Pair Instability and Proposed Mass Gaps

At sufficiently high entropy, a massive oxygen-rich core can convert energetic photons into electron–positron pairs. The resulting loss of pressure responsiveness destabilizes the core, triggers contraction, and can ignite explosive oxygen burning. That mechanism is robust. The frequently quoted lower and upper edges of a “pair-instability mass gap” are not: they depend on nuclear rates, mixing, winds, rotation, envelope treatment, dimensionality, and how an observed mass is connected to a birth mass. The scientifically useful statement is therefore a conditional sequence of outcomes, not a universal empty interval.

Helpful background. Black-Hole Geometry and Mechanics supplies the black-hole mass and spin conventions; Relativistic Matter and Compact Objects supplies equations of state and radial stability; Black-Hole Astrophysics supplies the distinction between a birth spectrum and a detected spectrum; and Black-Hole Dynamics and Collapse supplies the nonlinear collapse endpoint beyond the stellar-instability argument developed here.

Why pairs soften the core

For an approximately hydrostatic, radiation-dominated core, radial stability is controlled by the first adiabatic exponent

$$\Gamma_1=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{s}.$$

The Newtonian $4/3$ threshold captures the essential competition: pressure rises fast enough under compression when an appropriately averaged $\Gamma_1>4/3$, while a sufficiently extended region with $\Gamma_1<4/3$ is dynamically unstable. Near temperatures of order $10^9$ K, photons in the high-energy tail can make pairs. Rest-mass energy then absorbs energy that would otherwise increase radiation pressure, lowering $\Gamma_1$. Contraction heats the core until rapid oxygen burning releases nuclear energy.

The threshold can be understood with a homologous compression, in which every radius changes by the same factor at fixed mass. Density scales as $\rho\propto R^{-3}$ and an adiabatic pressure response as $P\propto R^{-3\Gamma_1}$. The pressure needed to oppose self-gravity scales as $P_{\rm grav}\sim GM^2/R^4$. If $\Gamma_1\lt4/3$, compression increases the actual pressure more slowly than the required pressure, encouraging further contraction. This argument explains the feedback; a stratified star requires the radial stability problem, not a test of $\Gamma_1$ at one grid point. Pair creation redistributes energy into rest mass; it does not simply remove all the photons or make the pressure vanish.

The outcome depends on whether this energy reverses collapse and on how strongly the star remains bound. Heger and Woosley follow the thermonuclear instability through disruption and collapse regimes in Heger and Woosley 2002, §§ 2–4. The $4/3$ argument identifies the mechanism, but it does not itself compute a remnant mass: general-relativistic corrections, composition, neutrino losses, photodisintegration, and hydrodynamic burning all matter near the boundaries.

Pulses, disruption, and renewed collapse

In the pulsational pair-instability regime, explosive burning halts contraction without unbinding the whole star. One or several pulses eject mass; the surviving core later resumes evolution and may collapse to a black hole. The pulse history matters because it controls both the retained mass and the circumstellar material through which a later explosion or outflow propagates.

At larger eligible core masses, a pair-instability supernova can release enough thermonuclear energy to disrupt the star, leaving no compact remnant. At still larger masses, photodisintegration and neutrino losses can absorb energy rapidly enough that collapse resumes rather than producing complete disruption. The ordering—pulses, complete disruption, then renewed collapse—is more durable than any numerical boundary attached to it. Woosley gives an explicit set of helium-star sequences, pulse histories, and remnant outcomes in Woosley 2017, § 3.1, Tables 1–3, and Figs. 1, 4–5.

Helium-core mass is usually a better organizing coordinate than zero-age main-sequence mass because winds and binary stripping can map many initial stars to similar cores. Even helium-core mass is not sufficient near an edge: the carbon-to-oxygen ratio, entropy profile, envelope pressure, rotation, and reaction network modify ignition and pulse energetics.

Why proposed gap edges move

Nuclear physics supplies a clear example. Changing the $^{12}{\rm C}(\alpha,\gamma)^{16}{\rm O}$ rate changes the carbon abundance after helium burning, hence the later burning structure and maximum black-hole mass below disruption. Farmer and collaborators quantify this sensitivity across stellar grids in Farmer et al. 2019, § 5.4 and Fig. 4(d). Convective-boundary mixing, wind mass loss, and numerical pulse handling introduce additional shifts.

There is also an astrophysical distinction between a birth gap and an observed gap. Hierarchical mergers can place a merger remnant inside a region avoided by first-generation stellar collapse. Stellar collisions, uncertain envelope retention, and later accretion provide other possible routes. Measurement uncertainty and detector selection then reshape the detected distribution. A black hole inside a proposed interval would test formation channels; it would not falsify pair production.

Results and limitations through 2026-08-18

The microphysical softening and the existence of pulse/disruption regimes are durable results. Exact edges, the degree of depletion, and the mix of objects that may populate the interval remain unresolved; competing values depend on the stellar parameter regime and source population.

Position or evidence	Result	Scope and limitation
Equation-of-state thermodynamics	Pair creation can lower $\Gamma_1$ and destabilize a hot, radiation-supported core	Robust local mechanism; does not determine a global outcome alone
Hydrodynamic stellar grids	Recover pulsations, complete disruption, and renewed-collapse regimes	Boundaries depend on reactions, mixing, rotation, winds, and grid controls
Nuclear-rate variations	Move pulse strengths and the maximum first-generation remnant mass	Laboratory constraints improve the input but do not remove stellar-model systematics
Compact-binary populations	Can test a depleted birth spectrum and identify candidates above it	Inference depends on selection, measurement uncertainty, and alternative channels such as hierarchical mergers

This treatment should be reviewed after a materially revised helium-burning rate changes published edge estimates, after independent multidimensional pulse calculations establish a different retained-mass relation, or after a selection-corrected compact-object population decisively distinguishes a stellar-birth cutoff from hierarchical or collisional gap filling.

Reading a mass-gap claim

A defensible claim states which mass is plotted—helium-core, pre-collapse, birth black-hole, detector-frame merger component, source-frame merger component after a declared redshift conversion, or merger remnant—and which stellar and binary assumptions generated it. It distinguishes an interval with a reduced formation probability from one with exactly zero probability. It also tests whether the same model reproduces lower-mass remnants and the rate normalization, rather than tuning only an edge.

This page stops before detailed nuclear networks, multidimensional pulse hydrodynamics, and gravitational-wave hierarchical likelihoods. Its role is to define the mechanism and the assumptions that must accompany a boundary. Pair instability is a causal explanation for structure in a birth spectrum; “the gap” is not an object with fixed endpoints independent of a model.

Common pitfalls

Using initial mass as the instability coordinate. Mass loss and stripping make the relation between initial mass and oxygen-core structure many-to-one and metallicity dependent.

Treating one massive detection as a microphysics verdict. The object may be a later-generation merger, and its inferred mass has a posterior distribution. Formation history and selection must be tested jointly.

Exercise

A model enters a phase in which a substantial oxygen-rich core has $\Gamma_1=1.31$. Explain the immediate dynamical tendency, the feedback that can reverse it, and why this information does not determine the final black-hole mass.

Solution

Because $1.31<4/3\simeq1.333$, pressure responds too weakly to compression for the idealized radiation-supported configuration, so contraction accelerates. Compression raises the temperature and can ignite explosive oxygen burning; nuclear energy may halt the contraction, launch a pulse, or unbind the star. The final remnant still depends on the mass participating in the unstable region, binding energy, composition, nuclear rates, pulse ejection, neutrino losses, and whether photodisintegration ultimately favors renewed collapse. Thus $\Gamma_1<4/3$ diagnoses instability, not a unique mass outcome.

References

• Farmer, R., Renzo, M., de Mink, S. E., Marchant, P., and Justham, S. (2019). “Mind the Gap: The Location of the Lower Edge of the Pair-instability Supernova Black Hole Mass Gap.” The Astrophysical Journal 887, 53. DOI.
• Heger, A., and Woosley, S. E. (2002). “The Nucleosynthetic Signature of Population III.” The Astrophysical Journal 567, 532–543. DOI.
• Woosley, S. E. (2017). “Pulsational Pair-instability Supernovae.” The Astrophysical Journal 836, 244. DOI.

Further reading

• Mapelli, M. (2021). “Formation Channels of Single and Binary Stellar-Mass Black Holes.” In Handbook of Gravitational Wave Astronomy. Open PDF.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• The Pair-Instability Mass-Gap Question