StarTrack: Modular Binary-Population Synthesis

• StarTrack is a binary-population synthesis code that simulates the evolution of zero-age binaries with detailed treatments of mass transfer, common-envelope phases, and supernova kicks.
• The code employs Monte Carlo methods and analytic prescriptions to evolve large ensembles of binaries, capturing key processes like wind mass loss and gravitational-wave inspiral times.
• Its versatility supports applications from predicting compact-object merger rates and X-ray binary properties to constructing synthetic Galactic remnant catalogs.

StarTrack is a modular, Monte Carlo binary-population synthesis code for evolving large ensembles of zero-age main-sequence binaries through stellar winds, Roche-lobe overflow, common-envelope evolution, supernovae, compact-object formation, and, where relevant, gravitational-wave inspiral. In the cited literature it is used to predict populations of double compact objects, X-ray binaries, ultraluminous X-ray sources, Galactic black holes, non-interacting black-hole binaries, and wide double white dwarfs, and to connect those populations to gravitational-wave, electromagnetic, and astrometric observables (Dominik et al., 2012, Biesiada et al., 2014, Tzanavaris et al., 2013, Périgois et al., 2020, Mondal et al., 2019, Wiktorowicz et al., 2020, Andrews et al., 2012).

1. Code identity and scope

Published descriptions characterize StarTrack as a binary-evolution population-synthesis framework that follows either $2\times 10^6$ zero-age binaries or, in other applications, ensembles of $\sim 10^6\!-\!10^7$ binaries, depending on the study design and the quantity being modeled (Dominik et al., 2012, Biesiada et al., 2014). The code is repeatedly described as modular, and its core remit is isolated stellar and binary evolution rather than $N$-body cluster dynamics or purely phenomenological rate fitting (Dominik et al., 2012).

The underlying workflow is consistent across applications. Primordial binaries are drawn from prescribed initial distributions, evolved through mass loss and binary interaction, subjected to compact-remnant formation prescriptions and natal-kick models, and then filtered according to the scientific target: compact binaries that merge within a Hubble time, accreting systems that contribute to X-ray luminosity functions, black-hole binaries detectable by Gaia or LAMOST, or wide white-dwarf pairs that survive Galactic perturbations (Tzanavaris et al., 2013, Wiktorowicz et al., 2020, Andrews et al., 2012).

StarTrack is also routinely embedded in larger population models. In cosmological calculations it is combined with metallicity-dependent star-formation histories and detector selection functions to produce redshift-dependent merger-rate densities and detectable-event rates (Biesiada et al., 2014, Périgois et al., 2020). In Milky Way applications it is convolved with disk, bulge, and halo star-formation histories and chemical-evolution models to produce synthetic present-day Galactic catalogs (Olejak et al., 2019, Wiktorowicz et al., 2020). This dual role—as an evolutionary engine and as the kernel of a synthetic-universe pipeline—is one of the defining features of the code.

2. Evolutionary prescriptions and governing equations

Across the cited studies, StarTrack uses analytic single-star evolution fits, metallicity-dependent wind prescriptions, stability criteria for Roche-lobe overflow, an $\alpha$-$\lambda$ treatment of common-envelope evolution, supernova remnant prescriptions of the “rapid” and “delayed” type, and natal kicks drawn from Maxwellian distributions with fallback-dependent suppression for black holes in many model families (Tzanavaris et al., 2013, Dominik et al., 2012, Banerjee et al., 2019). Several papers also include pair-instability and pulsational pair-instability prescriptions in the remnant-mass calculation (Banerjee et al., 2019, Olejak et al., 2019).

Initial conditions are not fixed by a single universal StarTrack setup; instead they vary by application. Recurring choices include Kroupa-like or three-segment Kroupa-type initial mass functions, flat mass-ratio distributions, orbital separations distributed either flat in $\log a$ or via $p(\log P)\propto(\log P)^{-0.55}$, and eccentricity distributions that are either thermal, $f(e)=2e$, or $p(e)\propto e^{-0.42}$ (Dominik et al., 2012, Kowalska et al., 2010, Olejak et al., 2019). The presence of multiple input families is not incidental: it reflects the code’s role as a parametric synthesis framework rather than a single fixed binary-evolution realization.

A central equation is the Webbink-style common-envelope energy balance. In one standard form used in StarTrack studies,

$$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$

where $\sim 10^6\!-\!10^7$0 and $\sim 10^6\!-\!10^7$1 are the donor masses before and after envelope ejection, $\sim 10^6\!-\!10^7$2 is the accretor mass, $\sim 10^6\!-\!10^7$3 and $\sim 10^6\!-\!10^7$4 are the pre- and post-common-envelope separations, $\sim 10^6\!-\!10^7$5 is the envelope mass, and $\sim 10^6\!-\!10^7$6 is the binding-energy parameter (Dominik et al., 2012).

Natal kicks are likewise encoded in compact formulas. For fallback-modulated black-hole kicks,

$\sim 10^6\!-\!10^7$7

with $\sim 10^6\!-\!10^7$8 drawn from a Maxwellian and $\sim 10^6\!-\!10^7$9 the fallback fraction (Dominik et al., 2012). Once a double compact object forms, gravitational-wave merger times can be computed with the Peters prescription,

$N$0

which links post-interaction orbital architecture directly to merger observability (Dominik et al., 2012).

These prescriptions give StarTrack its characteristic structure: binary-population outputs emerge from the coupling of mass loss, remnant formation, binary survival, and orbital shrinkage. The code’s scientific utility derives from this coupling, but so does its sensitivity to uncertain physics.

3. Common-envelope treatment and donor-structure criteria

Common-envelope evolution is the dominant structural uncertainty in much of the StarTrack literature. Earlier StarTrack-based compact-binary studies often bracketed the problem with two contrasting assumptions about Hertzsprung-gap donors: either a Hertzsprung-gap common envelope always leads to merger, or such donors are allowed to survive via the usual $N$1-$N$2 formalism (Biesiada et al., 2014, Dominik et al., 2012). This single modeling choice already shifts predicted merger rates by factors of a few to orders of magnitude, especially for BH–BH and BH–NS systems (Dominik et al., 2012).

A later revision made the criterion more restrictive for massive donors. In the implementation described in "The impact of common envelope development criteria on the formation of LIGO/Virgo sources" (Olejak et al., 2021), common-envelope onset for evolved H-rich giants with $N$3 depends on radius thresholds $N$4 and $N$5, together with a mass-ratio cutoff $N$6. In the three illustrative models reported there, the local BH–BH merger rate changes from $N$7 to $N$8 to $N$9, while NS–NS rates change from $\alpha$0 to $\alpha$1 to $\alpha$2 (Olejak et al., 2021). The point is not merely numerical spread; it is that modest modifications to the common-envelope development rule can move the dominant formation channel from common-envelope–assisted evolution to channels with no common-envelope phase at all.

An even more explicit donor-structure criterion appears in "Development of convective envelopes in massive stars: Implications for gravitational wave sources" (Romagnolo et al., 2024). There, a star is defined to have developed an outer convective envelope once the mass in outer convection zones exceeds $\alpha$3 of the total H-rich envelope mass. StarTrack then replaces the earlier evolutionary-type precheck in preCEOutcome() with an envelope-type precheck based on whether

$\alpha$4

If the condition is satisfied, the donor is treated as having a convective envelope and the system proceeds to the $\alpha$5-formalism; otherwise the radiative-envelope case is assigned an immediate merger, producing a Thorne–Żytkow object or quasi-star when the accretor is a neutron star or black hole (Romagnolo et al., 2024). The same study adds new data tables in /data/stars/, modifies src/common_envelope.F90 and src/binary_evolve.F90, and records the implementation under the commit tag conv_env_2024 (Romagnolo et al., 2024).

The astrophysical consequences are substantial. In the $\alpha$6 model, the local BH–BH merger rate for total masses $\alpha$7 is reduced by $\alpha$8, from $\alpha$9 to $\lambda$0, and the total-mass distribution becomes bimodal, with peaks at $\lambda$1 and $\lambda$2 (Romagnolo et al., 2024). Under the strong red-supergiant pulsation scenario, both the TŻO and quasi-star populations in the Galaxy collapse to $\lambda$3 object in the present epoch, making detection essentially unfeasible (Romagnolo et al., 2024). These results strongly suggest that StarTrack’s common-envelope channel is no longer well represented by any single Hertzsprung-gap rule; donor envelope structure has become a code-level state variable of primary importance.

4. Synthetic-universe construction and statistical inference

StarTrack outputs are often post-processed into cosmological rate models. In one formulation, each simulation yields a discrete merger population

$\lambda$4

where $\lambda$5 denotes binary parameters and $\lambda$6 are weights accounting for metallicity bins and cosmic assembly. Detector selection is then applied through

$\lambda$7

and expected counts follow by integration over masses and redshift (Delfavero et al., 2023). This framework turns StarTrack from a forward binary-evolution code into a generative model for observed compact-binary catalogs.

Earlier work uses an analogous construction for strong-lensing forecasts with the Einstein Telescope. There the redshift-dependent intrinsic merger density $\lambda$8 from StarTrack is combined with the detector selection function and a singular isothermal sphere lens population to obtain the yearly rate of strongly lensed, detectable mergers (Biesiada et al., 2014). The same logic reappears in stochastic-background calculations, where metallicity-resolved StarTrack outputs are convolved with cosmic star-formation and metallicity histories and then integrated over source redshift to compute $\lambda$9 (Périgois et al., 2020).

StarTrack has also been coupled directly to Bayesian population inference. In the inhomogeneous-Poisson formulation used for comparison with gravitational-wave catalogs,

$\log a$0

where $\log a$1 are expected counts by merger type and $\log a$2 are event-level likelihoods (Delfavero et al., 2023). Closely related work replaces direct posterior-sample handling with bounded multivariate normal “Normal Approximate Likelihood” fits, enabling fast evaluation of event likelihoods across a large StarTrack simulation bank (Favero, 2022).

These developments are methodologically significant because they transform StarTrack from a source of tabulated rates into a parameterized synthetic-universe engine. Once detector selection, cosmology, and event likelihoods are included, code parameters such as kick dispersion, mass-transfer efficiency, and wind scaling can be compared directly with gravitational-wave observations rather than only with population-level summary statistics.

5. Scientific applications

StarTrack’s most visible application is the prediction of compact-object merger populations. In strong-lensing forecasts for the Einstein Telescope, the expected rate is about $\log a$3 strongly lensed inspiral events per year, with the BH–BH channel dominating the lensed yield and the high-BH-kicks scenario reducing the prediction to only a few events per year (Biesiada et al., 2014). In stochastic-background calculations, the population I/II contribution reaches $\log a$4 at $\log a$5 Hz, making the background detectable at $\log a$6 after about $\log a$7 years of observation with current-generation ground-based detectors at design sensitivity; population III contributes about one order of magnitude less to the total background but dominates the residual background in 3G detectors after subtraction of resolvable sources (Périgois et al., 2020).

A second major application is X-ray binary and ultraluminous X-ray source modeling. In the SINGS comparison, StarTrack-based theoretical X-ray luminosity functions are convolved with galaxy star-formation histories, and the best global models are consistent with $\log a$8 and a $\log a$9 uniform–$p(\log P)\propto(\log P)^{-0.55}$0 “twins” initial mass-ratio distribution (Tzanavaris et al., 2013). In the ULX–double-compact-object study, typically $p(\log P)\propto(\log P)^{-0.55}$1 of merging BH–BH progenitor binaries are found to have evolved through a ULX phase, while the fraction of observed ULXs that will form merging double compact objects in the future varies between $p(\log P)\propto(\log P)^{-0.55}$2 and $p(\log P)\propto(\log P)^{-0.55}$3, depending on common-envelope model and metallicity (Mondal et al., 2019).

StarTrack is also used for Galactic census problems. A Milky Way synthetic catalog predicts that the present-day Galaxy contains about $p(\log P)\propto(\log P)^{-0.55}$4 single black holes with average mass about $p(\log P)\propto(\log P)^{-0.55}$5, and quotes current Galactic merger rates of $p(\log P)\propto(\log P)^{-0.55}$6 for BH–BH, $p(\log P)\propto(\log P)^{-0.55}$7 for BH–NS, and $p(\log P)\propto(\log P)^{-0.55}$8 for NS–NS systems across two common-envelope models (Olejak et al., 2019). In astrometric and spectroscopic detectability studies, Gaia is predicted to observe $p(\log P)\propto(\log P)^{-0.55}$9 non-interacting black-hole binaries, falling to $f(e)=2e$0 if the recent thin-disk star formation is low, while LAMOST is expected to detect $f(e)=2e$1 such systems (Wiktorowicz et al., 2020). For white-dwarf binaries, StarTrack-based calculations indicate a significant observable population of wide WDWDs with orbital separations $f(e)=2e$2 AU, and the corresponding SDSS search identified twelve high-confidence wide WDWD pairs (Andrews et al., 2012).

A plausible implication is that StarTrack’s scientific range is unusually broad not because the code solves one narrowly defined rate problem, but because its outputs can be reweighted, projected, and selection-filtered into many different observational spaces. The same binary-evolution core can therefore feed gravitational-wave cosmology, X-ray population synthesis, Galactic stellar-remnant cartography, and survey-yield forecasting.

6. Uncertainties, controversies, and name collisions

The most persistent conclusion across the StarTrack literature is that predictions are dominated by a small number of uncertain physical ingredients. Varying the common-envelope binding parameter $f(e)=2e$3 between $f(e)=2e$4 and $f(e)=2e$5 shifts merger rates by up to $f(e)=2e$6 orders of magnitude, and adopting full black-hole kicks suppresses BH–BH rates by $f(e)=2e$7 dex (Dominik et al., 2012). Later work on very massive binaries shows that different assumptions about common-envelope survival, mass and angular-momentum loss, stellar mixing, pair-instability mass loss, and supernova outbursts can send the same observed system toward a close BH–BH merger, a wide BH–BH binary, a Thorne–Żytkow object, or total disruption by pair-instability supernovae (Belczynski et al., 2021). The convective-envelope revision goes further, arguing that the common-envelope channel for BH–BH mergers with $f(e)=2e$8 has been considerably overestimated and that predictions for systems above $f(e)=2e$9 hinge heavily on limited understanding of stellar structure and mass loss close to the Eddington limit (Romagnolo et al., 2024).

Observation-driven inference does not eliminate this uncertainty. One StarTrack-based analysis that varies $p(e)\propto e^{-0.42}$0, $p(e)\propto e^{-0.42}$1, $p(e)\propto e^{-0.42}$2, and $p(e)\propto e^{-0.42}$3 finds a maximum-likelihood model $p(e)\propto e^{-0.42}$4 with $p(e)\propto e^{-0.42}$5, $p(e)\propto e^{-0.42}$6, $p(e)\propto e^{-0.42}$7, and $p(e)\propto e^{-0.42}$8 (Delfavero et al., 2023). Another isolated-binary analysis, using a different simulation family and likelihood approximation, finds a 4D likelihood peak at model $p(e)\propto e^{-0.42}$9 with $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$0, $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$1, $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$2, and $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$3 (Favero, 2022). These are not directly commensurate constraints, but together they show that StarTrack-based inference remains conditional on model family, parameterization, and post-processing assumptions.

A separate source of confusion is lexical rather than astrophysical. "S.T.A.R.-Track" (Doll et al., 2023) is an object-centric, transformer-based framework for end-to-end 3D object tracking; on the nuScenes validation split it reports AMOTA $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$4 and IDS $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$5, and on the test split with a stronger backbone it reports AMOTA $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$6. "The LSPE-Strip Pointing Reconstruction and Star Tracker" (Maris et al., 9 Jan 2025) describes an optical pointing instrument for a CMB telescope, with a prototype RMS accuracy of approximately $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$7 arcseconds and systematic errors below $$\alpha_{\rm CE} \Bigl[ \frac{G\,M_{\rm d,f}\,M_{\rm a}}{2\,a_f} - \frac{G\,M_{\rm d,i}\,M_{\rm a}}{2\,a_i} \Bigr] = \frac{G\,M_{\rm d,i}\,M_{\rm d,env}}{\lambda\,R_{\rm L}} ,$$8 arcseconds. These systems are unrelated to the astrophysical StarTrack population-synthesis code despite the near-identical naming.

Taken together, the literature presents StarTrack less as a finished predictive theory than as a highly structured computational hypothesis space for isolated binary evolution. Its continuing importance lies precisely in that role: it makes uncertain stellar and binary physics explicit, parameterized, and testable against increasingly heterogeneous data.