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

• Published in: Monthly notices of the Royal Astronomical Society 2004-07, Vol.351 (4), p.1365-1378
• Main Authors: Gilfanov, M., Grimm, H.-J., Sunyaev, R.
• Format: Article
• Language: English
• Subjects: methods: data analysis; methods: statistical; X-rays: binaries; X-rays: galaxies
• ISSN: 0035-8711; 1365-2966
• DOI: 10.1111/j.1365-2966.2004.07874.x

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 rmstot of 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 Ltot grows non-linearly with n, in apparent contradiction with the seemingly obvious prediction . In this non-linear regime, rmstot decreases 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, Ltot∝n2 and rmstot 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 . We give exact solutions and derive convenient analytical approximations for Ltot and rmstot. Using the total X-ray luminosity of a galaxy due to its X-ray binary population as an example, we show that the LX–star formation rate and LX–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 ≲ 4–5 M⊙ yr−1 and log(M*/M⊙) ≲ 10.0–10.5, respectively.