## FANDOM

155 Pages

Suppose there are astrophysical events that occur at random places in space. We denote the rate of such events per unit volume by $\dot{n}$. Suppose we observe for a time $T$ such that the density of events during that time $n = \dot{n} T$. Suppose further that the luminosity of each event is the same $L$. The observed flux from each event would be

$f = \frac{L}{4 \pi r^2}$

where $r$ is the distance to the observer. The number distribution of the fluxes would be

$dN = 4 \pi r^2 dr \cdot n \propto f^{-5/2} df$

If the threshold for detection is $f_{th}$, then the probability for each flux is

$\frac{dP}{dx} = -\frac{2}{3} x^{-5/2}$

where $x = \frac{f}{f_{th}}$ is the normalised flux.

For each event there's a maximal distance from which it can be detected

$r_{th} = \sqrt{\frac{L}{4 \pi f}}$

From each radius it is possible to construct a sphere around the observer. The ratio between the volume of the sphere from which an event was detected, and that of a sphere of maximum distance is $x^{-3/2}$. On average, it should be $\frac{1}{2}$ because an event has an equal opportunity to be in either half of the sphere. This can also be shown mathematically

$< x^{-3/2} > = \int_1^{\infty} x^{-3/2} dP = \frac{3}{2} \int_1^{\infty} x^{-4} dx = \frac{3}{2} \left[\frac{1}{3} x^{-3}\right]_1^{\infty} = \frac{1}{2}$