A Path to Constraints on Common Envelope Ejection in Massive Binaries: Full Evolutionary Reconstruction of Three Black Hole X-ray Binaries

Abstract: The massive binary common envelope (CE) phase plays a pivotal role in the formation of close black hole/neutron star (BH/NS) binaries, yet significant uncertainties remain in our understanding of this process. In this study, we aim to constrain the massive binary CE phase by systematically reconstructing three observed BH X-ray binaries (BHXBs): GRO J1655-40, SAX J1819.3-2525, and 4U 1543-47. Through comprehensive binary evolution simulations and parametric supernova (SN) modeling, we establish lower limits for the CE efficiency parameters under different energy considerations within the standard energy formalism. Specifically, we derive minimum values for three cases: $α<em>{\rm 0.5U}$ and $α</em>{\rm U}$ representing CE efficiencies with half and all of the internal energy contributing to the envelope ejection, respectively, and $α<em>{\rm H}$ accounting for the envelope's enthalpy. Our analysis reveals that the self-consistent formation of these three BHXBs requires CE efficiency parameters satisfying: $α</em>{\rm 0.5U}\gtrsim 6.7$, $α<em>{\rm U}\gtrsim 4.2$ and $α</em>{\rm H}\gtrsim 1.7$. Notably, we find no viable solutions with CE efficiency values below unity, even when considering the most extreme scenarios in which the envelope binding energy is significantly reduced through enthalpy inclusion. {Our results strongly imply that either additional energy sources are required, or the formalism itself must be revised.} Furthermore, we quantitatively assess the impact of BH natal kicks on our results. A key finding is that 4U 1543-47's formation requires substantial natal kicks ($\gtrsim 50 \;\rm km/s$), as lower kick velocities are incompatible with isolated binary evolution.

• The paper establishes robust lower limits on common envelope ejection efficiency, with α_CE values >1 across multiple energy prescriptions.
• It employs a hybrid framework using detailed MESA simulations and Monte Carlo natal kick models to reconstruct the full binary evolution.
• Results challenge canonical energy formalism assumptions, suggesting missing energy sources and impacting population synthesis and GW event predictions.

Constraining Common Envelope Ejection in Massive Binaries via Full Evolutionary Reconstruction of BH X-ray Binaries

Introduction and Motivation

The evolution of massive stellar binaries, particularly through the common envelope (CE) phase, is central to the formation of close compact object pairs such as black hole (BH) and neutron star (NS) binaries. However, the CE phase remains the least constrained aspect of binary evolution, and uncertainties in the CE ejection efficiency ($\alpha_{\rm CE}$) directly impact predictions related to supernova progenitors, compact object mergers, and GW event rates. Unlike the population of low-mass post-CE binaries, for which modest $\alpha_{\rm CE} < 1$ are inferred, the massive regime is theoretically more ambiguous and observationally less accessible. This study (2604.10440) targets three dynamically confirmed Galactic BH X-ray binaries (BHXBs)—GRO J1655-40, SAX J1819.3-2525, and 4U 1543-47—to set robust limits on $\alpha_{\rm CE}$ by reconstructing their full evolutionary histories.

Figure 1: Schematic evolutionary sequence for BH I/LMXBs, highlighting the CE and mass transfer phases.

Modeling Approach

A hybrid framework is employed, utilizing detailed single and binary stellar evolution models (MESA, version 12115) and a parametric SN fallback prescription, to evolve initial massive binaries through the relevant evolutionary stages:

• Determination of pre-CE and post-CE binary parameters using stellar evolution grids.
• Three energy formalism variants for binding energy: $\alpha_{\rm 0.5U}$ (half internal energy), $\alpha_{\rm U}$ (full internal energy), and $\alpha_{\rm H}$ (includes enthalpy).
• Natal kicks at BH formation are explicitly modeled via Monte Carlo realizations using the prescription from Hurley et al. (2002).

BH progenitor masses are mapped based on Schneider et al. (2021); grids cover $32-40\,M_\odot$ and $46-60\,M_\odot$ primaries to address the non-monotonic mapping between ZAMS and remnant mass in binary–stripped stars.

Reconstructing the Evolutionary Pathways

Self-consistent binary grids are generated for each of the three systems, matching all observed parameters within uncertainties.

Figure 3: Representative binary evolutionary tracks for GRO J1655, distinguishing between physically viable and forbidden post-CE solutions.

For each allowed endpoint, the pre- and post-CE states yield the required $\alpha_{\rm CE}$ via the classical energy formalism:

$E_{\rm bind} = \alpha_{\rm CE} \Delta E_{\rm orb}$

Allowable and forbidden regions in parameter space are demarcated by evaluating Roche lobe overflow post-CE, with tight constraints on the masses and periods.

Figure 2: Detailed parameter mapping for GRO J1655, with colors indicating $\alpha_{\rm CE} < 1$0 for viable grid points.

Effect of Binding Energy Treatment & Progenitor Structure

Binding energy calculations reveal significant sensitivity to (i) progenitor mass, (ii) envelope structure at CE onset, and (iii) adopted $\alpha_{\rm CE} < 1$1 prescription.

Figure 4: Example binding energy analysis of a $\alpha_{\rm CE} < 1$2 star, showing the impact of internal energy and enthalpy on the envelope's ejection ease.

The critical result is that, regardless of the adopted binding energy treatment, no solution exists with $\alpha_{\rm CE} < 1$3: typically, $\alpha_{\rm CE} < 1$4, $\alpha_{\rm CE} < 1$5, and $\alpha_{\rm CE} < 1$6, depending on system and BH progenitor mass.

Figure 6: Minimum required $\alpha_{\rm CE} < 1$7 for GRO J1655 as a function of progenitor mass and radius, showing only minor reduction with enthalpy.

This result is highly robust to the assumed evolutionary phase for envelope ejection, and is corroborated for all three analyzed targets.

Figure 8: Comparison of minimum $\alpha_{\rm CE} < 1$8 requirements among GRO J1655, SAX J1819, and 4U 1543 for three binding energy prescriptions.

Black Hole Natal Kicks and System Formation Probabilities

The BH natal kick (velocity imparted at BH formation) is a critical parameter. For 4U 1543-47, no viable formation pathway exists for $\alpha_{\rm CE} < 1$9 km/s; preferred configurations require $\alpha_{\rm CE}$0 km/s.

Figure 9: Fraction of synthetic binaries producing each target system as a function of natal kick, demonstrating sharp lower limits for 4U 1543.

For GRO J1655 and SAX J1819, non-zero kicks allow expansion into the so-called "mass gap" ($\alpha_{\rm CE}$1). The effect of the kick on the required $\alpha_{\rm CE}$2 is modest; even with increased mass loss, $\alpha_{\rm CE}$3 remains $\alpha_{\rm CE}$4 for all plausible initial conditions.

Figure 11: Distribution of $\alpha_{\rm CE}$5 as a function of imposed natal kick velocity. Systems with kicks in the regime $\alpha_{\rm CE}$6 are dynamically disfavored.

Theoretical and Practical Implications

The requirement for $\alpha_{\rm CE}$7 in all scenarios, even with maximal binding energy reduction, is both numerically robust and theoretically significant. It directly contradicts the physical expectation that $\alpha_{\rm CE}$8 should be $\alpha_{\rm CE}$9 in a closed energy budget and echoes the high values employed ad hoc in BH population synthesis studies (e.g., Dominik et al. 2012; Grichener 2023).

This necessitates revisiting the classical CE paradigm:

• Missing Energy Sources: Additional ejection energy may originate from accretion-driven jets (Soker 2015; Shiber et al. 2019), nuclear energy injection (Podsiadlowski et al. 2010), or other mechanisms not encapsulated in current formalisms.
• Structure Evolution: Time-dependent changes in envelope structure during CE and the role of the radiative intershell (Hirai & Mandel 2022) could alter the energetics and post-CE outcomes.
• Population Synthesis: The high $\alpha_{\rm 0.5U}$0 solutions demanded by empirical reconstruction must now be incorporated as constraints in GW event rate predictions and merger channel modeling.
• Alternative Formation Scenarios: Where isolated binary evolution faces fundamental energetic inconsistencies, dynamical or triple-induced paths may gain relative plausibility.

Future Directions

Key unresolved issues include the self-consistent incorporation of jet and nuclear energy into envelope ejection prescriptions, and a more comprehensive observational sample of dynamically confirmed BHXBs that traverse the relevant parameter space. Time-dependent 3D hydrodynamic and radiation MHD simulations, coupled to detailed evolutionary codes, are essential for translating the empirical $\alpha_{\rm 0.5U}$1 requirements into physical mechanisms.

Conclusion

By reconstructing the full parameter space for three Galactic BHXBs, this work establishes authoritative lower limits on the CE ejection efficiency in the massive binary regime: $\alpha_{\rm 0.5U}$2, $\alpha_{\rm 0.5U}$3, $\alpha_{\rm 0.5U}$4—all strictly exceeding unity regardless of reasonable physical assumptions. Isolated binary formation of 4U 1543-47 requires substantial BH natal kicks ($\alpha_{\rm 0.5U}$5 km/s). The empirical necessity for $\alpha_{\rm 0.5U}$6 highlights either missing physics (e.g., jet/nuclear-driven unbinding) or a fundamental deficiency in the canonical energy formalism for massive binary common envelope evolution. This result sets new benchmarks for future population synthesis, supernova modeling, and gravitational wave astrophysics.