  We report on spectroscopic studies of hot and ultracold RbSr molecules, and
combine the results in an analysis that allows us to fit a potential energy
curve (PEC) for the X(1)$^2\Sigma^+$ ground state bridging the
short-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 photoassociation
spectroscopy. The data yield the long-range dispersion coefficients $C_6$ and
$C_8$, along with the total number of supported bound levels. The hot RbSr
molecules are created in a $1000 \,$K gas mixture of Rb and Sr in a heat-pipe
oven and probed by thermoluminescence and laser-induced fluorescence
spectroscopy. We compare the hot molecule data with spectra we simulated using
previously published PECs determined by three different ab-initio theoretical
methods. We identify several band heads corresponding to radiative decay from
the B(2)$^2\Sigma^+$ state to the deepest bound levels of X(1)$^2\Sigma^+$. We
determine a mass-scaled high-precision model for X(1)$^2\Sigma^+$ by fitting
all data using a single fit procedure. The corresponding PEC is consistent with
all data, thus spanning short-to-long internuclear distances and bridging an
energy gap of about 75% of the potential well depth, still uncharted by any
experiment. We benchmark ab-initio PECs against our results, and give the PEC
fit parameters for both X(1)$^2\Sigma^+$ and B(2)$^2\Sigma^+$ states. As first
outcomes of our analysis, we calculate the $s$-wave scattering properties for
all stable isotopic combinations and corroborate the locations of Fano-Feshbach
resonances between alkali Rb and closed-shell Sr atoms recently observed
[Barb\'e et al., Nat. Phys., 2018, DOI:10.1038/s41567-018-0169-x]. These
results should greatly contribute to the generation of ultracold
alkali$-$alkaline-earth dimers, whose applications range from quantum
simulation to quantum chemistry.
