Can dwarf spheroidal galaxies host a central black hole ?

We construct mass models of Milky Way dwarf spheroidal galaxies to place constraints on the central black hole (BH) masses they can host. We model the galaxies as a three-component system consisting of the stars, dark matter halo, and a central black hole, using the Osipkov--Merritt--Cuddeford class of anisotropic distribution function. The posterior distribution of black hole mass remains flat toward the low-mass end, indicating that the kinematic data places an upper limit on the black hole mass. Our analysis yields a 95% credible upper limit of $\log(M_{\bullet}/M_{\odot}) < 6$. We combine our results with black hole mass measurements and upper limits from the literature to construct a unified $M_{\bullet}$--$\sigma_{}$ relation spanning $\sigma_{} \sim 10$--$300,\mathrm{km,s^{-1}}$, described by $\log(M_{\bullet}) = 8.32 + 4.08,\log\left(\sigma_{}/200,\mathrm{km,s^{-1}}\right)$, with an intrinsic scatter of $\sigma_{\rm int} = 0.55$. We compare the inferred limits to models of black hole growth via momentum-driven accretion and stellar capture, which predict black hole masses in the range $10^{3}$--$10^{4},M_{\odot}$ for the range $\sigma_{} \sim 6$--$12,\mathrm{km,s^{-1}}$, in close agreement with the $M_{\bullet}$--$\sigma_{*}$ relation within the 95% credible upper limits on the black hole masses derived in this work.