  The stellar binary $\eta$ Carinae has been observed during its full orbital
period in gamma rays by the Fermi-Large Area Telescope (LAT). The
shock-accelerated electrons in the colliding winds of the two stars radiate
synchrotron photons in the magnetic field of the shocked region and inverse
Compton photons, where the target photons are from the thermal emissions by the
more massive and luminous of the two stars. The inverse Compton emission
dominates the gamma-ray flux data from the $\eta$ Carinae, however the spectral
energy distribution shows signature of a hadronic component in the $\sim
10$-300 GeV range during the periastron passage. Current and future air
Cherenkov telescopes will be able to constrain this component at TeV energies.
Acceleration of cosmic-ray protons to $\gg 1$ TeV energies in the colliding
winds, required to explain the hadronic emission component through photopion
interactions, can lead to detectable signal of $\gtrsim 10$ TeV neutrino events
in large kilometer scale neutrino telescopes.
