In non-collisional magnetized astrophysical plasmas, vortices can form as it is the case of the Venus plasma wake where Lundin et al. (2013) identified a large vortex through the integration of data of many orbits from the Venus Express (VEX) spacecraft. On the one hand, our purpose is to develop a theoretical foundation in order to explain the occurrence and formation of vortices in non-collisional astrophysical plasmas. On the other hand, to apply the latter in order to study the vorticity in the wakes of Venus and Mars. We introduce two theorems and two corollaries, which may be applicable to any non-collisional plasma system, that relate the vorticity to electromagnetic variables such as the magnetic field and the electric current density. We also introduce a toy vortex model for the wakes of non-magnetized planetary bodies. From the proposed theorems and model and using magnetic data of the VEX and the Mars Global Surveyor (MGS) spacecraft, we identify vortices in the wakes of Venus and Mars in single spacecraft wake crossings. We also identify a spatial coincidence between current density and vorticity maxima confirming the consistency of our theorems and model. We conclude that vortices in non-collisional magnetized plasmas are always linked to electric currents and that both vortices and currents always coexist. This suggests that the mechanism that produces this type of vortices is the mutual interaction between the electric current and the magnetic field, that to a first approximation is explained considering that plasma currents due to a non-zero net charge density induce magnetic fields that modify the existing field and also produce a helical field configuration that drives charged particles along helical trajectories.
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