Statistical properties of the combined emission of a population of discrete sources: astrophysical implications

We study the statistical properties of the combined emission of a population of discrete sources (e.g. X-ray emission of a galaxy due to its X-ray binaries population). Namely, we consider the dependence of their total luminosity Ltot = ∑ Lk and of fractional rms tot of their variability on the number of sources n or, equivalently, on the normalization of the luminosity function. We show that due to small number statistics a regime exists, in which Ltot grows non-linearly with n, in an apparent contradiction with the seemingly obvious prediction 〈Ltot〉 = ∫ L dN dL dL ∝ n. In this non-linear regime, the rms tot decreases with n significantly more slowly than expected from the rms ∝ 1/√n averaging law. For example, for a power law luminosity function with a slope of α = 3/2, in the non-linear regime, Ltot ∝ n and the rms tot does not depend at all on the number of sources n. Only in the limit of n → ∞ do these quantities behave as intuitively expected, Ltot ∝ n and rms tot ∝ 1/ √ n. We give exact solutions and derive convenient analytical approximations for Ltot and rms tot. Using the total X-ray luminosity of a galaxy due to its X-ray binary population as an example, we show that the LX−SFR and LX −M∗ relations predicted from the respective “universal” luminosity functions of high and low mass X-ray binaries are in a good agreement with observations. Although caused by small number statistics the non-linear regime in these examples extends as far as SFR∼ 4 − 5 M⊙/yr and log(M∗/M⊙)∼ 10.0− 10.5, respectively.