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while $N_D$ gives the number of emissions being released at the present time, $N_G$ considers also the signals from no longer active emitters but whose emissions still occupy the Galaxy.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.

中文翻译：

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银河技术特征的人口统计学

关于地球以外技术生命存在的概率论据传统上是指德雷克方程，以得出对银河系中有意或无意释放电磁辐射的技术先进文明数量的可能估计。在这里，我们引入了 Drake 数字 $N_D$ 以外的其他指标来开发银河系中人工排放的人口统计数据。我们专注于统计独立信号的三个主要类别（各向同性、窄波束和旋转信标）来计算银河系中存在的发射过程的平均数 $N_G$ 以及它们穿过地球的平均数 $\bar{k} $，这是一个可以根据直接观察进行统计估计的数量。我们表明 $\bar{k}$ 仅在各向同性排放中与 $N_D$ 重合，而在高度定向信号的情况下，$\bar{k}$ 可以比 $N_D$ 小几个数量级。我们进一步表明，虽然 $N_D$ 给出了当前释放的排放量，但 $N_G$ 还考虑了来自不再活跃但排放量仍然占据银河系的信号。我们发现，只要排放的平均寿命短于每年约 $10^5$，$N_G$ 就完全由排放速率决定，而 $N_D$ 和 $\bar{k}$还取决于发射寿命。最后，使用为这里考虑的每种排放过程确定的 $N_G$、$N_D$ 和 $\bar{k}$ 的分析公式，我们提供了这些数量可能达到的作为排放函数的值的全面概述出生率、寿命和方向性。