MOBSE: Massive Objects in Binary Stellar Evolution Code

• MOBSE is a rapid binary population synthesis framework that models the evolution of massive stellar binaries and the formation of compact objects like black holes and neutron stars.
• It incorporates updated physical modules for stellar winds, supernova prescriptions, pair-instability phenomena, and detailed binary interactions to improve predictions.
• The code’s outputs, including remnant mass spectra and merger rates, are pivotal for interpreting gravitational-wave events and assessing metallicity effects on stellar evolution.

The Massive Objects in Binary Stellar Evolution (MOBSE) code is a state-of-the-art rapid binary population synthesis framework designed to model the evolution, interactions, and endpoints of massive stellar binaries, with particular emphasis on the formation pathways of compact objects such as black holes (BHs) and neutron stars (NSs). MOBSE is a major update to the Hurley et al. BSE code, incorporating up-to-date prescriptions for stellar winds, core-collapse supernova (SN) explosions, pair-instability phenomena, and the physics of binary interactions. Its outputs are directly relevant to interpreting gravitational-wave (GW) event catalogs and constraining the physics of massive star evolution and death in diverse galactic environments (Dabrowny et al., 2021, Giacobbo et al., 2018).

1. Code Architecture and Physical Modules

MOBSE inherits its fundamental architecture from Hurley, Tout & Pols (2000, 2002) BSE, retaining its analytic single-star evolution (including fitting formulae for stellar radii, luminosities, and core properties) and binary interaction engine (handling Roche-lobe overflow, common-envelope evolution, tidal effects, wind accretion, and gravitational-wave inspiral calculations). However, MOBSE features comprehensive updates and expansions:

• Stellar Wind Module: MOBSE employs mass-loss rates based on Vink et al. (2001), generalized to include explicit dependencies on surface metallicity ($Z$) and the Eddington factor $\Gamma_e$ as per Chen et al. (2015). The exponent of the metallicity scaling, $\beta(\Gamma_e)$, varies with the Eddington parameter, transitioning from standard OB-star values to nearly vanishing $Z$-dependence in the most luminous stars.
• Supernova and Remnant Mass Module: Implements multiple prescriptions for core-collapse SN (the Fryer et al. 2012 "rapid/delayed" models, and a physically motivated stellar compactness model), along with analytic fallback calculations and remnant mass determinations.
• Pair-Instability SN (PISN) and Pulsational Pair-Instability SN (PPISN): Adopts remnant recipes based on Woosley (2017) as implemented by Spera & Mapelli (2017), which sharply truncate the upper end of the BH mass spectrum at high He-core mass.
• Binary Evolution Processes: Up-to-date and customizable treatments of Roche-lobe overflow (using Eggleton’s Roche geometry), mass transfer stability, common-envelope physics (energy formalism with tunable $\alpha_{\mathrm{CE}}$ and $\lambda$ parameters), tides (equilibrium-tide model after Hut 1981), and wind accretion (Bondi–Hoyle scheme).

These modules enable MOBSE to model massive binary evolution across a wide parameter space, including variations in initial mass function (IMF), binary fraction, mass ratio, orbital parameters, and metallicity bins spanning $Z=2\times 10^{-4}$ to 0.02 (Giacobbo et al., 2018).

2. Compactness-Based Core-Collapse SN Prescription

MOBSE incorporates a novel compactness-based prescription for core-collapse SN outcomes, following O’Connor & Ott (2011) and Limongi (2018). The compactness parameter $\xi_M$ at core bounce is defined as:

$$\xi_M = \frac{M/M_\odot}{R(M_{\text{bary}}=M)/1000\,\text{km}}$$

For practical implementation, MOBSE adopts $M=2.5\,M_\odot$ ($\xi_{2.5}$). As detailed stellar-structure information is not tracked, $\xi_{2.5}$ is empirically estimated from the pre-supernova carbon-oxygen core mass $M_\mathrm{CO}$ using:

$$\xi_{2.5} \simeq 0.55 - 1.1\,\left(\frac{M_\mathrm{CO}}{1\,M_\odot}\right)^{-1}$$

The compactness model in MOBSE features two principal free parameters:

• Compactness Threshold ($\xi_{\mathrm{crit}}$): Sets the fate of the collapse. If $\xi_{2.5} > \xi_{\mathrm{crit}}$, the star collapses directly to a BH; if $\xi_{2.5} \leq \xi_{\mathrm{crit}}$, a successful SN produces a NS. Surveyed values include 0.20, 0.30, 0.365 (fiducial), and 0.40.
• Fallback Fraction ($f_H$, also denoted $f_\mathrm{fb}$): Fraction of ejected material that falls back onto the proto-compact object. MOBSE typically explores $f_H$ in $[0.1,0.9]$, with 0.9 as fiducial.

The outcome decision and remnant mass assignment follow:

• NS Formation: $$M_\mathrm{NS} \sim N(\mu=1.33\,M_\odot,\,\sigma=0.09\,M_\odot)$$, consistent with empirical NS mass distributions.
• BH Formation: $$M_\mathrm{BH} = M_\mathrm{He} + f_H (M_\mathrm{fin} - M_\mathrm{He})$$, where $M_\mathrm{He}$ is the Helium core mass and $M_\mathrm{fin}$ is the total pre-SN mass (Dabrowny et al., 2021).

3. Alternative SN Explosion Models: Rapid/Delayed Prescriptions

MOBSE also implements the Fryer et al. (2012) "rapid" and "delayed" explosion models as alternatives:

• Rapid Model: Classifies remnant outcome based on $M_\mathrm{CO}$ and $M_\mathrm{fin}$, with a sharp gap in remnant masses between $\sim 2$ and $5\,M_\odot$ and a pronounced dip at $20$–$30\,M_\odot$ ZAMS mass.
• Delayed Model: Yields a continuous NS–BH transition and fills in the low-mass gap, but lacks a strong remnant-mass drop at moderate ZAMS masses.

The compactness-based model, in contrast, produces a metallicity- and fallback-dependent mass gap with a tunable upper edge and does not exhibit the sharp mass drop of the rapid model (Dabrowny et al., 2021). This offers greater flexibility for confronting GW observations.

4. Population Synthesis Results and Parameter Sensitivities

MOBSE simulations reveal the dependencies of remnant and merger properties on core physical parameters, metallicity, and SN model choice:

• Remnant Mass Spectrum: Lower $\xi_{\mathrm{crit}}$ values allow lower-mass progenitors to form BHs, shifting the mass gap down; higher values move the NS–BH transition to higher ZAMS masses. The minimum BH mass grows roughly linearly with $f_H$, esp. at low $Z$.
• Merger Rates: The relative rates of double compact-object mergers (BBH, BH–NS, BNS) depend sensitively on $\xi_{\mathrm{crit}}$ but are nearly invariant ($\lesssim5\%$ variation) with respect to $f_H$. For example, at $Z=0.02$ and per $10^6$ binaries, BBH merger numbers increase from $\sim80$ to $\sim300$ as $\xi_{\mathrm{crit}}$ rises from 0.2 to 0.4.
• Metallicity Effects: Both the maximum BH mass and the location of the NS–BH mass gap display pronounced $Z$ dependence. At $Z=0.0002$, $f_H=0.1$ yields a minimum BH mass of $\sim5\,M_\odot$, while $f_H=0.9$ gives $\sim10\,M_\odot$.
• PPISN/PISN Constraints: At low $Z$, PISN and PPISN truncate the high-mass end, preventing the formation of BHs above $\sim60\,M_\odot$ (for isolated evolution), though binary mergers before SN may still yield heavier BHs (Giacobbo et al., 2018).

Table: ZAMS Mass for NS–BH Transition for Selected Metallicities (Dabrowny et al., 2021)

$\xi_{\mathrm{crit}}$	$Z=0.02$	$Z=0.0002$
0.20	14 $M_\odot$	13 $M_\odot$
0.30	24 $M_\odot$	16 $M_\odot$
0.365	29 $M_\odot$	20 $M_\odot$
0.40	37 $M_\odot$	23 $M_\odot$

5. Initial Conditions and Model Customization

MOBSE supports a range of initial binary and stellar parameters:

• IMF: Default is Kroupa (2001) ($\propto M^{-2.3}$ for $M>0.5\,M_\odot$).
• Binary fraction: Default 50%, adjustable by the user.
• Mass ratio ($q$), semi-major axis ($a$), and eccentricity ($e$): User-specified distributions; $q$ flat in $[0.1,1]$, $a$ log-uniform, $e$ thermal.
• Metallicity Grid: Discrete values from $2 \times 10^{-4}$ to $0.02$; each $Z$ bin run independently.
• CE Parameters: Common-envelope efficiency $\alpha_\mathrm{CE}$ (default 1.0), binding energy parameter $\lambda$ (tabulated or constant).
• Time Stepping: Adaptive, ensures high resolution near binary interaction events.
• Supernova Kicks: Drawn from Maxwellian, amplitude reduced by $(1-f_\mathrm{fb})$ for fallback SN (Giacobbo et al., 2018).

These settings allow systematic exploration of parameter space, relevant for statistical predictions of GW event populations.

6. Confronting Gravitational-Wave Observations

The transparency of MOBSE's compactness-based parameters enables direct confrontation with GW data:

• $\xi_{\mathrm{crit}}$: The frequency and position of lower-mass BH mergers and the sharpness of the NS–BH boundary can constrain this parameter.
• $f_\mathrm{fb}$: The upper edge of the mass gap between NSs and BHs, observable in LIGO/Virgo catalogs, tightly constrains this fallback fraction and its $Z$-dependence.
• Population rates: Predicted BBH, BH–NS, and BNS merger rates as functions of $Z$, cosmic history, and GW detector sensitivity can jointly constrain ($\xi_{\mathrm{crit}}$, $f_\mathrm{fb}$) and the relative importance of evolutionary channels.

A plausible implication is that further GW detections, particularly around the NS–BH mass gap and for lower-mass BHs, will enable calibration or falsification of the compactness-based prescription compared to the rapid/delayed alternatives (Dabrowny et al., 2021).

7. Benchmark Outcomes and Broader Implications

MOBSE predicts the formation of direct-collapse BHs up to $\sim120\,M_\odot$ at $Z=2\times10^{-4}$, merging BBH systems up to $\sim80\,M_\odot$ within a Hubble time, and a strong preference for metal-poor progenitors ($Z\lesssim0.002$) for events like GW150914. The code ties the heaviest compact mergers and the overall BBH event rate to metallicity-dependent wind and SN prescriptions, and demonstrates that only moderate to low-$Z$ environments yield the high-BH-mass mergers observed by LIGO/Virgo (Giacobbo et al., 2018).

The modular design and physically motivated SN/compactness models position MOBSE as a robust framework for interpreting compact-object merger data, mapping theoretical uncertainties to observable demography, and testing key aspects of massive stellar evolution in the GW era.