Gamma-Ray Bursts

For a few seconds a gamma-ray burst can outshine every other source of gamma rays in the observable universe combined. A compact engine no larger than a city — a newborn black hole or a millisecond magnetar — dumps of order a solar rest-mass of energy into a region light-milliseconds across, and the trapped radiation blows outward as an ultrarelativistic fireball. First recognised in 1967 by the Vela nuclear-test-monitoring satellites and revealed as cosmological only in 1997, GRBs turned out to be the death-cries of massive stars and the electromagnetic counterparts of neutron-star mergers. This article is the flagship synthesis of high-energy astrophysics: it takes the accretion and jet-launching physics of the preceding chapter and asks what happens when the accretion rate is a solar mass per second and the jet is relativistic enough to beat its own opacity. Three ideas organise everything. First, compactness: the enormous luminosity packed into a millisecond variability timescale would be hopelessly opaque to its own gamma rays unless the source races toward us at bulk Lorentz factor Γ of hundreds — the single most model-independent fact about GRBs. Second, two-stage dissipation: internal collisions inside the jet make the prompt gamma-rays, and the jet's deceleration against the surrounding gas drives a fading broadband synchrotron afterglow. Third, collimation: the outflow is a narrow cone, so the true energy is far below the isotropic-equivalent value, and the afterglow "jet break" lets us measure the opening angle directly. Together these turn a flash of gamma rays into a quantitative probe of black-hole birth, relativistic shocks, heavy-element nucleosynthesis, and the high-redshift universe.

Key takeaways

Gamma-ray bursts are the most luminous electromagnetic explosions known, releasing $10^{51}$ – $10^{54}\,\mathrm{erg}$ of isotropic-equivalent energy in seconds, mostly as hard X-rays and gamma rays.
The duration distribution is bimodal: long bursts ( $T_{90}\gtrsim2\,\mathrm{s}$ ) come from the collapse of rapidly rotating massive stars (collapsars); short bursts ( $T_{90}\lesssim2\,\mathrm{s}$ ) come from compact-object mergers.
The compactness problem forces the emitting outflow to be ultrarelativistic, $\Gamma\sim100$ – $1000$ : only a relativistic jet is transparent to its own MeV photons.
Prompt gamma-rays arise from internal dissipation (internal shocks or magnetic reconnection); the afterglow is synchrotron from the external forward shock ploughing into circumstellar gas.
Jets are narrow, so the true energy $E_\gamma=f_b E_{\rm iso}$ with $f_b=1-\cos\theta_j\ll1$ ; the achromatic jet break in the afterglow measures $\theta_j$ .
GW170817/GRB 170817A tied a short burst to a binary-neutron-star merger, confirming the merger origin and the r-process kilonova link.
Key idea: a stellar-mass compact object launches an ultrarelativistic collimated jet whose internal dissipation powers the prompt gamma-rays and whose external shock powers the synchrotron afterglow — relativistic beaming solves the compactness problem and cuts the true energy far below the isotropic-equivalent value.

The core insight

A stellar-mass compact object — a black hole or magnetar born in a massive-star collapse or a compact-object merger — launches an ultrarelativistic collimated jet ( $\Gamma\sim100$ – $1000$ ); internal dissipation powers the prompt gamma-rays and the external shock powers the synchrotron afterglow, while relativistic beaming resolves the compactness problem and reduces the true energetics far below the isotropic-equivalent value.

Where this lives today

Gamma-ray bursts are now a working tool rather than a puzzle, and the frontier is multimessenger. NASA's Swift and Fermi observatories catch and localise bursts within seconds, triggering robotic optical telescopes worldwide and feeding a real-time alert stream that the entire time-domain community watches. The landmark event was GW170817: on 2017 August 17 LIGO and Virgo detected gravitational waves from a binary-neutron-star inspiral, and $\sim1.7\,\mathrm{s}$ later Fermi-GBM and INTEGRAL recorded the short burst GRB 170817A, with the kilonova AT2017gfo emerging over the following days. That single coincidence confirmed the merger origin of short bursts, revealed mergers as forges of r-process heavy elements, and pinned the speed of gravitational waves to within $\sim10^{-15}$ of $c$ . On the theoretical side, general-relativistic magnetohydrodynamic simulations now follow jets from the hyperaccreting engine through the collapsing star, testing Blandford–Znajek launching and structured-jet geometry against the off-axis afterglow of 170817. Open problems remain sharp: the true prompt-emission mechanism (internal shocks versus magnetic reconnection versus photospheres) is unsettled, the highest-redshift bursts ( $z\gtrsim8$ ) probe reionisation but demand rapid infrared spectroscopy, and next-generation missions aim to make every neutron-star merger in the local universe a joint gravitational-wave and gamma-ray target.

The physical picture

Imagine depositing a solar rest-mass of energy, $\sim10^{54}\,\mathrm{erg}$ , into a region smaller than the Sun in under a second. The plasma is so hot and dense that photons cannot escape; radiation pressure detonates it outward, and as it expands and thins the internal energy converts to bulk kinetic energy. The flow coasts at a Lorentz factor of hundreds — a "fireball" moving at $99.999\%$ of light speed. Because the emitting shells chase us at nearly $c$ , relativistic beaming compresses arrival times and blueshifts photons: a shell radiating at radius $R$ produces variability on the far shorter timescale $R/(2\Gamma^2 c)$ . This is why we see millisecond substructure from a source physically light-seconds across, and why the comoving photon density is low enough to escape catastrophic pair-production opacity. The energy release is two-staged, like a bullet fired down a rifle and then striking a wall. First, faster shells overtake slower ones deep inside the jet — internal shocks, or magnetic reconnection — thermalising their relative kinetic energy into the prompt gamma-rays at radii $\sim10^{13}$ – $10^{15}\,\mathrm{cm}$ . Later the whole jet slams into the surrounding gas at the deceleration radius $\sim10^{16}$ – $10^{17}\,\mathrm{cm}$ , sweeping it up and radiating the fading multiwavelength afterglow as shock-accelerated electrons spiral in the amplified magnetic field.

Figure

Anatomy of a gamma-ray burst: a hyperaccreting black hole launches an ultrarelativistic jet, Γ ≈ 100–1000; internal shell collisions at R ≈ 10¹³–10¹⁵ cm make the prompt γ-rays, and the external forward shock at R_dec ≈ 10¹⁶–10¹⁷ cm makes the synchrotron afterglow, with observed variability δt ≈ R/(2Γ²c)

The jet is a narrow cone, not a sphere, so we detect a burst only when the cone points near us — the observed population is a small, biased slice of all events, and the real energy per burst is far below the naive isotropic estimate. As the blast wave decelerates, its beaming cone of half-angle $\sim1/\Gamma$ widens; when it reaches the physical jet edge $\theta_j$ , the afterglow light curve steepens achromatically. That "jet break" is the fingerprint of collimation and the observable that turns $E_{\rm iso}$ into the true $E_\gamma$ .

Figure

The canonical afterglow light curve, log flux versus log observer time: after onset near t_dec the slow-cooling flux fades as t^−0.98 for electron index p = 2.3, then steepens achromatically at the jet break t_j ≈ 1 day to t^−2.3; the break fixes θ_j ≈ 0.1 rad and beaming f_b ≈ 0.005

The core equations

The first thing an observer computes is the energy. A burst of measured fluence $S_\gamma$ at luminosity distance $d_L$ and redshift $z$ has an isotropic-equivalent energy — the energy it would carry if it radiated equally in all directions. This is an upper bound on the real budget, not the real budget.

E_{\rm iso} = \frac{4\pi d_L^2}{1+z}\, S_\gamma \qquad (1)

where: $E_{\rm iso}$ — isotropic-equivalent gamma-ray energy; $d_L$ — luminosity distance; $z$ — redshift; $S_\gamma$ — observed gamma-ray fluence (energy per unit area). Reading: the $4\pi d_L^2$ spreads the fluence over a sphere, and the single $(1+z)^{-1}$ is the k-corrected conversion from the observed band to the rest frame. The real energy is smaller because the outflow is collimated into two opposing cones of half-angle $\theta_j$ . Only the fraction of the sphere those cones cover is actually illuminated, and that fraction is the beaming factor.

E_\gamma = f_b\, E_{\rm iso},\qquad f_b = 1-\cos\theta_j \qquad (2)

where: $E_\gamma$ — true collimation-corrected energy; $f_b$ — beaming factor (fraction of the sphere covered by the two jets); $\theta_j$ — jet half-opening angle. Reading: for a narrow jet $f_b\approx\tfrac12\theta_j^2$ , so a few-degree cone cuts the energy by $10^{-3}$ – $10^{-2}$ . All the relativistic effects hang on two kinematic quantities: the bulk Lorentz factor $\Gamma$ , which sets how fast the flow moves, and the Doppler factor $\delta$ , which sets how strongly emission is boosted toward a given observer.

\Gamma = \frac{1}{\sqrt{1-\beta^2}},\qquad \delta = \frac{1}{\Gamma(1-\beta\cos\theta)} \qquad (3)

where: $\Gamma$ — bulk Lorentz factor; $\beta$ — bulk speed in units of $c$ ; $\delta$ — Doppler factor; $\theta$ — angle between the flow velocity and the line of sight. Reading: for an on-axis observer ( $\theta=0$ ) and $\Gamma\gg1$ , $1-\beta\approx1/2\Gamma^2$ gives $\delta\approx2\Gamma$ , and emission is beamed into a cone of half-angle $\sim1/\Gamma$ . The reason $\Gamma$ must be large is the compactness constraint. High-energy photons can annihilate in pairs, $\gamma\gamma\to e^+e^-$ , and the optical depth to this process is enormous for a stationary source of GRB luminosity and variability.

\tau_{\gamma\gamma} \approx \frac{\sigma_T\, f_{\gamma\gamma}\, L\, \Delta t}{4\pi R^2\, m_e c^3}\,\Gamma^{-2(\alpha+1)} \qquad (4)

where: $\tau_{\gamma\gamma}$ — optical depth to pair production; $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ — Thomson cross-section; $L$ — isotropic luminosity; $\Delta t$ — observed variability timescale; $R\sim2\Gamma^2 c\,\Delta t$ — emission radius from variability; $f_{\gamma\gamma}$ — fraction of pairs above threshold; $\alpha$ — high-energy photon spectral index. Reading: the strong $\Gamma^{-2(\alpha+1)}$ dependence is what drives $\tau_{\gamma\gamma}$ below unity once $\Gamma\gtrsim100$ . The afterglow begins when the jet has swept up enough external gas to have shared a substantial fraction of its energy. That happens at the deceleration radius, where the swept-up mass — inflated by its relativistic inertia — matches the ejecta.

R_{\rm dec} = \left(\frac{3\,E_{\rm iso}}{4\pi\, n\, m_p c^2\, \Gamma_0^2}\right)^{1/3} \qquad (5)

where: $R_{\rm dec}$ — deceleration radius (afterglow onset); $n$ — external number density; $m_p$ — proton mass; $\Gamma_0$ — initial Lorentz factor. Reading: the onset in observer time is $t_{\rm dec}\approx R_{\rm dec}/(2\Gamma_0^2 c)$ , typically seconds to minutes. Once the forward shock accelerates electrons into a power law $N(\gamma_e)\propto\gamma_e^{-p}$ , their synchrotron radiation produces a broken-power-law spectrum with two characteristic frequencies, the injection frequency $\nu_m$ and the cooling frequency $\nu_c$ .

F_\nu \propto \nu^{1/3}\ (\nu<\nu_m),\quad F_\nu\propto \nu^{-(p-1)/2}\ (\nu_m<\nu<\nu_c),\quad F_\nu\propto\nu^{-p/2}\ (\nu>\nu_c) \qquad (6)

where: $\nu_m$ — injection (characteristic) frequency; $\nu_c$ — cooling frequency; $p\approx2.2$ – $2.5$ — electron spectral index; $F_\nu$ — flux density. Reading: measuring the spectral slopes returns $p$ , and the same $p$ predicts the temporal decay through the closure relations. Empirically, brighter bursts peak at higher photon energy — the Amati relation, a diagnostic used to sift real bursts from instrumental artefacts and explored as a standardisable-candle relation.

E_{\rm peak}^{\rm rest} = (1+z)\,E_{\rm peak} \propto E_{\rm iso}^{1/2} \qquad (7)

where: $E_{\rm peak}$ — observed peak energy of the $\nu F_\nu$ spectrum; $E_{\rm peak}^{\rm rest}$ — rest-frame peak energy; $E_{\rm iso}$ — isotropic-equivalent energy. Reading: the roughly square-root scaling holds over four decades in energy for long bursts, though with real intrinsic scatter. Finally, the leading candidate for the central engine of a black-hole-powered burst extracts the hole's rotational energy electromagnetically — the Blandford–Znajek mechanism.

L_{\rm BZ} \approx \frac{1}{6c}\,\Omega_H^2\, B^2\, r_H^4 \qquad (8)

where: $\Omega_H$ — angular velocity of the horizon; $B$ — poloidal field threading the horizon; $r_H$ — horizon radius. Reading: for a maximally spinning stellar black hole threaded by $B\sim10^{15}\,\mathrm{G}$ , this reaches $10^{50}$ – $10^{51}\,\mathrm{erg\,s^{-1}}$ , ample to feed a GRB jet.

Derivation AThe compactness problem and the minimum Lorentz factor

The goal is to show, from little more than causality and the pair-production cross-section, that a GRB source of observed luminosity and millisecond variability would be catastrophically opaque to its own gamma-rays if it were stationary, and that only bulk motion with Lorentz factor $\Gamma$ of order a hundred or more can rescue transparency. This single argument is the strongest model-independent evidence that GRBs harbour relativistic jets, so we build it carefully from source size, to optical depth, to the resulting bound on $\Gamma$ .

Step 1. Variability sets the size. A source that flickers on timescale $\Delta t$ cannot be larger than $R\lesssim c\,\Delta t$ if it is stationary, because faster fluctuations would be washed out by light-crossing. With $\Delta t\sim10\,\mathrm{ms}$ this gives $R\sim3\times10^8\,\mathrm{cm}$ , a few Earth radii. Why: causality is the only assumption here, so the size bound is robust and independent of any burst model.

Step 2. Count the target photons. A luminosity $L$ sustained over $\Delta t$ supplies $N_\gamma\sim L\Delta t/(m_ec^2)$ photons near the MeV pair threshold, giving a number density $n_\gamma\sim L/(4\pi R^2 c\,m_ec^2)$ . For $L\sim10^{51}\,\mathrm{erg\,s^{-1}}$ this is $n_\gamma\sim4\times10^{28}\,\mathrm{cm^{-3}}$ . Why: pair production needs two photons above threshold, so the relevant opacity is set by how densely those hard photons are packed.

Step 3. Form the optical depth. With $\tau_{\gamma\gamma}\sim n_\gamma\sigma_T R$ and the numbers above, $\tau_{\gamma\gamma}\sim(4\times10^{28})(6.65\times10^{-25})(3\times10^8)\sim10^{13}$ . Such a source is opaque by thirteen orders of magnitude; its spectrum would thermalise to a Planck form, flatly contradicting the observed non-thermal power-law tails reaching GeV energies. Why: this is the paradox — the raw observations are internally inconsistent for any sub-relativistic source.

Step 4. Let the source move toward us with Lorentz factor $\Gamma$ . Two things change: the true emission radius grows to $R\sim2\Gamma^2 c\Delta t$ , boosting $R^2$ by $\Gamma^4$ ; and the photon energies in the comoving frame drop by a factor $\Gamma$ , so a smaller fraction $f_{\gamma\gamma}\propto\Gamma^{-2\alpha}$ exceeds the pair threshold. The net suppression of $\tau_{\gamma\gamma}$ is $\Gamma^{-(4+2\alpha)}$ . Why: relativistic beaming attacks the opacity on both fronts — geometric dilution and spectral de-boosting — and their product is a steep power of $\Gamma$ .

Step 5. Demand $\tau_{\gamma\gamma}\lesssim1$ . Reducing $10^{13}$ by $\Gamma^{4+2\alpha}$ with a typical high-energy index $\alpha\sim1$ – $2$ requires $\Gamma\gtrsim10^{13/(6\ \mathrm{to}\ 8)}\sim40$ – $160$ , i.e. of order $100$ , rising to $\gtrsim400$ for the brightest GeV-detected bursts. Why: the same inequality, run on individual bursts, yields the measured lower bounds on $\Gamma$ and cements the relativistic-jet picture. *Using $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ and $m_ec^2=0.511\,\mathrm{MeV}$ , the stationary estimate gives $\tau_{\gamma\gamma}\sim10^{13}$ and the bound $\Gamma\sim10^2$ – $10^3$ .*

Go deeper · Derivation B · 2 minInternal shocks and the origin of rapid variability

Derivation BInternal shocks and the origin of rapid variability

The aim is to explain how a central engine that lives for seconds can produce gamma-ray light curves with millisecond spikes, and to see why the natural efficiency of the mechanism is only of order ten percent. We treat the outflow as a train of shells with a spread of Lorentz factors, find the radius at which a fast shell overtakes a slow one, translate that radius back into an observer-frame timescale, and finally estimate how much relative kinetic energy the collision can dissipate.

Step 1. Set up the catch-up. The engine ejects a slow shell of Lorentz factor $\Gamma_s$ and, a time $\delta t$ later, a fast shell $\Gamma_f>\Gamma_s$ . In the ultrarelativistic limit the fast shell closes the gap and collides at radius $R_{\rm IS}\approx2\Gamma_s^2 c\,\delta t/[1-(\Gamma_s/\Gamma_f)^2]\approx2\Gamma^2 c\,\delta t$ . Why: the relative velocity of two nearly-luminal shells is tiny, so the collision happens far out, at exactly the radius scale where prompt emission is inferred.

Step 2. Translate to the observer. Photons emitted at $R_{\rm IS}$ from material moving with Lorentz factor $\Gamma$ arrive spread over $\delta t_{\rm obs}\approx R_{\rm IS}/(2\Gamma^2 c)\approx\delta t$ . The collision reproduces the engine's ejection timescale. Why: this is the crucial payoff — the observed variability directly samples the engine's activity, so millisecond spikes mean the engine itself fluctuates on milliseconds.

Step 3. Dissipate the energy. In the merged-shell frame the relative kinetic energy is thermalised. For two equal masses merging inelastically the converted fraction is $\eta\approx(\Gamma_f-\Gamma_s)^2/(2\Gamma_f\Gamma_s)$ , which follows from conserving energy and momentum of the two shells into one. Why: only the kinetic energy of relative motion is available to radiate; the bulk motion carries on and cannot be tapped by an internal collision.

Step 4. Read off the efficiency. A modest contrast, say $\Gamma_f/\Gamma_s\sim2$ , gives $\eta\sim0.1$ or less, and radiative losses shave it further. This low efficiency is a genuine tension, since some bursts appear to convert a large fraction of their energy to gamma-rays, and it is one motivation for magnetic-reconnection models such as ICMART. Why: the number matters — an order-ten-percent internal-shock efficiency is a testable prediction that pushes the field toward alternative dissipation mechanisms. *The internal-shock kinematics give $R_{\rm IS}\approx2\Gamma^2c\,\delta t$ and $\eta\approx(\Gamma_f-\Gamma_s)^2/(2\Gamma_f\Gamma_s)$ under ultrarelativistic velocity addition.*

Go deeper · Derivation C · 2 minAfterglow scaling from the decelerating blast wave

Derivation CAfterglow scaling from the decelerating blast wave

The purpose here is to derive, from energy conservation alone, how the bulk Lorentz factor of the afterglow blast wave falls with time, and hence how the synchrotron light curve decays — including the achromatic steepening at the jet break that lets us measure the opening angle. We link radius to Lorentz factor through the swept-up mass, convert radius to observer time through relativistic light-travel, and then feed the resulting $\Gamma(t)$ into synchrotron scalings.

Step 1. Conserve energy in the swept-up gas. For an adiabatic blast in a uniform medium of density $n$ , the energy $E\sim\Gamma^2 M(R)c^2$ with swept mass $M(R)=\tfrac43\pi R^3 n m_p$ stays constant, so $\Gamma^2 R^3=\mathrm{const}$ and $\Gamma\propto R^{-3/2}$ . Why: the factor $\Gamma^2$ is the relativistic inertia of shock-heated matter, and holding $E$ fixed is what ties deceleration to the growing swept-up mass.

Step 2. Convert radius to observer time. Photons from radius $R$ arrive at $t\approx R/(2\Gamma^2 c)\propto R\cdot R^3=R^4$ , so $R\propto t^{1/4}$ and therefore $\Gamma\propto R^{-3/2}\propto t^{-3/8}$ . Why: the strong $R^4$ compression of arrival time is the relativistic analogue of the Doppler clock, and it is what makes afterglows fade over hours-to-days rather than the light-crossing time.

Step 3. Scale the synchrotron flux. With fixed micro-physics fractions $\epsilon_e,\epsilon_B$ , the characteristic frequencies run as $\nu_m\propto t^{-3/2}$ and $\nu_c\propto t^{-1/2}$ , and in the slow-cooling band $\nu_m<\nu<\nu_c$ the flux decays as $F_\nu\propto t^{-3(p-1)/4}$ . For $p=2.3$ this is $F_\nu\propto t^{-0.98}$ . Why: these closure relations tie a measured temporal slope to a measured spectral slope through the single parameter $p$ , giving a falsifiable test of the whole blast-wave picture.

Step 4. Cross the jet break. While $\Gamma>1/\theta_j$ the observer sees only a beamed patch and cannot tell the jet from a sphere; once $\Gamma$ drops to $1/\theta_j$ the missing flux beyond the jet edge and the onset of lateral spreading steepen the decay achromatically to $F_\nu\propto t^{-p}$ . Reading $t_j$ back through the dynamics yields $\theta_j$ , and thence $f_b$ and $E_\gamma$ . Why: the jet break is the one afterglow feature that measures collimation directly, converting the enormous $E_{\rm iso}$ into a physically sane true energy. *The scalings $\Gamma\propto t^{-3/8}$ , $F_\nu\propto t^{-3(p-1)/4}$ , and the post-break $t^{-p}$ steepening are the Sari, Piran & Narayan (1998) closure relations, valid for $p\approx2.2$ – $2.5$ .*

Limiting cases

Case	Result	Meaning
Ultrarelativistic bulk flow	$\Gamma\gg1,\ \beta\approx1-\tfrac{1}{2\Gamma^2}$	on-axis $\delta\approx2\Gamma$ ; emission beamed into a cone $\sim1/\Gamma$ ; the regime that solves compactness
Narrow jet / strong beaming	$\theta_j\ll1,\ f_b\approx\tfrac12\theta_j^2$	typical $\theta_j\sim0.05$ – $0.2\,\mathrm{rad}$ cuts the true energy by $f_b\sim10^{-3}$ – $10^{-2}$
Newtonian late afterglow	$\Gamma\to1,\ R\propto t^{2/5}$	once swept mass dominates, the blast joins the classic Sedov–Taylor supernova-remnant scaling
Photosphere vs optically thin	$\tau_T\approx1\Rightarrow R_{\rm ph}$	below $R_{\rm ph}$ the flow is quasi-thermal; non-thermal spectra require dissipation above the photosphere
Spectral bands of the afterglow	$\nu_m<\nu<\nu_c$ (slow cooling)	the observed optical/X-ray band usually sits here, giving $F_\nu\propto t^{-3(p-1)/4}$

Common trapThe isotropic energy is not the real energy, and the prompt burst is not the afterglow

Two errors dominate GRB reasoning. The first is quoting $E_{\rm iso}$ as the energy budget: because the outflow is jetted, the true radiated energy is $E_\gamma=f_b E_{\rm iso}$ with $f_b=1-\cos\theta_j\ll1$ , so a burst with $E_{\rm iso}=10^{53}\,\mathrm{erg}$ may have radiated only $\sim10^{50}\,\mathrm{erg}$ — three orders of magnitude less. The second is conflating the two emission episodes: the prompt gamma-rays come from internal dissipation deep in the jet, while the broadband afterglow is external-shock synchrotron, and they have different radii, mechanisms, and light-curve behaviour. A third, subtler trap is treating the $T_{90}\approx2\,\mathrm{s}$ long/short boundary as a sharp physical divide — it is a statistical, detector-dependent split, and individual bursts near the boundary can be misclassified.

Worked examples

1 · From fluence to true energy (fully worked)

A long burst at redshift $z=1$ has gamma-ray fluence $S_\gamma=10^{-5}\,\mathrm{erg\,cm^{-2}}$ . Its afterglow shows a jet break implying $\theta_j=0.1\,\mathrm{rad}$ . Find the isotropic-equivalent energy and the true beamed energy.

Step 1. Fix the distance. For a standard flat cosmology ( $H_0=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ , $\Omega_m=0.3$ ), the luminosity distance at $z=1$ is $d_L\approx6.7\,\mathrm{Gpc}=2.07\times10^{28}\,\mathrm{cm}$ . This is the single external input; everything else is arithmetic.

Step 2. Apply equation (1). $E_{\rm iso}=4\pi d_L^2 S_\gamma/(1+z)$ . Compute $4\pi d_L^2=4\pi(2.07\times10^{28})^2=5.4\times10^{57}\,\mathrm{cm^2}$ ; multiply by $S_\gamma=10^{-5}$ to get $5.4\times10^{52}\,\mathrm{erg}$ ; divide by $(1+z)=2$ . $E_{\rm iso}\approx2.7\times10^{52}\,\mathrm{erg}$ — a typical bright long burst.

Step 3. Beam-correct with equation (2). $f_b=1-\cos(0.1)=1-0.99500=5.0\times10^{-3}$ , consistent with $\tfrac12\theta_j^2=\tfrac12(0.1)^2=5.0\times10^{-3}$ . Then $E_\gamma=f_b E_{\rm iso}=(5.0\times10^{-3})(2.7\times10^{52})$ . $E_\gamma\approx1.4\times10^{50}\,\mathrm{erg}$ .

Step 4. Interpret. The isotropic value looks like a fraction of a solar rest-mass in gamma rays; the true value is $\sim10^{-4}\,M_\odot c^2$ , comfortably within what a hyperaccreting stellar-mass engine can supply. Collimation buys back a factor $1/f_b\approx200$ in the energy budget.

2 · Deceleration radius and afterglow onset (fully worked)

A burst has $E_{\rm iso}=10^{53}\,\mathrm{erg}$ , initial Lorentz factor $\Gamma_0=300$ , expanding into a uniform medium of $n=1\,\mathrm{cm^{-3}}$ . When does the afterglow peak?

Step 1. Evaluate equation (5). With $m_p c^2=1.50\times10^{-3}\,\mathrm{erg}$ and $\Gamma_0^2=9\times10^4$ , the denominator is $4\pi n m_p c^2\Gamma_0^2=4\pi(1)(1.50\times10^{-3})(9\times10^4)=1.70\times10^3$ . The bracket is $3E_{\rm iso}/1.70\times10^3=3\times10^{53}/1.70\times10^3=1.77\times10^{50}$ .

Step 2. Take the cube root. $R_{\rm dec}=(1.77\times10^{50})^{1/3}\approx5.6\times10^{16}\,\mathrm{cm}$ , squarely in the expected $10^{16}$ – $10^{17}\,\mathrm{cm}$ range.

Step 3. Convert to observer time. $t_{\rm dec}\approx R_{\rm dec}/(2\Gamma_0^2 c)=5.6\times10^{16}/(2\times9\times10^4\times3\times10^{10})=5.6\times10^{16}/5.4\times10^{15}$ . $t_{\rm dec}\approx10\,\mathrm{s}$ .

Step 4. Sanity-check the scaling. Because $t_{\rm dec}\propto\Gamma_0^{-8/3}$ , halving $\Gamma_0$ to $150$ would push the onset later by $2^{8/3}\approx6.3$ , to about a minute — which is why higher- $\Gamma$ bursts show earlier, sharper afterglow onsets. The afterglow peaks seconds to minutes after the prompt phase.

3 · Why merely "fast" is not enough (fully worked)

Show numerically that a stationary GRB source is opaque, and estimate the Lorentz factor needed to make it transparent.

Step 1. Stationary optical depth. With $L=10^{51}\,\mathrm{erg\,s^{-1}}$ , $\Delta t=10\,\mathrm{ms}$ , and $R\sim c\Delta t=3\times10^8\,\mathrm{cm}$ , the hard-photon density is $n_\gamma\sim L/(4\pi R^2 c\,m_ec^2)\approx4\times10^{28}\,\mathrm{cm^{-3}}$ , so $\tau_{\gamma\gamma}\sim n_\gamma\sigma_T R\approx(4\times10^{28})(6.65\times10^{-25})(3\times10^8)$ . $\tau_{\gamma\gamma}\approx8\times10^{12}$ — utterly opaque.

Step 2. Introduce bulk motion. Boosting to Lorentz factor $\Gamma$ suppresses $\tau_{\gamma\gamma}$ by $\Gamma^{4+2\alpha}$ . Taking a representative $\alpha=1$ gives a suppression $\Gamma^6$ .

Step 3. Solve for the threshold. Requiring $\Gamma^6\gtrsim8\times10^{12}$ gives $\Gamma\gtrsim(8\times10^{12})^{1/6}\approx1.4\times10^2$ . $\Gamma\gtrsim140$ , i.e. of order $100$ .

Step 4. Interpret. A "fast" flow at $\Gamma\sim10$ falls short by orders of magnitude; only genuinely ultrarelativistic motion clears the opacity. This is the cleanest evidence that GRB outflows are relativistic jets.

Physics of the portrait

Gamma-ray bursts spent three decades as the greatest mystery in high-energy astrophysics — nobody even knew whether they were nearby or cosmological. The single figure who did the most to force the answer, and to build the physical framework that turned the mystery into a quantitative science, was Bohdan Paczyński.

Origins · 1 minBohdan Paczyński and how this was discovered

Bohdan Paczyński

1940–2007 · cosmological gamma-ray bursts, the fireball model, and gravitational lensing

Bohdan Paczyński was a Polish-American astrophysicist of extraordinary range who, more than anyone, dragged gamma-ray bursts into the cosmological camp. Through the 1980s and early 1990s the community was split between "local" models — bursts on Galactic-halo neutron stars — and cosmological ones at redshifts of order unity. Paczyński argued forcefully, from the isotropy that BATSE was revealing and from the absence of a disk signature, that the bursts must lie at cosmological distances, and he and his students developed the "fireball" reasoning that showed how such vast energies could escape as non-thermal radiation only if the outflow were ultrarelativistic. In a famous 1995 public debate with Donald Lamb the two laid out the local-versus-cosmological cases; within two years the BeppoSAX afterglow redshifts vindicated Paczyński decisively. He went further, proposing (with Kevin Hurley and others) that long bursts might be "hypernovae" — the collapse of very massive rotating stars — anticipating the collapsar model that Woosley made concrete, and he pushed early ideas on gamma-ray-burst beaming and afterglows. Beyond GRBs he pioneered gravitational microlensing as a tool to hunt dark matter and planets, co-founding the OGLE survey, and made deep contributions to stellar evolution and accretion. His style — bold, physically transparent, willing to be wrong in public — set the template for a field that learned to test extreme physics against real data.

History

1967

— Vela nuclear-test-monitoring satellites record the first gamma-ray bursts

1973

— Klebesadel, Strong and Olson publish the discovery in the astronomical literature

1991

— BATSE on the Compton Gamma Ray Observatory finds the bursts isotropically distributed

1997

— BeppoSAX localises an X-ray afterglow and redshifts confirm cosmological distances

2017

— GW170817 and GRB 170817A link a short burst to a binary-neutron-star merger

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §25.4 (Gamma-Ray Bursts) · Ryden, Introduction to Cosmology, 2nd ed. (high-z probes and standard candles) · Dodelson, Modern Cosmology (relativistic kinematics and distances) · Mo, van den Bosch & White, Galaxy Formation and Evolution (massive-star progenitors and host environments) · Binney & Tremaine, Galactic Dynamics (stellar-dynamical context of merger progenitors).

Reviews. Piran, Rev. Mod. Phys. 76, 1143 (2004), "The physics of gamma-ray bursts" · Mészáros, Rep. Prog. Phys. 69, 2259 (2006) · Kumar & Zhang, Phys. Rep. 561, 1 (2015), "The physics of gamma-ray bursts and relativistic jets" · Sari, Piran & Narayan, ApJL 497, L17 (1998) (afterglow closure relations) · Abbott et al. (LIGO/Virgo and partners), ApJL 848, L13 (2017), GW170817/GRB 170817A. Key numerical benchmarks: (i) stationary compactness $\tau_{\gamma\gamma}\sim10^{13}$ and bound $\Gamma\gtrsim100$ , using $\sigma_T=6.65\times10^{-25}\,\mathrm{cm^2}$ , $m_ec^2=0.511\,\mathrm{MeV}$ , $L\sim10^{51}\,\mathrm{erg\,s^{-1}}$ , $\Delta t\sim10\,\mathrm{ms}$ ; (ii) $E_{\rm iso}\approx2.7\times10^{52}\,\mathrm{erg}$ for $z=1$ , $S_\gamma=10^{-5}\,\mathrm{erg\,cm^{-2}}$ , $d_L\approx6.7\,\mathrm{Gpc}$ ; (iii) $R_{\rm dec}\approx5.6\times10^{16}\,\mathrm{cm}$ and $t_{\rm dec}\approx10\,\mathrm{s}$ for $E_{\rm iso}=10^{53}\,\mathrm{erg}$ , $n=1\,\mathrm{cm^{-3}}$ , $\Gamma_0=300$ ; (iv) closure relation $F_\nu\propto t^{-3(p-1)/4}$ and post-break $t^{-p}$ steepening for $p\approx2.3$ ; (v) GW170817/GRB 170817A delay of $\sim1.7\,\mathrm{s}$ . The $T_{90}\approx2\,\mathrm{s}$ long/short boundary is the standard BATSE-based classification.

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

Ask the tutor: Ask me to walk through the compactness bound for a specific burst, or to turn a measured jet-break time into an opening angle and true energy.

Why were gamma-ray bursts a mystery for so long?

For nearly thirty years nobody could measure a distance to a burst. The prompt flash lasts seconds, gamma-ray detectors of the era localised sources only to degrees, and by the time any telescope looked the burst was gone. Without a distance, the energy could be anything from a modest Galactic flare to a cosmological cataclysm, and models spanned twenty orders of magnitude in energy. The deadlock broke in 1997 when BeppoSAX caught a fading X-ray afterglow precisely enough for optical telescopes to find the host galaxy and measure its redshift, proving the bursts lie billions of light-years away and therefore release enormous energies.

What is the difference between the prompt emission and the afterglow?

They are two distinct episodes with different physics. The prompt emission is the gamma-ray flash itself, lasting milliseconds to minutes, produced by dissipation deep inside the jet — internal shocks between shells of different speed, or magnetic reconnection — at radii of $10^{13}$ – $10^{15}\,\mathrm{cm}$ . The afterglow is the slow, smooth, fading glow across X-ray, optical, and radio bands that follows for hours to weeks. It is synchrotron radiation from electrons accelerated at the external forward shock, where the decelerating jet ploughs into the surrounding gas at $10^{16}$ – $10^{17}\,\mathrm{cm}$ . The prompt phase tells you about the engine; the afterglow tells you about the jet and its environment.

How do we actually measure the jet opening angle?

Through the "jet break" in the afterglow light curve. Early on, relativistic beaming lets the observer see only a small patch of the blast wave of angular size $1/\Gamma$ , which is narrower than the jet, so the source is indistinguishable from a sphere and the flux fades gently. As the blast decelerates, $\Gamma$ falls, and once $1/\Gamma$ grows to the physical jet half-angle $\theta_j$ the observer suddenly "sees" the jet's edge and the missing flux; lateral spreading adds to the effect. The light curve then steepens achromatically — the same break appears at all frequencies at the same time. The break time $t_j$ , run through the blast-wave dynamics, yields $\theta_j$ , and hence the beaming factor and the true energy.

What is the central engine, and how does it stay on for the burst?

The engine is a newborn compact object surrounded by a hyperaccreting reservoir. In long bursts, the collapsing core of a rapidly rotating massive star forms a black hole fed by a torus at $0.1$ – $1\,M_\odot\,\mathrm{s^{-1}}$ ; in short bursts, a neutron-star merger leaves a black hole or magnetar with a similar accretion disk. Two channels can launch the jet: neutrino–antineutrino annihilation above the torus, and — more likely for the most powerful bursts — the Blandford–Znajek mechanism, which taps the black hole's spin electromagnetically through magnetic field lines threading the horizon. The engine's lifetime is set by how long the accretion reservoir lasts, which naturally spans the observed range from a fraction of a second to tens of seconds.

Could a gamma-ray burst threaten Earth?

Only a very nearby, on-axis burst would matter, and such an alignment is extremely unlikely. A long burst within a few thousand light-years, beamed straight at us, could deplete the ozone layer through nitrogen-oxide chemistry and raise ultraviolet exposure for years; some authors have speculated that a burst contributed to a past mass extinction, though the evidence is circumstantial. The key protections are geometry and rarity: jets are narrow, so only a tiny fraction of the sky is illuminated by any given burst, and the Milky Way's present low star-formation rate and high metallicity make long-burst progenitors uncommon locally. No known burst poses a threat, and the nearest ones detected are cosmologically distant.

Why is the electron index p around 2.3, and does it matter?

The value $p\approx2.2$ – $2.5$ is what diffusive (Fermi) acceleration at ultrarelativistic collisionless shocks is predicted to produce, and it is what afterglow spectra and light curves actually require. It matters because $p$ ties together the spectrum and the temporal decay through the closure relations: the same $p$ that sets the spectral slope $F_\nu\propto\nu^{-(p-1)/2}$ also sets the decay $F_\nu\propto t^{-3(p-1)/4}$ before the jet break and $t^{-p}$ after. When observed spectral and temporal slopes are mutually consistent for a single $p$ , the external-shock synchrotron model passes a non-trivial test — and GRB afterglows have passed it many times, which is why the blast-wave picture is trusted.