A burst and its afterglow probe different physical stages

In a baryonic fireball model, expansion converts initial internal energy into bulk motion; the energy per baryonic rest mass limits the achievable Lorentz factor. Magnetically dominated outflows accelerate and dissipate through a different energy reservoir. Internal shocks, magnetic dissipation and photospheric emission are possible prompt-radiation mechanisms. A measured prompt spectrum does not by itself select one of them. Its observed variability constrains the dimensions of a varying patch only after specifying Doppler factors, geometry and whether variability is intrinsic.

A high comoving photon density permits gamma–gamma pair production. With photon energies E′ and ε′ and angle ψ between their directions, the threshold follows from the two-photon invariant mass. For head-on photons the energy product must exceed the electron rest-energy squared. This is a kinematic permission, not an optical depth. Absorption requires integrating the energy-dependent cross section over the target spectrum, angles and path. A target field concentrated away from the necessary energies or directions can remain transparent despite a large total radiation energy density.

After substantial deceleration, a relativistic outflow can drive an external forward shock. In a spherical or locally spherical adiabatic stage, the approximately conserved isotropic-equivalent kinetic energy scales as ambient rest-mass density times c², bulk Lorentz factor squared, and swept volume. Here Γ denotes the fluid Lorentz factor immediately behind the shock; exact coefficients differ if one uses the shock-front Lorentz factor. Radiation from a growing shock radius reaches the observer over compressed arrival intervals, rather than the full host-frame expansion time.

For a uniform external density, combining the energy and arrival-time relations gives radius proportional to observer time to the one-quarter power and Γ proportional to time to the minus three-eighths power at fixed redshift. A wind density proportional to radius to the minus second power instead gives Γ proportional to time to the minus one-quarter power. These are regime-dependent dynamics before jet-edge effects, major energy injection or nonrelativistic transition. The isotropic-equivalent energy is a spherical inference; a collimated outflow’s true angle-integrated energy requires an angular model.

A common afterglow model assigns fractions of post-shock internal energy to accelerated electrons and magnetic fields, with an electron power-law index. Synchrotron break frequencies then distinguish injection, cooling and absorption regimes. A light-curve slope depends on which spectral segment a fixed observing band occupies, not just the bulk deceleration exponent. Compton cooling must satisfy its Thomson or Klein–Nishina regime. Pair-created electrons can radiate and feed a cascade, transferring absorbed gamma-ray energy to lower bands instead of simply erasing it from the energy budget.

The function lab checks the squared Lorentz-factor dependence of the energy relation while holding radius and density fixed. Identify x with Γ divided by a reference Γ₀. Reset and set both controls to a=1,x=0.5, predicting y=0.25. Reset and set a=1,x=1, predicting y=1. This comparison is an algebraic sensitivity at one radius. It is not the time evolution of an energy-conserving blast wave, which must increase radius while Γ declines, and it computes no prompt spectrum or afterglow emission.