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We present a multi-wavelength (X-ray to optical) analysis, based on non-localthermodynamic equilibrium photospheric+wind models, of the B0 Ia-supergiant:$\epsilon$~Ori. The aim is to test the consistency of physical parameters, suchas the mass-loss rate and CNO abundances, derived from different spectralbands. The derived mass-loss rate is$\dot{M}/\sqrt{f_\infty}\sim$1.6$\times$10$^{-6}$ M$_\odot$ yr$^{-1}$ where$f_\infty$ is the volume filling factor. However, the S IV$\lambda\lambda$1062,1073 profiles are too strong in the models; to fit theobserved profiles it is necessary to use $f_\infty<$0.01. This value is afactor of 5 to 10 lower than inferred from other diagnostics, and implies$\dot{M} \lesssim1 \times 10^{-7}$ M$_\odot$ yr$^{-1}$. The discrepancy couldbe related to porosity-vorosity effects or a problem with the ionization ofsulfur in the wind. To fit the UV profiles of N V and O VI it was necessary toinclude emission from an interclump medium with a density contrast($\rho_{cl}/\rho_{ICM}$) of $\sim$100. X-ray emission in H-He like and Fe Llines was modeled using four plasma components located within the wind. Wederive plasma temperatures from $1 \times 10^{6}$ to $7\times 10^{6}$ K, withlower temperatures starting in the outer regions (R$_0\sim$3-6 R$_*$), and ahot component starting closer to the star (R$_0\lesssim$2.9 R$_*$). From X-rayline profiles we infer $\dot{M} <\, 4.9\times10^{-7}$ M$_\odot$ yr$^{-1}$. TheX-ray spectrum ($\geq$0.1 kev) yields an X-ray luminosity $L_{\rm X}\sim2.0\times10^{-7} L_{\rm bol}$, consistent with the superion line profiles.X-ray abundances are in agreement with those derived from the UV and opticalanalysis: $\epsilon$ Ori is slightly enhanced in nitrogen and depleted incarbon and oxygen, evidence for CNO processed material.

X-ray, UV and optical analysis of supergiants: $\epsilon$ Ori