Parameters of the Hercules-like (first and second columns) and Hat-like (third and fourth columns) overdensities representatives in the models. The parameter space is swept around the fiducial models β and Ω_b. First row: V_ϕ of the overdensity. Second row: V_R of the overdensity. Third row: ∂V_ϕ/∂ϕ of the overdensity. In black, we provide the statistical deviation to the measurements in Gaia DR3 (Table 2). Fourth row: Same as the third row but with ∂V_ϕ/∂R.