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We report on spectroscopic studies of hot and ultracold RbSr molecules, andcombine the results in an analysis that allows us to fit a potential energycurve (PEC) for the X(1)$^2\Sigma^+$ ground state bridging theshort-to-long-range domains. The ultracold RbSr molecules are created in a$\mu$K sample of Rb and Sr atoms and probed by two-colour photoassociationspectroscopy. The data yield the long-range dispersion coefficients $C_6$ and$C_8$, along with the total number of supported bound levels. The hot RbSrmolecules are created in a $1000 \,$K gas mixture of Rb and Sr in a heat-pipeoven and probed by thermoluminescence and laser-induced fluorescencespectroscopy. We compare the hot molecule data with spectra we simulated usingpreviously published PECs determined by three different ab-initio theoreticalmethods. We identify several band heads corresponding to radiative decay fromthe B(2)$^2\Sigma^+$ state to the deepest bound levels of X(1)$^2\Sigma^+$. Wedetermine a mass-scaled high-precision model for X(1)$^2\Sigma^+$ by fittingall data using a single fit procedure. The corresponding PEC is consistent withall data, thus spanning short-to-long internuclear distances and bridging anenergy gap of about 75% of the potential well depth, still uncharted by anyexperiment. We benchmark ab-initio PECs against our results, and give the PECfit parameters for both X(1)$^2\Sigma^+$ and B(2)$^2\Sigma^+$ states. As firstoutcomes of our analysis, we calculate the $s$-wave scattering properties forall stable isotopic combinations and corroborate the locations of Fano-Feshbachresonances between alkali Rb and closed-shell Sr atoms recently observed[Barb\'e et al., Nat. Phys., 2018, DOI:10.1038/s41567-018-0169-x]. Theseresults should greatly contribute to the generation of ultracoldalkali$-$alkaline-earth dimers, whose applications range from quantumsimulation to quantum chemistry.

The RbSr $^2\Sigma^+$ ground state investigated via spectroscopy of hot & ultracold molecules