Stellar Origins and Remnant Populations

Stellar-origin black holes are the end products of coupled histories, not a one-parameter sequence labeled by initial mass. Burning and mixing build a core; winds and binary interaction remove or add mass and angular momentum; collapse, fallback, and pair instability determine what can become a remnant; kicks decide which binaries and cluster members survive; population synthesis then combines those conditional outcomes with cosmic formation and observational selection. This chapter develops that chain while keeping durable mechanisms separate from prescription-dependent boundaries.

Helpful background. Black-Hole Geometry and Mechanics supplies the remnant mass and spin conventions; Relativistic Matter and Compact Objects supplies stellar structure and compact-remnant physics; Black-Hole Astrophysics supplies the observational and population setting; and Black-Hole Dynamics and Collapse supplies the dynamical formation and horizon-scale endpoint represented by the remnant mappings used here.

The causal chain from star to population

The central state passed from stellar evolution to collapse is a set of profiles, not merely a zero-age main-sequence mass:

$$\{\rho(m),s(m),Y_e(m),X_i(m),j(m)\}_{t_{\rm coll}}.$$

Here $m$ labels enclosed baryonic mass; $\rho$, $s$, $Y_e$, $X_i$, and $j$ are density, entropy per baryon, electron fraction, isotope mass fractions, and specific angular momentum at collapse. Their density and entropy structure governs accretion onto a proto-compact object; their composition and binding energy influence explosive burning and ejection; their angular-momentum profile determines whether infall remains nearly radial or circularizes. Heger and collaborators show how winds, metallicity, core structure, and instability regimes combine in a conditional initial-mass fate map Heger et al. 2003, §§ 2–3 and Fig. 1. The map is a useful summary of a stated model grid, not an immutable lookup table.

Collapse then supplies a remnant transfer function. In schematic form,

$$\boldsymbol\theta_{\rm rem} =F\!\left(\boldsymbol\theta_{\rm pre}; \hbox{explosion, fallback, pair physics, kick, outflow}\right).$$

For a binary, this function must also update mass, orbital energy, angular momentum, eccentricity, tilt, and center-of-mass velocity. A population calculation integrates the function over initial conditions and cosmic history. Every arrow is conditional, so a black-hole mass or spin usually admits more than one progenitor explanation.

Quantities that must not be conflated

Several masses coexist. Initial stellar mass differs from final pre-collapse mass; helium- or carbon–oxygen-core mass differs from total mass; baryonic remnant mass differs from gravitational mass; and a black hole can gain mass after birth. Pair instability is best organized by core thermodynamics. A gravitational-wave detector first constrains redshifted, detector-frame mass combinations through a waveform model; source-frame component masses follow only after a redshift and cosmology are supplied. A proposed mass gap must say which mass, frame, and epoch it concerns.

Spin requires similar care. The stellar profile contains specific angular momentum $j(m)$, while the hole has $a_*=J/(GM^2)$ in the site convention $c=1$. Winds can preferentially remove high-$j$ surface material, internal torques can couple the core to the envelope, and disk outflows can prevent high-$j$ fallback from joining the hole. Natal spin, effective binary spin, and a post-merger spin are not interchangeable.

Finally, distinguish natal kick, symmetric mass-loss recoil, and later dynamical acceleration. The kick is a vector impulse to the remnant; the systemic velocity is a mass-weighted binary response; cluster retention additionally compares that velocity with a local escape speed. These distinctions turn qualitative channel labels into checkable conservation problems.

A six-stage route through the chapter

The pages form a model-building sequence rather than a literal stellar timeline. The first three establish pre-collapse structure and mass outcomes; the fourth isolates the mass and angular momentum removed before collapse; the fifth propagates the remnant into a binary or cluster; and the last assembles the branches into a population.

1. Massive-Star Evolution Toward Collapse develops burning, shell structure, mixing, winds, rotation, and compactness, and shows how a pre-collapse profile is numerically checked.
2. Fallback, Direct Collapse, and Remnant Mass converts that profile into a conditional mass budget, separating baryonic mass, gravitational mass, weak ejection, delayed black-hole formation, and nearly direct collapse.
3. Pair Instability and Proposed Mass Gaps derives pressure softening from pair creation, then distinguishes the robust pulse/disruption sequence from moving mass boundaries and alternative ways to populate a proposed gap.
4. Metallicity, Winds, and Angular Momentum connects line-driven loss and binary stripping to retained mass and $j(m)$, with an executable scaling and integration benchmark.
5. Natal Kicks and Binary Survival derives the post-collapse energy criterion and follows survivors into eccentricity, tilt, systemic recoil, and cluster retention.
6. Stellar-Remnant Population Synthesis assembles the earlier mappings into weighted intrinsic and selected populations, with a reproducible quadrature diagnostic and an explicit uncertainty map.

A productive reading strategy is to ask the same three questions at every stage: What state enters? Which physical or statistical rule changes it? Which limit or independent calculation checks the output? That discipline exposes where an apparently precise final distribution inherits an uncertain branch.

From a birth distribution to observed evidence

This overview’s literature and observational status is assessed through 2026-08-18. It does not analyze a particular detector dataset, catalog, or data release and claims no new posterior. To confront a predicted birth population with observations, one must declare the catalog version, event-quality thresholds, waveform and calibration or instrument-response boundary, search-pipeline version, and selection function. Completeness cannot be assumed merely because every published event is included.

The statistical layer must also state its likelihood, population model, inference prior, and treatment of measurement uncertainty. Statistical credible intervals do not include all systematic uncertainty: stellar prescriptions, cosmic metallicity history, waveform model dependence, redshift evolution, and channel degeneracy can dominate. A normalized detected shape also discards the absolute rate unless the selection-corrected normalization is retained.

No software reproduction of an observational result is claimed here. A reproducible extension would need machine-readable event posterior samples and injection-based selection data, exact release identifiers, environment and code versions, and commands that regenerate both rate and shape diagnostics. The model archive accompanying the dense progenitor grid of Sukhbold and collaborators is an example of a stable underlying stellar data product Sukhbold et al. 2016, § 2.3 and model archive, but it is not itself a detector catalog.

Scope and model boundaries

The chapter includes mass loss, metallicity, collapse outcomes, pair instability, natal kicks, binary interactions, and remnant distributions. It emphasizes mass and angular-momentum budgets, limiting cases, convergence, dated evidence boundaries, and failure modes. It uses schematic transfer functions only when their conditional nature is visible.

Detailed neutrino-radiation supernova transport is excluded: it belongs with collapse microphysics rather than a short remnant prescription. Full gravitational-wave population likelihoods are also excluded: those require event-level posterior samples, selection injections, a waveform and cosmological model, calibration choices, and a separately reviewable statistical pipeline. The chapter prepares those calculations without pretending to perform them.

Review the chapter

A circular binary contains a $24M_\odot$ pre-collapse star and a $12M_\odot$ companion. A toy prescription uses $M_{\rm proto}=1.4M_\odot$, $f_{\rm fb}=0.50$, and $M_{\rm rem}=0.9M_{\rm rem,bar}$. Assume the remaining mass is lost instantaneously and symmetrically, with no natal kick. Find the remnant masses, decide whether the binary survives, and identify two reasons the answer is conditional.

Solution

The fallback and baryonic masses are

$$M_{\rm fb}=0.50(24-1.4)M_\odot=11.3M_\odot, \qquad M_{\rm rem,bar}=12.7M_\odot.$$

The adopted conversion gives $M_{\rm rem}=11.43M_\odot$. The binary total changes from $M_{\rm pre}=36M_\odot$ to $M_{\rm post}=23.43M_\odot$. Because $23.43>36/2=18$, the instantaneous no-kick criterion predicts a bound survivor. Its idealized eccentricity would be

$$e=\frac{36-23.43}{23.43}\simeq0.536.$$

The result is conditional first on the fallback and baryonic-to-gravitational mass prescriptions, and second on the instantaneous, symmetric, zero-kick orbital model. Disk outflows, a different explosion mapping, anisotropic recoil, or finite-duration mass loss could change both the remnant and the orbit.

References

• Heger, A., Fryer, C. L., Woosley, S. E., Langer, N., and Hartmann, D. H. (2003). “How Massive Single Stars End Their Life.” The Astrophysical Journal 591, 288–300. DOI.
• Sukhbold, T., Ertl, T., Woosley, S. E., Brown, J. M., and Janka, H.-T. (2016). “Core-collapse Supernovae from 9 to 120 Solar Masses Based on Neutrino-powered Explosions.” The Astrophysical Journal 821, 38. DOI; model archive.