BPS Model Assumptions — Complete Parameter Table

B22-A-fiducial

https://ui.adsabs.harvard.edu/abs/2021arXiv210302608B/abstract

2022

COMPAS

COMP

SSE

Stellar evolution scheme. SSE=Hurley et al. (2000) analytic fitting formulae. hybrid=MESA for key phases + analytic elsewhere. MESA=full stellar structure grids. PARSEC=precomputed look-up tables. STARS=Eggleton detailed binary code.

"COMPAS uses the rapid analytic stellar and binary evolution algorithms described in Hurley et al. (2000, 2002)" — Riley et al. (2022). PARSEC: "SEVN interpolates stellar properties from precomputed PARSEC look-up tables" — Iorio et al. (2023) Sec. 6.

fiducial

Primary short description of what distinguishes this model from the fiducial, as written in figure captions or table headers.

As written in figure captions, appendix table headers, or panel labels of the paper.

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Secondary variant descriptor for models with multiple varied parameters.

As written in figure captions or table headers. N/A if fewer than two variant dimensions.

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Tertiary variant descriptor.

As written in figure captions or table headers.

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Quaternary variant descriptor.

As written in figure captions or table headers.

BRO22

Short citation tag (AuthorYY) grouping models by their origin paper. E.g. BRO22=Broekgaarden et al. (2022).

Constructed as first-author surname + 2-digit year. Groups all models from the same paper.

265

1D velocity dispersion of the Maxwellian natal kick distribution for core-collapse SNe applied to NS and BH (before fallback reduction). Standard 265 km/s from pulsar proper motions.

"we draw kick velocities with random isotropic orientations and draw the kick magnitudes from a Maxwellian distribution (Hobbs et al. 2005)" — van Son et al. (2022) Sec. 2.1. Hobbs et al. (2005) MNRAS 360 974 Table 1: sigma=265 km/s from 233 radio pulsars.

30

Natal kick dispersion for stripped-star (case BB / He-star) SNe. Lower than CCSN sigma because reduced ejecta mass imparts less momentum. Critical for BNS/NSBH formation.

"BH kicks are reduced by the amount of mass falling back onto the newly-formed BH during the explosion mechanism" — van Son et al. (2022) Sec. 2.1. sigma_stripped=30 km/s in COMPAS reflects lower ejecta mass for stripped progenitors.

Hobbs 2005

Additional details on the kick distribution: source reference, bimodal parameters, or mass-scaling recipe.

"The distribution of Verbunt, Igoshev & Cator (2017) is a combination of two Maxwellian functions with sigma=75 km/s and sigma=315 km/s, and weights 0.42 and 0.58" — Dorozsmai & Toonen (2024). GM20: v_kick = f_H05*(m_ej/<m_ej>)*(<m_NS>/m_rem) — Giacobbo & Mapelli (2020) Eq. 1.

1

CE efficiency: fraction of released orbital energy used to eject the common envelope. alpha=1 is 100% efficient. alpha<1 gives tighter post-CE orbit. alpha>1 allows extra energy sources.

"How efficiently this orbital energy can be used to eject the envelope is parameterized by the alpha_CE parameter, which is set to one in this work." — van Son et al. (2022) Sec. 2.1. Webbink (1984): E_bind = alpha_CE * DeltaE_orb = alpha_CE * [G*M_d,core*M_a/(2*a_f) - G*M_d*M_a/(2*a_i)].

Standard CE energy formalism; αCE=1 is fiducial (Webbink 1984)

Additional context on CE efficiency: whether recombination energy (alpha_th) is included, code-specific formalism details, or notes on which phases alpha applies.

"we fix alpha_CE=1 (we do not include external energy terms as discussed in Fragos et al. 2019; Santoliquido et al. 2021)" — Pellouin et al. (2025) Sec. 2.1 L45-47. alpha_th=0: no recombination energy.

thermal

Fraction of donor mass accreted by companion in stable RLOF. beta=1 fully conservative, beta=0 fully non-conservative. "thermal"=capped at 10x the thermal accretion rate of the accretor.

"during stable mass transfer onto a stellar companion we assume that the accretion rate is limited to ten times the thermal rate of the accreting star" — van Son et al. (2022) Sec. 2.1. M_dot_acc <= 10 * M_acc / t_KH.

specific AM of accreting star (Soberman et al. 1997)

Angular momentum carried away per unit mass lost in non-conservative MT. gamma=specific orbital AM of accretor (isotropic re-emission). Numeric values encode COSMIC prescription flags.

"Material lost from the system ... assumed to carry away the specific orbital angular momentum of the accreting component ... orbit shrinks when M_acc/M_donor <= 0.79" — van Son et al. (2022) Sec. 2.1 Appendix A. Soberman et al. (1997) Eq. A1.

pessimistic

Pessimistic: HG donors do NOT survive CE (no clear core-envelope boundary, assumed to merge). Optimistic: HG donors CAN survive CE. Strongly impacts BBH/BNS CE channel yields.

"We adopt the pessimistic CE scenario from Dominik et al. (2012), that is, we assume that Hertzsprung Gap donor stars do not survive a CE event." — van Son et al. (2022) Sec. 2.1. "stellar companions without a clear core-envelope boundary automatically lead to a merger (cemergeflag=1)" — Pellouin et al. (2025) Sec. 2.1 L48-50.

standard

Mathematical framework for CE evolution. standard=Webbink (1984) alpha-lambda energy balance. BPASS=detailed stellar structure. BSE=COSMIC alpha-lambda with cemergeflag logic.

"We assume that ejecting the envelope shrinks the binary orbit following the energy considerations proposed by Webbink (1984) and de Kool (1990)" — van Son et al. (2022) Sec. 2.1. "COSMIC treats CE in a simple parametric way using the alpha lambda formalism (Webbink 1984)" — Pellouin et al. (2025) Sec. 2.1 L42-45.

Nanjing

Dimensionless envelope structure parameter in CE binding energy: E_bind = G*M_env*M_star / (lambda*R). Encodes how tightly the envelope is bound. Nanjing=Xu & Li (2010) polynomial fits.

"For the binding (and internal) energy of the envelope, we use the Nanjing prescription (Dominik et al. 2012)" — van Son et al. (2022) Sec. 2.1. "a variable lambda that depends on the stellar type (see Claeys et al. 2014)" — Pellouin et al. (2025) Sec. 2.1 L47-48.

F12 delayed

Prescription converting pre-collapse CO core mass to final compact remnant mass. F12 delayed/rapid=Fryer et al. (2012). N20=Muller et al. (2016) neutrino-driven engine. MM=Mandel & Muller (2022).

"The remnant mass is modelled as a function of the estimated M_CO at the moment of core collapse following Fryer et al. (2012)." — van Son et al. (2022) Sec. 2.1. F12 delayed: no mass gap. F12 rapid Eq. 14: M_proto~1 Msun, creates NS-BH gap ~2-5 Msun.

Fryer 2012 delayed http://adsabs.harvard.edu/abs/2012ApJ...749...91F

Full citation and supplementary notes: fallback mechanism, mass gap creation, baryonic-to-gravitational conversion factor.

Fryer et al. (2012) ApJ 749 91 doi:10.1088/0004-637X/749/1/91. Delayed Eq. 16: M_rem=M_proto+(1-a2)*M_fb. Rapid Eq. 14: M_rem=M_proto+(1-a1)*M_fb. Full fallback when M_CO>11 Msun in delayed: no natal kick for that BH.

fit based on Marchant et al. (2019)

Pair-instability and pulsational PISN prescription setting the upper BH mass limit. PPISN for M_He>35 Msun (pulsational mass loss). PISN (full disruption, no remnant) for M_He 60-135 Msun.

"Stars with helium cores above 35 Msun ... assumed to experience pulsational-pair instability following Farmer et al. (2019). Stars with helium core masses between 60-135 Msun ... expected to be completely disrupted by pair instability" — van Son et al. (2022) Sec. 2.1. Farmer et al. (2019) ApJ 887 53.

zeta prescription

Prescription deciding whether MT is dynamically stable (RLOF) or unstable (CE). Zeta-prescription compares donor adiabatic radius response zeta*=(d log R/d log M)_ad to Roche lobe response. qcrit=critical mass ratio.

"COMPAS determines stability by comparing estimates of the adiabatic response of the donors radius and the response of the donors Roche-lobe radius" — van Son et al. (2022) Sec. 2.1. zeta_ad=2 for MS, zeta_ad=6.5 for HG (Ge et al. 2015), Soberman et al. (1997) post-He ignition.

non-conservative MT mass lost with specific orbital AM of accreting star (Soberman et al. 1997; van den Heuvel et al. 2017)

Physical mechanism by which non-accreted mass removes angular momentum from the orbit, determining whether the orbit shrinks or widens during non-conservative stable MT.

"Material lost from the system ... assumed to carry away the specific orbital angular momentum of the accreting component ... orbit shrinks when M_acc/M_donor <= 0.79 (van Son et al. 2020 Appendix A)" — van Son et al. (2022) Sec. 2.1. j_lost = sqrt(G*M_tot*a)*(M_acc/M_tot).

True

Whether accretion onto a compact object is capped at the Eddington rate. TRUE=Eddington-limited, strongly non-conservative MT onto BH, wide post-MT separations, long delay times (stable RLOF channel).

"If the accreting component is a BH, the accretion is assumed to be Eddington limited. ... most of the mass that is transferred is lost from the system." — van Son et al. (2022) Sec. 2.1; Sec. 3.2. L_Edd=4*pi*G*M*c/kappa, M_dot_Edd=L_Edd*eta^{-1}*c^{-2}.

1

Multiplicative scaling factor on Wolf-Rayet wind mass-loss rates. f_WR=1.0 fiducial. f_WR<1 reduced winds, heavier BHs. f_WR>1 enhanced winds, lighter BHs.

"For hot Wolf-Rayet-like stars, we use the empirical mass loss prescription from Belczynski et al. (2010), that is adapted from Hamann & Koesterke (1998) but scaled by metallicity following Vink & de Koter (2005)" — van Son et al. (2022) Sec. 2.1. f_WR multiplies the base rate: M_dot_WR,eff = f_WR * M_dot_WR,fid.

Nanjing prescription: λ computed from polynomial fits to stellar structure as function of stellar type, mass, and radius (Xu & Li 2010a,b; Dominik et al. 2012). Includes internal energy (αth contribution). λ varies from ~0.01–10 depending on evolutionary state.

Full description of envelope binding energy computation: lambda source, whether thermal/recombination energy (alpha_th) is included, and the physical form E_bind = G*M_env*M/(lambda*R).

"E_bind = G*M_env*M/(lambda_CE*R_star)" — standard CE formalism. Nanjing (Xu & Li 2010a,b): lambda from polynomial fits to stellar structure models. Combined alpha*lambda=0.05 in SeBa (Toonen & Nelemans 2013). "a variable lambda that depends on the stellar type (see Claeys et al. 2014)" — Pellouin et al. (2025).

Kroupa (2001) broken power-law; M1: 10–150 M☉

Initial mass function for the primary star. Sets the mass distribution of stellar progenitors. Kroupa (2001) broken power-law is near-universal.

"We assume the masses of the initially more massive stellar components (the primary, M1) are universally distributed following a Kroupa (2001) initial mass function and draw masses in the range 10-150 Msun" — van Son et al. (2022) Sec. 2.2. Kroupa (2001) Science 295 82: xi(M) proportional to M^{-2.3} for M>0.5 Msun.

Öpik (1924) flat-in-log; 0.01–1000 AU; circular at birth

Distribution of initial orbital separations/periods. Opik (1924)=flat-in-log in separation. Sana et al. (2012)=power-law in log-period from O-star survey: pi(log P) proportional to (log P)^{-0.55}.

"The initial binary separations are assumed to follow a distribution of orbital separations that is flat in the logarithm (Opik 1924) in the range 0.01-1000 AU. All binary orbits are assumed to be circular at birth." — van Son et al. (2022) Sec. 2.2. Sana et al. (2012) Science 337 444.

Uniform in q = M2/M1; q ∈ [0.01, 1); M2 ≥ 0.1 M☉ (Sana et al. 2012)

Distribution of initial mass ratios q=M2/M1. Flat-in-q=uniform. Sana et al. (2012): pi(q) proportional to q^{-0.1} from O-star observations.

"binary systems are assumed to follow a uniform distribution of mass ratios (0.01 <= q = M2/M1 < 1.0 ...) We require M2 >= 0.1 Msun" — van Son et al. (2022) Sec. 2.2. Sana et al. (2012) Science 337 444: pi(q) proportional to q^{-0.1}.

0.7 (Sana et al. 2012)

Fraction of stars in binary systems. Re-normalisation factor converting per-binary simulation yields to absolute cosmic merger rate densities.

"we assume that the Universe has a constant binary fraction of f_bin=0.7 (Sana et al. 2012)" — van Son et al. (2022) Sec. 4. "We assume that the fraction of stars in binary systems is 50% (Sana et al. 2012)" — Pellouin et al. (2025) Sec. 2.4 L13-14 Table A.1 L10.

Flat-in-log; 10⁻⁴ ≤ Z ≤ 0.03; re-weighted to IllustrisTNG100 S(Z,z)

Range and sampling of stellar birth metallicities Z. Low Z gives less wind mass loss and more massive BHs. Convolved with SFRD model to compute the cosmic merger rate.

"We sample birth metallicities with a probability distribution that is flat-in-log in the range 1e-4 <= Z <= 0.03." — van Son et al. (2022) Sec. 2.2. "We create a grid of metallicities ranging from Z=9.5e-5 to Z=0.028" — Pellouin et al. (2025) Sec. 2.2 L73-80.

IllustrisTNG100 cosmological simulation; skewed log-normal S(Z,z); z_first=10 (van Son et al. 2022)

Star formation rate density S(Z,z) model used to convolve binary yields over cosmic time and metallicity to compute the observable merger rate density R(z).

"We extract the amount of star formation ongoing at each redshift and metallicity in the IllustrisTNG100 simulations and use this to derive S(Z,z) [Eq. C6]" — van Son et al. (2022) Appendix C. "We use the functional form for the SFR from Springel & Hernquist (2003) [Eq. 2]... nu=0.178, zm=2, a=2.37, b=1.80" — Pellouin et al. (2025) Sec. 2.4 L2-14.

2.5 (fiducial; alternatives 2.0 and 3.0 tested in sub-models J, K)

Maximum gravitational NS mass before collapse to BH. Sets the NS/BH boundary for remnant classification.

mxns=3.0 Msun in COSMIC (Pellouin et al. 2025 Table A.1 L46). Default ~2.5 Msun in COMPAS/POSYDON (inferred from code defaults). Alternatives: 2.0 Msun (B22-M) and 3.0 Msun (B22-N) in Broekgaarden et al. (2022) Appendix F Fig. 15.

30 (same Maxwellian as σ_strippedSN; no separate ECSN σ in COMPAS)

Natal kick dispersion for electron-capture SNe (ECSN), collapsing ONeMg cores near the Chandrasekhar mass. Very low kicks expected due to near-symmetric collapse.

"For electron-capture SNe, ultra-stripped SNe and AIC events, we assume that the kicks are reduced and have a dispersion sigma_{k,low}=20 km/s. This parameter is especially important as most evolutionary tracks involve one or several of these events with lower kicks, and therefore higher probabilities for the binary to survive" — Pellouin et al. (2025) Sec. 2.1 L57-63 Table A.1 L37.

No (pessimistic; HG donors do not survive CE — Dominik et al. 2012)

Binary flag: can a Hertzsprung Gap star survive as CE donor? HG stars lack a strong density gradient between core and envelope, making CE ejection physically uncertain.

"Hertzsprung Gap donor stars do not survive a CE event" — pessimistic (Dominik et al. 2012), applied in van Son et al. (2022) Sec. 2.1. "stellar companions without a clear core-envelope boundary automatically lead to a merger during the CE phase (cemergeflag=1)" — Pellouin et al. (2025) Sec. 2.1 L48-50.

ζ_ad=2 for MS donors; ζ_ad=6.5 for HG donors (Ge et al. 2015); Soberman et al. (1997) post-He ignition; stripped stars always stable

Full numerical values of the MT stability criterion: zeta_ad per stellar type or qcrit thresholds per donor type, with source references.

"We assume zeta_ad=2 for main sequence donors, zeta_ad=6.5 for Hertzsprung gap donor stars (Ge et al. 2015) and follow Soberman et al. (1997) for donor stars post helium ignition." — van Son et al. (2022) Sec. 2.1. "qcflag=5 (Neijssel et al. 2019 / COMPAS values). mass transfers from stripped donors are always dynamically stable" — Pellouin et al. (2025) Sec. 2.1 L38-40.

Fryer et al. (2012) delayed (fiducial); gravitational mass from CO core mass; ECSN track for ONeMg cores

Detailed notes on NS mass computation: gravitational vs baryonic conversion, ECSN channel, code-specific remnant flags (remnantflag, rembar_massloss).

"We infer the remnant mass following the rapid mechanism for the SN explosion (Fryer et al. 2012), with updates from Giacobbo & Mapelli (2020) (remnantflag=3). This leads to a mass gap between NSs and BHs" — Pellouin et al. (2025) Sec. 2.1 L66-69. rembar_massloss=0.5 (baryonic-to-gravitational conversion, Table A.1 L47).

Hot stars (Teff≥12500K): Vink et al. (2000a,b) metallicity-dependent; Cool stars: max(Kudritzki & Reimers 1978, Nieuwenhuijzen & de Jager 1990) × metallicity factor (Kudritzki et al. 1989); WR: Belczynski et al. (2010) adapted from Hamann & Koesterke (1998) × Vink & de Koter (2005); LBV enhanced winds above Humphreys-Davidson limit (metallicity-independent)

Complete stellar wind prescription: hot OB winds, cool/giant/AGB winds, WR winds, and LBV enhanced winds. Metallicity scaling throughout. Dominant factor setting pre-SN masses.

"For hot O and B type stars (Teff>=12500K), we follow Vink et al. (2000a,b) to account for metallicity-dependent stellar wind mass loss" — van Son et al. (2022) Sec. 2.1. LBV: "(R/Rsun)*(L/Lsun)^0.5 > 1e5 ... enhanced mass loss rates ... metallicity independent. ... In our simulations this is the dominant reason for the suppression of the CE channel at higher masses." — van Son et al. (2022).

Included via COMPAS/BSE framework; not separately varied in this study

Prescription for tidal synchronization and circularization. Determines whether close binaries circularize before MT onset and affects spin-orbit alignment.

Hurley et al. (2002) BSE Sec. 2.7: tidal interactions following Hut (1981) equilibrium tides. "tflag=1 / ST_tide=1 (Hut 1981 / BSE formalism)" — Pellouin et al. (2025) Table A.1 L55-56. POSYDON: tides included self-consistently in MESA binary grids for MS and HeMS phases.

B22-B-beta0-25

https://ui.adsabs.harvard.edu/abs/2021arXiv210302608B/abstract

2022

COMPAS

COMP

SSE

Stellar evolution scheme. SSE=Hurley et al. (2000) analytic fitting formulae. hybrid=MESA for key phases + analytic elsewhere. MESA=full stellar structure grids. PARSEC=precomputed look-up tables. STARS=Eggleton detailed binary code.

"COMPAS uses the rapid analytic stellar and binary evolution algorithms described in Hurley et al. (2000, 2002)" — Riley et al. (2022). PARSEC: "SEVN interpolates stellar properties from precomputed PARSEC look-up tables" — Iorio et al. (2023) Sec. 6.

$\beta = 0.25$

Primary short description of what distinguishes this model from the fiducial, as written in figure captions or table headers.

As written in figure captions, appendix table headers, or panel labels of the paper.

—

Secondary variant descriptor for models with multiple varied parameters.

As written in figure captions or table headers. N/A if fewer than two variant dimensions.

—

Tertiary variant descriptor.

As written in figure captions or table headers.

—

Quaternary variant descriptor.

As written in figure captions or table headers.

BRO22

Short citation tag (AuthorYY) grouping models by their origin paper. E.g. BRO22=Broekgaarden et al. (2022).

Constructed as first-author surname + 2-digit year. Groups all models from the same paper.

265

1D velocity dispersion of the Maxwellian natal kick distribution for core-collapse SNe applied to NS and BH (before fallback reduction). Standard 265 km/s from pulsar proper motions.

"we draw kick velocities with random isotropic orientations and draw the kick magnitudes from a Maxwellian distribution (Hobbs et al. 2005)" — van Son et al. (2022) Sec. 2.1. Hobbs et al. (2005) MNRAS 360 974 Table 1: sigma=265 km/s from 233 radio pulsars.

30

Natal kick dispersion for stripped-star (case BB / He-star) SNe. Lower than CCSN sigma because reduced ejecta mass imparts less momentum. Critical for BNS/NSBH formation.

"BH kicks are reduced by the amount of mass falling back onto the newly-formed BH during the explosion mechanism" — van Son et al. (2022) Sec. 2.1. sigma_stripped=30 km/s in COMPAS reflects lower ejecta mass for stripped progenitors.

Hobbs 2005

Additional details on the kick distribution: source reference, bimodal parameters, or mass-scaling recipe.

"The distribution of Verbunt, Igoshev & Cator (2017) is a combination of two Maxwellian functions with sigma=75 km/s and sigma=315 km/s, and weights 0.42 and 0.58" — Dorozsmai & Toonen (2024). GM20: v_kick = f_H05*(m_ej/<m_ej>)*(<m_NS>/m_rem) — Giacobbo & Mapelli (2020) Eq. 1.

1

CE efficiency: fraction of released orbital energy used to eject the common envelope. alpha=1 is 100% efficient. alpha<1 gives tighter post-CE orbit. alpha>1 allows extra energy sources.

"How efficiently this orbital energy can be used to eject the envelope is parameterized by the alpha_CE parameter, which is set to one in this work." — van Son et al. (2022) Sec. 2.1. Webbink (1984): E_bind = alpha_CE * DeltaE_orb = alpha_CE * [G*M_d,core*M_a/(2*a_f) - G*M_d*M_a/(2*a_i)].

Standard CE energy formalism; αCE=1 is fiducial (Webbink 1984)

Additional context on CE efficiency: whether recombination energy (alpha_th) is included, code-specific formalism details, or notes on which phases alpha applies.

"we fix alpha_CE=1 (we do not include external energy terms as discussed in Fragos et al. 2019; Santoliquido et al. 2021)" — Pellouin et al. (2025) Sec. 2.1 L45-47. alpha_th=0: no recombination energy.

0.25