The hot component of the circum-galactic medium (CGM) around star forming galaxies is detected as diffuse X-ray emission. The X-ray spectra from the CGM depend on the temperature and metallicity of the emitting plasma, providing important information about the feeding and feedback of the galaxy. The observed spectra are commonly fitted using simple 1-Temperature (1-T) or 2-T models. However, the actual temperature distribution of the gas can be complex because of the interaction between galactic outflows and halo gas. Here we demonstrate this by analysing 3-D hydrodynamical simulations of the CGM with a realistic outflow model. We investigate the physical properties of the simulated hot CGM, which shows a broad distribution in density, temperature, and metallicity. By constructing and fitting the simulated spectra, we show that, while the 1-T and 2-T models are able to fit the synthesized spectra reasonably well, the inferred temperature(s) bear little physical meaning. Instead, we propose a log-normal distribution as a more physical model. The log-normal model better fits the simulated spectra while reproducing the gas temperature distribution. We also show that when the star formation rate is high, the spectra inside the bi-conical outflows are distinct from that outside, as outflows are generally hotter and more metal-enriched. Finally, we produce mock spectra for future missions with the eV-level spectral resolution, such as Athena, Lynx, and HUBS.
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