The connection between stellar granulation and oscillation as seen by the Kepler mission

Context. The long and almost continuous observations by Kepler show clear evidence of a granulation background signal in a large sample of stars, which is interpreted as the surface manifestation of convection. It has been shown that its characteristic timescale and rms intensity fluctuation scale with the peak frequency (nu(max)) of the solar-like oscillations. Various attempts have been made to quantify the observed signal, to determine scaling relations for its characteristic parameters, and to compare them to theoretical predictions. Even though they are consistent on a global scale, large systematic differences of an unknown origin remain between different methods, as well as between the observations and simulations. Aims. We aim to study different approaches to quantifying the signature of stellar granulation and to search for a unified model that reproduces the observed signal best in a wide variety of stars. We then aim to define empirical scaling relations between the granulation properties and nu(max) and various other stellar parameters. Methods. We use a probabilistic method to compare different approaches to extracting the granulation signal. We fit the power density spectra of a large set of Kepler targets, determine the granulation and global oscillation parameter, and quantify scaling relations between them. Results. We establish that a depression in power at about nu(max)/2, known from the Sun and a few other main-sequence stars, is also statistically significant in red giants and that a super-Lorentzian function with two components is best suited to reproducing the granulation signal in the broader vicinity of the pulsation power excess. We also establish that the specific choice of the background model can affect the determination of nu(max), introducing systematic uncertainties that can significantly exceed the random uncertainties. We find the characteristic frequency (i.e., inverse timescale) and amplitude of both background components to tightly scale with nu(max) for a wide variety of stars (about 2-2000 mu Hz in nu(max)), and quantify a mass dependency of the latter. To enable comparison with theoretical predictions (which do not include the observed power depression), we computed effective timescales and bolometric intensity fluctuations and found them to approximately scale as tau(eff) proportional to g(-0.85) T-0.4 and A(gran) proportional to (g(2)M)(-1/4) (or more conveniently R/M-3/4), respectively. Similarly, the bolometric pulsation amplitude scales approximately as A(puls) proportional to (g(2)M)(-1/3) (or R-4/3/M), which implicitly verifies a separate mass and luminosity dependence of Apuls. We have also checked our scaling relations with solar reference values and find them in good agreement. Conclusions. We provide a thorough analysis of the granulation background signal in a large sample of stars, from which we establish a unified model that allows us to accurately extract the granulation and global oscillation parameter. The resulting scaling relations allow a simple estimate of the overall spectral shape of any solar-type oscillator and might serve as a starting point for future large-sample studies or as a reference for theoretical modelling of granulation.