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By modelling the expansion of a cloud of electrons and positrons with thetemperature 400 keV that propagates at the mean speed 0.9c ($c:$ speed oflight) through an initially unmagnetized electron-proton plasma with aparticle-in-cell (PIC) simulation, we find a mechanism that collimates the paircloud into a jet. A filamentation instability develops between the protons atrest and the moving positrons. Its magnetic field collimates the positrons anddrives an electrostatic shock into the electron-proton plasma. The magneticfield acts as a discontinuity that separates the protons of the shocked ambientplasma, known as the outer cocoon, from the jet's interior region. The outercocoon expands at the speed 0.15c along the jet axis and at 0.03cperpendicularly to it. The filamentation instability converts the jet'sdirected flow energy into magnetic energy in the inner cocoon. The magneticdiscontinuity cannot separate the ambient electrons from the jet electrons.Both species rapidly mix and become indistinguishable. The spatial distributionof the positive charge carriers is in agreement with the distributions of theambient material and the jet material predicted by a hydrodynamic model apartfrom a dilute positronic outflow that is accelerated by the electromagneticfield at the jet's head.

Cocoon formation by a mildly relativistic pair jet in unmagnetized collisionless electron-proton plasma