2004MNRAS.351.1365G

We study the statistical properties of the combined emission of a population of discrete sources (for example, the X-ray emission of a galaxy due to its population of X-ray binaries). Namely, we consider the dependence of their total luminosity and of the fractional rms_totof their variability on the number of sources n or, equivalently, on the normalization of the luminosity function. We show that, as a result of small number statistics, a regime exists in which L_totgrows non-linearly with n, in apparent contradiction with the seemingly obvious prediction. In this non-linear regime, rms_totdecreases with n significantly more slowly than expected from the averaging law. For example, for a power-law luminosity function with a slope of α= 3/2, in the non-linear regime, L_tot∝n² and rms_totdoes not depend at all on the number of sources n. Only in the limit of n⟶∞ do these quantities behave as intuitively expected, L_tot∝n and. We give exact solutions and derive convenient analytical approximations for L_totand 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 L_X-star formation rate and L_X-M_*relations predicted from the respective `universal' luminosity functions of high- and low-mass X-ray binaries are in good agreement with observations. Although caused by small number statistics, the non-linear regime in these examples extends as far as SFR ≲ 4-5M_☉/yr and log(M_*/M_☉) ≲ 10.0-10.5, respectively.