Constraining the Sub-Galactic Relationship Between Star Formation and the Hot Interstellar Medium in NGC 4254

We investigate the relationship between star formation and X-ray emission from the hot interstellar medium (ISM) on $\sim$kpc scales in NGC 4254 (M99) by combining spatially resolved star formation histories (SFHs) and Bayesian X-ray spectral fitting. We measure sub-galactic star formation rates (SFR) by modeling spectrophotometric UV-IR data with flexible SFHs, and we produce point-source-subtracted maps of the diffuse X-ray emission using Chandra data. We extract and fit the spectra of 5 regions selected by their SFR density $?_{\rm SFR}$, deriving hot gas luminosities and plasma temperatures. We examine the sub-galactic $kT-?_{\rm SFR}$ and $L^{\rm gas}_X-?_{\rm SFR}$ scaling relations in NGC 4254, and compare to predictions from simple models of the feedback into the ISM from core collapse supernovae (CCSNe). The hot gas emission from NGC 4254 is consistent with thermalization of $\approx 40-50\%$ of the energy from CCSNe in the ISM, and mass-loading of the CCSNe ejecta which decreases as $?_{\rm SFR}^{-1/3}$. Our optimized model implies a temperature and X-ray production efficiency that scale as $kT = (0.72^{+0.26}_{-0.18}~{\rm keV}) ?_{\rm SFR}^{0.34\pm0.10}$ and $?= (0.03^{+0.02}_{-0.01}) ?_{\rm SFR}^{0.34\pm0.18}$, respectively, for $?_{\rm SFR} = 0.01-0.13~{\rm M_{\odot}~yr^{-1}~kpc^{-2}}$. We also compare the properties of the hot ISM to other ISM phases using data from the PHANGS program. The diffuse X-ray emission of a given region is on average 200 times fainter than the H$?$ emission, and we see evidence that the hot ISM is over-pressurized compared to the large-scale dynamical equilibrium pressure of the galaxy, consistent with expansion of the hot ISM into the ambient medium.