Demography of galactic technosignatures
  Probabilistic arguments about the existence of technological life beyond
Earth traditionally refer to the Drake equation to draw possible estimates of
the number of technologically advanced civilizations releasing, either
intentionally or not, electromagnetic emissions in the Milky Way. Here, we
introduce other indicators than Drake's number $N_D$ to develop a demography of
artificial emissions populating the Galaxy. We focus on three main categories
of statistically independent signals (isotropic, narrow beams, and rotating
beacons) to calculate the average number $N_G$ of emission processes present in
the Galaxy and the average number of them crossing Earth, $\bar{k}$, which is a
quantity amenable to statistical estimation from direct observations. We show
that $\bar{k}$ coincides with $N_D$ only for isotropic emissions, while
$\bar{k}$ can be orders of magnitude smaller than $N_D$ in the case of highly
directional signals. We further show that while $N_D$ gives the number of
emissions being released at the present time, $N_G$ considers also the signals
from no longer active emitters but whose emissions still occupy the Galaxy. We
find that as long as the average longevity of the emissions is shorter than
about $10^5$ yr, $N_G$ is fully determined by the rate of emissions alone, in
contrast to $N_D$ and $\bar{k}$ which depend also on the emission longevity.
Finally, using analytic formulas of $N_G$, $N_D$, and $\bar{k}$ determined for
each type of emission processes here considered, we provide a comprehensive
overview of the values these quantities can possibly achieve as functions of
the emission birthrates, longevities, and directionality.
