of this paper are applied to the model in Moreno Méndez et al.
(2008); Moreno Méndez (2011) to acheive row 6.

## Appendix F: Evolutionary paths and implications for LIGO sources.

Systems like M33 X-7, Cyg X-1, LMC X-1, IC 10 X-1, and likely NGC 300 X-1 and
Cyg X-3, are likely formed with a massive primary which transfers a
substantial amount of its envelope mass to their less massive companion star
(i.e., up to a few tens of M⊙) and rejuvenates it (thus, increasing its
lifespan by up to a few Myrs Braun & Langer (1995)) in either, late-Case A
(late during MS 111111Let us remember that a very massive Fe core must be
allowed to form before CC; so this scenario is not highly favored, especially
in the larger metallicity systems, as winds from a stripped stellar He core
will probably allow the star to lose too much mass and produce a neutron star
as opposed to the massive BHs observed; however, this could explain the
apparent BH-mass diference in Cyg X-3 (Table 2) with respect to the other
HMXBs.), case B (during the RG phase; once most of the He mass of the star has
been produced, $\sim 90\%$, and the formation of a BH is more likely; also,
close to 90% of the radial expansion occurs at this stage, so this has the
largest probability considering the possible initial orbital separations
between these binaries), or even better, Case C (here the He core has grown
even larger), favoring the BH-mass observations (Figure 1 shows a general
schematic outline of this stellar evolutionary set of channels for the
formation of HMXBs until they reach said stage; Table 2, columns (2,3) show
the masses in the HMXBs). The latter mechanism has the two MT runaway
processes active, so it may also be the one that better explains the final,
small orbital separations. This is the case discussed in Moreno Méndez et al.
(2008); Moreno Méndez (2011); Brown et al. (2007, 2008); Moreno Méndez et al.
(2011); Moreno Méndez (2014) given that it allows LMXB and IMXB to obtain and
retain the largest amount of angular momentum for HN/GRB central engines
(Appendix A). Prior to the mass transfer, $q>1$ (the primary star, or
progenitor of the current BH in said systems, being more massive); then, the
nearly conservative MT via RLOF and CE strips the H envelope from the primary
star and onto the secondary, inverting $q$ (helping bring this first CE phase
to an end), decreasing the original orbital separation which can be as large
as several-hundred to over-a-thousand R⊙, to only a few R⊙, an orbit not much
wider than those currently observed in these HMXBs (particularly, the O-star
systems). The two stars in these systems, being rather massive and not-too-
compact, do not fit in orbits with periods shorter than several days, thus,
even if they tidally synchronize, not enough angular momentum can be
transferred from their orbit to their spins Moreno Méndez et al. (2008);
Moreno Méndez (2011). Thus, a massive-stellar SBH forms in a rather dark CCSN
or by direct collapse. The system is hardly affected due to the mass loss
during the SN (from Blaauw-Boersma -BB- kicks, Blaauw (1961); Boersma (1961),
we know the system cannot lose more than half the mass before the SN,
otherwise the system is unbound) mostly because most of the envelope has been
transferred onto the companion star which helps keep the binary bound (along
with the substantial decrease in orbital separation). The fact that the core
is very massive helps by reducing the likeliness of the SN being successful,
thus favouring a direct collapse into a massive stellar SBH.

After CC of the primary into a massive-stellar SBH, the secondary star will
wRLOF, and eventually RLOF staring a CE/GE/pulsed-GE phase (the one where we
have descibed, in section 4, how to avoid or stop the merger; the case of the
pulsed-GE will alternate with extended KH-periods of wRLOF and it is likely
this is the current situation in most of these HMXBs) of MT back onto the BH
which spins it up to a KBH (to the presently observed values) by increasing
the BH mass by a factor of 1.5 to 2.5121212This depends on the final spin, as
well as the location of the ISCO: at the marginally-stable or marginally-bound
orbit; Fig. 6 Brown et al. (2000)., and which may further decrease the orbital
separation (as $q>1$, again, although, note that here the originally secondary
star has become the massive donor component of the system). If this is the
case, and if the orbits shrink to orbital periods within 1 to 2 days (or
less), the secondary stars will probably synchronize with the orbital period
within a few kyrs (Zahn, 1977, 1975, 1989; Zahn & Bouchet, 1989; Moreno Méndez
et al., 2011; Brown et al., 2007, 2008; Moreno Méndez, 2014). In the case of
the WR HMXBs in this study, it is likely that these systems underwent a CE
phase towards a merger, which was accretion-separated, thus MT was nearly
conservative, allowing for a further decrease of the orbital separation (the
radius of the BH no longer being an issue) up to the point where $q\sim 1$ and
are either synchrinizing or fully synchronized. If this is the case, they will
most likely produce a central engine and, thus, explode as HN/GRB explosions
when the second BH forms. However, if the mass transferred onto the massive
KBHs is large, or if little mass is lost during the HN (again, the three WRs
are much less massive than the O stars in the other three HMXBs), the BB kicks
will not unbind these systems (the HN kicks could still do this if not
properly aligned, though; but the orbits are more bound); the tighter orbital
separation may also help keep the binary from breaking appart (but the
eccentricity may become extreme). In any case, if these binaries survive the
formation of the secondary BHs (regardless of whether or not they produce
HN/GRB explosions) and if the SNe and BB kicks are favorable, they have the
possibility, if the final orbital separatations are within a couple of tens of
R⊙, to form future BBH mergers detectable as gravitational waves (GWs) events.

Cyg X-1 has a spin which is extremely close to $a_{\star}\simeq 1$ and the
cosmic censorship hypothesis may be at play, it also has an orbital period
which is currently the largest among the six HMXBs, so it is questionable
whether the BH may easily accept any more mass (unless forced into a CE) and
thus, whether the orbital separation may grow any smaller (e.g., driven into a
merger attempt which separates by CE-separation by accretion). Hence, it is
unlikely that a central engine may form. Thus, a massive SBH may form when the
secondary collapses. Due to the large mass of the companion star in Cyg X-1,
it may produce another BH around $M_{BH}\sim 20\mbox{M${}_{\odot}$}$, not
unlike the BBH systems in GWs $190408\\_181802$, $190512\\_180714$,
$190708\\_232457$, $190719\\_215514$, or $190828\\_065509$ Abbott et al.
(2021). However, its current orbital situation does not favor it within a
Hubble timescale (HT); nonetheless, a correctly aimed SN kick may help.

Given the current mass and apparent age of the companion in M33 X-7, the BH
may accrete a further 10 to 15 M⊙ from its secondary star spinning it up to
$a_{\star}\simeq 1$, and this may still allow for the secondary to collapse
into a massive 30-M⊙ BH. If so, it could lead to the formation of a system
analogue to that which gave rise to GW150914. Given the size, mass, $q$, and
MT necessary to spin the BH, it is unlikely that the secondary BH will form as
a KBH (unless a merger attempt with CE-separation by accretion phase can take
place). Thus, the binary has big chances of surviving such an event. Now,
unless the orbital period can manage to get shorter than a couple days (by MT
or SN/BB kicks) it may not merge by GW emission within a HT.

The BH in LMC X-1 could very well double its mass before it reaches its
maximum spin and brings $q\lesssim 1$. Its orbit like that of Cyg-1 and M33
X-7 does not currently help predict a high-likelyhood merger within a HT; but
once more, SN kicks or a CE-separation by accretion scenario could help
produce such an outcome.

The cases of IC 10 X-1, and probably NGC 300 X-1, dependending on their actual
masses, may be somewhat similar (as their companion types and orbital periods
are), they may form HN/GRB events, but their strongly bound orbits may help
them survive, and in the case of low mass loss, they are likelier candidates
for BBHs that may merge by GW emission within a HT.

The outlayer among the systems we study is Cyg X-3. It may have the smallest
masses, but these still remain uncertain. What is clearly different is that
the system is extremely bound at $P_{orb}\sim 0.2$ days, allowing little doubt
of tidal synchronization and, most likely, a HN/GRB local (Galactic) candidate
within the next Myrs (even if the CC mass were low, it could esaily produce a
millisecond magnetar as a central engine). Its tight orbit will most likely
allow it to also produce a GW event in the not-so-distant future, unless the
HN/GRB event is powerful enough to break it appart even in such a tight
binary; formation of a BH rather than a magnetar might also help keep the
system bound.

An interesting insight here is that, given the KBHs already present in the
O-star systems, we can infer such KBHs present as well in the WR systems, and
that the latter WR stars are probably synchronizing/synchronized. Central
engines will develop during the CC of the secondaries of at least the WR
systems, but being tight binaries having lost the H envelopes, and not being
too massive, they may turn into BBH with two KBHs which may merge within a
Hubble timescale. However, the BH spins, and orbital AM vector, may be
arbitrarily oriented, even if they are originally parallel (as may be expected
due to the formation channel here suggested).

If MT can proceed close to conservative while still preventing a merger
(through the CE accretion-separation mechanism) in the three O-star-BH HMXBs
(M33 X-7, Cyg X-1, and LMC X-1) and the orbital separations decrease, they may
evolve into WR-star HMXBs similar to the other three systems (at larger masses
and orbital separations, albeit). If these are indeed two stages of a similar
evolutionary track, this is another reason to suspect that the expected Kerr
parameters of the current BHs in NGC 300 X-1 and Cyg X-3 may be as high as
those in the rest of the HMXBs we have discussed.

Klencki et al. (2021); Soberman et al. (1997); Shore et al. (1994) argue that
in order to remove a CE phase, it is much more favorable to have a convective
envelope (like in Case C mass transfer) as it expands at the loss of material,
so it is less bound. If the envelope is radiative, it contracts, and becomes
more bound and it is harder to remove, possibly leading to a merger in the
case of HMXBs under the $\alpha\lambda$ CE prescription, which would otherwise
evolve into the BBHs which merge by GW emission and are detectable by LIGO.
However, in our BHL HCA regime the BH may fully accrete an envelope if it is
not unbound, i.e., if it is radiative, (during late MS or RG, case A or B mass
transfer), and instead, case C will remove a larger portion of them (which
could result in lower accretion into the BH). Assuming a merger is regularly
prevented, a bifurcation in masses and spins could occur depending on whether
Case A or B vs Case C mass transfer are the cause of a CE/GE phase. By
preventing mergers the prescriptions presented in this article turn around the
conclusions of the $\alpha\lambda$ CE prescription that BBHs cannot be formed
by CE in a radiative envelope (Cases A or B), and replaces it with the
possible bifurcation in BH mass distribution. The BH masses and BH spins in
the predictions of many binary evolution channels for BHBs (one BH) and BBHs
(two BHs) might need to be revisited and modified accordingly as well.

Out of the binary stellar evolutionary channels to produce BBH LIGO sources,
HMXBs are among the main contenders. LMXBs will not form a second BH by
definition. IMXBs, may produce CO-WD, CO-WD (CO stands for compact object;
these may not be LIGO sources but are GW sources nonetheless), or BH-NS
sources which have been observed in a few events (see, e.g., GW200105 and
GW200115131313Where no electromagnetic counterparts were detected, see e.g.,
Becerra et al. (2021), which is, by itself, an interesting result given the
masses, $q$, and, hence NS and BH radii involved. Abbott et al. (2021)),
however, having the main HN/GRB sweetspot in the BH- spin/energy range, many
binaries might get disrupted instead. HMXBs, which currently are BHBs, can
naturally evolve into smaller-orbit BH-NSs or BBHs which may merge in a Hubble
timescale. The ones with the most bound orbits during the BHB-stage could
potentially develop HN/GRB central engines which will produce a bright
tranient source and could break appart (but given the masses involved are less
prone than those in IMXBs), the rest of them may be well poised to explain a
large portion of the LIGO sources (along with the channels coming from
multiple systems and/or dynamical exchanges in clusters).

## Appendix G: Possible observational tests.

A possible observational confirmation of a KH-mediated HCA/SAE regime at work
in HMXB or ULX systems would consist on identifying these by observing their
nebula in search for clumps or nodules, if their velocity is fast with respect
to the ambient medium, or bubbles within bubbles if they are static.
Estimating the energy required to inflate each bubble/nodule, and their ages
(from their velocities) can help differntiate HCA/SAE stages from wind feeding
stages. If accretion rates are close to 10$\%$ of BHL ($\sim
1\mbox{M${}_{\odot}$}$ day-1), as expected from the dynamical, unstable MT due
to their $q$, and jets form to separate the CE by ablation with an efficiency
of 0.1% (acounting for accretion efficiency of 10% plus photon to neutrino
efficiency of 1%) we obtain $L_{jet}\sim 10^{46}$ erg s-1 over $10^{5}$ s,
thus lifting a $10^{51}$ erg bound CE in about a day. KH timescale is of order
$10^{4-5}$ years for the massive companions in the HMXBs in Tables 2 and 3,
so, in a Myr timescale there should be 10 to a hundred day-long outbursts
(depending on actual sizes of the MT.

ULXs are interesting since, at $10^{39}\lesssim L_{x}\sim\lesssim 10^{42}$ erg
s-1 they cover the SEA range for a $10-\mbox{M${}_{\odot}$}$ BH. So orbital
modulation of days and orbital velocities of $v_{orb}\sim 10^{7}$ cm sec-1
could confirm the existence of HMXBs within the accretion range we explore in
this article. Obesrvartions of sources with larger luminosities, e.g., up to 4
or 5 orders of magnitude larger hyperluminous X-ray sources (HLXs) but similar
orbital modulation and/or orbital velocities, extend that range well into the
HCA range. Lasota et al. (2011) have observed variability in HLX-1 (at $P\sim
10^{7}$ s) and proposed a model for an Eddington-limited BH of $M_{BH}\sim
10^{4}$ which is fed by a star in an eccentric ($e\sim 0.3$) $P_{orb}\sim 380$
day orbital period. We note here that a Cyg X-1-like or heavier system,
undergoing a HCA/SEA GE during Case B or C (the latter would probably provide
for larger accretion rates), with orbital separation of 6 to 7 AU may fit the
dynamical model as well. Thus HLX-1 could be the transition of a massive
stellar SBH into a KBH in a HMXB. Further study would be interesting in this
direction.

Systems where CE are observed may also help, but in such cases, the CE itself
may interfere with obtaining $L_{x}$, nonetheless, if orbital parameters,
velocities, and energies can be extracted, progress can be made as well.

## Appendix H: If the KBHs were natal.

By disregarding again, and for the last time, only for the sake of argument,
the whole (fundamental and most important part of this whole excercise) energy
argument against the natal large spins of the BHs in these HMXBs, we may still
ask, why would such exotic mechanisms as I present in this article (HCA, CE,
and Jets and/or full CE accretion) be required to explain the formation of
such systems if something like the Case A mass transfer, or even CHRM systems
explained in, e.g., Qin et al. (2019) seem to work? Well, because they do not
really solve the problem of forming such large natal BH spins satifactorily
either. They also need assumptions which do not match well with the
observations. It is further, highly unlikely that, inside such a kinetically-
energetic star, differential rotation, convection, and thus, dynamos, will not
actively use the TS mechanism to transfer angular momentum within it during
their MS. This situation must be maintained with this rotation profile for a
few million years. The scenarios in their parameter space run into trouble
even considering chemically-homogeneous, rotationally-mixed (CHRM) stars as
they do not reach the observed orbital parameters either (some of the problems
for CHRMs to produce such BH spins in HMXBs were studied and pointed out as
early as Moreno Méndez et al. (2008) and later in Moreno Méndez (2011)).

To finish this hypothetical arc let us finally return to answer one of the
questions we left pending: If these BHs were born with such large spins and
energies, what would the expected observable consequences be for CC? The
precollapsing stars in these HMXBs systems had their $10^{53}$ energy budget
from a typical CCSN, but, if the observed BH spins where natal, they had
between $10^{54-55}$ ergs of extra kinetic energy available during collapse
(so, it would not be as easily lost to neutrinos). During CC, a large fraction
of the material will not be able to fall directly onto the PNS or BH in the
center, but it will rather form a massive accretion disk (as discussed in
MacFadyen & Woosley, 1999; Moreno Méndez et al., 2011; Brown et al., 2007,
2008; Moreno Méndez, 2014, as well as other literature on collapsars) and a
large magnetic field; a Penrose process Penrose & Floyd (1971) will most
likely emerge, e.g., a BZ (Blandford and Znajek, Blandford & Znajek (1977))
and/or a BP (Blandford-Payne, Blandford & Payne (1982)) central engine,
producing a GRB and a HN, blowing the rest of the star appart and, likely,
preventing the formation of a very massive stellar BH (so, it is to be
expected that $M_{BH}<10\mbox{M${}_{\odot}$}$) . This is not consistent with
what observations show, all BHs in these HMXBs are rather heavy (with the
possible exception of CygX-3, all have $M_{BH}>10\mbox{M${}_{\odot}$}$, see
(3) in Table 1); Cyg X-1 had a dark explosion or no explosion at all Mirabel &
Rodrigues (2003). None of the companion stars show strong HN pollution,
although, this is difficult to corroborate. The eccentricity seems to be small
in all these systems (eventhough the orbital separations are large so tidal
circularization is of order Myrs Zahn (1977, 1975, 1989); Zahn & Bouchet
(1989)). No strong evidence of large spin (or jets) missalignment with the
orbital-angular-momentum vector. Thus, it is highly unlikely that the
evolutionary paths which assume a large natal spin are correct. This is not to
say that all of these systems came out of direct collapse onto BHs nor from
dark explosions, only that, at most, the SNe explosions were modest at best,
the natal spins were originally much smaller than they currently are
($a_{\star}<0.2$ as pointed out in Moreno Méndez & Cantiello (2016); Moreno
Méndez et al. (2008); Moreno Méndez (2011)) and it is unlikely that a central
engine formed in any of these HMXB systems during the collapse that formed
their, originally SBHs, currently KBHs. It is important to note then, that if
a CE/GE phase has occurred, with the implied mass and angular-momentum
transfer, this naturally explains the spin-orbit alignment (even if it was not
the case after BH formation), the low eccentricity, the low peculiar velocity
(if SN is small or abscent), and perhaps the low pollution of companion (with
caveats).

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