Pair-Instability and Pulsational Supernovae

Every star is a truce between two enormous numbers — the inward pull of gravity and the outward push of the pressure it makes by burning fuel. For most stars the truce is comfortable, because compressing an ordinary gas heats it and makes it push back harder. But in the most massive stars the pressure is supplied not by particles but by photons, and a pure radiation gas pushes back with an adiabatic index of exactly $\Gamma_1 = 4/3$ — the precise value below which a self-gravitating sphere can no longer defend itself. Such a core is not stable but marginally stable, balanced on a knife-edge, and any process that softens the gas even slightly tips it into dynamical collapse. The softening process is electron-positron pair production. When a radiation-dominated core reaches $T \sim 10^9\,$ K at modest density, the hot tail of the blackbody spectrum begins to convert photons into rest-mass pairs at the threshold $2 m_e c^2 = 1.022\,$ MeV. Energy that would have raised the pressure is diverted into making rest mass, the effective index dips below $4/3$ across a wide region of the core, and gravity wins. The star contracts on the free-fall time, heating until oxygen and silicon ignite explosively — and unlike an iron core, oxygen releases energy when it burns. For the most massive cores that release exceeds the entire gravitational binding energy of the star, blowing it completely apart and leaving no remnant at all: a pair-instability supernova. For somewhat lighter cores the burn only partly reverses the collapse, and the star pulsates, shedding shells before finally forming a black hole. This frontier physics matters far beyond stellar death because it sculpts the black-hole mass function: fully disrupted stars leave nothing, and pulsational mass loss caps what lighter stars can leave, carving a predicted "upper mass gap" between roughly $50$ and $130\,M_\odot$ — a sharp prediction now tested directly by the masses of black holes seen merging in gravitational waves.

Key takeaways

Radiation-dominated cores of very massive stars ( $M_{\mathrm{ZAMS}} \gtrsim 100\,M_\odot$ , helium cores $M_{\mathrm{He}} \gtrsim 30\,M_\odot$ ) sit at the marginal-stability index $\Gamma_1 = 4/3$ , barely able to resist gravity.
Near $T \sim 10^9\,$ K electron-positron pair production drains energy into rest mass, dropping the effective $\Gamma_1$ below $4/3$ over much of the core and triggering dynamical contraction.
Contraction ignites explosive oxygen and silicon burning; because O and Si release nuclear energy, the burn can exceed the star's binding energy and unbind it entirely — a pair-instability supernova (PISN) that leaves no compact remnant.
The fate is set by helium-core mass: $30$ – $60\,M_\odot$ pulsates (PPISN) and forms a black hole; $60$ – $130\,M_\odot$ is fully disrupted, synthesizing up to $\sim 40\,M_\odot$ of $^{56}\mathrm{Ni}$ ; above $\sim 130\,M_\odot$ photodisintegration wins and the core collapses directly to a black hole.
The instability carves an upper black-hole mass gap of roughly $50$ – $130\,M_\odot$ , a prediction now probed by LIGO-Virgo-KAGRA binary-black-hole masses.
Key idea: a radiation-dominated stellar core hovers at the marginal-stability threshold $\Gamma_1 = 4/3$ ; copious electron-positron pair production near $T\sim10^9$ K pushes $\Gamma_1$ below this value, triggering dynamical contraction, explosive nuclear burning, and either complete disruption (PISN) or violent pulsations (PPISN).

Where this lives today

For half a century the pair instability was untestable — a prediction about stars too rare and too distant to catch. Gravitational-wave astronomy changed that. Because full-disruption PISNe leave no remnant and pulsational mass loss trims the black holes from lighter cores, theory predicts a dearth of black holes between roughly $50$ and $130\,M_\odot$ , and LIGO-Virgo-KAGRA now weighs merging black holes by the dozen. The observed population piles up near $35\,M_\odot$ , close to where PPISN mass loss is expected to bank them, and the sharpness of that edge constrains the still-uncertain $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ rate that sets the core's carbon-to-oxygen ratio. Yet GW190521 merged two black holes of which at least one, near $85\,M_\odot$ , sits squarely inside the forbidden gap — hinting that hierarchical mergers in dense clusters can populate a region single stars cannot reach. On the electromagnetic side, PISNe are a most-wanted target of high-redshift surveys. Metal-free Population III stars should reach the extreme masses PISNe require, so a genuine pair-instability supernova would be a direct signature of the first stellar generation; the James Webb Space Telescope hunts these superluminous, months-long transients at the edge of the observable Universe. Stellar archaeologists complement this by searching the most metal-poor Galactic stars for the distinctive PISN fingerprint — a strong odd-even abundance pattern with essentially nothing beyond the iron peak. The microphysics of a $1\,$ MeV photon making a pair has become, improbably, a probe of the first light in the Universe.

The physical picture

Picture the core as a spring holding up the weight of the star. Compress an ordinary gas and it heats, pushing back harder — that restoring stiffness, measured by the adiabatic index $\Gamma_1$ , is stability. A self-gravitating sphere needs $\Gamma_1 > 4/3$ for a genuine restoring push; at exactly $4/3$ the internal and gravitational energies scale identically with radius, so the spring has no preferred length and is only marginally stable. A pure radiation gas sits at precisely this value — the peril of very massive stars, whose cores are held up mostly by photon pressure and so balance on the edge before anything goes wrong. What goes wrong is pair production. As the core heats toward $10^9\,$ K, the most energetic photons in the Planck tail exceed $2m_ec^2 = 1.022\,$ MeV and convert into electron-positron pairs. Creating rest mass is an energy sink: on compression, part of the heat that should have raised the temperature — and hence $P_{\mathrm{rad}} \propto T^4$ — is spent making pairs instead. The pressure rises less than adiabatically, the effective index dips below $4/3$ over a wedge of the temperature-density plane, and the marginal spring goes soft exactly when the star leans on it hardest. The core does not settle gently; it contracts on the free-fall time.

Figure

Where Γ₁ dips below 4/3: the pair-instability region in the log T – log ρ plane. Above kT ≈ mₑc² (T ≈ 6×10⁹ K) pairs form copiously from the 2mₑc² = 1.022 MeV threshold; in the wedge Γ₁ < 4/3 and the core loses pressure support, while a 100 M☉ helium-core center tracks straight into it

The dramatic twist is what halts the fall: nuclear fuel. As the core shrinks it heats, and the oxygen and silicon left from earlier burning stages ignite explosively — the opposite of an iron core, which absorbs energy when it photodisintegrates, whereas oxygen fusing to the iron peak releases about $0.7\,$ MeV per nucleon. If enough burns, the release overwhelms the star's gravitational binding and the entire object is detonated, leaving no remnant behind. Whether the star is fully disrupted or merely pulsates is one competition: nuclear energy released versus binding energy. Heavier cores are bound too tightly for one burn to unbind them, so they pulse, shedding mass each cycle before finally collapsing to a black hole. The second figure is the consequence astronomers actually measure — the map from helium-core mass to the object left behind, and the gap it opens.

Figure

The punchline: compact-remnant mass versus helium-core mass. Black-hole mass grows with core mass, is capped near ≈45 M☉ by pulsational mass loss, drops to zero across the full-disruption PISN band 60–130 M☉, then jumps back for direct collapse above 130 M☉ — leaving an upper black-hole mass gap of roughly 50–130 M☉

The core equations

The whole phenomenon turns on one inequality: a self-gravitating sphere is dynamically stable only if its pressure-weighted adiabatic index exceeds $4/3$ . The index measures how stiffly pressure responds to an adiabatic squeeze; below the critical value the restoring push is too weak and contraction runs away.

\Gamma_1 \equiv \left(\frac{\partial \ln P}{\partial \ln \rho}\right)_{\!S}, \qquad \Gamma_1 < \frac{4}{3} \;\Rightarrow\; \text{dynamical instability} \qquad (1)

where: $\Gamma_1$ — first adiabatic index (pressure response to adiabatic compression); $P$ — pressure; $\rho$ — mass density; $S$ — specific entropy, held constant along the adiabat. Read it as: squeeze the gas by one per cent in density; if the pressure rises by more than $4/3$ per cent, the star recovers; if by less, it falls. A pure radiation gas obeys $P_{\mathrm{rad}} = \tfrac13 aT^4$ and $\rho \propto T^3$ along an adiabat, so its pressure scales as $\rho^{4/3}$ — its index is exactly the critical value. This is why radiation-dominated cores are dangerous: they start life at the boundary, not safely above it.

P_{\mathrm{rad}} = \frac{1}{3} a T^4, \qquad \Gamma_1 = \frac{4}{3} \;\; (\text{pure radiation}) \qquad (2)

where: $a = 7.566\times10^{-16}\,\mathrm{J\,m^{-3}\,K^{-4}}$ — radiation constant; $T$ — temperature; $P_{\mathrm{rad}}$ — radiation pressure. A pure photon gas is marginal by construction. Real cores are mixtures of gas and radiation, and the exact index — the Chandrasekhar formula — interpolates between the ideal-gas value $5/3$ and the radiation value $4/3$ as the gas fraction $\beta$ falls, showing how a radiation-dominated core creeps toward the cliff.

\Gamma_1 = \beta + \frac{(4-3\beta)^2 (\Gamma_{\mathrm{gas}}-1)}{\beta + 12(\Gamma_{\mathrm{gas}}-1)(1-\beta)}, \qquad \beta \equiv \frac{P_{\mathrm{gas}}}{P} \qquad (3)

where: $\beta$ — ratio of gas pressure to total pressure ( $0\le\beta\le1$ ); $\Gamma_{\mathrm{gas}}$ — adiabatic index of the gas component ( $5/3$ for an ideal monatomic gas); $P$ — total pressure $P_{\mathrm{gas}}+P_{\mathrm{rad}}$ . At $\beta=1$ it returns $5/3$ ; at $\beta\to0$ it returns $4/3$ ; at $\beta=0.1$ it gives $1.350$ — a margin of only $0.017$ above the cliff. The softening agent is pair production, switched on when the photon bath contains photons above the pair threshold. Since the mean photon energy is $\sim k_B T$ , pairs become common once $k_B T$ approaches $m_e c^2$ .

E_\gamma \gtrsim 2 m_e c^2 = 1.022\ \mathrm{MeV} \;\Leftrightarrow\; k_B T \gtrsim m_e c^2, \quad T \gtrsim \frac{m_e c^2}{k_B} \approx 6\times10^{9}\ \mathrm{K} \qquad (4)

where: $m_e c^2 = 0.511\,$ MeV — electron rest energy; $k_B$ — Boltzmann constant; $E_\gamma$ — photon energy. The nominal threshold is $6\times10^9\,$ K, but the exponential Planck tail seeds significant pairs already at $T\sim10^9\,$ K — exactly the O/Si-burning regime. Below the threshold the pair abundance follows a Saha-like Boltzmann factor, exponentially sensitive to temperature — which is why the softening switches on sharply over a narrow wedge rather than drifting on.

n_{\pm} \propto (m_e T)^{3/2} \, e^{-m_e c^2 / k_B T} \quad (k_B T \ll m_e c^2) \qquad (5)

where: $n_{\pm}$ — number density of electron-positron pairs; $T$ — temperature. The steep factor makes $\mathrm{d}n_\pm/\mathrm{d}\ln T$ enormous near the knee, so the effective index plunges through $4/3$ . Once the core contracts and reignites, the outcome is decided by an energy inequality: full disruption occurs when the explosive nuclear yield exceeds the gravitational binding energy of the core.

E_{\mathrm{nuc}} \gtrsim |E_{\mathrm{bind}}| \approx \frac{3}{5}\,\alpha\,\frac{GM^2}{R}\;\Rightarrow\; \text{full disruption} \qquad (6)

where: $E_{\mathrm{nuc}}$ — total nuclear energy released in explosive O/Si burning; $E_{\mathrm{bind}}$ — gravitational binding energy of the core; $\alpha$ — structure factor of order unity ( $\alpha\approx1$ for uniform density); $M,R$ — mass and radius of the helium core. Because binding grows as $M^2/R$ , heavier cores are harder to unbind — the origin of the pulsational regime. The nuclear side of that inequality is dominated by oxygen fusing to the iron group, with a fixed specific yield that turns a burned mass directly into an energy.

q_{\mathrm{O}} = 6.7\times10^{17}\ \mathrm{erg\,g^{-1}}, \qquad E_{\mathrm{nuc}} \sim q_{\mathrm{O}}\, M_{\mathrm{O}} \qquad (7)

where: $q_{\mathrm{O}}$ — specific energy from burning $^{16}$ O to iron-group ( $\approx 0.7\,$ MeV/nucleon); $M_{\mathrm{O}}$ — mass of oxygen consumed. Ten solar masses of oxygen liberate $\sim1.3\times10^{52}\,$ erg, ten times the kinetic energy of an ordinary supernova. Finally, the light we see from a disruption is powered not by the explosion itself but by the radioactive decay of the freshly minted nickel, whose two-stage decay chain sets the shape and duration of the light curve.

L_{\mathrm{peak}} \propto M_{^{56}\mathrm{Ni}}\, \frac{\epsilon_{\mathrm{Ni}}}{\tau_{\mathrm{Ni}}} e^{-t/\tau_{\mathrm{Ni}}}, \quad \tau_{^{56}\mathrm{Ni}} = 8.8\ \mathrm{d}, \;\; \tau_{^{56}\mathrm{Co}} = 111\ \mathrm{d} \qquad (8)

where: $M_{^{56}\mathrm{Ni}}$ — synthesized nickel mass (up to $\sim40\,M_\odot$ in PISNe); $\tau_{\mathrm{Ni}}$ — e-folding time of $^{56}$ Ni $\to{}^{56}$ Co; $\tau_{\mathrm{Co}}$ — e-folding time of $^{56}$ Co $\to{}^{56}$ Fe. The huge nickel mass and the long cobalt tail are why PISNe can shine as superluminous transients for many months.

Derivation AThe four-thirds stability threshold from an energy argument

The aim is to prove from first principles why $\Gamma_1 = 4/3$ is the exact boundary between a star that springs back when squeezed and one that collapses. The cleanest tool is the total energy as a function of a single scale, the radius: write it as gravitational plus internal, express each as a power of $R$ under a homologous compression $R\to\lambda R$ , and ask whether the energy has a genuine minimum (a restoring equilibrium) or slides monotonically toward smaller $R$ (runaway collapse). The exponent comparison delivers the magic number.

Step 1. For a polytrope of index $n$ , the gravitational energy is $E_{\mathrm{grav}} = -\tfrac{3}{5-n}\,GM^2/R \equiv -k_1/R$ , with $k_1>0$ absorbing the structure factor. It scales as $R^{-1}$ : squeeze the star and gravity releases energy $\propto 1/R$ . Why: gravity always favours contraction, so it can only be resisted by an internal term that grows faster on compression.

Step 2. For a gas obeying $P\propto\rho^{\Gamma_1}$ the specific internal energy is $u = P/[\rho(\Gamma_1-1)]$ ; integrating over the star with $\rho\propto R^{-3}$ gives $E_{\mathrm{int}} = k_2\,R^{3(1-\Gamma_1)}$ with $k_2>0$ . Why: this is the only place the equation of state enters — the whole stability question is encoded in the single exponent $3(1-\Gamma_1)$ .

Step 3. The total energy is therefore $E(R) = -k_1/R + k_2 R^{3(1-\Gamma_1)} = R^{-1}\big[-k_1 + k_2 R^{\,4-3\Gamma_1}\big]$ . Equilibrium sits at the extremum $\mathrm{d}E/\mathrm{d}R = 0$ ; stability requires it to be a minimum, $\mathrm{d}^2E/\mathrm{d}R^2>0$ . Why: a minimum restores perturbations either way; anything shallower lets the star fall downhill by contracting.

Step 4. The internal term overtakes gravity on compression only if its exponent is positive, $4-3\Gamma_1>0$ , i.e. $\Gamma_1>4/3$ . At exactly $\Gamma_1=4/3$ the bracket becomes a constant and $E(R)\propto 1/R$ — no extremum at all, only marginal stability. Below $4/3$ , $E(R)$ decreases monotonically as $R$ shrinks. Why: this is the entire result — the sign of $4-3\Gamma_1$ is the sign of stability, and it flips at $4/3$ .

Step 5. Physically, when pair production drops $\Gamma_1$ below $4/3$ across a large fraction of the core, the star lowers its total energy by contracting, so it accelerates inward on the free-fall time $t_{\mathrm{ff}}\sim(G\rho)^{-1/2}$ rather than oscillating about equilibrium. Why: a system that releases energy by moving one way has no reason to stop — the instability is dynamical, not a gentle readjustment.

Go deeper · Derivation B · 2 minPair production as an effective adiabatic-index sink

Derivation BPair production as an effective adiabatic-index sink

Here the goal is to show quantitatively why pair creation is the process that softens the equation of state, and to locate the region of the temperature-density plane where it does the most damage. The strategy is to follow the energy budget of a radiation-dominated gas compressed adiabatically at a temperature where pairs form, tracking where the heat goes. When some of it is diverted into the rest mass of new pairs instead of raising the temperature, the pressure response weakens and the effective index falls below $4/3$ by an amount $\Delta$ we can map on the $(\log T,\log\rho)$ plane.

Step 1. Start from the radiation-only baseline: $P=\tfrac13 aT^4$ , internal energy density $u=aT^4$ , and hence $\Gamma_1=4/3$ exactly. A pure photon gas is marginal, with no restoring surplus. Why: this fixes the reference index the pair term will subtract from — the core is already at the edge before pairs appear.

Step 2. Add a pair channel. On compression the temperature tries to rise, but a share of the added heat is spent creating pairs, each costing the rest energy $2m_ec^2$ . The temperature — and therefore $P_{\mathrm{rad}}\propto T^4$ — rises by less than it would adiabatically. Why: pressure in a radiation-dominated gas is a steep function of $T$ , so any suppression of the temperature rise directly suppresses the pressure response.

Step 3. The reduced response is captured by writing $(\partial\ln P/\partial\ln\rho)_S = \tfrac43 - \Delta(T,\rho)$ , where $\Delta>0$ wherever pairs are actively created. The size of $\Delta$ tracks $\mathrm{d}n_\pm/\mathrm{d}\ln T$ , which by equation (5) peaks sharply at the pair-creation knee near $T\sim1$ – $3\times10^9\,$ K. Why: the effective-index deficit $\Delta$ is largest where the pair number is changing fastest with temperature — that is where the energy sink is deepest.

Step 4. The deficit $\Delta$ is largest at high $T$ and low $\rho$ , so that the pairs are non-degenerate and free to be created. This maps to a wedge in the $(\log T,\log\rho)$ plane — the green region of the first figure — and larger helium cores have lower central density at a given temperature, so they drive deeper into it. Why: this makes the instability a mass threshold — only cores massive enough to reach low-ρ, high-T conditions enter the danger region.

Step 5. When enough of the core has $\Gamma_1^{\mathrm{eff}} = \tfrac43 - \Delta < \tfrac43$ , the pressure-weighted mean index drops below $4/3$ and, by the previous derivation, the core contracts dynamically — heating until explosive oxygen and silicon burning intervenes to decide its fate. Why: global stability is set by the pressure-weighted average, not any single shell — it triggers once the softened region dominates.

Go deeper · Derivation C · 2 minEnergy balance — disruption versus pulsation

Derivation CEnergy balance — disruption versus pulsation

The purpose here is to estimate, from a single energy comparison, where the boundary lies between a star blown completely apart and one that survives to pulsate and eventually collapse. The method weighs the explosive nuclear yield released when the contracting core reignites its oxygen and silicon against the gravitational binding energy holding the core together. Because the yield scales with the fuel burned while binding scales faster, as $M^2/R$ , the balance tips with core mass — and following it across the mass range reproduces the full sequence of fates from ordinary collapse through pulsation, full disruption, and direct collapse.

Step 1. Estimate the nuclear yield: $E_{\mathrm{nuc}} \approx q_{\mathrm{O}} M_{\mathrm{burn}}$ with $q_{\mathrm{O}}\approx6.7\times10^{17}\,\mathrm{erg\,g^{-1}}$ . Burning $M_{\mathrm{burn}}\sim10\,M_\odot\approx2\times10^{34}\,$ g of oxygen liberates $E_{\mathrm{nuc}}\sim1.3\times10^{52}\,$ erg. Why: this is the energy budget available to reverse the collapse — everything downstream is a comparison against it.

Step 2. Estimate the binding energy: $|E_{\mathrm{bind}}|\approx\tfrac35 GM_{\mathrm{He}}^2/R_{\mathrm{He}} \propto M^2/R \propto M^{5/3}\rho_c^{1/3}$ . It grows faster than linearly with mass, so heavier cores are disproportionately harder to unbind. Why: this super-linear scaling is why the outcome depends so sharply on core mass.

Step 3. Balance them. For $M_{\mathrm{He}}\lesssim60\,M_\odot$ the released $E_{\mathrm{nuc}}$ only marginally exceeds $|E_{\mathrm{bind}}|$ per burn: the core bounces, ejects a shell, cools, and re-contracts, repeating until a final collapse — the pulsational pair-instability supernova (PPISN). Why: a burn that is only marginally super-binding cannot disperse the whole star at once, so it vents in pulses instead.

Step 4. For $60\lesssim M_{\mathrm{He}}\lesssim130\,M_\odot$ a single explosive burn releases $\sim10^{52}$ – $10^{53}\,$ erg, comfortably exceeding the binding energy and unbinding the entire star, leaving no remnant — the full-disruption PISN. Why: once the yield clears the binding energy with margin, the shock unbinds every shell before it can fall back, so nothing is left.

Step 5. For $M_{\mathrm{He}}\gtrsim130\,M_\odot$ the post-contraction temperatures climb so high that photodisintegration of the newly formed heavy nuclei ( $\gamma+\mathrm{nucleus}\to$ lighter fragments) absorbs the nuclear energy, so the contraction is never reversed and the core collapses directly to a black hole of $\gtrsim130\,M_\odot$ . Why: photodisintegration is endothermic — above this mass the energy that would have exploded the star is swallowed instead, closing the mass gap from above.

Limiting cases

The fate of a very massive star is governed almost entirely by its helium-core mass at the end of central helium burning — the four branches of the second figure, read from left to right:

Case	Result	Meaning
No instability	$M_{\mathrm{He}} \lesssim 30\,M_\odot$	Cores stay too dense and cool for significant pair production; the star ends as an ordinary core-collapse supernova or collapses to a black hole of ordinary mass.
Pulsational PISN	$30\,M_\odot \lesssim M_{\mathrm{He}} \lesssim 60\,M_\odot$	Partial burning reverses collapse; the star ejects mass in one or more pulses, then collapses to a black hole. Caps the black-hole mass near $\sim 40$ – $50\,M_\odot$ .
Full disruption (PISN)	$60\,M_\odot \lesssim M_{\mathrm{He}} \lesssim 130\,M_\odot$	Explosive O/Si burning unbinds the entire star; no remnant. Ejects up to $\sim 40\,M_\odot$ of $^{56}$ Ni for the most massive cases, powering superluminous transients.
Direct collapse	$M_{\mathrm{He}} \gtrsim 130\,M_\odot$	Temperatures rise so high that photodisintegration absorbs the nuclear release; the core collapses directly to a very massive black hole ( $\gtrsim 130\,M_\odot$ ), reopening the black-hole spectrum above the gap.

Common trapThe pair instability is just another name for core-collapse

It is not — and conflating the two gets the physics backwards. In an ordinary core-collapse supernova the softening at $\Gamma_1<4/3$ comes from photodisintegration of iron and electron capture in a degenerate, iron-rich core; the collapse proceeds all the way to nuclear density and leaves a neutron star or black hole, with the visible explosion driven by a neutrino-heated bounce shock. The pair instability strikes a non-degenerate, oxygen/silicon-rich core that has not yet made iron. Its contraction is reversed by explosive burning of that O and Si — which releases energy, whereas iron photodisintegration absorbs it — and for the right mass range the star is blown completely apart, leaving nothing at all. The two share only the symptom, $\Gamma_1$ dipping below $4/3$ ; the cause, the fuel, and the outcome are opposites.

Worked examples

1 · Can burning oxygen unbind the core? (fully worked)

A contracting helium core of $M_{\mathrm{He}}=30\,M_\odot$ and radius $R_{\mathrm{He}}\approx10^{10}\,$ cm reignites and burns $10\,M_\odot$ of oxygen to the iron group. Compare the nuclear energy released with the gravitational binding energy, and decide whether the star is unbound.

Step 1. Nuclear yield. With $q_{\mathrm{O}}=6.7\times10^{17}\,\mathrm{erg\,g^{-1}}$ and $M_{\mathrm{burn}}=10\,M_\odot=10\times1.989\times10^{33}=1.989\times10^{34}\,$ g, equation (7) gives $E_{\mathrm{nuc}} = 6.7\times10^{17}\times1.989\times10^{34} = \mathbf{1.3\times10^{52}\ erg}$ . This already dwarfs the $\sim10^{51}\,$ erg kinetic energy of a normal supernova.

Step 2. Binding energy. Using equation (6) with $\alpha\approx1$ , $M_{\mathrm{He}}=30\,M_\odot=5.97\times10^{34}\,$ g and $R_{\mathrm{He}}=10^{10}\,$ cm: $|E_{\mathrm{bind}}| = \tfrac35\,G M^2/R = 0.6\times(6.674\times10^{-8})\times(5.97\times10^{34})^2 / 10^{10}$ . Evaluating, $G M^2 = 6.674\times10^{-8}\times3.56\times10^{69} = 2.38\times10^{62}$ , so $|E_{\mathrm{bind}}| = 0.6\times2.38\times10^{62}/10^{10} = \mathbf{1.4\times10^{52}\ erg}$ .

Step 3. Compare. The two energies are within a factor of order unity: $E_{\mathrm{nuc}}\approx1.3\times10^{52}\,$ erg versus $|E_{\mathrm{bind}}|\approx1.4\times10^{52}\,$ erg. A $30\,M_\odot$ core therefore sits right at the pulsational boundary — a single burn barely fails to unbind it, so it vents a shell and re-contracts. The burn is marginal: this core pulsates. For a larger core that burns more oxygen while binding grows only as $M^{5/3}$ , $E_{\mathrm{nuc}}$ pulls decisively ahead — which is why the $60$ – $130\,M_\odot$ range is fully disrupted with no remnant.

2 · How marginal is a radiation-dominated core?

A very massive stellar core is radiation-dominated with a gas-pressure fraction $\beta=0.10$ and an ideal monatomic gas component ( $\Gamma_{\mathrm{gas}}=5/3$ ). Compute its adiabatic index and comment on its stability margin.

Step 1. Insert into the Chandrasekhar mixture formula (3). The numerator is $(4-3\beta)^2(\Gamma_{\mathrm{gas}}-1) = (4-0.3)^2\times\tfrac23 = (3.7)^2\times0.6667 = 13.69\times0.6667 = 9.13$ . The denominator is $\beta + 12(\Gamma_{\mathrm{gas}}-1)(1-\beta) = 0.10 + 12\times0.6667\times0.90 = 0.10 + 7.20 = 7.30$ .

Step 2. Combine: $\Gamma_1 = \beta + 9.13/7.30 = 0.10 + 1.250 = \mathbf{1.350}$ . The critical value is $4/3 = 1.333$ , so the margin is only $\Delta\Gamma_1 = 1.350 - 1.333 = \mathbf{0.017}$ — barely one per cent above the cliff.

Step 3. Interpret. Such a core is stable, but only just: a pair-production deficit of $\Delta\gtrsim0.017$ over enough of the core is all it takes to drive the pressure-weighted index below $4/3$ and trigger dynamical contraction. For comparison, a gas-dominated core ( $\beta=1$ ) would have $\Gamma_1=5/3=1.667$ , an enormous $0.33$ margin — utterly safe. The margin collapses from $0.33$ to $0.017$ as the core becomes radiation-dominated, which is precisely why only the most massive stars are vulnerable.

Physics of the portrait

The pair-instability supernova is not one person's discovery but a sixty-year chain of theorists chasing what happens when a star is simply too big — from the first hint that pairs could destabilize a core to the realization that they carve a gap in the black-hole spectrum.

Origins · 1 minpair-instability supernova and how this was discovered

pair-instability supernova

the Γ₁ = 4/3 instability of very massive radiation-dominated cores · the thermonuclear deaths of the most massive stars

The pair-instability supernova is a theorist's object: predicted in detail decades before any candidate was seen. The seed was planted in 1964 by William Fowler and Fred Hoyle, who realized while cataloguing the equation of state of hot stellar matter that copious pair creation near $10^9\,$ K would soften the adiabatic index of a radiation-dominated core and destabilize very massive stars. The physics was made rigorous in 1967 by Zalman Barkat, Giora Rakavy and Nathan Sack, who showed the softening was strong enough to drive genuine dynamical collapse; that same year Rakavy and Gideon Shaviv mapped the unstable region onto the temperature-density plane, giving the wedge still drawn in every textbook. In 1968 Gary Fraley computed the first explicit explosion, showing that for the right masses the reignition of oxygen and silicon does not merely halt the collapse but unbinds the entire star. For decades this remained a beautiful prediction in search of a star. The modern era belongs to Alexander Heger and Stan Woosley, whose 2002 survey computed PISN nucleosynthesis and pinned the fiducial helium-core boundaries, and to Woosley's 2017 treatment of the pulsational regime, which tied the whole picture to the black-hole mass gap now tested by gravitational waves. What began as an oddity in a table of opacities became a prediction about the census of black holes in the Universe.

History

1964

— Fowler and Hoyle identify electron-positron pair creation as a stellar instability

1967

— Barkat, Rakavy and Sack show pair production drives dynamical collapse of massive cores

1968

— Fraley computes the first pair-instability supernova explosion, unbinding the whole star

2002

— Heger and Woosley map PISN outcomes and nucleosynthesis across helium-core mass

2017

— Woosley develops the pulsational pair-instability supernova and the black-hole mass gap

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 10 & 13 (stellar stability, advanced burning, deaths of massive stars) · Kippenhahn, Weigert & Weiss, Stellar Structure and Evolution, 2nd ed. (adiabatic index, pair thermodynamics, the $\Gamma_1$ instability) · Fowler & Hoyle (1964), ApJS 9, 201 (original pair-instability treatment) · Barkat, Rakavy & Sack (1967), Phys. Rev. Lett. 18, 379 (dynamical pair instability) · Fraley (1968), Astrophys. Space Sci. 2, 96 (first PISN explosion model) · Heger & Woosley (2002), ApJ 567, 532 (PISN nucleosynthesis) · Woosley (2017), ApJ 836, 244 (pulsational pair-instability supernovae). The threshold $\Gamma_1=4/3$ is exact and structure-independent. The pair threshold $2m_ec^2=1.022\,$ MeV corresponds to $T\approx6\times10^9\,$ K (more precisely $5.93\times10^9\,$ K). The helium-core boundaries ( $\sim30,60,130\,M_\odot$ ), $^{56}$ Ni yields (up to $\sim40\,M_\odot$ ), decay times ( $\tau_{\mathrm{Ni}}=8.8\,$ d, $\tau_{\mathrm{Co}}=111\,$ d, derived from the $6.08$ -d and $77.2$ -d half-lives), and oxygen yield ( $\approx0.7\,$ MeV/nucleon $\approx6.7\times10^{17}\,$ erg/g) are the fiducial values used throughout. Key numerical checks: $\Gamma_1(\beta=0.1)=1.350$ ; burning $10\,M_\odot$ of oxygen releases $1.3\times10^{52}\,$ erg, to be compared with the $1.4\times10^{52}\,$ erg binding energy of a $30\,M_\odot$ helium core.

Prerequisites. Advanced Burning Stages in Massive Stars · Helium Burning and the Triple-Alpha Process · White Dwarfs and Electron Degeneracy Pressure · Mass-Luminosity Relation. Leads to. Asymptotic Giant Branch and Thermal Pulses · Advanced Burning Stages in Massive Stars.

Ask the tutor: Ask me to walk through why $\Gamma_1=4/3$ is the exact tipping point, or to trace a $120\,M_\odot$ star from the pair-instability wedge to the black-hole mass gap.

Why is 4/3 the magic number, and why does radiation give exactly that value?

It falls out of how the two energies scale with radius. Gravity scales as $1/R$ , while the internal energy of a gas with index $\Gamma_1$ scales as $R^{3(1-\Gamma_1)}$ ; the internal term stops being able to out-fight gravity on compression exactly when the exponent reaches $-1$ , i.e. $\Gamma_1=4/3$ . Radiation lands there because $P=\tfrac13 aT^4$ and $u=aT^4$ give $P\propto\rho^{4/3}$ along an adiabat. A radiation gas is thus marginally stable by nature — it has no restoring stiffness to spare, which is why radiation-dominated cores are the ones that fall.

If the mean photon energy at 10⁹ K is far below 1.022 MeV, how do pairs form at all?

Through the exponential tail of the blackbody spectrum. At $T\sim10^9\,$ K the mean photon energy $k_BT\approx0.086\,$ MeV is more than ten times below the $1.022\,$ MeV threshold, but the Planck tail still holds a small fraction of photons above it. Because the pair density carries a factor $e^{-m_ec^2/k_BT}$ , it is fiercely sensitive to temperature — a modest rise multiplies the pair population enormously. The core needs not most of its photons above threshold but only enough pairs, made fast enough, to drain the pressure response over a wide region, and the exponential sensitivity makes the softening switch on abruptly once the knee is reached.

How is this different from an ordinary core-collapse supernova?

Both begin with $\Gamma_1$ dropping below $4/3$ , but for opposite reasons and with opposite endings. Core-collapse happens in a degenerate iron core: iron cannot release energy by burning, so once photodisintegration and electron capture soften it, the collapse runs to nuclear density and forms a neutron star or black hole via a neutrino-heated bounce. The pair instability strikes a non-degenerate, oxygen-rich core that has not made iron yet; its collapse is halted — and for the right masses reversed — by explosive O/Si burning, which releases energy. The extreme outcome has no analogue in core-collapse: the whole star can be unbound, leaving no remnant at all.

Why does a heavier core pulsate instead of exploding outright?

Because binding energy grows faster with mass than the nuclear yield does. The energy released scales roughly as the oxygen burned, $E_{\mathrm{nuc}}\propto M_{\mathrm{burn}}$ , but the binding energy scales as $M^2/R\propto M^{5/3}$ . As the core mass climbs, binding outpaces the available release, so one burn can no longer disperse the whole star: the reignition only ejects an outer shell and halts the collapse, and the remaining core cools, re-contracts, and repeats, shedding a little mass each pulse. Only after several cycles, or once the fuel is spent, does it finally collapse to a black hole — capping the mass pulsational progenitors leave at the lower edge of the gap.

What actually makes a pair-instability supernova so luminous?

Radioactive nickel, in enormous quantity. The burn that unbinds the star can synthesize up to $\sim40\,M_\odot$ of $^{56}$ Ni — ten to a hundred times what an ordinary supernova makes. That nickel decays to $^{56}$ Co ( $\tau=8.8\,$ d) and then to stable $^{56}$ Fe ( $\tau=111\,$ d), and the gamma rays and positrons keep the expanding ejecta hot for months. Since peak luminosity scales with nickel mass, a PISN can outshine a normal supernova by orders of magnitude and stay bright far longer, producing the superluminous, slowly fading light curves that surveys hunt as signatures of the most massive stars.

Does the mass gap definitely exist, and what would break the prediction?

The gap is a robust prediction of single-star evolution, but not inviolable. Its lower edge depends on the $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ rate, wind mass loss, rotation and mixing — uncertainties that shift the boundaries by tens of solar masses. More fundamentally, the prediction concerns black holes from isolated single stars: black holes built by other channels — hierarchical mergers in dense clusters, disk accretion, or exotic Population III remnants — can land inside the gap without contradicting the physics. GW190521, with a component near $85\,M_\odot$ , is exactly such a case. So the gap is a statement about one formation pathway; a black hole inside it does not falsify the pair instability but shows nature has more than one way to build a heavy black hole.