Collimating, relativistic, magnetic jets from rotating disks - The axisymmetric field structure of relativistic jets and the example of the M 87 jet

We investigate the axisymmetric structure of collimating, relativistic, strongly magnetized (force-free) jets. In particular, we include the differential rotation of the foot points of the held lines in our treatment. The magnetic flux distribution is determined by the solution of the Grad-Shafranov equation and the regularity condition along the light surface. With differential rotation, i.e. the variation of the iso-rotation parameter Omega (F), the shape of the light surface is not known a priori and must be calculated in an iterative way. For the first time, we have calculated the force-free magnetic structure of truly two-dimensional, relativistic jets, anchored in a differentially rotating disk. Such an approach allows for a direct connection between parameters of the central source (mass, rotation) and the extension of the radio jet. In particular, this can provide a direct scaling of the location of the asymptotic jet light cylinder in terms of the central mass and the accretion disk magnetic flux distribution. We demonstrate that differentially rotating jets must be collimated to a smaller radius in terms of the light cylinder if compared to jets with rigid rotation. Also, the opening angle is smaller. Further we present an analytical estimate for the jet opening angle along the asymptotic branches of the light surface. In general, differential rotation of the iso-rotation parameter leads to an increase of the jet opening angle. Our results are applicable for highly magnetized, highly collimated, relativistic jets from active galactic nuclei and Galactic superluminal jet sources. Comparison to the M 87 jet shows agreement in the collimation distance. We derive a light cylinder radius of the M 87 jet of 50 Schwarzschild radii.