We study the collimation of relativistic hydrodynamic jets by the pressure of an ambient medium in the limit where the jet interior has lost causal contact with its surroundings .
For a jet with an ultrarelativistic equation of state and external pressure that decreases as a power of spherical radius , p \propto r ^ { - \eta } , the jet interior will lose causal contact when \eta > 2 .
However , the outer layers of the jet gradually collimate toward the jet axis as long as \eta < 4 , leading to the formation of a shocked boundary layer .
Assuming that pressure-matching across the shock front determines the shape of the shock , we study the resulting structure of the jet in two ways : first by assuming that the pressure remains constant across the shocked boundary layer and looking for solutions to the shock jump equations , and then by constructing self-similar boundary-layer solutions that allow for a pressure gradient across the shocked layer .
We demonstrate that the constant-pressure solutions can be characterized by four initial parameters that determine the jet shape and whether the shock closes to the axis .
We show that self-similar solutions for the boundary layer can be constructed that exhibit a monotonic decrease in pressure across the boundary layer from the contact discontinuity to the shock front , and that the addition of this pressure gradient in our initial model generally causes the shock front to move outwards , creating a thinner boundary layer and decreasing the tendency of the shock to close .
We discuss trends based on the value of the pressure power-law index \eta .