Area Coverage of Expanding E.T. Signals in the Galaxy: SETI and Drake’s N

The Milky Way Galaxy contains an unknown number, $N$, of civilizations that emit electromagnetic radiation (of unknown wavelengths) over a finite lifetime, $L$. Here we are assuming that the radiation is not produced indefinitely, but within $L$ as a result of some unknown limiting event. When a civilization stops emitting, the radiation continues traveling outward at the speed of light, $c$, but is confined within a shell wall having constant thickness, $cL$. We develop a simple model of the Galaxy that includes both the birthrate and detectable lifetime of civilizations to compute the possibility of a SETI detection at the Earth. Two cases emerge for radiation shells that are (1) thinner than or (2) thicker than the size of the Galaxy, corresponding to detectable lifetimes, $L$, less than or greater than the light-travel time, $\sim 100,000$ years, across the Milky Way, respectively. For case (1), each shell wall has a thickness smaller than the size of the Galaxy and intersects the galactic plane in a donut shape (annulus) that fills only a fraction of the Galaxy's volume, inhibiting SETI detection. But the ensemble of such shell walls may still fill our Galaxy, and indeed may overlap locally, given a sufficiently high birthrate of detectable civilizations. In the second case, each radiation shell is thicker than the size of our Galaxy. Yet, the ensemble of walls may or may not yield a SETI detection depending on the civilization birthrate. We compare the number of different electromagnetic transmissions arriving at Earth to Drake's $N$, the number of currently emitting civilizations, showing that they are equal to each other for both cases (1) and (2). However, for $L < 100,000$ years, the transmissions arriving at Earth may come from distant civilizations long extinct, while civilizations still alive are sending signals yet to arrive.