  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.
