2013A&A...559A..40S

A large set of stars observed by CoRoT and Kepler shows clear evidence for the presence of a stellar background, which is interpreted to arise from surface convection, i.e., granulation. These observations show that the characteristic time-scale (τ_eff) and the root-mean-square (rms) brightness fluctuations (σ) associated with the granulation scale as a function of the peak frequency (ν_max) of the solar-like oscillations. We aim at providing a theoretical background to the observed scaling relations based on a model developed in Paper I. We computed for each 3D model the theoretical power density spectrum (PDS) associated with the granulation as seen in disk-integrated intensity on the basis of the theoretical model published in Paper I. For each PDS we derived the associated characteristic time (τ_eff) and the rms brightness fluctuations (σ) and compared these theoretical values with the theoretical scaling relations derived from the theoretical model and the measurements made on a large set of Kepler targets. We derive theoretical scaling relations for τ_eff and σ, which show the same dependence on ν_max as the observed scaling relations. In addition, we show that these quantities also scale as a function of the turbulent Mach number (M_a) estimated at the photosphere. The theoretical scaling relations for τ_eff and σ match the observations well on a global scale. Quantitatively, the remaining discrepancies with the observations are found to be much smaller than previous theoretical calculations made for red giants. Our modelling provides additional theoretical support for the observed variations of σ and τ_eff with ν_max. It also highlights the important role of M_a in controlling the properties of the stellar granulation. However, the observations made with Kepler on a wide variety of stars cannot confirm the dependence of our scaling relations on M_a. Measurements of the granulation background and detections of solar-like oscillations in a statistically sufficient number of cool dwarf stars will be required for confirming the dependence of the theoretical scaling relations with M_a.