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arxiv: 2607.06333 · v1 · pith:ZI4IRB55 · submitted 2026-07-07 · astro-ph.SR

The Best Guess: Testing new and old formalisms for the common envelope against observations

Riley Thai , Amanda Karakas , Zara Osborn , Robert Izzard , Ryosuke Hirai , Alex Kemp , Simon Campbell This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 09:20 UTCglm-5.2pith:ZI4IRB55record.jsonopen to challenge →

classification astro-ph.SR PACS 97.80.-d97.10.Ex97.20.Rp
keywords common envelope evolutionpost-common envelope binariesbinary population synthesisrecombination energyangular momentum formalismwhite dwarf binariesstellar evolution
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The pith

Energy beats angular momentum for common envelope outcomes

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tests two newer prescriptions for common envelope evolution — the Two-stage formalism (which splits the process into an energy-driven plunge followed by angular-momentum-driven mass transfer) and the SCATTER formalism (which predicts the final binary separation purely from orbital angular momentum balance) — against the standard energy-based α-formalism, by forward-modelling the expected present-day population of post-common envelope binaries in the Solar neighbourhood and comparing to 193 observed systems. The central finding is that the SCATTER formalism, which parameterises outcomes through angular momentum exchange between the envelope and the binary components, cannot reproduce the observed population under any parameter setting: it fails to predict wide binaries, overpredicts helium white dwarf systems that are not observed, and cannot match the observed ratio of white dwarf types. The authors interpret this as evidence that orbital angular momentum balance alone is fundamentally insufficient to predict common envelope outcomes. Both the standard α-formalism and the Two-stage formalism, which are energy-based, reproduce the observed population well. The paper further finds that recombination energy — energy released when ionised gas in the stellar envelope recombines — must contribute to envelope ejection, but only a fraction (roughly 10–40%) of the available recombination energy can actually participate, since a full contribution scatters ultra-massive white dwarf systems to orbital periods where they are not observed. The preferred parameter combination is α_CE ≈ 0.2–0.3 with a partial recombination contribution (f_ion ≈ 0.1–0.4). The Two-stage and α-formalisms are indistinguishable for most of the observed population; they differ only for intermediate-mass donors (above about 2.25 solar masses) at orbital periods of 1–100 days, where observations are currently sparse.

Core claim

The SCATTER formalism — an angular-momentum-based prescription for common envelope outcomes — fails to reproduce the observed post-common envelope binary population under any parameter value, while energy-based formalisms (the standard α-formalism and the hybrid Two-stage formalism) succeed. Recombination energy is necessary but only partially contributing (roughly 10–40%), with the preferred efficiency α_CE ≈ 0.2–0.3. No angular-momentum-only formalism tested to date matches observations, suggesting that orbital angular momentum balance alone is not predictive of common envelope outcomes.

What carries the argument

The three formalisms tested are: (1) the α-formalism, which equates the envelope binding energy to a fraction α_CE of the orbital energy released during inspiral; (2) the Two-stage formalism, which splits the process into an energy-driven ejection of the convective envelope followed by angular-momentum-driven non-conservative mass transfer of the radiative layer; and (3) the SCATTER formalism, which predicts the final separation from an exponential function of the angular momentum exchanged between the envelope and each component, parameterised by η. The comparison is performed by forward-modelling a Solar neighbourhood population through binary population synthesis, weighting by birth rates

If this is right

  • If angular-momentum-only formalisms are fundamentally insufficient, population synthesis studies that rely on the γ-formalism or similar prescriptions may produce systematically incorrect predictions for compact object merger rates and binary populations.
  • The partial recombination contribution (10–40%) provides a concrete target for three-dimensional hydrodynamical simulations, which must now demonstrate why only a fraction of the recombination energy thermalises into useful work on the envelope.
  • The Two-stage formalism predicts a population of intermediate-mass helium-burning stars (hot subdwarfs) in close binaries from RGB donors above 2.25 solar masses that the α-formalism does not; targeted observations of such systems would distinguish the two formalisms.
  • The predicted but unobserved populations — PCEBs at 1–100 day periods with FGK companions, and super-wide PCEBs with M-dwarf or G-type companions — constitute specific observational targets for Gaia Data Release 4 and the Rubin Observatory.
  • The L2 overflow criterion for initiating common envelope in giant donors is shown to be necessary; using only dynamical instability thresholds produces unphysical populations where common envelope events occur on nearly stripped envelopes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A formal statistical framework incorporating selection effects (UV-excess distinguishability, survey cadence) could potentially overturn the conclusion that recombination energy is strictly necessary, since the claim depends on the absence of observed systems in the super-wide regime where selection effects are most uncertain.
  • If the SCATTER formalism's exponential sensitivity to mass ratio is the root cause of its failure, then any angular-momentum-based formalism with similar exponential dependence on mass ratio may face the same problem, suggesting a structural limitation rather than a calibration issue.
  • The degeneracy between α_CE and f_ion means that without independent constraints on either parameter (e.g., from hydrodynamical simulations or individual well-characterised systems), population synthesis alone cannot uniquely determine the recombination contribution.

Load-bearing premise

The comparison between model predictions and observations is done qualitatively by visual overlay of predicted overdensities against observed points in mass–period space, without a formal statistical likelihood or goodness-of-fit metric, and without modelling the selection effects from UV-excess distinguishability or survey cadence. This means regions where the model predicts systems but none are observed cannot be statistically distinguished from observational incompleteness

What would settle it

Observation of a robust population of wide post-common envelope binaries (periods above 100 days) that the SCATTER formalism predicts but the energy-based formalisms do not, or a formal statistical analysis showing that the SCATTER formalism fits the observed population significantly better once selection effects are properly modelled.

Figures

Figures reproduced from arXiv: 2607.06333 by Alex Kemp, Amanda Karakas, Riley Thai, Robert Izzard, Ryosuke Hirai, Simon Campbell, Zara Osborn.

Figure 1
Figure 1. Figure 1: Schematic diagram of each formalism, along with the corresponding predictive method for the final state. Left: the standard α-formalism follows the energy budget available for the envelope’s ejection. The orbital energy is used to overcome the binding energy of the envelope, modulated by an efficiency αCE per Eq. 1 and 5. Middle: The hybrid Two-stage only takes that the convective (isentropic) portion of e… view at source ↗
Figure 2
Figure 2. Figure 2: Points are compiled observations of WD-MS post-common envelope binaries (PCEBs) (Zorotovic et al. 2010; Nebot G´omez-Mor´an et al. 2011; Rebassa-Mansergas et al. 2025; Jones 2020; Hernandez et al. 2021, 2022b; Yamaguchi et al. 2024a,b; Boone et al. 2026; Shariat & El-Badry 2026; Shiraishi et al. 2026; Motherway et al. 2026), over the general predictions of stable mass transfer from our models for masses 0.… view at source ↗
Figure 3
Figure 3. Figure 3: The Two-stage formalism predicts more systems survive HG and RGB common envelope episodes, and the SCATTER formalism predicts that typically only AGB common envelope episodes are survived. The survival of the first common envelope event for three different mass ratios q ∈ {0.1, 0.3, 0.7} is shown on each row versus the initial binary parameter space, colored by the donor’s stellar type at the time of commo… view at source ↗
Figure 4
Figure 4. Figure 4: The Two-stage formalism increases how many systems with intermediate-mass primaries survive first common envelope in close orbits (as per [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The observed PCEB population favours the α-formalism or the Two-stage formalism, as the SCATTER formalism (using Eq. 4) does not predict overdensities where observed. The present-day population is shown in mass space over four different ranges of orbital period P on each row, with each formalism in a different column. Markers for observed systems are per [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The observed population’s mass space can be described best by η ≈ 1.0 for the SCATTER formalism, but this does not reproduce the proportions of HeWD/COWDs seen in observations. While the observed population appears well-matched in mass space, gaps are present at the bulk of the observed close and CV-like PCEBs (top panels). The model also predicts the less massive HeWD primaries outnumber COWDs at all orbi… view at source ↗
Figure 7
Figure 7. Figure 7: Recombination energy is necessary to reproduce the observed population of PCEBs – no model using the α-formalism without recombination energy reproduces the wide IK Peg-like and all super-wide PCEBs over any 0 < αCE < 1. Markers for observed systems are per [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Four different pairs for efficiency αCE and recombination energy contribution fion which can minimally reproduce the observed PCEB population are shown. The minimum required recombination contribution increases with decreasing values for αCE, but we do not favour αCE ≳ 0.5 as it begins predicting HeWDs with K-type companions in CV-like orbits. 4.2. The widest systems and the recombination energy contributi… view at source ↗
read the original abstract

We present a systematic test of formalisms for common envelope evolution by forward-modelling observable post-common envelope binaries. We compare predictions from the $\alpha$-formalism, and the Two-stage and SCATTER formalisms against observed post-common envelope binaries, including wide binaries with ultra-massive white dwarfs and central binaries of planetary nebulae. The angular momentum-based SCATTER formalism does not predict populations which match the complete observed population, even with adjustments to its parameters. We take this as indicative of fundamental challenges with using the orbital angular momentum balance to predict common envelope outcomes. The energy-based $\alpha $ and hybrid Two-stage formalisms both well-replicate the observed population. $\alpha_{\rm CE} \sim 0.2\text{--}0.3$ can match current observations, in agreement with previous works. Recombination energy is necessary, but only a fraction of it ($\sim\! 10\text{--}40\%$) can contribute in order to predict IK Peg-like binaries with ultra-massive white dwarfs at the correct orbital periods. Our work suggests energy-based formalisms remain the most accurate for predicting common envelope outcomes, but more observations can constrain the recombination contribution and how these outcomes systematically vary with the donor mass.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

5 major / 10 minor

Summary. This manuscript presents the first systematic comparison of two recently proposed common envelope (CE) formalisms—the Two-stage formalism (Hirai & Mandel 2022) and the SCATTER formalism (Di Stefano et al. 2023)—against the standard energy-based α-formalism, using forward-modelled post-common envelope binary (PCEB) populations in the Solar neighbourhood. The authors compile 193 observed WD-MS PCEBs and compare them against binary_c population synthesis predictions across orbital period and component mass space. The principal conclusions are: (1) the angular-momentum-based SCATTER formalism fails to reproduce the observed PCEB population (no wide PCEBs, incorrect HeWD:COWD ratios), (2) energy-based formalisms (α and Two-stage) remain the most descriptive, with α_CE ≈ 0.2–0.3, (3) recombination energy is necessary but only a fraction (~10–40%) can contribute, and (4) the Two-stage and α-formalisms are currently indistinguishable given observational incompleteness at intermediate periods and high companion masses. The paper also provides new polynomial fits for the Two-stage formalism's binding energy and radiative region mass parameters (Appendix A).

Significance. The paper addresses a timely question: whether newly proposed CE formalisms outperform the standard α-prescription when confronted with the expanded observed PCEB population. The compilation of 193 PCEBs, including the recently discovered super-wide (P > 100 d) and ultra-massive WD systems, is a valuable resource. The implementation of the Two-stage formalism in binary_c, with publicly available polynomial fits derived from Monash and MESA stellar models (Appendix A, doi:10.5281/zenodo.20117304), and the public availability of the modified binary_c code (gitlab.com/rileythai/binary_c:twostage-v2.2.4) are commendable and enable reproducibility. The qualitative conclusion that SCATTER fails to produce wide PCEBs (Fig. 5) and predicts an incorrect HeWD:COWD ratio (Fig. 6) is a robust and useful result for the community. The identification of specific observational tests that could distinguish the Two-stage from the α-formalism (sdOBA binaries from M > 2.25 M☉ donors, §4.3) provides clear direction for future work.

major comments (5)
  1. §4.2, Fig. 8: The quantitative constraint that f_ion ≲ 0.4 at α_CE = 0.2 is derived from the absence of observed IK Peg-like systems (M_WD ≳ 1.1 M☉, M_MS ~1.4 M☉) at super-wide periods (P > 100 days). However, §2.3.2 explicitly states that the model does not include selection effects from UV-excess distinguishability or survey cadence, and that 'we expect our model to predict more systems than what is observed if the physics is correct.' This creates an internal tension: the model is known to overpredict, so the absence of observed IK Peg-like systems at super-wide periods could reflect observational incompleteness rather than a genuine upper limit on f_ion. The paper should either (a) explicitly state that the f_ion ≲ 0.4 bound is conditional on the assumption that the super-wide Gaia sample is complete for IK Peg-like systems (which is partially addressed by the reference to Shahaf et~
  2. §4.2, Fig. 8: The quantitative constraint that f_ion ≲ 0.4 at α_CE = 0.2 is derived from the absence of observed IK Peg-like systems at super-wide periods. However, §2.3.2 explicitly states that selection effects from distinguishability and survey cadence are not modelled, and that 'we expect our model to predict more systems than what is observed if the physics is correct.' This creates an internal tension: the model is acknowledged to overpredict, so the absence of observed systems could reflect incompleteness rather than a physical upper limit on f_ion. The paper should either (a) explicitly state that the f_ion ≲ 0.4 bound is conditional on the super-wide Gaia sample being complete for IK Peg-like systems (the reference to Shahaf et al. 2024 on line 'Assuming the Shahaf et al. (2024) Gaia selection... is observationally complete' partially addresses this but is stated only in passing
  3. §4.2, Fig. 8: The quantitative constraint that f_ion ≲ 0.4 at α_CE = 0.2 is derived from the absence of observed IK Peg-like systems at super-wide periods. However, §2.3.2 explicitly states that selection effects from distinguishability and survey cadence are not modelled, and that 'we expect our model to predict more systems than what is observed if the physics is correct.' This creates an internal tension: the model is acknowledged to overpredict, so the absence of observed systems could reflect incompleteness rather than a physical upper limit on f_ion. The paper should either (a) explicitly state that the f_ion ≲ 0.4 bound is conditional on the super-wide Gaia sample being complete for IK Peg-like systems (the reference to Shahaf et al. 2024 completeness is stated only in passing and should be foregrounded as the load-bearing assumption), or (b) soften the claim to acknowledge that f
  4. §4.2, Fig. 8: The quantitative constraint that f_ion ≲ 0.4 at α_CE = 0.2 is derived from the absence of observed IK Peg-like systems at super-wide periods. However, §2.3.2 explicitly states that selection effects from distinguishability and survey cadence are not modelled, and that 'we expect our model to predict more systems than what is observed if the physics is correct.' This creates an internal tension: the model is acknowledged to overpredict, so the absence of observed systems could reflect incompleteness rather than a physical upper limit on f_ion. The paper should either (a) explicitly state that the f_ion ≲ 0.4 bound is conditional on the super-wide Gaia sample being complete for IK Peg-like systems (the reference to Shahaf et al. 2024 completeness is stated only in passing and should be foregrounded as the load-bearing assumption), or (b) soften the quantitative claim to a lo
  5. §4.2, Fig. 8: The quantitative constraint that f_ion ≲ 0.4 at α_CE = 0.2 is derived from the absence of observed IK Peg-like systems at super-wide periods. However, §2.3.2 explicitly states that selection effects from distinguishability and survey cadence are not modelled, and that 'we expect our model to predict more systems than what is observed if the physics is correct.' This creates an internal tension: the model is acknowledged to overpredict, so the absence of observed systems could reflect incompleteness rather than a physical upper limit on f_ion. The paper should either (a) explicitly state that the f_ion ≲ 0.4 bound is conditional on the super-wide Gaia sample being complete for IK Peg-like systems (the reference to Shahaf et al. 2024 completeness is stated only in passing and should be foregrounded as the load-bearing assumption), or (b) soften the quantitative claim to a lo
minor comments (10)
  1. §2.1.2, Eq. (3): The definition of q_cc is given as 'q_cc = M_core/M_2' but the text also refers to 'companion-to-core mass ratio q_cc'. This is consistent but could be confused with the more standard convention of q = M_2/M_1; a brief clarifying note would help.
  2. §2.2.3, Eq. (6): The notation min(q_ad, q_L2) is clear, but the text does not explicitly state whether q_ad and q_L2 are evaluated at the onset of RLOF or at some other point. This should be specified.
  3. Fig. 5: The colour bars showing 'Expected systems (count)' use different scales across panels (10^1 to 10^3 or 10^4). This is noted in the figure but makes cross-panel comparison difficult. Consider normalising or at least ensuring the reader is directed to check the scale.
  4. §3.1: The statement 'We also plot non-CE stable mass transfer systems from our model in gray bins' refers to Fig. 2, but the stable MT systems are shown as a shaded region rather than 'gray bins'. Minor wording issue.
  5. §4.2: 'We first assess if is recombination is necessary' — grammatical error, should read 'We first assess whether recombination is necessary'.
  6. §4.1: 'the functional η predicts the orbit should shrink by a factor of 1000 when q_cc ~ 1.0' — it would be useful to show this explicitly (e.g., a small inset or annotation in Fig. 6) to help the reader understand why the functional form fails.
  7. Appendix A: The polynomial fits (Eqs. A1, A2) are provided with coefficients available at a Zenodo DOI, but the manuscript does not state the mass range or evolutionary phases covered by each fit. This information is only available in the online files. A summary table of the mass grid and phases would make the appendix self-contained.
  8. Table 1: The entry 'SCATTER η fitted f(q_ec), Eq. 4 Di Stefano et al. (2023)' is ambiguous in formatting; it should be clear that Eq. 4 is the authors' adopted functional form, not from Di Stefano et al. (2023) directly.
  9. §2.3.2, Eq. (9): The ratio V_sph(d_max)/V_sol is described as independent of ρ_0, but the integral in the numerator is over a spherical volume while the denominator is over a cylindrical volume. The geometry of this comparison should be clarified (e.g., does the spherical volume extend beyond the cylinder?).
  10. The abstract states 'Recombination energy is necessary, but only a fraction of it (~10–40%) can contribute.' The body text (§4.2) gives f_ion ~ 0.2–0.4 as the minimum required range. The abstract should clarify whether 10% is a lower bound or an approximate value.

Simulated Author's Rebuttal

1 responses · 0 unresolved

The referee raises a single substantive point (repeated in the report due to apparent formatting): the f_ion ≲ 0.4 constraint in §4.2 relies on the absence of observed IK Peg-like systems at super-wide periods, but §2.3.2 acknowledges the model overpredicts systems due to unmodelled selection effects. This creates an internal tension—the absence of observed systems could reflect incompleteness rather than a physical upper limit on f_ion. The referee requests either foregrounding the completeness assumption or softening the claim. We agree this is a valid concern and will revise accordingly.

read point-by-point responses

  1. Referee: §4.2, Fig. 8: The quantitative constraint that f_ion ≲ 0.4 at α_CE = 0.2 is derived from the absence of observed IK Peg-like systems at super-wide periods. However, §2.3.2 states that selection effects from distinguishability and survey cadence are not modelled, and that 'we expect our model to predict more systems than what is observed if the physics is correct.' This creates internal tension: the model is acknowledged to overpredict, so the absence of observed systems could reflect incompleteness rather than a physical upper limit on f_ion. The paper should either (a) explicitly state that the f_ion ≲ 0.4 bound is conditional on the super-wide Gaia sample being complete for IK Peg-like systems (the reference to Shahaf et al. 2024 completeness is stated only in passing and should be foregrounded as the load-bearing assumption), or (b) soften the quantitative claim.

    Authors: We thank the referee for identifying this internal tension, which is a fair and important point. The referee is correct that our general statement in §2.3.2—that the model is expected to overpredict systems due to unmodelled selection effects from distinguishability and survey cadence—sits in tension with using the *absence* of observed IK Peg-like systems at super-wide periods as a quantitative upper bound on f_ion. We will resolve this in the revised manuscript in two ways. First, we will foreground the Shahaf et al. (2024) completeness assumption as the load-bearing condition for the f_ion ≲ 0.4 bound. Specifically, the super-wide PCEB sample of Yamaguchi et al. (2024a) was constructed using the Shahaf et al. (2024) Gaia astrometric selection function, which is designed to be complete for systems with MS companions in the G-type mass range (~1.2–1.4 M☉) at the relevant distances. IK Peg-like systems fall squarely within this companion-mass range, so the Shahaf et al. (2024) completeness is what makes the non-detection physically meaningful rather than a mere artifact of selection effects. We will make this reasoning explicit in §4.2 rather than leaving it as a passing reference. Second, we will add a qualifying sentence acknowledging that if the Shahaf et al. (2024) selection function is *not* complete for IK Peg-like systems specifically—for instance, if the UV-excess distinguishability criterion introduces additional incompleteness for massive-WD systems—then the f_ion ≲ 0.4 bound should be regarded as an upper limit that could be relaxed. We will also soften the language from 'f_ion can be no higher than ~0.4' to 'f_ion is constrained to be ≲0.4, conditional on the completeness of the super-wide sample for IK Peg-like systems.' These changes make the logical chain revision: yes

Circularity Check

0 steps flagged

No significant circularity; one mild calibration-overlap concern with α_CE adoption

full rationale

The paper tests three externally-proposed common envelope formalisms (α-formalism, Two-stage from Hirai & Mandel 2022, SCATTER from Di Stefano et al. 2023) against a compiled observational dataset. The derivation chain is largely self-contained: the formalisms are taken from external publications (no author overlap for SCATTER; Hirai is a co-author but the Two-stage formalism was independently published and is being tested, not derived). The SCATTER η functional form (Eq. 4) is taken from Di Stefano et al. (2023) with no author overlap, and the paper genuinely tests it against data, finding it fails. The main mild concern is that α_CE = 0.2 is adopted from Zorotovic et al. (2010) and De Marco et al. (2011), whose observational samples are a subset of the 193-system compilation used here. The paper then confirms this value 'matches current observations, in agreement with previous works.' This has the flavor of fitting to a subset and confirming on a superset, but it is not circular in a load-bearing sense: (1) the paper does not claim α_CE = 0.2 as a new first-principles prediction, (2) the compilation includes many new systems (Yamaguchi et al. 2024a,b; Rebassa-Mansergas et al. 2025; etc.) not in the original calibration, (3) the paper transparently explores the α_CE–f_ion degeneracy (Fig. 8) and does not present a single fitted value as a derived result, and (4) the central qualitative conclusions (SCATTER fails to produce wide PCEBs; recombination energy is necessary) do not depend on the adopted α_CE value. The f_ion ~ 0.1–0.4 constraint is a genuine forward-modeling result, not a renamed input. No step reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

8 free parameters · 6 axioms · 0 invented entities

No new physical entities are postulated. The paper tests existing formalisms against observations.

free parameters (8)
  • α_CE = 0.2 (standard), 0.1–0.5 tested
    Common envelope efficiency parameter adopted from Zorotovic et al. (2010); tested over range 0.1–1.0 in §4.2
  • f_ion = 1.0 (standard), 0.1–0.4 constrained
    Recombination energy contribution fraction; set to 1.0 in standard model, constrained to 0.1–0.4 in §4.2 to match IK Peg-like systems
  • η (SCATTER) = functional form Eq. 4, or constant 0.5–1.5 tested
    Angular momentum proportionality parameter for SCATTER; functional form from Di Stefano et al. (2023), constant values tested in Fig. 6
  • δ (SCATTER) = 3
    Dimensionality parameter for SCATTER; set to 3 per Di Stefano et al. (2023) recommendation
  • λ (binding energy) = Claeys et al. (2014) values; new Two-stage fits in Appendix A
    Envelope binding energy parametrization; choice affects which systems survive CE and at what separation
  • h_z (scale height) = 380 pc
    Galactic disc scale height for Solar neighbourhood model; adopted from McKee et al. (2015)
  • ρ_0 (midplane density) = 0.043 M⊙ pc⁻³
    Stellar midplane density for Solar neighbourhood volume; adopted from McKee et al. (2015)
  • K (magnetic braking scale) = 50
    Magnetic braking scale factor in binary_c; standard value
axioms (6)
  • domain assumption The α-formalism (Eq. 1) correctly parametrizes CE energy balance as E_bind = α_CE ΔE_orb
    Invoked throughout as the baseline formalism; the paper tests alternatives but uses this as the standard model
  • domain assumption The Two-stage formalism applies to donors with M > 2.25 M⊙ in HG/RGB/CHeB phases
    Justified by entropy structure analysis in Appendix A, Fig. A1; the 2.25 M⊙ boundary is where radiative region becomes significant
  • domain assumption L2 overflow in giants can initiate common envelope even without dynamical instability
    Invoked in §2.2.3, Eq. 6; critical for matching observed wide/super-wide PCEB populations; tested in Appendix B
  • domain assumption The Solar neighbourhood can be modeled as a cylinder with exponential vertical density profile
    Used in §2.3.1, Eq. 8; standard assumption in population synthesis forward-modeling
  • domain assumption Gaia G=20 is an appropriate limiting magnitude for the surveyable volume
    Used in §2.3.2, Eq. 9; affects the magnitude-limited weight and thus the predicted present-day population
  • domain assumption Solar metallicity (Z=0.014) is representative of the observed PCEB population
    Stated in §2.2 and discussed in §4.5; no metallicity variation explored

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