We study models of the \gamma -ray emission of Cyg X-3 observed by Fermi .
We calculate the average X-ray spectrum during the \gamma -ray active periods .
Then , we calculate spectra from Compton scattering of a photon beam into a given direction by isotropic relativistic electrons with a power-law distribution , both based on the Klein-Nishina cross section and in the Thomson limit .
Applying the results to scattering of stellar blackbody radiation in the inner jet of Cyg X-3 , we find that a low-energy break in the electron distribution at a Lorentz factor of \sim 300 – 10 ^ { 3 } is required by the shape of the observed X-ray/ \gamma -ray spectrum in order to avoid overproducing the observed X-ray flux .
The electrons giving rise to the observed \gamma -rays are efficiently cooled by Compton scattering , and the power-law index of the acceleration process is \simeq 2.5 –3 .
The bulk Lorentz factor of the jet and the kinetic power before the dissipation region depend on the fraction of the dissipation power supplied to the electrons ; if it is \simeq 1 / 2 , the Lorentz factor is \sim 2.5 , and the kinetic power is \sim 10 ^ { 38 } erg s ^ { -1 } , which represents a firm lower limit on the jet power , and is comparable to the bolometric luminosity of Cyg X-3 .
Most of the power supplied to the electrons is radiated .
The broad band spectrum constrains the synchrotron and self-Compton emission from the \gamma -ray emitting electrons , which requires the magnetic field to be relatively weak , with the magnetic energy density \la a few times 10 ^ { -3 } of that in the electrons .
The actual value of the magnetic field strength can be inferred from a future simultaneous measurement of the IR and \gamma -ray fluxes .