The classical definition of the virial temperature of a galaxy halo excludes a fundamental contribution to the energy partition of the halo: the kinetic energy of non-thermal gas motions. Using simulations of low-redshift, $sim L^*$ galaxies from the FOGGIE project (Figuring Out Gas & Galaxies In Enzo) that are optimized to resolve low-density gas, we show that the kinetic energy of non-thermal motions is roughly equal to the energy of thermal motions. The simulated FOGGIE halos have $sim 2times$ lower bulk temperatures than expected from a classical virial equilibrium, owing to significant non-thermal kinetic energy that is formally excluded from the definition of $T_mathrm{vir}$. We derive a modified virial temperature explicitly including non-thermal gas motions that provides a more accurate description of gas temperatures for simulated halos in virial equilibrium. Strong bursts of stellar feedback drive the simulated FOGGIE halos out of virial equilibrium, but the halo gas cannot be accurately described by the standard virial temperature even when in virial equilibrium. Compared to the standard virial temperature, the cooler modified virial temperature implies other effects on halo gas: (i) the thermal gas pressure is lower, (ii) radiative cooling is more efficient, (iii) O VI absorbing gas that traces the virial temperature may be prevalent in halos of a higher mass than expected, (iv) gas mass estimates from X-ray surface brightness profiles may be incorrect, and (v) turbulent motions make an important contribution to the energy balance of a galaxy halo.
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