Empirical anchoring of the Galactic rates and observability of merging neutron-star-black-hole and binary-neutron-star systems: the five model uncertainties flagged by Bethe and Brown (1998)

The 1998 Bethe-Brown analytical evolution of massive binary stars predicted that low-mass ($\sim 2.4\,M_\odot$) black-hole-neutron-star (BH-NS) binaries form and merge at $\sim 10^{-4}$ per year per Galaxy, an order of magnitude above tight binary neutron stars, and would therefore dominate the gravitational-wave merger signal; the authors flagged five model-dependent inputs as limiting these predictions: the initial separation distribution of massive binaries, the supernova natal-kick distribution and its correlation with the orbit (including an explicitly deferred eccentricity-distribution computation), the common-envelope ejection efficiency and hypercritical accretion onto the neutron star, unquantified radio 'observability penalties' for unrecycled pulsars in binaries, and the Galactic supernova-rate and massive-star wind physics. Precise, input-by-input-quantified merger-rate and observability predictions remain unachieved.

Background

Two stars massive enough to undergo core-collapse supernovae can, if born as a close pair, end as a compact binary: two stellar remnants, neutron stars (NSs, the $\sim 1.4\,M_\odot$ objects left when a massive star's core collapses) or black holes (BHs), bound in orbit. After both supernovae the binary shrinks by emitting gravitational waves, and the standard quadrupole formula for this orbital decay says the time to merger grows as the fourth power of the separation, so only pairs that finish their stellar evolution separated by of order a solar radius merge within the age of the Universe. Predicting how many such merging compact binaries the Galaxy produces matters because they are the primary targets of gravitational-wave detectors such as LIGO.

The route from a massive binary to a merging compact binary passes through several poorly controlled stages. Each supernova gives the newborn neutron star a natal kick, a birth velocity of order hundreds of km/s inferred from the measured space motions of young radio pulsars, which can unbind the binary and always leaves it eccentric. When the initially heavier star explodes first, its companion later swells into a giant and can engulf the neutron star: the two then share a common envelope (CE), a phase in which the compact object spirals in through the giant's extended hydrogen envelope and the orbital energy released is invoked to eject that envelope, leaving a tight binary of the compact object plus the giant's helium core. How much of the released orbital energy actually unbinds the envelope (the CE efficiency) and how much envelope gas the neutron star accretes while spiralling in (the Bondi-Hoyle-Lyttleton rate, controlled by a drag coefficient $c_d$, with neutrino cooling argued to permit accretion far above the Eddington limit, i.e. hypercritical accretion) are the classic uncertain inputs. Population-synthesis codes follow large ensembles of binaries through prescriptions for all of these stages.

Bethe and Brown (1998) estimated the Galactic output of merging compact binaries analytically rather than with such a code. Their new ingredient was a quantitative calculation of hypercritical accretion in the common envelope, which they found grows the first-born neutron star from $1.4$ to $\sim 2.4\,M_\odot$, collapsing it into a low-mass black hole while the envelope is ejected. This fixed their chief conclusions: a Galactic production rate of $\sim 10^{-4}$ per year of BH-NS binaries containing a young, unrecycled pulsar (one never spun up by accretion, hence radio-loud for only $\lesssim 5\times 10^6$ years), ten times the rate of tight binary neutron stars such as the Hulse-Taylor pulsar PSR B1913+16; a predicted dominance of BH-NS signals in LIGO; a bimodal chirp-mass distribution (the chirp mass $\mathcal{M}=(m_1 m_2)^{3/5}(m_1+m_2)^{-1/5}$ is the combination best measured from a gravitational-wave signal) concentrated near $1.7\,M_\odot$ for BH-NS and $1.2\,M_\odot$ for NS-NS; and the puzzle that no BH-NS binary had been seen in radio surveys although the authors argued that several such systems should already have been detectable.

The authors explicitly flagged five model-dependent inputs as the limits on these numbers, and these constitute the problem. (1) The initial distribution of separations of massive binaries: they assumed a distribution flat in $\ln a$ over a chosen interval, while stating it was not well supported empirically for massive stars and calling it the most important assumption of the whole estimate, because the predicted rate scales directly with the fraction of binaries in the narrow separation window that neither merges inside the common envelope nor ends too wide to merge in a Hubble time. (2) The supernova natal-kick distribution, taken from the two-Gaussian fit to young-pulsar velocities of Cordes and Chernoff, and its directional correlation with the pre-supernova orbit, which controls binary survival and the post-supernova eccentricity; the paper deferred a complete eccentricity-distribution computation, to be done with the population code of Wettig and Brown (1996), needed to fix the $\sim 40\%$ eccentricity enlargement of the mergeable window. (3) The common-envelope treatment: the one-half fraction of released orbital energy assumed available to unbind the envelope and the fiducial drag coefficient $c_d\approx 6$, which together set both the accreted mass ($\sim 2.4\,M_\odot$) and the final separation. (4) The 'observability penalties' for unrecycled pulsars in binaries, the suggestion, offered without computation, that such systems are missed disproportionately because their higher systemic mass keeps them near the dusty, electron-rich Galactic plane, stellar winds obscure the radio pulses, and orbital acceleration hinders the Fourier searches used in pulsar surveys. (5) The Galactic supernova-rate normalization (two per century) and the wind mass loss and helium-star evolution that set pre-supernova helium-star masses. The paper positioned its analytic rates against the detailed numerical population syntheses available at the time and argued that their agreement validated both.

Problem Statement

Determine, with empirically anchored or theoretically justified inputs rather than fiducial assumptions, the Galactic production and merger rates, the mass (chirp-mass) distribution, and the radio observability of merging compact binaries, in the source paper's frame: tight binary neutron stars and low-mass black-hole-neutron-star binaries. The common scientific objective is to firm up, or refute, the source paper's quantitative predictions by resolving the five model-dependent inputs it explicitly flagged as limiting them: (i) the initial separation (period) distribution of massive binaries, that is, whether the log-flat $d\ln a$ distribution and its adopted interval are empirically supported, and what fraction of massive binaries consequently occupies the separation window that leads to merger within a Hubble time; (ii) the distribution of supernova natal kicks and their directional correlation with the pre-supernova orbital motion, including the full post-supernova eccentricity distribution and its effect on gravitational-wave merger times, the computation the source paper explicitly deferred; (iii) the common-envelope ejection efficiency and the accretion onto the neutron star during the common-envelope phase, in particular whether the neutron star accretes enough hypercritically to collapse into a $\sim 2.4\,M_\odot$ black hole or only negligibly; (iv) the 'observability penalties' for unrecycled pulsars in binaries, that is, the quantified radio-survey selection effects (Galactic-plane confinement, wind absorption, orbital acceleration) that determine how many black-hole-neutron-star and binary-neutron-star systems pulsar surveys should detect; and (v) the Galactic supernova-rate normalization together with the massive-star wind and helium-star evolution that set pre-supernova masses and hence survival and accretion outcomes. These five results jointly support one objective: merger-rate and observable-population predictions whose uncertainty is quantified input by input and which can be confronted with radio pulsar surveys and gravitational-wave observations. Resolving the problem requires each input to be measured or bounded, the rates and observability predictions to follow from those inputs, and the sensitivity of the predictions to each input to be established. Each of the five inputs is an independently refutable sub-claim that may be resolved, bounded, or refuted on its own; the five are posed jointly with the rate, chirp-mass, and observability predictions and their sensitivity to each input. The cited works are background and candidate inputs for the five flagged inputs.

References

1. Hans A. Bethe and G. E. Brown, 'Evolution of Binary Compact Objects That Merge', The Astrophysical Journal 506, 780-789 (1998). DOI: 10.1086/306265; arXiv:astro-ph/9802084. https://arxiv.org/abs/astro-ph/9802084
2. T. Wettig and G. E. Brown, 'The evolution of relativistic binary pulsars', New Astronomy 1, 17 (1996). DOI: 10.1016/S1384-1076(96)00002-4. https://doi.org/10.1016/S1384-1076(96)00002-4
3. J. M. Cordes and David F. Chernoff, 'Neutron Star Population Dynamics. II. Three-dimensional Space Velocities of Young Pulsars', The Astrophysical Journal 505, 315-338 (1998). DOI: 10.1086/306138. https://doi.org/10.1086/306138
4. G. Hobbs, D. R. Lorimer, A. G. Lyne and M. Kramer, 'A statistical study of 233 pulsar proper motions', Monthly Notices of the Royal Astronomical Society 360, 974 (2005). DOI: 10.1111/j.1365-2966.2005.09087.x. https://doi.org/10.1111/j.1365-2966.2005.09087.x
5. H. Sana, S. E. de Mink, A. de Koter, N. Langer, C. J. Evans, M. Gieles, E. Gosset, R. G. Izzard, J.-B. Le Bouquin and F. R. N. Schneider, 'Binary Interaction Dominates the Evolution of Massive Stars', Science 337, 444-446 (2012). DOI: 10.1126/science.1223344; arXiv:1207.6397. https://arxiv.org/abs/1207.6397
6. N. Ivanova, S. Justham, X. Chen, O. De Marco, C. L. Fryer, E. Gaburov, H. Ge, E. Glebbeek, Z. Han, X.-D. Li, G. Lu, T. Marsh, Ph. Podsiadlowski, A. Potter, N. Soker, R. Taam, T. M. Tauris, E. P. J. van den Heuvel and R. F. Webbink, 'Common envelope evolution: where we stand and how we can move forward', The Astronomy and Astrophysics Review 21, 59 (2013). DOI: 10.1007/s00159-013-0059-2; arXiv:1209.4302. https://arxiv.org/abs/1209.4302
7. Morgan MacLeod and Enrico Ramirez-Ruiz, 'On the accretion-fed growth of neutron stars during common envelope', The Astrophysical Journal Letters 798, L19 (2015). DOI: 10.1088/2041-8205/798/1/L19; arXiv:1410.5421. https://arxiv.org/abs/1410.5421
8. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), 'GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral', Physical Review Letters 119, 161101 (2017). DOI: 10.1103/PhysRevLett.119.161101. https://doi.org/10.1103/PhysRevLett.119.161101
9. R. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), 'Observation of Gravitational Waves from Two Neutron Star-Black Hole Coalescences', The Astrophysical Journal Letters 915, L5 (2021). DOI: 10.3847/2041-8213/ac082e; arXiv:2106.15163. https://arxiv.org/abs/2106.15163
10. R. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), 'Population of merging compact binaries inferred using gravitational waves through GWTC-3', Physical Review X 13, 011048 (2023). DOI: 10.1103/PhysRevX.13.011048; arXiv:2111.03634. https://doi.org/10.1103/PhysRevX.13.011048
11. Ilya Mandel and Floor S. Broekgaarden, 'Rates of compact object coalescences', Living Reviews in Relativity 25, 1 (2022). DOI: 10.1007/s41114-021-00034-3; arXiv:2107.14239. https://arxiv.org/abs/2107.14239
12. Debatri Chattopadhyay, Simon Stevenson, Jarrod R. Hurley, Matthew Bailes and Floor Broekgaarden, 'Modelling neutron star-black hole binaries: future pulsar surveys and gravitational wave detectors', Monthly Notices of the Royal Astronomical Society 504, 3682 (2021). DOI: 10.1093/mnras/stab973; arXiv:2011.13503. https://arxiv.org/abs/2011.13503
13. Veome Kapil, Ilya Mandel, Emanuele Berti and Bernhard Mueller, 'Calibration of neutron star natal kick velocities to isolated pulsar observations', Monthly Notices of the Royal Astronomical Society 519, 5893 (2023). DOI: 10.1093/mnras/stad019. https://doi.org/10.1093/mnras/stad019
14. A. G. Abac et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), 'GWTC-4.0: Population Properties of Merging Compact Binaries', The Astrophysical Journal Letters 1005, L51 (2026). DOI: 10.3847/2041-8213/ae771e; arXiv:2508.18083. https://arxiv.org/abs/2508.18083
15. T. Bulik, D. Gondek-Rosinska and K. Belczynski, 'Expected masses of merging compact object binaries observed in gravitational waves', Monthly Notices of the Royal Astronomical Society 352, 1372 (2004). DOI: 10.1111/j.1365-2966.2004.08028.x; arXiv:astro-ph/0310544. https://arxiv.org/abs/astro-ph/0310544