Context : Pulsar wind nebulae ( PWNe ) represent the most prominent population of Galactic very-high-energy gamma-ray sources and are thought to be an efficient source of leptonic cosmic rays .
Vela X is a nearby middle-aged PWN , which shows bright X-ray and TeV gamma-ray emission towards an elongated structure called the cocoon .

Aims : Since TeV emission is likely inverse-Compton emission of electrons , predominantly from interactions with the cosmic microwave background , while X-ray emission is synchrotron radiation of the same electrons , we aim to derive the properties of the relativistic particles and of magnetic fields with minimal modelling .

Methods : We used data from the Suzaku XIS to derive the spectra from three compact regions in Vela X covering distances from 0.3 pc to 4 pc from the pulsar along the cocoon .
We obtained gamma-ray spectra of the same regions from H.E.S.S .
observations and fitted a radiative model to the multi-wavelength spectra .

Results : The TeV electron spectra and magnetic field strengths are consistent within the uncertainties for the three regions , with energy densities of the order 10 ^ { -12 } erg cm ^ { -3 } .
The data indicate the presence of a cutoff in the electron spectrum at energies of \sim 100 TeV and a magnetic field strength of \sim 6 \upmu G .
Constraints on the presence of turbulent magnetic fields are weak .

Conclusions : The pressure of TeV electrons and magnetic fields in the cocoon is dynamically negligible , requiring the presence of another dominant pressure component to balance the pulsar wind at the termination shock .
Sub-TeV electrons can not completely account for the missing pressure , which may be provided either by relativistic ions or from mixing of the ejecta with the pulsar wind .
The electron spectra are consistent with expectations from transport scenarios dominated either by advection via the reverse shock or by diffusion , but for the latter the role of radiative losses near the termination shock needs to be further investigated in the light of the measured cutoff energies .
Constraints on turbulent magnetic fields and the shape of the electron cutoff can be improved by spectral measurements in the energy range \gtrsim 10 keV .