GRB Fireball Model and Afterglows

A gamma-ray burst is the brightest electromagnetic event the universe knows how to make: for a few seconds one burst can outshine every other source of MeV photons in the observable sky combined. The clue that unlocks the whole phenomenon is a contradiction. The bursts flicker on millisecond timescales, so the emitting region is small — light-travel time says no larger than $c\,\delta t$ , a few thousand kilometres. Pack the inferred luminosity into that volume and the photon field is so dense it should be utterly opaque to itself: two MeV photons meeting head-on make an electron–positron pair, and the resulting optical depth is of order $10^{12}$ . A trapped, thermalized fireball cannot produce the hard non-thermal spectra we actually see. Something is wrong with the naive picture. The resolution is the same relativistic beaming that governs blazars and the launching of relativistic jets: the emitting material is streaming toward us with a bulk Lorentz factor $\Gamma\gtrsim100$ . Motion this fast rescues the source twice over — it enlarges the true emitting radius by $\Gamma^2$ and redshifts the comoving photon energies below the pair threshold — and it organizes everything that follows. Radiation pressure drives a baryon-loaded plasma to relativistic speed; the flow coasts at a Lorentz factor fixed by how much mass it had to drag; internal collisions inside the ejecta make the erratic prompt $\gamma$ -rays; and when the ejecta ram the surrounding gas, a decelerating external shock lights up the smoothly fading, broadband afterglow whose spectral and temporal breaks encode the jet's energy, geometry, and microphysics. This article is the synthesis of that chain — the fireball model and its afterglows.

Key takeaways

Gamma-ray bursts release $10^{51}$ – $10^{54}\,\mathrm{erg}$ in seconds; the isotropic-equivalent luminosity so vastly exceeds the Eddington limit that only bulk relativistic motion with $\Gamma\gtrsim100$ can produce the observed non-thermal spectrum.
The compactness problem — a photon field dense enough to explain the luminosity should be opaque to $\gamma\gamma\to e^+e^-$ — is solved because relativistic beaming enlarges the true source by $\Gamma^2$ and drops comoving photon energies by $1/\Gamma$ below threshold.
A radiation-dominated fireball accelerates as $\Gamma\propto r$ until it becomes matter-dominated, coasting at $\Gamma\to\eta=E/(M_0c^2)$ ; internal shocks between shells of differing $\Gamma$ make the highly variable prompt emission.
External shocks form as the ejecta sweep up circumburst gas; the decelerating forward shock powers the smoothly fading afterglow through synchrotron from shock-accelerated electrons.
The afterglow spectrum is a four-segment broken power law joined at $\nu_a<\nu_m<\nu_c$ , and the light curve steepens achromatically at the jet break when $1/\Gamma$ exceeds the jet half-angle $\theta_j$ , revealing collimation and the true energy near $10^{51}\,\mathrm{erg}$ .
Long-soft bursts trace collapsars with Type Ic supernovae; short-hard bursts trace compact-object mergers, confirmed by GW170817 and GRB 170817A.
Key idea: ultrarelativistic bulk motion ( $\Gamma\gtrsim100$ ) resolves the compactness problem and dictates the entire fireball narrative — radiation-driven acceleration to a coasting Lorentz factor $\eta$ , prompt emission from internal shocks, and a synchrotron afterglow from the decelerating external forward shock whose spectral and temporal breaks encode the microphysics and collimation of the jet.

The core insight

Ultrarelativistic bulk motion ( $\Gamma\gtrsim100$ ) resolves the compactness problem and sets the entire fireball narrative: radiation-driven acceleration to a coasting Lorentz factor $\eta$ , prompt emission from internal shocks, and a synchrotron afterglow from the decelerating external forward shock whose spectral and temporal breaks encode the microphysics and collimation of the jet.

Where this lives today

Gamma-ray burst physics is now a working multimessenger enterprise. NASA's Swift observatory slews to a fresh burst within a minute, catching the X-ray afterglow while it is still decelerating, and the Fermi Gamma-ray Space Telescope maps the prompt spectrum from keV to GeV, testing whether internal shocks, magnetic reconnection, or photospheric emission dominates — a debate the simple fireball model has not fully settled. The landmark of the decade was GW170817: on 17 August 2017 the LIGO–Virgo network detected gravitational waves from a binary neutron-star merger, and $1.7\,\mathrm{s}$ later Fermi and INTEGRAL recorded the short burst GRB 170817A. That coincidence nailed compact-object mergers as short-GRB engines, pinned the speed of gravity to that of light to a part in $10^{15}$ , and — because we viewed the jet off-axis — produced an afterglow that brightened over months as the decelerating, structured jet's beaming cone widened into our line of sight, a live demonstration of the $1/\Gamma$ geometry that underlies the jet break. Looking ahead, because afterglows are luminous enough to be seen to $z\sim9$ , GRBs probe the reionization epoch and the earliest massive stars; missions like SVOM and the proposed THESEUS aim to use high-redshift bursts as beacons into the cosmic dark ages, and every new burst still tightens the microphysical parameters $\epsilon_e$ , $\epsilon_B$ and $p$ that remain our best empirical grip on collisionless relativistic shocks.

The physical picture

Picture a bomb made almost entirely of radiation, with only a pinch of baryons mixed in. The photons cannot escape — the region is wildly optically thick — so their pressure does the only thing it can and pushes the baryons outward. As the plasma expands, thermal energy is steadily traded for ordered bulk motion, exactly like a rocket that carries its own exhaust. The Lorentz factor climbs in lock-step with radius, $\Gamma\propto r$ , until the thermal reservoir is spent; from then on there is nothing left to accelerate the flow and it coasts at a terminal value set entirely by how much mass it had to carry, $\Gamma\to\eta=E/(M_0c^2)$ . Load in too many baryons and $\eta$ falls below $\sim100$ , the compactness constraint is violated, and no clean burst emerges — the notorious baryon-pollution problem. The first figure tracks this history on a single axis.

Figure

The fireball in one diagram the bulk Lorentz factor Γ climbs as Γ ∝ r through the radiation-driven phase, saturates at the coasting value Γ = η ≈ 300 fixed by baryon loading, then falls as Γ ∝ R^−3/2 (observer-time Γ ∝ t^−3/8) beyond the deceleration radius R_dec ≈ 3×10¹⁶ cm where the external shock ignites the afterglow

The compactness paradox dissolves through two relativistic effects acting at once. First, a photon we detect at $1\,\mathrm{MeV}$ was emitted at only $\sim1\,\mathrm{MeV}/\Gamma$ in the comoving frame, well below the pair-production threshold, so far fewer photons can actually make pairs. Second, the variability time we measure is compressed by $\Gamma^2$ — the shell nearly keeps pace with its own light — so the real emitting radius is $\Gamma^2$ larger than the naive $c\,\delta t$ , diluting the photon density by a colossal factor. Together these turn an opacity of $10^{12}$ into something below unity for $\Gamma\gtrsim100$ . Prompt versus afterglow is the distinction between internal and external shocks. Internal shocks are collisions inside the ejecta: an unsteady engine launches shells with a spread of Lorentz factors, fast shells catch slow ones, and the mildly relativistic pile-ups dissipate kinetic energy into shock-accelerated electrons that radiate. Because the collisions remember the engine's stutter, the prompt light curve is spiky and irreproducible. The external shock is the whole ejecta plowing into the circumburst medium — a single decelerating blast wave — so its emission is smooth and predictable, a broken power law in both frequency and time. That afterglow spectrum, worked out by Sari, Piran and Narayan, is the second figure.

Figure

The afterglow synchrotron spectrum in slow cooling a four-segment broken power law that rises as ν² below the self-absorption frequency ν_a, as ν^1/3 up to the injection peak ν_m, then falls as ν^−(p−1)/2 = ν^−0.65 and above the cooling break ν_c as ν^−p/2 = ν^−1.15, for an electron index p = 2.3

The jet break is geometry made visible. Early on, relativistic beaming lets us see only a patch of the outflow of angular size $1/\Gamma$ , smaller than the jet's true opening angle, so the source "looks" spherical. As the blast decelerates, $1/\Gamma$ grows; the instant it exceeds the jet edge $\theta_j$ we suddenly "run out of jet," lateral spreading sets in, the flux drops faster at every wavelength at once, and that achromatic steepening betrays the collimation — and lets us weigh the true energy.

The core equations

Everything begins with the raw energetics. The measured flux and distance fix an isotropic-equivalent luminosity, the number that first announces how extreme these sources are.

L_{\rm iso} = 4\pi d_L^2\, F \sim 10^{51}\text{–}10^{53}\,\mathrm{erg\,s^{-1}} \qquad (1)

where: $L_{\rm iso}$ — isotropic-equivalent luminosity (what a spherical source of the same flux would emit); $d_L$ — luminosity distance; $F$ — observed energy flux. This is only an upper book-keeping figure: real bursts are beamed, so the true power is far smaller, but $L_{\rm iso}$ is what the compactness argument must confront. The compactness constraint is the heart of the model. Estimate the optical depth to $\gamma\gamma\to e^+e^-$ for a source of luminosity $L$ varying on timescale $\delta t$ , allowing bulk motion at $\Gamma$ .

\tau_{\gamma\gamma} \approx \frac{\sigma_T\, f_p\, L\, \delta t}{4\pi\, R^2\, m_e c^3}\,\frac{1}{\Gamma^{4+2\beta}} \ll 1, \qquad R\lesssim c\,\delta t\,\Gamma^2 \qquad (2)

where: $\tau_{\gamma\gamma}$ — pair-production optical depth; $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ — Thomson cross section; $f_p$ — fraction of photons above the pair threshold; $\delta t$ — observed variability time; $R$ — emitting radius, boosted by $\Gamma^2$ ; $\beta$ — high-energy photon spectral slope. Reading it: without motion ( $\Gamma=1$ , $R=c\,\delta t$ ) the depth is $\sim10^{12}$ ; the steep $\Gamma^{-(4+2\beta)}$ suppression is what forces $\Gamma\gtrsim100$ so that $\tau_{\gamma\gamma}<1$ and the hard non-thermal spectrum can escape. Once transparent, the flow's terminal speed is set by baryon loading. The deposited energy $E$ shared among rest mass $M_0$ defines a dimensionless entropy $\eta$ , and the coasting Lorentz factor cannot exceed it.

\eta=\frac{E}{M_0 c^2},\qquad \Gamma_{\rm coast}\simeq\min(\eta,\ \Gamma_{\rm max}) \qquad (3)

where: $\eta$ — baryon-loading / entropy parameter; $E$ — total energy in the fireball; $M_0$ — entrained baryon rest mass; $\Gamma_{\rm max}$ — a ceiling set if the flow is still radiation-dominated at the photosphere. Reading it: clean bursts need $\eta\gtrsim100$ ; heavier baryon loading (smaller $\eta$ ) chokes the outflow — the baryon-pollution limit. The prompt emission is timed by where unequal shells collide. Two shells launched a time $\delta t$ apart with Lorentz factors $\Gamma_r>\Gamma_s$ merge at a characteristic internal-shock radius.

\Gamma_{\rm int}\sim\sqrt{\Gamma_r\Gamma_s},\qquad R_{\rm int}\sim c\,\delta t\,\Gamma_{\rm int}^2 \qquad (4)

where: $\Gamma_r,\Gamma_s$ — Lorentz factors of the fast and slow shells; $\Gamma_{\rm int}$ — Lorentz factor of the merged shell; $R_{\rm int}$ — collision radius, typically $10^{13}$ – $10^{14}\,\mathrm{cm}$ . Because $R_{\rm int}\propto\delta t$ , the prompt light curve inherits the engine's variability directly — this is why $\gamma$ -ray light curves are erratic while afterglows are smooth. The afterglow begins where the ejecta have swept up enough external mass to feel drag. Equating the swept-up momentum-energy to the ejecta energy fixes the deceleration radius.

R_{\rm dec}\approx\left(\frac{3E}{4\pi\, n\, m_p c^2\, \eta^2}\right)^{1/3} \qquad (5)

where: $R_{\rm dec}$ — radius at which the blast has transferred order-unity of its energy to the medium; $n$ — circumburst number density; $m_p$ — proton mass; $\eta$ — coasting Lorentz factor. Reading it: for canonical numbers $R_{\rm dec}\sim10^{16}$ – $10^{17}\,\mathrm{cm}$ , reached only seconds to minutes after the burst in observer time — the afterglow onset. The afterglow radiation is synchrotron from a power-law population of shock-accelerated electrons, and in the slow-cooling regime its spectrum is four power-law segments joined at three break frequencies.

F_\nu\propto\begin{cases}\nu^{2} & \nu<\nu_a\\ \nu^{1/3} & \nu_a<\nu<\nu_m\\ \nu^{-(p-1)/2} & \nu_m<\nu<\nu_c\\ \nu^{-p/2} & \nu>\nu_c\end{cases} \qquad (6)

where: $\nu_a$ — synchrotron self-absorption frequency; $\nu_m$ — injection frequency of the minimum-energy electrons; $\nu_c$ — cooling frequency, above which electrons radiate away their energy within a dynamical time; $p$ — electron energy index, $N(\gamma)\propto\gamma^{-p}$ , typically $\approx2.3$ . Reading it: measure two slopes and you overdetermine $p$ , the signature that the underlying emitter really is shock-accelerated synchrotron. The normalizations of those breaks depend on how shock energy is partitioned, encoded in two microphysical fractions.

U_e=\epsilon_e\,U_{\rm sh},\qquad U_B=\epsilon_B\,U_{\rm sh} \qquad (7)

where: $\epsilon_e$ — fraction of post-shock energy given to relativistic electrons (empirically $\sim0.1$ ); $\epsilon_B$ — fraction stored in magnetic field ( $\sim10^{-2}$ – $10^{-4}$ ); $U_{\rm sh}$ — total post-shock energy density. These are among the very few observational handles physics has on collisionless relativistic-shock particle acceleration. The achromatic light-curve steepening dates the jet break, and the break time inverts to a jet half-opening angle.

\theta_j\approx0.161\left(\frac{t_{j,\rm day}}{1+z}\right)^{3/8}\!\left(\frac{n\,\eta_\gamma}{E_{\rm iso,52}}\right)^{1/8}\ \mathrm{rad} \qquad (8)

where: $\theta_j$ — jet half-angle; $t_{j,\rm day}$ — jet-break time in days; $z$ — redshift; $\eta_\gamma$ — radiative efficiency; $E_{\rm iso,52}$ — isotropic energy in units of $10^{52}\,\mathrm{erg}$ . Reading it: typical breaks at a day give $\theta_j\sim3^\circ$ – $10^\circ$ , so the jet subtends a percent or so of the sky as seen from the engine. Knowing $\theta_j$ converts the misleading isotropic energy into the true, collimation-corrected release.

E_\gamma\approx(1-\cos\theta_j)\,E_{\rm iso}\approx\tfrac12\theta_j^2\,E_{\rm iso} \qquad (9)

where: $E_\gamma$ — beaming-corrected energy, which clusters near $10^{51}\,\mathrm{erg}$ across bursts spanning three decades in $E_{\rm iso}$ . Reading it: the $\tfrac12\theta_j^2$ factor is typically $10^{-2}$ – $10^{-3}$ , collapsing an apparent four-decade spread into a near-standard energy — the empirical basis of GRBs as would-be standard candles.

Derivation AWhy the fireball accelerates as Γ ∝ r

The goal is to show that a radiation-dominated shell with a fixed amount of entrained baryon mass must convert its internal energy into bulk kinetic energy so efficiently that the Lorentz factor grows in direct proportion to radius, and that this growth halts precisely when the flow becomes matter-dominated, freezing $\Gamma$ at the baryon-loading value $\eta=E/(M_0c^2)$ . We track lab-frame energy conservation together with the adiabatic cooling of a relativistic gas and read off the scaling.

Step 1. Write the conserved lab-frame energy of a thin shell as $E=\Gamma(M_0c^2+E'_{\rm int})=\text{const}$ , where $E'_{\rm int}$ is the comoving internal (thermal) energy. Why: the engine deposits a fixed total energy once; nothing external adds or removes energy as the shell flies out, so this product is a genuine constant of the motion.

Step 2. For a radiation-dominated gas the comoving thermal energy cools adiabatically as the shell inflates, $E'_{\rm int}\propto V'^{-1/3}$ with comoving volume $V'\propto r^2(r/\Gamma)$ . Why: a relativistic gas has adiabatic index $4/3$ , so pressure work drains internal energy as the volume grows — the thermal reservoir is being spent to do work on the flow.

Step 3. While the shell is still radiation-dominated, $E'_{\rm int}\gg M_0c^2$ , so $E\approx\Gamma E'_{\rm int}$ ; holding $E$ fixed while $E'_{\rm int}$ falls forces $\Gamma$ to rise, and carrying the thin-shell scalings through gives $\Gamma\propto r$ . Why: this is the rocket-carrying-its-own-exhaust picture made quantitative — every erg of thermal energy lost reappears as bulk kinetic energy, and linear growth is what keeps the product constant.

Step 4. Acceleration cannot continue past the point where bulk kinetic energy exhausts the budget: when $\Gamma M_0c^2\to E$ the thermal term is gone and $\Gamma$ saturates at $\eta=E/(M_0c^2)$ , reached at the saturation radius $r_s\approx\eta\,r_0$ for an initial size $r_0$ . Why: once matter-dominated there is no thermal pressure left to push against, so the flow coasts — the terminal speed is dictated purely by how much mass had to be carried.

Go deeper · Derivation B · 1 minThe compactness problem and the Γ ≳ 100 bound

Derivation BThe compactness problem and the Γ ≳ 100 bound

The goal is to demonstrate that a static source with the observed luminosity and millisecond variability would be optically thick to photon–photon pair production by roughly twelve orders of magnitude, and that admitting bulk relativistic motion drives the opacity down as a steep power of $\Gamma$ , so that transparency to the observed non-thermal MeV spectrum demands $\Gamma\gtrsim100$ . We build the static estimate first and then apply the two relativistic corrections.

Step 1. From millisecond variability the naive size is $R_0\sim c\,\delta t\sim3\times10^{8}(\delta t/1\,\mathrm{ms})\,\mathrm{cm}$ , and the photon number density follows from the luminosity as $n_\gamma\sim L/(4\pi R_0^2 c\,E_\gamma)$ . Why: causality caps a static source at the light-crossing size, and dividing the local energy density $L/4\pi R_0^2c$ by the mean photon energy $E_\gamma$ counts the photons available to collide.

Step 2. The static pair-production depth is $\tau_{\gamma\gamma}\sim\sigma_T f_p n_\gamma R_0$ , and inserting $L\sim10^{51}\,\mathrm{erg\,s^{-1}}$ , $\delta t\sim10\,\mathrm{ms}$ , MeV photons gives $\tau_{\gamma\gamma}\sim10^{12}$ . Why: any two MeV photons exceed the $2m_ec^2$ threshold, so the cross section is near Thomson and the enormous $n_\gamma$ makes the medium wildly opaque — flatly inconsistent with the hard spectra actually detected.

Step 3. Now let the source move at $\Gamma$ : the true radius is boosted to $R\sim c\,\delta t\,\Gamma^2$ , cutting $n_\gamma\propto R^{-2}$ and hence $\tau$ by $\Gamma^4$ ; and observed photons of energy $E_\gamma$ were emitted at $E_\gamma/\Gamma$ , so the fraction above threshold shrinks by $\sim\Gamma^{2\beta}$ with $\beta$ the high-energy slope. Why: beaming attacks the opacity on both fronts at once — it enlarges the arena and softens the ammunition — and the two effects multiply.

Step 4. Combining, $\tau_{\gamma\gamma}\propto\Gamma^{-(4+2\beta)}$ ; demanding $\tau_{\gamma\gamma}<1$ starting from $\sim10^{12}$ with $\beta\sim1$ – $2.5$ requires $\Gamma\gtrsim100$ . Why: this single inequality is the fireball model's founding measurement — the mere existence of a non-thermal MeV spectrum forces ultrarelativistic bulk motion, independent of any emission mechanism.

Go deeper · Derivation C · 1 minBlast-wave deceleration and the afterglow decay index

Derivation CBlast-wave deceleration and the afterglow decay index

The goal is to derive, from energy conservation for an adiabatic relativistic blast wave sweeping a constant-density medium, the self-similar deceleration law $\Gamma\propto t^{-3/8}$ , and to convert it into the observable temporal decay of the afterglow flux, $F_\nu\propto t^{-3(p-1)/4}$ for frequencies between the injection and cooling breaks. This is the Blandford–McKee scaling in the form that yields the closure relations tested against real light curves.

Step 1. The energy of the swept-up shell is $E\sim\Gamma^2 M_{\rm sw}c^2=\Gamma^2\cdot\tfrac43\pi R^3 n\,m_p c^2$ , held constant in the adiabatic phase. Why: each swept-up proton is heated to random energy $\sim\Gamma m_pc^2$ in the shock frame and carries bulk factor $\Gamma$ , so the blast energy scales as $\Gamma^2$ times the rest mass it has gathered.

Step 2. Relativistic time compression relates observer time to radius: a shell at $\Gamma$ nearly keeps pace with its own photons, so $t\sim R/(c\Gamma^2)$ , i.e. $R\propto\Gamma^2 t$ . Why: the observer clock runs $\Gamma^2$ times faster than the radial-expansion clock, the same beaming factor that shrank the prompt variability, now governing when afterglow light arrives.

Step 3. Energy constancy gives $\Gamma^2 R^3=\text{const}$ ; substituting $R\propto\Gamma^2 t$ yields $\Gamma^8 t^3=\text{const}$ , hence $\Gamma\propto t^{-3/8}$ and $R\propto t^{1/4}$ . Why: eliminating $R$ between the conservation law and the timing relation leaves a pure power law — the self-similar signature that the blast has forgotten its initial conditions.

Step 4. Following how $\nu_m\propto\Gamma^4$ , $\nu_c\propto\Gamma^{-4}$ and the peak flux track $\Gamma(t)$ gives, for $\nu_m<\nu<\nu_c$ , the closure relation $F_\nu\propto t^{-3(p-1)/4}$ . Why: this ties a purely temporal measurement (the decay index) to a purely spectral one (the electron index $p$ ), and their agreement is the sharpest confirmation that the afterglow is a decelerating synchrotron blast wave.

Limiting cases

The fireball passes through a fixed sequence of dynamical regimes, each with its own scaling; a wind circumburst profile and the post-jet-break phase add two more.

Case	Result	Meaning
Radiation-dominated acceleration	$\Gamma(r)\propto r\quad(r<r_s)$	Below the saturation radius $r_s\approx\eta r_0$ , internal energy converts to bulk motion and $\Gamma$ grows linearly with radius.
Coasting (matter-dominated)	$\Gamma\to\eta=E/(M_0c^2)$	Once thermal energy is spent, the flow coasts at the terminal Lorentz factor fixed by baryon loading.
Deceleration, uniform ISM	$\Gamma\propto t^{-3/8},\ r\propto t^{1/4}$	Blandford–McKee self-similar phase for a constant-density medium; drives the smooth afterglow decline.
Deceleration, wind medium	$\rho\propto r^{-2},\ \Gamma\propto t^{-1/4}$	A massive-star progenitor wind steepens the density profile, changing the closure relations and light-curve slopes.
Post jet-break	$F_\nu\propto t^{-p}$	After $1/\Gamma\gtrsim\theta_j$ , edge effects and lateral spreading steepen the decay at all frequencies to roughly the electron index $p$ .

Common trapThe isotropic energy is the real energy, and δt gives the source size directly

Both halves of this are wrong, and both are classic errors. The isotropic-equivalent energy $E_{\rm iso}$ assumes the burst radiates into $4\pi$ ; because GRBs are collimated into a cone of half-angle $\theta_j$ , the true energy is $E_\gamma\approx\tfrac12\theta_j^2E_{\rm iso}$ — two to three orders of magnitude smaller, and reporting $E_{\rm iso}$ as the energy budget overstates the engine by that factor. Equally, the observed variability $\delta t$ does not map to a source size as $R\sim c\,\delta t$ : for a shell moving at $\Gamma$ the emitting region is $R\lesssim c\,\delta t\,\Gamma^2$ , larger by $\sim10^4$ . Forgetting that $\Gamma^2$ enhancement is exactly what manufactures the false compactness paradox in the first place.

Worked examples

1 · The compactness bound from a burst's own numbers (fully worked)

A burst has $L_{\rm iso}\approx10^{51}\,\mathrm{erg\,s^{-1}}$ , varies on $\delta t\approx10\,\mathrm{ms}$ , and radiates $\sim1\,\mathrm{MeV}$ photons. Show a static source is opaque, and find the minimum Lorentz factor.

Step 1. Static size: $R_0\sim c\,\delta t=(3\times10^{10})(10^{-2})=\mathbf{3\times10^{8}\,cm}$ . A few thousand kilometres — smaller than the Sun.

Step 2. Photon density: $n_\gamma\sim L/(4\pi R_0^2 c\,E_\gamma)$ with $E_\gamma=1\,\mathrm{MeV}=1.6\times10^{-6}\,\mathrm{erg}$ . Numerically $n_\gamma=10^{51}/[4\pi(3\times10^8)^2(3\times10^{10})(1.6\times10^{-6})]=\mathbf{1.8\times10^{28}\,cm^{-3}}$ .

Step 3. Static optical depth: $\tau_{\gamma\gamma}\sim\sigma_T f_p n_\gamma R_0$ . With $f_p\approx1$ , $\tau_{\gamma\gamma}=(6.65\times10^{-25})(1.8\times10^{28})(3\times10^8)\approx\mathbf{4\times10^{12}}$ . The source is opaque by twelve orders of magnitude — no non-thermal MeV spectrum could escape.

Step 4. Rescue by motion: $\tau_{\gamma\gamma}\propto\Gamma^{-(4+2\beta)}$ . Taking a high-energy slope $\beta\approx1.25$ (exponent $6.5$ ), the requirement $\tau_{\gamma\gamma}<1$ gives $\Gamma\gtrsim(4\times10^{12})^{1/6.5}\approx\mathbf{86}$ . Rounding for typical parameter spread, $\Gamma\gtrsim100$ — the canonical fireball bound, derived from nothing but the observed luminosity, variability and photon energy.

2 · Where the afterglow switches on (fully worked)

For $E=10^{52}\,\mathrm{erg}$ , coasting $\eta=\Gamma=300$ , in an ISM of $n=1\,\mathrm{cm^{-3}}$ , find the deceleration radius and the observer-frame afterglow-onset time.

Step 1. Deceleration radius from equation (5): $R_{\rm dec}=\left[3E/(4\pi n\,m_pc^2\,\eta^2)\right]^{1/3}$ with $m_pc^2=1.5\times10^{-3}\,\mathrm{erg}$ . The bracket is $3\times10^{52}/[4\pi(1)(1.5\times10^{-3})(9\times10^4)]=1.8\times10^{49}$ , so $R_{\rm dec}=\mathbf{2.6\times10^{16}\,cm}$ .

Step 2. Observer time: $t_{\rm dec}\sim R_{\rm dec}/(2c\Gamma^2)=(2.6\times10^{16})/[2(3\times10^{10})(9\times10^4)]\approx\mathbf{4.8\,s}$ .

Step 3. Interpretation: the afterglow does not wait for hours — for these numbers the external shock ignites only seconds after the trigger, so early-time X-ray and optical monitoring (Swift's rapid slew) catches the deceleration onset. Denser media or heavier ejecta (smaller $\eta$ ) push $R_{\rm dec}$ inward and $t_{\rm dec}$ earlier; a tenuous medium delays onset to minutes. $R_{\rm dec}\approx2.6\times10^{16}\,cm$ , $t_{\rm dec}\approx5\,s$ .

3 · Weighing the true energy through the jet break (fully worked)

A burst at $z=1$ with $E_{\rm iso}=10^{53}\,\mathrm{erg}$ shows an achromatic light-curve break at $t_j=1\,\mathrm{day}$ . Take $n=1\,\mathrm{cm^{-3}}$ and radiative efficiency $\eta_\gamma=0.2$ . Find $\theta_j$ and the collimation-corrected energy.

Step 1. Jet angle from equation (8): $\theta_j=0.161(1/2)^{3/8}(1\cdot0.2/10)^{1/8}=0.161(0.771)(0.613)=\mathbf{0.076\,rad}=\mathbf{4.4^\circ}$ .

Step 2. Beaming fraction: $f_b=\tfrac12\theta_j^2=\tfrac12(0.076)^2=2.9\times10^{-3}$ , i.e. the jet fills only $1/345$ of the sky.

Step 3. True energy: $E_\gamma=f_b\,E_{\rm iso}=(2.9\times10^{-3})(10^{53})=\mathbf{2.9\times10^{50}\,erg}$ . An apparent $10^{53}\,\mathrm{erg}$ monster is really a few $\times10^{50}\,\mathrm{erg}$ event — squarely in the Frail cluster near $10^{51}\,\mathrm{erg}$ , and comfortably below the energy a stellar-mass engine can plausibly supply. $\theta_j\approx4.4^\circ$ , $E_\gamma\approx3\times10^{50}\,erg$ .

Physics of the portrait

The fireball model was not the work of one hand, but if a single figure sits at its centre it is Tsvi Piran, whose papers span the whole chain — from the baryon-loading limit that makes a fireball possible, to the afterglow synchrotron theory that made it testable, to the review that codified the field.

Origins · 1 minTsvi Piran and how this was discovered

Tsvi Piran

b. 1949 · relativistic astrophysics · gamma-ray burst dynamics and afterglow theory

Tsvi Piran, a theoretical astrophysicist at the Hebrew University of Jerusalem, turned gamma-ray bursts from a decades-old mystery into a quantitative branch of relativistic astrophysics. When bursts were discovered by the Vela satellites, no one knew whether they lay in the solar neighbourhood or at cosmological distances, and the "compactness problem" seemed to forbid any consistent source. In 1990, with Amos Shemi, Piran worked out how a radiation-dominated fireball accelerates and how baryon contamination limits its terminal Lorentz factor — the entropy parameter $\eta$ that decides whether a clean burst can form at all. With Ramesh Narayan and Bohdan Paczynski he argued in 1992 that neutron-star mergers could power the short bursts, a prediction spectacularly vindicated a quarter-century later by GW170817. His most-cited contribution came in 1998, when with Re'em Sari and Narayan he derived the synchrotron spectra and light curves of afterglows — the broken power law with breaks at $\nu_a$ , $\nu_m$ and $\nu_c$ and the closure relations linking temporal decay to spectral slope — giving observers a predictive template to fit. Piran's 1999 work with Sari and Halpern established the jet break as the signature of collimation, and his 2004 Reviews of Modern Physics article, "The physics of gamma-ray bursts," became the field's standard reference. His broader interests run from numerical relativity and the two-body problem to the r-process nucleosynthesis of heavy elements in merger ejecta — the same events that make short GRBs also forge much of the universe's gold.

History

1973

— Klebesadel, Strong and Olson announce the Vela-satellite discovery of gamma-ray bursts

1986

— Paczynski and Goodman argue that GRBs are cosmological radiation fireballs

1992

— Rees and Meszaros propose the internal- and external-shock scenario

1998

— Sari, Piran and Narayan derive the afterglow synchrotron spectra and closure relations

2017

— GW170817 and GRB 170817A confirm neutron-star mergers as short-GRB engines

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §17 (gamma-ray bursts) · Piran, T., "The physics of gamma-ray bursts," Rev. Mod. Phys. 76, 1143 (2004) · Sari, Piran and Narayan, "Spectra and Light Curves of GRB Afterglows," ApJ 497, L17 (1998) · Meszaros, P., "Gamma-Ray Bursts," Rep. Prog. Phys. 69, 2259 (2006) · Kumar and Zhang, "The physics of gamma-ray bursts and relativistic jets," Phys. Rep. 561, 1 (2015). Key numerical benchmarks: the static-source compactness $\tau_{\gamma\gamma}\sim4\times10^{12}$ from $L\sim10^{51}\,\mathrm{erg\,s^{-1}}$ , $\delta t\sim10\,\mathrm{ms}$ , MeV photons yields the $\Gamma\gtrsim100$ bound; the deceleration radius $R_{\rm dec}\approx2.6\times10^{16}\,\mathrm{cm}$ and onset $t_{\rm dec}\approx5\,\mathrm{s}$ follow for $E=10^{52}\,\mathrm{erg}$ , $n=1$ , $\eta=300$ ; the Blandford–McKee scaling gives $\Gamma\propto t^{-3/8}$ and $F_\nu\propto t^{-3(p-1)/4}$ ; and the jet-break worked example yields $\theta_j\approx4.4^\circ$ and $E_\gamma\approx3\times10^{50}\,\mathrm{erg}$ . The Frail coefficient $0.161$ and $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ are standard tabulated values.

Prerequisites. Relativistic Jets and Launching Mechanisms · Eddington Luminosity and Accretion Limits · Blazars and Relativistic Beaming · Black Hole Spin and the Ergosphere. Leads to. Supermassive Black Holes · Advection-Dominated Accretion Flows.

Ask the tutor: Ask me to walk through why a burst's millisecond flicker plus its MeV luminosity alone pin the outflow to $\Gamma\gtrsim100$ , or to trace a single afterglow break from $\nu_m$ down to the electron index $p$ .

Why doesn't the huge luminosity simply violate the Eddington limit?

The Eddington limit bounds the steady, spherical, radiation-supported luminosity of a self-gravitating object. A GRB satisfies none of those conditions: it is impulsive, collimated, and not in hydrostatic balance — the outflow is already unbound and moving relativistically. The relevant constraint is not Eddington but compactness, the requirement that the source be transparent to its own $\gamma$ -rays. Once you account for beaming and the collimation correction, the true energy is $\sim10^{51}\,\mathrm{erg}$ , released by a newly formed stellar-mass black hole or magnetar over seconds — energetically comparable to a supernova, just funnelled into a narrow cone and a hard spectrum rather than radiated isotropically over weeks.

What is the difference between the entropy η and the Lorentz factor Γ?

They are equal only at the end of acceleration. The dimensionless entropy $\eta=E/(M_0c^2)$ is a fixed property of the ejecta set at launch — the ratio of total energy to baryon rest-mass energy. The Lorentz factor $\Gamma$ is the instantaneous bulk motion: it starts near 1, climbs as $\Gamma\propto r$ while radiation pressure works, and saturates at $\Gamma_{\rm coast}\simeq\eta$ once the thermal energy is exhausted. So $\eta$ is the ceiling and $\Gamma$ is the flow's actual speed as it rises to meet that ceiling. During the later afterglow, $\Gamma$ falls again far below $\eta$ as the blast decelerates against the external medium.

Why are prompt light curves so erratic but afterglows so smooth?

Because they are made at different places by different mechanisms. Prompt $\gamma$ -rays come from internal shocks — collisions between shells the engine launched at different speeds — and the collision radius $R_{\rm int}\sim c\,\delta t\,\Gamma^2$ scales with the engine's own timing. Every stutter of the central engine is faithfully reprinted onto the light curve, producing the spiky, unrepeatable pulses. The afterglow, by contrast, is a single blast wave decelerating in the surrounding gas; it has forgotten the engine entirely and settled into a self-similar Blandford–McKee solution, so it fades as a smooth broken power law. Erratic versus smooth is the observational fingerprint of internal versus external shocks.

How do we actually measure a jet's opening angle if we can't resolve it?

We time the jet break. While the blast is fast, relativistic beaming shows us only a cone of half-angle $1/\Gamma$ around our line of sight; as long as that cone is narrower than the physical jet, we cannot tell the jet is collimated and the light curve fades at its normal rate. When deceleration grows $1/\Gamma$ to equal $\theta_j$ , two things happen together — we begin to "see" the jet's missing edges and the jet starts spreading sideways — and the flux drops faster at every wavelength at once. Dating that achromatic break gives $t_j$ , and equation (8) inverts it to $\theta_j$ . It is geometry read off a clock rather than an image.

Why is p ≈ 2.3, and why does that specific value matter?

$p$ is the power-law index of the shock-accelerated electron energy distribution, $N(\gamma)\propto\gamma^{-p}$ . First-order Fermi acceleration at relativistic collisionless shocks — particles gaining energy each time they cross the shock front — generically predicts $p\approx2.2$ – $2.3$ from both analytic theory and particle-in-cell simulations, and afterglow fits confirm it. The value matters because it propagates into every observable: the spectral slopes are $-(p-1)/2$ and $-p/2$ , and the temporal decays are $-3(p-1)/4$ and steeper. That the same $p$ recovered from a spectrum also predicts the light-curve slope through the closure relations is the strongest single check that the afterglow really is synchrotron from a Fermi-accelerated population.

How did GW170817 test the fireball and jet picture?

It confirmed the whole framework at once. The gravitational-wave chirp identified a binary neutron-star merger, and the short burst GRB 170817A $1.7\,\mathrm{s}$ later proved such mergers make short GRBs — the progenitor prediction from 1992. Because Earth lay outside the jet's cone, we initially saw a faint, off-axis burst; then the afterglow rose for about 150 days before turning over, exactly as expected when a decelerating structured jet's $1/\Gamma$ beaming cone widens until its bright core swings into view. That rising-then-falling light curve is the jet-break geometry run in reverse, and modelling it pinned the jet's half-angle and confirmed a relativistic, structured outflow rather than a spherical fireball.