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

TL;DR

This paper analyzes the statistical behavior of the total emission from populations of discrete sources, revealing non-linear growth and variability patterns due to small number statistics, with implications for astrophysical observations.

Contribution

It provides exact solutions and analytical approximations for the combined luminosity and variability of source populations, highlighting non-linear regimes caused by small number effects.

Findings

Total luminosity can grow non-linearly with number of sources due to small number statistics.

RMS variability decreases more slowly than 1/sqrt(N) in the non-linear regime.

Predicted X-ray luminosity relations agree with observations for certain galaxy populations.

Abstract

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 L_tot=SUM(L_k) 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 L_tot grows non-linearly with N, in an apparent contradiction with the seemingly obvious prediction <L_tot>=integral(dN/dL*L*dL) ~ N. In this non-linear regime, the rms_tot decreases with N significantly more slowly than expected from the rms ~ 1/sqrt(N) averaging law. For example, for a power law luminosity function with a slope of a=3/2, in the non-linear regime, L_tot ~ N^2 and the rms_tot does not depend at all on the number of sources N. Only in the limit of N>>1 do these quantities behave as intuitively expected, L_tot ~ N and rms_tot ~ 1/sqrt(N). We give exact solutions and derive convenient analytical approximations for L_tot 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 Msun/yr and log(M*/Msun)<10.0-10.5, respectively.