*2.1. General Definition of Jet Structure*

At a fixed time *t*, assuming axisymmetry, radial expansion and a relativistic equation of state *p* = *e*int/3 (where *p* is the pressure and *e*int is the internal energy density, both measured in the comoving frame of the fluid) for simplicity, the jet structure can be described by four functions of the radial coordinate *r* and polar angle *θ* (measured from the jet axis), namely the modulus of the four-velocity Γ*β* = *u*(*r*, *θ*, *t*) (where Γ is the bulk Lorentz factor and *β* = (1 − Γ −2 ) 1/2), the comoving rest-mass density *ρ* 0 (*r*, *θ*, *t*), the dimensionless enthalpy *h*(*r*, *θ*, *t*) = 1 + 4*e*int/3*ρ* 0 *c* <sup>2</sup> and the magnetization *B* <sup>2</sup>/4*πρ*<sup>0</sup> *c* <sup>2</sup> = *σ*(*r*, *θ*, *t*) (where *B* is the comoving magnetic field strength, assumed transverse with respect to the expansion). Often, it is possible to limit the discussion to cold (*h* ∼ 1) and highly relativistic (*u* ∼ Γ) parts of the fluid, in which case the rest-mass density, magnetization and Lorentz factor Γ(*r*, *θ*, *t*) are sufficient. If the radial structure is unimportant and the focus is on kinetic energy, then the description of the jet structure can be accomplished by two angular functions: the kinetic energy per unit solid angle. If the time *t* at which the expression is valuated is such that the outflow is still in an acceleration phase, the appropriate Lorentz factor here

is the "terminal" one, i.e., the one that can be estimated assuming the available internal (and/or magnetic) energy will be eventually converted to kinetic energy.

$$\frac{\mathrm{d}E}{\mathrm{d}\Omega}(\theta,t) = \int\_0^\infty (\Gamma(r,\theta,t) - 1)\Gamma(r,\theta,t)\rho'(r,\theta,t)c^2r^2\mathrm{d}r\tag{1}$$

and the average Lorentz factor

$$
\Gamma(\theta, t) = \left(\frac{\text{dE}}{\text{d}\Omega}\right)^{-1} \int\_0^\infty (\Gamma(r, \theta, t) - 1) \Gamma^2(r, \theta, t) \rho'(r, \theta, t) c^2 r^2 \text{d}r,\tag{2}
$$

The latter description, applied to the "coasting" phase (see below), is the most widely adopted one, as it is sufficient for a basic description of the link between the prompt and afterglow emission observables and the jet structure in many contexts. The purpose of the above definitions is to clarify the connection between the three-dimensional physical properties of the outflow and the functions that are customarily used to describe its angular structure. A variety of similar, but not identical, definitions can be found in the literature: the essence of the arguments presented here does not depend on the precise definition.
