Generalized investigation of the rotation-activity relation: Favouring rotation period instead of Rossby number

Magnetic activity in Sun-like and low-mass stars causes X-ray coronal emission, which is stronger for more rapidly rotating stars. This relation is often interpreted in terms of the Rossby number, i.e., the ratio of rotation period to convective overturn time. We reconsider this interpretation on the basis of the observed X-ray emission and rotation periods of 821 stars with masses below 1.4 Msun. A generalized analysis of the relation between X-ray luminosity normalized by bolometric luminosity, L_X/L_bol, and combinations of rotational period, P, and stellar radius, R, shows that the Rossby formulation does not provide the solution with minimal scatter. Instead, we find that the relation L_X/L_bol ~ P^{-2}R^{-4} optimally describes the non-saturated fraction of the stars. This relation is equivalent to L_X ~ P^{-2}, indicating that the rotation period alone determines the total X-ray emission. Since L_X is directly related to the magnetic flux at the stellar surface, this means that the surface flux is determined solely by the star's rotation and is independent of other stellar parameters. While a formulation in terms of a Rossby number would be consistent with these results if the convective overturn time scales exactly as L_bol^{-1/2}, our generalized approach emphasizes the need to test a broader range of mechanisms for dynamo action in cool stars.