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

By reconstructing the evolutionary histories of three black hole X-ray binaries, this study demonstrates that standard common envelope ejection models require unphysically high efficiency parameters (significantly greater than unity), implying that either additional energy sources are necessary or the current theoretical formalism must be revised.

The Gist

Imagine the universe as a giant cosmic dance floor. Most stars don't dance alone; they dance in pairs, holding hands and spinning around a common center. Sometimes, this dance gets a little too wild. One star swells up, gets too big, and accidentally swallows its partner. This chaotic event is called a Common Envelope (CE) phase.

The big question astronomers have is: How does the partner star escape?

Usually, the swallowed star spirals inward, and the energy from this spiral is supposed to blow the giant outer layers (the "envelope") off the massive star, leaving behind a tight pair of compact objects (like Black Holes). But there's a problem: our current math says this escape is almost impossible. The energy required to blow the layers off seems much higher than the energy available from the spiral.

This paper is like a group of cosmic detectives trying to solve this mystery by looking at three specific, real-life examples of Black Hole pairs that did survive this chaotic dance: GRO J1655-40, SAX J1819.3-2525, and 4U 1543-47.

Here is the story of their investigation, broken down simply:

1. The "Energy Budget" Problem

Think of the Common Envelope phase as a budget crisis.

• The Cost: To blow off the giant star's outer layers, you need a massive amount of energy (like paying a huge bill).
• The Income: The only money in the bank is the energy released when the two stars spiral closer together (like selling a house).

For decades, astronomers have used a "magic number" called efficiency ($\alpha$) to see if the income covers the cost. If the number is less than 1, it means the energy released was enough to pay the bill. If it's greater than 1, it means we are short on cash, and we need a "loan" (extra energy) to make it work.

2. The Investigation: Reconstructing the Past

The authors took the three surviving Black Hole pairs and ran a massive computer simulation to rewind the clock. They asked: "What had to happen in the past for these specific pairs to exist today?"

They looked at three different ways to calculate the "Cost" (the energy needed to blow the layers off):

• Case A (Strict): Only counting the heat inside the star.
• Case B (Moderate): Counting the heat plus some internal pressure.
• Case C (Generous): Counting the heat, pressure, and a term called "enthalpy" (which accounts for how the gas moves and expands).

3. The Shocking Discovery

No matter which way they calculated the "Cost," the math came back with a shocking result: The "Income" was never enough.

To make these three Black Hole pairs exist, the efficiency number had to be much higher than 1.

• In the strictest case, they needed an efficiency of 6.7.
• Even in the most generous case (including enthalpy), they still needed an efficiency of 1.7.

The Analogy: Imagine you are trying to blow up a balloon. You have a tiny straw (the orbital energy). You try to blow, but the balloon won't pop. The math says you need to blow 6.7 times harder than your lungs can possibly allow. Since that's impossible, something else must be helping you blow.

4. The "Kick" Factor

There was another twist. Black holes are born from supernova explosions, which are like cannon shots. Sometimes, the new black hole gets "kicked" out of the system at high speed.

The team found that for one of the systems (4U 1543-47), the black hole must have received a significant kick (at least 50 km/s, likely around 160 km/s) to survive. Without this kick, the math says this system couldn't have formed in isolation. It's like the black hole needed a running start to jump over a wall that was too high to clear from a standstill.

5. The Conclusion: We Need a New Theory

The paper concludes that our current understanding of how stars shed their layers is incomplete. The standard "energy budget" model doesn't work for massive stars.

So, what's the solution? The authors suggest two possibilities:

1. Extra Energy Sources: Maybe there are hidden helpers we aren't counting. For example, jets (powerful beams of particles shooting out from the stars) might provide the extra "push" needed to blow off the layers. Or maybe nuclear explosions inside the star help out.
2. Rewrite the Rules: Maybe the way we calculate the energy (the formula itself) is wrong for massive stars and needs a complete overhaul.

Summary

In simple terms, this paper says: "We looked at three successful Black Hole couples. Our current physics says they shouldn't have survived the 'swallowing' phase because they didn't have enough energy to escape. Since they did survive, nature must have a secret weapon (like jets) or a rulebook we haven't written yet."

This is a crucial step because if we don't understand how these pairs form, we can't accurately predict how many gravitational waves (the ripples in spacetime) the universe is producing.

Technical Summary

1. Problem Statement

The Common Envelope (CE) phase is a critical mechanism in the formation of close compact object binaries (such as Black Hole/Neutron Star binaries). However, the physics governing the ejection of the stellar envelope remains poorly understood, particularly for massive binaries ($M \gtrsim 8 M_\odot$).

• The Core Issue: The standard energy formalism relies on a free parameter, the CE efficiency ($\alpha_{CE}$), which represents the fraction of orbital energy used to eject the envelope.
• The Discrepancy: While post-CE systems with low-mass donors often suggest $\alpha_{CE} < 1$, massive binary evolution models frequently require $\alpha_{CE} \gg 1$ (up to 10) to explain observed systems.
• The Gap: There is a lack of robust, observationally constrained lower limits for $\alpha_{CE}$ in massive systems. Existing constraints are often derived from population synthesis with large uncertainties or from low-mass systems that do not scale to massive progenitors.

2. Methodology

The authors employed a full evolutionary reconstruction approach to constrain $\alpha_{CE}$ by reverse-engineering the formation of three specific, dynamically confirmed Black Hole Intermediate-Mass X-ray Binaries (BH IMXBs):

1. GRO J1655-40
2. SAX J1819.3-2525
3. 4U 1543-47

Key Technical Steps:

• Simulation Framework: Used the MESA (Modules for Experiments in Stellar Astrophysics) code (v12115) with solar metallicity ($Z=0.0142$).
• Progenitor Modeling:
  • Assumed BH progenitors in the mass range 32–60 $M_\odot$.
  • Utilized parametric supernova (SN) models (based on F. R. N. Schneider et al. 2021) to determine BH formation via fallback accretion.
  • Assumed the CE phase occurs during Case B stripping (after core H-burning, before core He-ignition).
• Binding Energy Calculation: Calculated the envelope binding energy ($E_{bind}$) using three distinct energy formalisms to test sensitivity:
  1. $\alpha_{0.5U}$: Includes 50% of internal energy ($U$).
  2. $\alpha_{U}$: Includes 100% of internal energy.
  3. $\alpha_{H}$: Includes enthalpy ($H = U + P/\rho$), which redistributes heat but does not add new energy sources.
• Backward Evolution:
  • Started from observed parameters (BH mass, companion mass, orbital period) at the X-ray binary stage.
  • Evolved the system backward through the SN event (accounting for mass loss and potential natal kicks) and the post-CE wind mass loss phase to determine the pre-CE orbital separation ($a_{pre-CE}$) and post-CE separation ($a_{post-CE}$).
  • Solved Equation (1) (the standard energy equation) to derive the required $\alpha_{CE}$ for each viable evolutionary track.
• Parameter Grids: Systematically scanned initial BH masses, companion masses, and orbital periods, filtering out "forbidden" configurations where the companion would prematurely fill its Roche lobe.

3. Key Contributions

• Systematic Reconstruction: This is one of the first studies to systematically reconstruct the entire evolutionary history of three specific massive BHXBs to derive model-independent lower limits on $\alpha_{CE}$.
• Inclusion of Enthalpy: Rigorously tested the impact of including enthalpy in the binding energy calculation, a factor often debated in CE physics.
• Natal Kick Analysis: Quantitatively assessed the impact of Black Hole natal kicks on the viability of isolated binary evolution, specifically for 4U 1543-47.
• Robust Lower Limits: Established that even under the most favorable conditions (including enthalpy and natal kicks), the efficiency must exceed unity.

4. Key Results

A. Constraints on CE Efficiency ($\alpha_{CE}$)

The study derived minimum efficiency values required to successfully eject the envelope for all three systems. Even when accounting for enthalpy (which significantly reduces the binding energy), the efficiencies remain high:

• $\alpha_{0.5U} \gtrsim 6.7$ (50% internal energy contribution)
• $\alpha_{U} \gtrsim 4.2$ (100% internal energy contribution)
• $\alpha_{H} \gtrsim 1.7$ (Enthalpy included)

Crucial Finding: No viable solutions exist with $\alpha_{CE} < 1$. This holds true even when considering the most extreme scenarios where the envelope binding energy is minimized via enthalpy inclusion.

B. Impact of Natal Kicks

• 4U 1543-47: Under the assumption of zero natal kick, no solution exists for 4U 1543-47 within the isolated binary evolution framework. The post-CE orbit is too tight, causing the companion to fill its Roche lobe immediately.
  • Requirement: A substantial natal kick of $\gtrsim 50$ km/s is required to widen the orbit sufficiently.
  • Preferred Kick: The formation probability peaks at $\sim 160$ km/s.
• GRO J1655 & SAX J1819: While viable without kicks, the inclusion of natal kicks expands the allowable parameter space and allows for BH formation in the "mass gap" ($3-5 M_\odot$).
• Efficiency vs. Kick: Counter-intuitively, higher natal kicks tend to increase the required median $\alpha_{CE}$ values slightly, though the constraint $\alpha_{CE} > 1$ remains robust regardless of kick magnitude.

C. Progenitor Dependence

The minimum required $\alpha_{CE}$ is not strictly monotonic with progenitor mass. It depends on the interplay between the envelope binding energy (which increases with mass) and the post-CE orbital separation (which is influenced by the mass of the resulting He star and wind mass loss).

5. Significance and Implications

• Challenge to Standard Formalism: The finding that $\alpha_{CE} > 1$ is required for massive binaries strongly implies that the standard energy formalism is insufficient. Orbital energy alone (even with internal energy or enthalpy) cannot explain the ejection of massive envelopes.
• Need for Additional Physics: The results suggest the necessity of additional energy sources or modified mechanisms, such as:
  • Jets: Energy injection from accretion-driven jets (a leading candidate).
  • Nuclear Energy: Runaway nuclear burning triggered by mass transfer.
  • Modified Evolution: A revised description of the CE phase where the donor star's structure evolves dynamically during the spiral-in, or alternative ejection mechanisms (e.g., two-stage ejection).
• Population Synthesis: These derived lower limits ($\alpha_{H} \gtrsim 1.7$) provide essential, observationally grounded benchmarks for future population synthesis studies of massive binaries and gravitational wave sources. Current models assuming $\alpha_{CE} < 1$ for massive systems are likely physically inconsistent with these specific observed systems.

In conclusion, this paper provides strong evidence that the common envelope phase in massive binaries is more energetic than previously thought, necessitating a paradigm shift in how we model envelope ejection in binary evolution.