Dynamics of dwarf galaxies in $f(R)$ gravity

We use the kinematic data of the stars in eight dwarf spheroidal galaxies to assess whether $f(R)$ gravity can fit the observed profiles of the line-of-sight velocity dispersion of these systems without resorting to dark matter. Our model assumes that each galaxy is spherically symmetric and has a constant velocity anisotropy parameter $\beta$ and constant mass-to-light ratio consistent with stellar population synthesis models. We solve the spherical Jeans equation that includes the Yukawa-like gravitational potential appearing in the weak field limit of $f(R)$ gravity, and a Plummer density profile for the stellar distribution. The $f(R)$ velocity dispersion profiles depend on two parameters: the scale length $\xi^{-1}$, below which the Yukawa term is negligible, and the boost of the gravitational field $\delta>-1$. $\delta$ and $\xi$ are not universal parameters, but their variation within the same class of objects is expected to be limited. The $f(R)$ velocity dispersion profiles fit the data with a value $\xi^{-1}= 1.2^{+18.6}_{-0.9}$ Mpc for the entire galaxy sample. On the contrary, the values of $\delta$ show a bimodal distribution that picks at $\bar{\delta}=-0.986\pm0.002$ and $\bar{\delta}=-0.92\pm0.01$. These two values disagree at $6\sigma$ and suggest a severe tension for $f(R)$ gravity. It remains to be seen whether an improved model of the dwarf galaxies or additional constraints provided by the proper motions of stars measured by future astrometric space missions can return consistent $\delta$'s for the entire sample and remove this tension.