Accretion Disk Assembly During Common Envelope Evolution: Implications for Feedback and LIGO Binary Black Hole Formation

We will consider flow around a secondary object of mass, M₂, and radius, R₂, that is engulfed by its evolving companion (denoted here as the primary star) with total mass M₁ and radius R₁ ≫ R₂. The pair has a mass ratio, q = M₂/M₁. The embedded object, separated by a distance a ≲ R₁, will move within the CE with a characteristic orbital velocity ${v}_{{\rm{k}}}^{2}(a)=G[{M}_{2}+{M}_{1}(a)]/a$, where M₁(a) is the enclosed mass inside the orbit of the secondary. The orbital motion of the embedded object is likely to be desynchronized from the envelope of M₁ and the relative velocity can be written as ${v}_{\infty }={f}_{{\rm{k}}}{v}_{{\rm{k}}}$, where f_k is the fraction of Keplerian velocity representing the relative motion between the gas in M₁'s envelope and M₂.

Studies of CE often make use of Hoyle–Lyttleton accretion (HLA), a simple framework for understanding flow around an embedded secondary (e.g., Iben & Livio 1993; MacLeod & Ramirez-Ruiz 2015a, 2015b; MacLeod et al. 2017). In this case, the object moves supersonically through the envelope and gravitationally focuses the surrounding gas. Accretion is envisioned to take place if the impact parameter of the incoming gas is less than the accretion radius,

$$\begin{eqnarray}&&{R}_{{\rm{a}}}=\displaystyle \frac{2{{GM}}_{2}}{{v}_{\infty }^{2}},\end{eqnarray} \tag{ 1 }$$

where ${v}_{\infty }$ is assumed to be supersonic (Hoyle & Lyttleton 1939; Bondi & Hoyle 1944; Bondi 1952). The corresponding mass accretion rate onto the embedded companion can then be written as

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{HLA}}=\pi {R}_{{\rm{a}}}^{2}{\rho }_{\infty }{v}_{\infty },\end{eqnarray} \tag{ 2 }$$

where ${\rho }_{\infty }$ is the density of the incoming gas (for a recent comprehensive review, the reader is referred to Edgar 2004). The deflection of material will result in a reconfiguration of the flow, which in turn generates a net dynamical friction, drag on the secondary (Ostriker 1999).

HLA was first investigated numerically by Hunt (1971) to determine whether ${\dot{M}}_{\mathrm{HLA}}$ can provide an accurate estimate of the rate of mass accretion, and it was concluded that it was indeed reasonable ($\dot{M}\approx 0.88\,{\dot{M}}_{\mathrm{HLA}}$). This pioneering work laid the ground for several hydrodynamical studies for HLA in two (Shima et al. 1985; Blondin & Pope 2009; Blondin 2013) and three (Ruffert 1994a, 1994b, 1995, 1996; Ruffert & Arnett 1994; Blondin & Raymer 2012) dimensions. Of particular relevance to our work are the studies of Ruffert (1994a, 1994b, 1995, 1996) and Ruffert & Arnett (1994) as they explored in great detail the effects of varying the properties of the background gas, in particular, the role of the compressibility of the flow. If the flow is more compressible, the loss of pressure support will result in a standing shock that resides closer to the accretor. The higher post-shock densities in addition to the steep pressure gradients, were shown to produce higher mass accretion rates. Simulations in two dimensions showed that for highly compressible gas, the flow structure becomes significantly less stable, resulting in large variations in the mass accretion rate (Blondin & Pope 2009; Blondin 2013).

The HLA formalism has been widely used to describe the flow around objects embedded within a common envelope, but it fails to provide an accurate description of the flow. The formalism assumes a homogeneous background, which does not reflect the steep density profiles of evolving stars. Studies of HLA with vertical density and velocity gradients have been tackled by several groups (Livio et al. 1986a, 1986b; Soker et al. 1986; Fryxell et al. 1987; Fryxell & Taam 1988; Taam & Fryxell 1989; Ruffert 1997, 1999; Armitage & Livio 2000; MacLeod & Ramirez-Ruiz 2015a, 2015b), although in most cases the assumed density gradients are shallow and thus not representative of those found in stellar envelopes (MacLeod & Ramirez-Ruiz 2015a). Symmetry breaking generated by the vertical gradient gives the flow net angular momentum relative to the accreting object. Thus, even small gradients can have large-scale impacts on the flow, leading to rotational support for material instead of radial infall as envisioned in HLA.

The radial inflow approximation breaks down when the gas reaches a radius ${R}_{\mathrm{circ}}={{l}_{z}}^{2}/{{GM}}_{2}$, where l_z is the specific angular momentum (see, e.g., Section 4.2 of MacLeod & Ramirez-Ruiz 2015a). Armitage & Livio (2000) performed two-dimensional simulations with an exponentially decreasing density gradient for radiation pressure dominated (γ = 4/3) flows; they use a cylindrical geometry and assume a reflective inner boundary condition. Under these conditions, they found a stable centrifugally supported structure forming in their simulations. However, in recent three-dimensional calculations using similar density gradients, MacLeod & Ramirez-Ruiz (2015a) and MacLeod et al. (2017) failed to produce rotationally supported structures for radiation pressure dominated flows and only saw disks when considering a softer equation of state.

Generally, envelope gas may have a different response to compression under varying density and temperature conditions as determined by its equation of state. The thermodynamic description of the flow can be characterized by adiabatic exponents

$$\begin{eqnarray}&&{\gamma }_{1}={\left(\displaystyle \frac{d\mathrm{ln}P}{d\mathrm{ln}\rho }\right)}_{\mathrm{ad}},\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&{\gamma }_{3}=1+{\left(\displaystyle \frac{d\mathrm{ln}T}{d\mathrm{ln}\rho }\right)}_{\mathrm{ad}},\end{eqnarray} \tag{ 4 }$$

where the subscript signals partial derivatives along a particular adiabat. γ₁ is relevant for calculating the sound speed of the gas, ${c}_{{\rm{s}}}^{2}={\gamma }_{1}P/\rho$, while γ₃ is related to the equation of state,

$$\begin{eqnarray}&&P=({\gamma }_{3}-1)\rho e,\end{eqnarray} \tag{ 5 }$$

where e is the internal energy. In general, γ₁ will be greater than γ₃ when radiation plays a prominent role because in that case pressure increases faster than temperature in response to compression.

Material in a disk dissipates its motion perpendicular to the orbital plane, forming a differentially rotating structure. A net flow of material inward results when a viscosity-like stress transports angular momentum content outward in the shear flow. A dynamo process of some kind is commonly believed to work and simple physical considerations suggest that fields generated in this way would have a length scale of the order of the disk thickness and could drive a strong hydromagnetic wind (Balbus & Hawley 1991). As discussed in the Introduction, if the embedded object is a compact object, this outflow could have enough kinetic energy to substantially alter the structure of the envelope.

In the remainder of this work, we discuss how the properties of the stellar envelope have a decisive effect on whether a rotationally supported structure can form around an object embedded within a CE.

We perform idealized simulations of the flow around an embedded object using the CE Wind Tunnel (CEWT) formalism presented by MacLeod et al. (2017). The inviscid hydrodynamic equations are solved using FLASH (Fryxell et al. 2000), an Eulerian, adaptive mesh refinement code. This setup models the embedded secondary companion as a sink point particle of radius R_s at the origin. A wind, representing the gaseous envelope with a vertical profile of density and pressure, is fed in the +x direction from the −x boundary. This profile is in hydrostatic equilibrium with a vertical ($\hat{y}$), external y⁻² gravitational acceleration representing the gravity of the enclosed mass of the primary star. The code and methodology are described in detail in MacLeod et al. (2017), but we include a few key points here for context.

2.2.1. Local Description of the CE

The vertical profile of density and pressure within the envelope are locally approximated with a polytropic profile of a massless envelope, which assumes that the enclosed mass is small compared to the total mass of the primary across the region simulated ∼R_a. In this case, the pressure and density profiles of the surrounding envelope are described by

$$\begin{eqnarray}&&\displaystyle \frac{d\rho }{{dr}}=-g\displaystyle \frac{{\rho }^{2}}{{{\rm{\Gamma }}}_{{\rm{s}}}P},\end{eqnarray} \tag{ 6 }$$

and,

$$\begin{eqnarray}&&\displaystyle \frac{{dP}}{{dr}}=-g\rho ,\end{eqnarray} \tag{ 7 }$$

where g = GM₁/r². The structural polytropic index of the stellar profile is ${{\rm{\Gamma }}}_{{\rm{s}}}={\left(\tfrac{d\mathrm{ln}P}{d\mathrm{ln}\rho }\right)}_{\mathrm{env}}$.

The gas envelope might have a different response to compression (as characterized by γ₁ and γ₃) than the one implied by the polytropic index of the stellar profile. This is because rearrangements induced by the embedded object will happen on a timescale much shorter than the thermal timescale of the evolving primary. For example, a fully convective envelope might have Γ_s ≈ γ₁ whereas a radiative envelope might have Γ_s < γ₁. In the case of an ideal gas, as considered in our FLASH calculations, we have the simplification γ = γ₁ = γ₃. Regions where gas pressure dominates can be described by a γ = 5/3 while regions where radiation pressure dominates are well characterized by a γ = 4/3.

Locally, within this polytropic stellar envelope, the flow is described by dimensionless parameters such as the Mach number,

$$\begin{eqnarray}&&{ \mathcal M }={v}_{\infty }/{c}_{s,\infty },\end{eqnarray} \tag{ 8 }$$

and the density gradient

$$\begin{eqnarray}&&{\epsilon }_{\rho }={R}_{{\rm{a}}}/{H}_{\rho }.\end{eqnarray} \tag{ 9 }$$

Here, H_ρ = −ρdr/dρ, and _ρ represents the number of scale heights across the accretion radius within the primary star's envelope, with ${\epsilon }_{\rho }\to 0$ describing an homogeneous density structure and ${\epsilon }_{\rho }\to \infty$ describing a very steep density gradient. MacLeod et al. (2017) show that the expression

$$\begin{eqnarray}&&{{ \mathcal M }}^{2}={\epsilon }_{\rho }\displaystyle \frac{{(1+q)}^{2}}{2q}{f}_{{\rm{k}}}^{4}\left(\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{s}}}}{{\gamma }_{1}}\right),\end{eqnarray} \tag{ 10 }$$

relates these flow parameters and describes pairings of ${ \mathcal M }$ and _ρ for a given binary mass ratio and envelope structure.

2.2.2. Wind Tunnel Domain, Conditions, and Diagnostics

2.2.3. Simulation Parameters

The key parameter that we vary across our simulations is the gas adiabatic index, γ. In doing so, we represent portions of the CE material with different compressibility, and as we will show, different susceptibility to the formation of dense, rotationally supported disk structures.

The simulations adopt a sink boundary of R_s = 0.02 R_a, a central density gradient of _ρ = 2, a velocity fraction f_k = 1, and a mass ratio of q = 0.1. These parameters may be compared with stellar envelope structures shown in MacLeod & Ramirez-Ruiz (2015a) and MacLeod et al. (2017). We will discuss the properties of typical stellar envelopes in Section 4, but for context, these conditions could represent those found in a M₁ = 1 M_⊙ giant branch star with R₁ = 140 R_⊙ engulfing a M₂ = 0.1 M_⊙ star at a separation of a = 0.9 R₁, or alternatively, a M₁ = 80 M_⊙ red giant with R₁ = 740 R_⊙ engulfing a M₂ = 8 M_⊙ star at a separation of a = 0.85 R₁.

We vary the adiabatic index across four simulations using (a) γ = Γ_s = 5/3, (b) γ = Γ_s = 4/3, (c) γ = 1.2 with Γ_s = 4/3, and (d) γ = 1.1 with Γ_s = 4/3. We adopt Γ_s ≥ 4/3 to have a polytropic index that is stable to perturbations in pressure (Bonnor 1958).

The circularization radius depends on the density profile and can be integrated numerically. We make use of ${R}_{\mathrm{circ}}={l}_{z,\infty }^{2}/{{GM}}_{2}$, where ${l}_{z,\infty }$ is the initial specific angular momentum

$$\begin{eqnarray}&&{l}_{z,\infty }={\dot{L}}_{z}(\lt {R}_{{\rm{a}}})/\dot{M}(\lt {R}_{{\rm{a}}}).\end{eqnarray} \tag{ 11 }$$

Here, ${\dot{L}}_{z}(\lt {R}_{{\rm{a}}})$ is the accreted momentum inside the accretion radius, and $\dot{M}(\lt {R}_{{\rm{a}}})$ is the accreted mass.

Numerical integration in the vertical direction from the initial density profile can then provide

$$\begin{eqnarray}&&\dot{M}(\lt {R}_{{\rm{a}}})={v}_{\infty }{\int }_{\lt {R}_{{\rm{a}}}}\rho (y)\,{dA},\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&\dot{{L}_{z}}(\lt {R}_{{\rm{a}}})={v}_{\infty }^{2}{\int }_{\lt {R}_{{\rm{a}}}}\rho (y)y\,{dA}.\end{eqnarray} \tag{ 13 }$$

Therefore, the circularization radius depends solely on the initial density profile, which in turn is set by Γ_s and _ρ. For the conditions of our numerical simulations, with _ρ = 2, we find R_circ = 0.35 R_a for Γ_s = 4/3 and R_circ = 0.33 R_a for Γ_s = 5/3.

In this work, we found centrifugally supported structures only for highly compressible flows γ ≲ 1.2. This differs from Armitage & Livio (2000), who reported disk formation in radiation-dominated (γ = 4/3) flows. The main reason for this discrepancy is undoubtedly because they carried out simulations in two dimensions. In three dimensions, the flow can be deflected in the z direction and is not restricted to the orbital plane. This additional degree of freedom hinders disk formation (Ruffert 1997). MacLeod & Ramirez-Ruiz (2015a) argued that pressure support under compression in two dimensions with an adiabatic equation of state (γ = 5/3) is very similar to that in a three-dimensional simulation with a nearly isothermal equation of state (γ = 1). This is because P ∝ ρ^γ ∝ V^−γ, where V is the volume term. In two dimensions, we then have P_2d ∝ r^−2γ, while in three dimensions we can instead write P_3d ∝ r^−3γ. As a result, P_2d ∝ r^−10/3 (P_2d ∝ r^−8/3) for γ = 5/3 (γ = 4/3) and P_3d ∝ r⁻⁵ (P_3d ∝ r⁻³) for γ = 5/3 (γ = 1).

Our analysis in Section 3 indicates that the radial component of the pressure gradient is more important than the pressure itself, because this is the quantity that enters into the gas momentum equation, as ∇P/ρ. If we consider the idealized case of spherical compression, the pressure gradient term, in three dimensions with a nearly isothermal equation of state or in two dimensions with an adiabatic one, $\tfrac{1}{\rho }\tfrac{{dP}}{{dr}}\propto {r}^{-1}$. In contrast, the pressure gradient in three dimensions is $\tfrac{1}{\rho }\tfrac{{dP}}{{dr}}\propto {r}^{-3}$ for an adiabatic flow. Thus, near the embedded object, the resistance to compression due to the pressure gradient is much stronger for the adiabatic case than for the isothermal one. Figure 6 compares simulations in three and two dimensions with γ = 4/3. The flow in two dimensions is rotationally supported, as also seen by Armitage & Livio (2000), while the increase in pressure support in three dimensions does not allow the flow to circularize.

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A sufficiently strong pressure gradient can act effectively against the gravitational force of the embedded object, which goes as ∝r⁻². Returning to our three-dimensional flow structures, this leads to larger deflections of the flow in the γ = 5/3 case, as observed in Figures 1 and 2, which prevent the formation of a dense disk. If the resistance of a pressure gradient against gravity is the controlling parameter, we find that these both scale as r⁻² for γ = 4/3, implying that in initial ratio of pressure support to gravitational acceleration is preserved at all radii under spherical compression. To settle into a disk, we can imagine that fluid needs to have a pressure-gradient scaling shallower than r⁻², so that gravity can become dominant at some radii and a rotationally supported flow can develop. This logic predicts a bifurcation inflow structure above and below γ ≈ 4/3. Our results in Section 3 support that prediction; only in calculations with γ < 4/3 (γ ≲ 1.2) did we find dense disks on the scale of R_circ.

As discussed previously, during a CE event, a rotationally supported structure could form around the embedded object in the presence of highly compressible gas. Two natural questions then arise: where in a stellar envelope can this occur and what is the scale of the associated disk?

The highest compressibility environment found in stars is within partial ionization zones. In these zones where the gas is partially ionized, a fraction of the energy released during a layer's compression can be used for further ionization, rather than raising the temperature of the gas (Harpaz 1984). The partial ionization produces an opacity bump and a considerable decrease in the adiabatic exponents. As a result, a steeper temperature gradient is required for radiative diffusion to transport energy through these regions.

Such partial ionization zones are located in the outer layers of evolving stars. To illustrate this, we calculate stellar models with MESA (version 7624; Paxton et al. 2011, 2013, 2015) for stars of different mass and evolutionary stages.⁶

As the star evolves into the giant branch, the partial ionization regions occupy a progressively larger fraction of the mass of the star. This can be seen in Figure 7, where we map the compressibility that enters into the equation of state, γ₃ (Equation (5)), in the ρ–T plane using the equation of state module in MESA (Paxton et al. 2011, 2013, 2015). Overplotted are the evolutionary tracks for 1 M_⊙ stars and 20 M_⊙ stars at various evolutionary stages and solar abundance. The dashed area represents the region where crystallization occurs and the equation of state is not well determined.

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The almost horizontal (constant T) white bands in Figure 7 represent regions in which partial ionization of various species takes place and, as a result, the gas is highly compressible. The regions of high compressibility are more prominent in low-mass stars, whose envelopes cross through larger portions of these regions. High-mass giants approach their Eddington limit and show profiles in the ρ–T plane that straddle the gas-radiation pressure transition. These profiles touch the partial ionization regions in ρ–T space only at their extreme limbs, occasionally in regions of density inversion.

Figure 8 shows the fractional radius of stars that have γ₃ < 1.2. As can be clearly seen in Figure 8, low-mass stars have significantly more extended partial ionization zones in their outer layers. Higher-mass stars, above ≈3 M_⊙, exhibit radially narrow partial ionization zones with ≲1% of their radius occupied by these regions.

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Next, we address the scale of a disk that might result from passage of a secondary object through one of these regions of high gas compressibility. Figure 9 illustrates how the gas circularization radius, R_circ, changes as the embedded object, here characterized by R_a, spirals deeper into the star. This figure adopts q = 0.1. As the embedded companion spirals deeper into the primary, density gradients, as parameterized by _ρ, become shallower, and the circularization radius decreases relative to R_a. Rotationally supported structures will have scale similar to R_a only in the outer portions of stellar envelopes, in similar regions to where zones of partial ionization (and high compressibility) are found.

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We expect that only in cases where the radial extent of highly compressible gas (γ < 4/3) is sufficiently large, ΔR₁ ≳ R_a, is it possible for a large-scale disk structure at the R_circ scale to be formed around the embedded object. This figure indicates that, given the unperturbed structures of stellar envelopes, disk assembly might be restricted to objects embedded in the outer envelopes of low-mass giant stars. Furthermore, in comparison with Figure 8, for R_a ≲ ΔR₁ to occur, the encounter must be one with a low-mass secondary object and correspondingly low-mass ratio, q, such that R_a ≪ a. Taken together, these considerations suggest that disk formation in CE is rare and is probably only a brief phase during the inspiral in cases in which it does occur.

Several caveats affect the firmness of this conclusion. CE structures are undoubtedly expanded by interaction with the secondary star, perhaps even prior to the phase when an object plunges through a given radial coordinate. This expansion, and associated adiabatic degradation of the temperature of the expanded envelope, could lead larger portions of the ρ–T trajectories of massive stars to cross through partial ionization zones. A second concern relates to the extension of our wind-tunnel results to the realistic CE process. In particular, in a full equation of state, such as that shown in Figure 7, γ₃ is a strong function of density and especially temperature. This might lead to different structures (and degrees of pressure support) as the gas compresses through various phase transitions, perhaps differentiating the dynamics of the system under a realistic equation of state from that with a constant γ. For now, we can speculate that the important scale is the circularization radius scale, where the angular momentum budget is dominated, but performing more complex simulations is beyond the scope of the current work. An additional caveat is that although this paper mainly considers the formation of accretion disks during the CE phase, in principle, the secondary could enter the CE with a pre-existing disk (Staff et al. 2016; Chen et al. 2017) formed due to mass transfer before the CE phase.

A disk would transport mass, energy, and angular momentum to small scales from which it might generate accretion-driven winds and collimated outflows that aid in ejecting the envelope (Armitage & Livio 2000; Voss & Tauris 2003; Soker 2004, 2015). There are some interesting results that accompany the collimated outflow formation. As the jet traverses through the star, the jet will deposit some of its energy into a cocoon that surrounds the jet (Moreno Méndez et al. 2017). The cocoon will expand laterally and, in doing so, quench the mass accretion onto the companion. Another interesting effect is that because this energetic feedback occurs preferentially in the outer regions of the star, the close binary could eject a portion of the envelope early and enter a grazing envelope phase (Soker 2015; Shiber et al. 2017). If the feedback is strong enough, the CE phase might not fully occur.

The first detection of gravitational waves was catalyzed by the existence of moderately massive, stellar-mass black holes in binary systems (Abbott et al. 2016). One of the preferred channels for the formation of this type of binary black holes necessitates a CE stage (Dominik et al. 2012; Belczynski et al. 2016; Kruckow et al. 2016). This channel involves a massive stellar binary (40–100 M_⊙), likely formed in a low-metallicity environment, in which the first-born black hole is engulfed by an evolving massive companion (Belczynski et al. 2016). For the merger to occur within the age of the universe, the black hole needs to tighten its orbit. This tightening could occur due to the CE phase or through some mechanisms, such as tidal torques from circumbinary disks (Chen & Podsiadlowski 2017), after the envelope is ejected.

Here, we argue that accretion feedback is not likely to effectively operate during the CE phase when involving massive stars. As was shown in Figures 7 and 8, extended zones of sufficiently high gas compressibility for disk formation exist in the envelopes of low-mass giants, and are found in zones of partial ionization. The extent of these zones is drastically reduced and they are only found in the outermost envelopes of the high-mass stars that are relevant to the formation of binary black holes. In Figure 10, we additionally illustrate that this conclusion is not sensitive to varying metallicity.

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Because no extended regions of sufficiently low γ exist to allow disks to form in CE events involving high-mass giants, there is a lack of a mechanism (such as a disk outflow) to couple the accretion energy lost to an embedded black hole with the large-scale flow. This implies that significant tightening of the orbit can take place without significant feedback energy injection from the embedded black hole. By avoiding strong feedback, CE events may serve as a mechanism to drive substantial orbital tightening as originally envisioned (Webbink 1984), and are a natural channel to the formation of merging binary black holes (e.g., Belczynski et al. 2016; Kruckow et al. 2016).

In this paper, we studied the conditions required to form a disk around the embedded companion during a CE phase. We studied the flows using the idealized CEWT setup of MacLeod et al. (2017). Some key conclusions of our study are as follows.

• 1.  
  The introduction of a density gradient in HLA allows for angular momentum to be introduced to the flow, which in turn opens the possibility for the formation of a disk around the embedded companion.
• 2.  
  The formation of disk structures in the context of a CE phase is linked to the thermal properties of the envelope. In envelope gas with higher compressibility (γ < 4/3), the gravitational force dominates over the pressure support near the accretor, allowing for effective circularization of the material into a disk. On the other hand, in lower compressibility gas environments (γ ≳ 4/3), the pressure support dominates as the gas compresses toward the accretor. We find that a disk does not form and the flow will be advected away from the embedded object, typically completing less than one full rotation.
• 3.  
  Within stellar envelopes extended regions of sufficiently compressible gas to allow disk formation around embedded objects are found only within zones of partial ionization, where the additional (ionization) degrees of freedom reduce γ significantly.
• 4.  
  These partial ionization zones always comprise a small fraction of a stellar envelope radius or mass. They are more extended in the outer layers of low-mass stars than in the exteriors of high-mass stars. We therefore expect that disk formation around embedded objects in CE, is, at most, a transitory phase.
• 5.  
  The lack of regions conducive to disk formation in high-mass stellar envelopes suggests that CE episodes involving these stars, such as those in the assembly history of merging binary black holes, are not subject to strong disk-outflow powered accretion feedback. Without overwhelming feedback from accretion, we suggest that CE events in massive systems should proceed with significant orbital tightening as they draw on orbital energy as an CE ejection mechanism rather than accretion energy. The lack of feedback implies that CE events remain a natural channel for the formation of LIGO-source binaries that must be assembled into tight orbits from which they merge under the influence of gravitational radiation.