# Demography of galactic technosignatures

Claudio Grimaldi,1
1Laboratory of Physics of Complex Matter, Ecole Polytechnique Fédérale de
Lausanne, Station 3, CP-1015 Lausanne, Switzerland E-mail:
<EMAIL_ADDRESS>
###### Abstract

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considers also the signals from no longer active emitters but whose emissions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.

###### keywords:

Extraterrestrial intelligence – Astrobiology – Methods: statistical – Galaxy:
disk

††pubyear: 2020††pagerange: Demography of galactic technosignatures–A.0.4

##  1 Introduction

As new extrasolar worlds are being routinely discovered, there is an ever
mounting evidence that a significant fraction of exoplanets may have
environmental conditions suitable for developing life (Dressing & Charbonneau,
2013; Petigura et al., 2013; Zink & Hansen, 2019). In the hunt for signs of
life beyond the solar system, the search for biosignatures from the atmosphere
and surface of extrasolar Earth-like planets is moving its first steps and
will likely dominate the exoplanet science in the next decades (Kiang et al.,
2018). The prospects of life elsewhere in the Galaxy have also reinvigorated
the longstanding search for putative signs of technologically savvy life from
beyond Earth. The search for such technosignatures is going indeed through a
phase of intense activity, boosted primarily by large scale private
initiatives such as the Breakthrough Listen program (Enriquez et al., 2017;
Isaacson et al., 2017; Price et al., 2019) and benefiting of significant
advances in detector technologies.

The searches for bio- and technosignatures are two complementary strategies in
the general quest of finding life elsewhere as they probe different remotely
detectable byproducts of life. Future searches for spectroscopic biosignatures
will however target exoplanets up to only a few tens of light years from Earth
(Seager, 2018), while current telescopes are potentially capable of detecting,
for example, radio emissions from artificial sources of technological level
comparable to our own located well beyond 100 ly from the Earth (Gray, 2020).
Moreover, searches for remotely detectable technosignatures probe a parameter
search space so large that the absence of detection to date does not warrant
any firm conclusion about the existence of potentially detectable
exocivilizations (Wolfe et al., 1981; Tarter, 2001). This point has been
recently emphasized by Wright et al. (2018) who compare the tiny fraction of
the parameter space explored so far to the ratio of the volume of a small
swimming pool to that of the Earth’s oceans.

On the theoretical side, the prospects of technological life existing
elsewhere and the probability of its detection are undetermined as well. In
this context, the famous Drake equation has traditionally inspired the search
for electromagnetic (EM) technosignatures (Drake, 1961, 1965). In its compact
form, the Drake equation equates the mean number of active emitters, $N_{D}$,
with the product between the average rate of emergence of communicating
civilizations, $\Gamma$, and the average longevity, $\bar{L}$, of the emission
processes:

$N_{D}=\Gamma\bar{L}.$ (1)

In the original formulation of Eq. (1), $\Gamma$ was expressed as a product of
probability factors, later grouped in astrophysical and
biological/evolutionary probability events (Prantzos, 2013), though to be
conducive to the emergence of technological life capable of releasing EM
emissions. A vast literature has been devoted to the analysis of (1)
(Ćirković, 2004; Maccone, 2010; Frank & Sullivan III, 2016; Glade et al.,
2016; Balbi, 2018) and to the discussion of each entry of the original Drake
equation thought, nowadays, only the astrophysical contributions to $\Gamma$
are known with some confidence (Anchordoqui et al., 2018; Prantzos, 2019). Not
surprisingly, lacking any empirical knowledge about the rate of abio- and
technogenesis beyond Earth or the size of $\bar{L}$, estimates of $N_{D}$ span
several orders of magnitude, even for the Milky Way galaxy alone (Forgan,
2009).

Although Eq. (1) is the most celebrated equation in the field of
technosignatures, it is not the only vehicle to study their statistical
properties. For example, of particular importance for assessing the
probability of detection are the emission processes that cross our planet, as
only those can be potentially detected (Grimaldi, 2017). Their average number,
denoted $\bar{k}$, is thus a quantity that can be empirically estimated from
observations, at least in principle. In practice, however, the limited
sensitivity of current telescopes and the aforementioned vastness of the
parameter search space permits only a probabilistic inference of the range of
possible values of $\bar{k}$ compatible with observations (Grimaldi & Marcy,
2018; Flodin, 2019).

Even if $\bar{k}$ is sometimes confused with the Drake number, it actually
coincides with $N_{D}$ only by assuming a constant birthrate of emissions that
are either entirely isotropic, that is, radiating in all directions, or
otherwise all directed towards our planet. In more general scenarios that
contemplate anisotropic EM emissions, such as randomly directed beam-like
signals or beacons sweeping across space, only the fraction of emissions
directed towards the Earth can be potentially detected, implying
$N_{D}\geq\bar{k}$.

Another quantity of interest discussed here is $N_{G}$, the average total
number of emission processes present in the Galaxy. According to this
definition, $N_{G}$ contains both the processes generated by emitters that are
still transmitting, whose average number is $N_{D}$, and those that come from
emitters that are no longer active, but whose emissions still occupy the
Galaxy (regardless of whether or not they intersect the Earth). In full
generality, $N_{G}\geq N_{D}$, where the equality sign holds true either if
there are no galactic emissions ($N_{G}=0$ and $N_{D}=0$) or if the only
emissions present in the Galaxy come from emitters currently radiating.

The three quantities $N_{G}$, $N_{D}$, and $\bar{k}$ are the main statistical
parameters characterizing the demography of technosignatures in the Milky Way,
from which other quantities and properties of interest can be derived. For
example $\bar{k}/N_{G}$ gives the fraction of galactic emissions intersecting
Earth, while it can be demonstrated that $1-\exp(-\bar{k})$ yields the
fraction of the galactic volume occupied by the emissions (Grimaldi, 2017).
Furthermore, we see from the discussion above that the sequence of nested
inequalities,

$N_{G}\geq N_{D}\geq\bar{k},$ (2)

holds true for all types of EM emissions (isotropic, anisotropic) and for any
combination of them, implying that $\bar{k}$, the quantity that could be
possibly estimated from observations, sets a lower limit to the population
($N_{G}$) of emissions filling the Galaxy.

Here, we present a detailed study of $N_{G}$, $N_{D}$, and $\bar{k}$ to
ascertain their dependence on the birthrate, the longevity, and the geometry
of the emission processes. We base our analysis on the presumption that the
rate of technogenesis in the Milky Way has not changed significantly during
the recent history of the Galaxy (a few million years) and that the population
of artificial EM sources can be described by a collection of statistically
independent emitters.

Table 1: Legend of symbols used in the text and their meaning. The subscript $i$ refers to the different types of emissions considered here: isotropic signals ($i=$ iso), random beams ($i=$ rb), and rotating lighhouses ($i=$ lh). symbol | meaning
---|---
$N_{D}^{i}$ | Drake’s number of active emitters of type $i$
$\bar{k}_{i}$ | average number of $i$-emissions crossing Earth at the present time
$N_{G}^{i}$ | average number of $i$-emission intersecting the galactic plane
$\Gamma_{i}$ | birthrate of emissions of type $i$
$\bar{L}_{i}$ | mean longevity of emissions of type $i$
$R_{G}$ | radius of the galactic plane ($\sim 60$ kly)
$t_{G}=2R_{G}/c$ | travel time of a photon between two opposite edges of the galaxy ($\sim 10^{5}$ yr)
$\alpha_{0}$ | angular aperture of a conical beam signal

##  2 Emission processes

We start by defining our model of emission processes generated by artificial
emitters in the Milky Way galaxy. We focus on the thin disk component of the
Galaxy containing roughly $10^{10}$ potentially habitable planets within a
radius $R_{G}\simeq 60$ kly from the galactic centre, located at the origin of
a cartesian reference frame with axes $x$, $y$, and $z$. We approximate the
thin disk by an effectively two-dimensional disk of radius $R_{G}$ on the
$x$-$y$ plane. We make the assumption that the artificial emitters are located
at random sites $\mathbf{r}=(x,y)$ relative to the galactic centre and that
they are statistically independent of each other, meaning that their
birthrates and longevities are random variables uncorrelated with
$\mathbf{r}$.

In the following, we shall employ the generic term “emission process" to
indicate an artificial EM radiation of any wavelength and power spectrum that
is emitted either continuously or not during a time duration $L$. We shall
however distinguish the emission processes according to their
isotropy/anisotropy and to the geometry of the volume occupied by their
radiations by assigning to them a distinct type or class labelled by the index
$i$. In particular, here we shall focus on three prototypical types of
emissions: isotropic radiations ($i=$ iso), randomly directed narrow beams
(denoted as “random beams", $i=$ rb), and narrow beams emitted by rotating
lighthouses (denoted simply as “lighthouses", $i=$ lh). Furthermore, we shall
assume that the radiations propagate unperturbed throughout the Galaxy at the
speed of light $c$. In so doing, we are neglecting scattering and absorption
processes by the interstellar medium, which is a highly idealized setup meant
to illustrate more clearly the effects of the emission geometries.

We allow the possibility that the different types of emission processes can
have correspondingly different birthrates and longevities. For example, a
continuous isotropic emission in the infrared could be resulting from the
waste heat produced by a civilization exploiting the energy of its sun (Dyson,
1960), as for type II civilizations of the Kardashev scale (Kardashev, 1964).
The corresponding emission birthrate would be presumably lower, and the
emission longevity longer, than that of a less technologically developed (or
less energy harvesting) civilization targeting other planets with radio
signals to just advertise its existence. We therefore introduce the rate of
appearance per unit area for emitters of type $i$, $\gamma_{i}(\mathbf{r},t)$,
defined so that $\gamma_{i}(\mathbf{r},t)d\mathbf{r}dt$ gives the expected
number of $i$-emitters within an area element $d\mathbf{r}$ about $\mathbf{r}$
that started emitting within a time interval $dt$ centered at a time $t$
before present. Likewise, we associate to all processes of type $i$ a common
probability distribution function (PDF) of the longevity, denoted
$\rho_{i}(L)$, such that $\int_{0}^{\infty}\\!dL\,\rho_{i}(L)=1$ for each $i$.

The emitter rate of emergence vanishes for distances on the galactic disk
larger than $R_{\textrm{G}}$, so that integrating $\gamma_{i}(\mathbf{r},t)$
over $\mathbf{r}$ gives the birthrate of the emission processes of type $i$ in
the entire Galaxy:

$\Gamma_{i}(t)=\int\\!d\mathbf{r}\,\gamma_{i}(\mathbf{r},t).$ (3)

Finally, owing to the statistical independence of the emission processes, the
sum of the birthrates of each type of emission gives the total rate of
appearance of all emission processes:

$\Gamma(t)=\sum_{i}\Gamma_{i}(t).$ (4)

###  2.1 Average number of active emitters (Drake’s $N_{D}$)

The Drake equation can be directly derived from considering the number of
emitters that are currently transmitting (Grimaldi et al., 2018). To see this,
we note that for any galactic emitter that started an emission process at a
time $t$ in the past, the necessary condition that at present time the emitter
is still active is that the time elapsed since its birth is shorter than the
emission longevity, that is, $t\leq L$. The time integral of $\Gamma_{i}(t)$
from $t=0$ to $t=L$ gives therefore the expected number of emitters of type
$i$ and longevity $L$ that are still emitting. The average number of active
$i$-emitters is obtained by marginalizing $L$ with respect to the PDF
associated to the processes of type $i$:

$N_{D}^{i}=\int_{0}^{\infty}\\!dL\,\rho_{i}(L)\int_{0}^{L}\\!dt\,\Gamma_{i}(t).$
(5)

We take the time-scale over which the rate $\Gamma_{i}(t)$ is expected to show
appreciable variations to be much larger than $L$, even for longevity values
distributed over several million years. In so doing, we are assuming that the
emitter birthrates did not change significantly during the recent history of
the Galaxy and can be taken constant in Eq. (5), leading to:

$N_{D}^{i}=\Gamma_{i}\bar{L}_{i},$ (6)

where $\bar{L}_{i}=\int_{0}^{\infty}\\!dL\,\rho_{i}(L)L$ is the mean longevity
of the emission processes of type $i$ (see Table 1 for a list of symbols used
in this paper and their meaning). Equation (6) is the Drake equation relative
to signals of type $i$ under the steady-state hypothesis. By setting
$x_{i}=\Gamma_{i}/\Gamma$ with $\sum_{i}x_{i}=1$, and defining

$\bar{L}=\sum_{i}x_{i}\bar{L}_{i}$ (7)

as the longevity averaged over all types of emission processes, the sum of
$N_{D}^{i}$ over all $i$’s leads to the usual Drake equation in the compact
form:

$N_{D}=\Gamma\bar{L}.$ (8)

It is worth stressing that the steady-state hypothesis upon which the
derivation of Eq. (8) rests would be less justifiable if we were considering
active emitters from a region extending over Giga light-years, as in this case
the temporal dependence of the emission birthrates should be taken into
account.

###  2.2 Average number of emission processes at Earth ($\bar{k}$)

While in deriving $N_{D}$ we only needed to count the number of active
emitters without specifying the characteristics of their emission processes,
to calculate the mean number of emission processes intersecting Earth,
$\bar{k}$, we have to specify the conditions under which such intersections
occur. First, we note that for an emission process that started at a time $t$
in the past, the emitted EM radiations, traveling through space at the speed
of light $c$, fill at the present time a more or less extended region of space
that depends on the longevity and directionality of the emission process. This
region can be continuous, as in the case of an emission process (either
isotropic or anisotropic) lasting a time $L$ without interruptions, or
discontinuous as for an emitter having sent during $L$ a sequence of
intermittent signals. In the latter case, if we assume that the train of
signals is crossing Earth, there is a finite probability of the Earth not
being illuminated at a given instant of time (Gray, 2020), which may lead us
to overlook this process in the calculation of $\bar{k}$.

Similar considerations apply also to intrinsically continuous emission
processes that appear discontinuous or intermittent from the Earth’s
viewpoint, as it is the case of a rotating beacon whose beamed signal crosses
Earth periodically. Also, the signal intermittency may be due to variations in
the emitted power with minimum flux at the receiver below the detection
threshold (Gray, 2020) or to scintillation effects due to the interstellar
medium (Cordes et al., 1997).

To avoid ambiguities in determining $\bar{k}$, we shall treat any intermittent
(as seen from the Earth) emission of total longevity $L$ as an effectively
continuous process lasting the same amount of duration time. Operatively, we
could think of a periodic signal of period $T$ and duty cycle $w$ impinging
upon the Earth during an observational time interval $\Delta t$. The condition
$\Delta t/T\geq 1-w$ ensures that the “on" phase of the emission crosses Earth
at least once during $\Delta t$, so that the process is “detectable" with
probability one and can be added to the list of processes crossing Earth. We
further note that among the requisites an intermittent signal should have to
be recognized as a bona fide technosignature, the recurrence of detection is
one of the most important (Forgan et al., 2019).

####  2.2.1 Isotropic emissions

Let us consider an emitter located at $\mathbf{r}$ that started emitting an
isotropic process at a time $t$ in the past and for a duration $L$. If the
process is intrinsically continuous, at the present time the region of space
filled by the EM waves is a spherical concentric shell of outer radius $ct$
and thickness $cL$, centered on the emitter position $\mathbf{r}$. In the case
of a intermittent isotropic process of period $T$ and duty cycle $w$, this
region encompasses a sequence of nested concentric spherical shells, each of
thickness $cwT$, of consecutive outer radii differing by $cT$. Conforming to
the above prescription for intermittent emissions, we ignore the internal
structure of the encompassing shell by treating it as an effectively
continuous spherical shell of width $cL$.

The condition that the shell intersects the Earth is fulfilled by the
requirement (Grimaldi, 2017; Balbi, 2018)

$ct-cL\leq|\mathbf{r}-\mathbf{r}_{\textrm{E}}|\leq ct,$ (9)

where $\mathbf{r}_{\textrm{E}}$ is the vector position of the Earth. If
$\gamma_{\textrm{iso}}(\mathbf{r},t)$ (where the subscript “iso" stands for
isotropic) is the process birthrate per unit volume and
$\rho_{\textrm{iso}}(L)$ is the PDF of the longevity, the average number of
spherical shells at Earth is obtained by marginalizing the condition (9) over
$\mathbf{r}$, $L$, and $t$:

$\bar{k}_{\textrm{iso}}=\int\\!dL\,\rho_{\textrm{iso}}(L)\int\\!d\mathbf{r}\int_{|\mathbf{r}-\mathbf{r}_{\textrm{E}}|/c}^{|\mathbf{r}-\mathbf{r}_{\textrm{E}}|/c+L}\\!\\!\\!dt\,\gamma_{\textrm{iso}}(\mathbf{r},t).$
(10)

As done in Sec. 2.1, we neglect the temporal dependence of the birthrate,
$\gamma_{\textrm{iso}}(\mathbf{r},t)\simeq\gamma_{\textrm{iso}}(\mathbf{r})$,
so that Eq. (10) reduces to:

$\bar{k}_{\textrm{iso}}=\int\\!dL\,\rho_{\textrm{iso}}(L)L\int\\!d\mathbf{r}\,\gamma_{\textrm{iso}}(\mathbf{r})=\Gamma_{\textrm{iso}}\bar{L}_{\textrm{iso}},$
(11)

which, as anticipated in Sec. 2.1, coincides with the Drake number
$N_{D}^{\textrm{iso}}$ relative to isotropic emission processes. Note that
since we have taken a time independent birthrate, $\mathbf{r}_{E}$ has dropped
off Eq. (11), meaning that $\bar{k}_{\textrm{iso}}$ actually gives the mean
number of emissions crossing any given point in the Galaxy. This holds true
also for other types of emission processes as long as the corresponding
birthrates do not depend on $t$.

####  2.2.2 Anisotropic emissions: random beams and lighthouses

In the case of anisotropic signals, the region of space filled by the EM
radiation does not cover all directions and, therefore, only the fraction of
signals that are directed towards the Earth can contribute to $\bar{k}$. For
example, a prototypical anisotropic signal often discussed in the literature
is that of a conical beam of opening angle $\alpha_{0}$ pointing to a given
direction over the full lifetime of the emission process. As shown in Appendix
A.0.1, if such beamed signals are generated with a constant birthrate and
their orientation is distributed uniformly over the unit sphere (3D case), the
average number of random beams (rb) intersecting Earth will be proportional to
the solid angle subtended by the beams, that is:

$\begin{array}[]{lll}\bar{k}_{\textrm{rb}}=\dfrac{\langle\alpha_{0}^{2}\rangle}{16}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}},&\textrm{(3D
random beams)}\end{array}$ (12)

where $\langle\cdots\rangle$ denotes an average over the beam apertures
(assumed to be narrow), $\bar{L}_{\textrm{rb}}$ is the average longevity of
the beams, and $\Gamma_{\textrm{rb}}$ is their birthrate. We see therefore
that contrary to the case of isotropic emission processes, the mean number of
beams crossing Earth can be many orders of magnitude smaller than the
corresponding Drake’s number
$N_{D}^{\textrm{rb}}=\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}}$. Taking for
example beam apertures comparable to that of the Arecibo radar ($\sim
2\arcmin\simeq 6\times 10^{-4}$ rad) Eq. 12 yields
$\bar{k}_{\textrm{rb}}/N_{D}^{\textrm{rb}}\sim 2\times 10^{-8}$, and even
smaller values of $\bar{k}_{\textrm{rb}}/N_{D}^{\textrm{rb}}$ are obtained by
assuming optical or infrared laser emissions of apertures under an arcsecond
(Howard et al., 2004; Tellis & Marcy, 2017).

Instead of pointing towards random directions in space, another hypothetical
scenario is that in which the emitters generate beams directed preferably
along the galactic plane in order to enhance the probability of being detected
by other civilizations. In this two dimensional (2D) case, the mean number of
beams illuminating the Earth becomes (see A.0.1):

$\begin{array}[]{ll}\bar{k}_{\textrm{rb}}=\dfrac{\langle\alpha_{0}\rangle}{2\pi}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}},&\textrm{(2D
random beams)}\end{array}$ (13)

so that $\bar{k}_{\textrm{rb}}/N_{D}^{\textrm{rb}}\sim 10^{-4}$ for
$\langle\alpha_{0}\rangle$ comparable to that of the Arecibo radar.

Another type of anisotropic signal is that of a narrow beam emitted by a
rotating source, like a lighthouse rotating with constant angular velocity.
This kind of process generates a continuous, radiation-filled spiralling beam
revolving around the emitter and expanding at the speed of light. The spiral
cross section increases with the beam aperture $\alpha_{0}$ and the distance
from the source. Even if this kind of process is intrinsically continuous, an
expanding spiral impinging upon Earth will be perceived as a periodic sequence
of signals. For example, if the spin axis is perpendicular to the galactic
plane, the spiral generated by a rotating conical beam of angle aperture
$\alpha_{0}$ will periodically cross Earth’s line-of-sight with duty cycle
$\alpha_{0}/2\pi$.

In a manner similar to what we have done for the case of intrinsically
discontinuous signals, we discard the signal intermittency perceived at Earth
by introducing an effective volume encompassing the spiralling beam, whose
construction is detailed in the Appendix A.0.2. As long as the rotation axes
are oriented uniformly over the unit sphere (3D case), we find that the mean
number of lighthouse (lh) signals crossing Earth reduces to:

$\begin{array}[]{cc}\bar{k}_{\textrm{lh}}=\dfrac{\displaystyle\langle\alpha_{0}\rangle}{\displaystyle
2}\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}},&\textrm{(3D
lighthouses)}\end{array}$ (14)

while when the spin axes are perpendicular to the galactic plane (2D case),
$\bar{k}_{\textrm{lh}}$ becomes:

$\begin{array}[]{lr}\bar{k}_{\textrm{lh}}=\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}},&\textrm{(2D
lighthouses)}.\end{array}$ (15)

The signals generated by lighthouses have therefore much larger values of
$\bar{k}$ than those of random beams with comparable birthrates and
longevities, as shown in Fig. 1. In particular, from Eqs. (12)-(15) we see
that $\bar{k}_{\textrm{rb}}/\bar{k}_{\textrm{lh}}$ scales as
$\sim\langle\alpha_{0}\rangle N_{D}^{\textrm{rb}}/N_{D}^{\textrm{lh}}$ for 2D
or 3D anisotropic signals of similar apertures, meaning for example that over
$\sim 10^{3}$ active Arecibo-like beams must thus be added to each active
lighthouse of comparable $\langle\alpha_{0}\rangle$ to have analogous values
of $\bar{k}$. Furthermore, $\bar{k}_{\textrm{lh}}$ for the 2D case turns out
to be independent of $\alpha_{0}$, as for isotropic processes. This is an
interesting result, implying that an observer on Earth has the same chances of
being illuminated by a 2D lighthouse as by a (pulsed) isotropic signal if the
two have equal Drake’s numbers, Fig. 1.

Figure 1: Average number of emissions of type $i$ crossing Earth,
$\bar{k}_{i}$, over the corresponding Drake number $N_{D}^{i}$ plotted as a
function of the beam aperture $\alpha_{0}$ for the different types $i$ of
processes considered in this work.

###  2.3 Average number of emission processes intersecting the Galaxy
($N_{G}$)

So far, the only temporal variable required to calculating $N_{D}^{i}$ and
$\bar{k}_{i}$ has been the signal longevity $L$. In deriving $N_{G}$, we shall
introduces an additional time-scale,

$t_{G}=\frac{2R_{G}}{c},$ (16)

defined as the time required by a photon to travel, unperturbed, across two
opposite edges of the Milky Way ($\approx 10^{5}$ yr). Contrary to $L$ and
$\Gamma_{i}$, $t_{G}$ is an astrophysical quantity, specific to our Galaxy,
that is independent of any assumption about the existence and/or the
properties of the artificial emissions.

Figure 2: Top view of the galactic disk of radius $R_{G}$ with superimposed
areas covered by emissions of different geometries originated by a source
located at $(r,0)$. (a): the intersection with the plane $x$-$y$ of the
spherical shell generated by an isotropic emission forms an annulus of inner
radius $c(t-L)$, which intersects the galactic disk if $c(t-L)\leq R_{G}+r$.

####  2.3.1 Isotropic emissions

The necessary condition for an isotropic signal intersecting the Galaxy is
that there is a non-null intersection between the spherical shell and the
galactic disk. As shown in Fig. 2, this is fulfilled by requiring the inner
radius of the shell to be smaller than the maximum distance of the emitter
from the edge of the galactic disk, that is, $c(t-L)\leq r+R_{G}$. The number
of isotropic emissions intersecting the Galaxy is therefore:

$\displaystyle N_{G}^{\textrm{iso}}$
$\displaystyle=\int\\!dL\rho_{\textrm{iso}}(L)\int\\!d\mathbf{r}\\!\int_{0}^{(r+R_{G})/c+L}\\!dt\,\gamma_{\textrm{iso}}(\mathbf{r})$
$\displaystyle=\Gamma_{\textrm{iso}}\bar{L}_{\textrm{iso}}+\frac{1}{c}\int\\!d\mathbf{r}\gamma_{\textrm{iso}}(\mathbf{r})(r+R_{G}).$
(17)

Using a birthrate that is uniform over the entire galactic disk,
$\gamma_{\textrm{iso}}(\mathbf{r})=\theta(R_{G}-r)\Gamma_{\textrm{iso}}/\pi
R_{G}^{2}$, the above expression reduces to:

$N_{G}^{\textrm{iso}}=\Gamma_{\textrm{iso}}\left(\bar{L}_{\textrm{iso}}+\frac{5}{6}t_{G}\right),$
(18)

where $t_{G}$ is the time-scale given in Eq. (16). Other functional forms of
$\gamma_{\textrm{iso}}(\mathbf{r})$ affects only the prefactor of $t_{G}$. For
example, taking
$\gamma_{\textrm{iso}}(\mathbf{r})\propto\theta(R_{G}-r)\exp(-r/r_{s})$ with
$r_{s}=8.15$ kly (Grimaldi & Marcy, 2018), the numerical factor $5/6$ ($\simeq
0.833$) in (18) becomes $\simeq 0.646$.

An interesting feature of Eq. (18) is that using (11) we can replace
$\Gamma_{\textrm{iso}}$ by $\bar{k}_{\textrm{iso}}/\bar{L}_{\textrm{iso}}$,
yielding:

$N_{G}^{\textrm{iso}}=\bar{k}_{\textrm{iso}}\left(1+\frac{5}{6}\frac{t_{G}}{\bar{L}_{\textrm{iso}}}\right),$
(19)

so that for $\bar{L}_{\textrm{iso}}\ll t_{G}\approx 10^{5}$ yr the expected
number of the emissions intersecting the Galaxy can be much larger than that
of the emissions crossing Earth. For example, even if $\bar{k}_{\textrm{iso}}$
is only $\approx 0.1$ and $\bar{L}_{\textrm{iso}}\approx 10^{2}$ yr,
$N_{G}^{\textrm{iso}}$ is nevertheless of the order $10^{2}$. As we shall see
below, the directionality of the signals can amplify even more the difference
between $\bar{k}_{i}$ and $N_{G}^{i}$.

####  2.3.2 Anisotropic emissions: random beams and lighthouses

The calculations of the average number of random beams present in the Galaxy,
$N_{G}^{\textrm{rb}}$, and the one relative to the lighthouse signals,
$N_{G}^{\textrm{lh}}$, are detailed in the Appendixes A.0.3 and A.0.4,
respectively. Here we report only the final expressions obtained under the
assumption of small opening angles and a spatially uniform birthrate of the
emitters:

$N_{G}^{\textrm{rb}}=\left\\{\begin{array}[]{lr}\Gamma_{\textrm{rb}}\left(\bar{L}_{\textrm{rb}}+\dfrac{\langle\alpha_{0}\rangle}{2}\dfrac{4}{3\pi}t_{G}\right),&\textrm{3D
random beams}\\\\[8.0pt]
\Gamma_{\textrm{rb}}\left(\bar{L}_{\textrm{rb}}+\dfrac{4}{3\pi}t_{G}\right),&\textrm{2D
random beams},\end{array}\right.$ (20)
$N_{G}^{\textrm{lh}}=\left\\{\begin{array}[]{lr}\Gamma_{\textrm{lh}}\left(\bar{L}_{\textrm{lh}}+\dfrac{2}{\pi}t_{G}\right),&\textrm{3D
lighthouses}\\\\[8.0pt]
\Gamma_{\textrm{lh}}\left(\bar{L}_{\textrm{lh}}+\dfrac{5}{6}t_{G}\right),&\textrm{2D
lighthouses}.\end{array}\right.$ (21)

The relevant result of these calculations is that for all but one case (that
is, the 3D random beams) the mean number of anisotropic emissions intersecting
the galactic plane is independent of the angular aperture, and is therefore
comparable to that obtained for the case of isotropic emissions with similar
birthrates and longevities.

##  3 Discussion

Table 2: Analytic expressions of the average number of emissions intersecting the Galaxy, $N_{G}$, and the expected number of processes at Earth, $\bar{k}$, for the different types of emissions considered in this article. $\Gamma_{i}$ is the birthrate of emissions of type $i$ (with $i=$ iso for isotropic signals, $i=$ rb for random beams, and $i=$ lh for rotating lighthouses) and $\bar{L}_{i}$ is the corresponding mean longevity. $\alpha_{0}$ denotes the beam aperture. For all cases, the corresponding Drake’s number is $N_{D}^{i}=\Gamma_{i}\bar{L}_{i}$.  type | $N_{G}$ | $\bar{k}$
---|---|---
isotropic | $\Gamma_{\textrm{iso}}\\!\left(\bar{L}_{\textrm{iso}}+\frac{5}{6}t_{G}\right)$ | $\Gamma_{\textrm{iso}}\bar{L}_{\textrm{iso}}$
3D random beams | $\Gamma_{\textrm{rb}}\\!\left(\bar{L}_{\textrm{rb}}+\frac{\langle\alpha_{0}\rangle}{2}\frac{4}{3\pi}t_{G}\right)$ | $\frac{\langle\alpha_{0}^{2}\rangle}{16}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}}$
2D random beams | $\Gamma_{\textrm{rb}}\\!\left(\bar{L}_{\textrm{rb}}+\frac{4}{3\pi}t_{G}\right)$ | $\frac{\langle\alpha_{0}\rangle}{2\pi}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}}$
3D lighthouses | $\Gamma_{\textrm{lh}}\\!\left(\bar{L}_{\textrm{lh}}+\frac{2}{\pi}t_{G}\right)$ | $\frac{\langle\alpha_{0}\rangle}{2}\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}}$
2D lighthouses | $\Gamma_{\textrm{lh}}\\!\left(\bar{L}_{\textrm{lh}}+\frac{5}{6}t_{G}\right)$ | $\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}}$

Table 2 summarizes the analytic expressions of $N_{G}^{i}$ and $\bar{k}_{i}$
derived in the previous section. For each type of emission process, the Drake
number $N_{D}^{i}=\Gamma_{i}\bar{L}_{i}$ is the only quantity that does not
depend on the geometry of the emission process and we shall therefore focus
our discussion primarily on $\bar{k}_{i}$ and $N_{G}^{i}$.

Figures 3-5 show $\bar{k}_{i}$ as a function of the mean signal longevity
$\bar{L}_{i}$ and the population of signals in the Galaxy $N_{G}^{i}$, with
$i=\textrm{iso}$ (isotropic, Fig.3), $i=\textrm{rb}$ (random beams,Fig.4), and
$i=\textrm{lh}$ (lighthouses, Fig.5). The results have been obtained by taking
$R_{G}=60$ kly for the galactic radius, corresponding to
$t_{G}=2R_{G}/c=1.2\times 10^{5}$ yr. The red solid lines demarcate the
boundary between $\bar{k}_{i}>1$ (red colour scale) and $\bar{k}_{i}<1$ (blue
colour scale), while the black solid lines indicate $N_{G}^{i}$ calculated for
constant values of the emission birthrate $\Gamma_{i}$. The results shown in
Figs. 4 and 5 have been obtained assuming
$\langle\alpha_{0}^{2}\rangle\simeq\langle\alpha_{0}\rangle^{2}$ and an
average beam aperture of $2\arcmin$, corresponding to
$\langle\alpha_{0}\rangle\simeq 6\times 10^{-4}$ rad. Results for different
beam apertures can be easily obtained using the expressions in Table 2.

Figure 3: Expected number of isotropic emission processes at Earth,
$\bar{k}_{\textrm{iso}}$, as a function of the mean signal longevity
$\bar{L}_{\textrm{iso}}$ and the average number $N_{G}^{\textrm{iso}}$ of
isotropic emissions intersecting the Galaxy. The red solid line indicates the
special value $\bar{k}_{\textrm{iso}}=1$, while the black solid lines show
$N_{G}^{\textrm{iso}}$ calculated for different values of the emission
birthrate $\Gamma_{\textrm{iso}}$.

A first interesting feature is the behaviour of the galactic population of
technosignatures, $N_{G}^{i}$, as a function of $\bar{L}_{i}$ for fixed
$\Gamma_{i}$ (black solid lines). While $N_{G}^{i}$ increases proportionally
to the signal longevity for $\bar{L}_{i}\gtrsim t_{G}$, reaching
asymptotically the corresponding Drake’s number
$N_{D}^{i}=\bar{L}_{i}\Gamma_{i}$, for $\bar{L}_{i}\lesssim t_{G}\sim 10^{5}$
yr it reduces to

$N_{G}^{i}\simeq\Gamma_{i}t_{G}\simeq\Gamma_{i}\times(10^{5}\textrm{yr}),$
(22)

for all types of emission processes with the exception of random beams in 3D.
In this case $N_{G}^{\textrm{rb}}$ scales as
$\langle\alpha_{0}\rangle\Gamma_{\textrm{rb}}t_{G}$ for
$\bar{L}_{\textrm{rb}}\lesssim\langle\alpha_{0}\rangle t_{G}$.

Figure 4: Expected number of random narrow beams crossing Earth,
$\bar{k}_{\textrm{rb}}$, as a function of their mean longevity
$\bar{L}_{\textrm{rb}}$ and the average number $N_{G}^{\textrm{rb}}$ of beams
intersecting the Galaxy. In panel (a) the orientations of the beam axes are
distributed uniformly along the galactic disk (2D case), while in panel (b)
they are distributed uniformly over the 3D space. The beam aperture is fixed
at $2\arcmin$, corresponding to $\alpha_{0}=6\times 10^{-4}$ rad. The red
solid lines indicate the special value $\bar{k}_{\textrm{rb}}=1$, while the
black solid lines show $N_{G}^{\textrm{rb}}$ calculated for different values
of the emission birthrate $\Gamma_{\textrm{rb}}$. Figure 5: Expected number of
emissions from rotating beacons crossing Earth, $\bar{k}_{\textrm{lh}}$, as a
function of their mean longevity $\bar{L}_{\textrm{lh}}$ and the average
galactic population $N_{G}^{\textrm{lh}}$. In panel (a) the orientations of
the spin axes are kept perpendicular to the galactic plane (2D case), while in
panel (b) they are distributed uniformly over the 3D space. The red solid
lines indicate the special value $\bar{k}_{\textrm{lh}}=1$, while the black
solid lines show $N_{G}^{\textrm{lh}}$ calculated for different values of the
emission birthrate $\Gamma_{\textrm{lh}}$.

Equation (22) is remarkable because it prescripts the galactic population of
emission processes to be proportional to only the birthrate $\Gamma_{i}$,
regardless of the signal longevity as long as it is assumed $\bar{L}_{i}$ to
be less than $\sim 10^{5}$ yr. This is an advantage compared to the Drake’s
number $N_{D}^{i}$, where in addition to $\Gamma_{i}$ the longevity of the
signals is a further object of speculations. For a wide range of $\bar{L}_{i}$
values, we can thus conjecture about the size of $N_{G}^{i}$ by reasoning only
in terms of the signal birthrate. To this end it is instructive to compare
$\Gamma_{i}$ with the rate of formation of habitable planets in the Milky Way,
$\Gamma_{\textrm{P}}$, whose estimates place it in the range $0.01$-$0.1$
planet per year (Behroozi & Peebles, 2015; Zackrisson et al., 2016; Gobat &
Hong, 2016; Anchordoqui et al., 2018).

Let us first make the hypothesis that each habitable planet can be the
potential source of no more than one artificial emission. This would
correspond to $\Gamma_{\textrm{P}}$ being a theoretical upper limit of
$\Gamma_{i}$. Under this assumption, the resulting galactic population of both
isotropic emissions (Fig. 3) and rotating beacons (Fig. 5) would be bounded
from above by $\max(N_{G}^{i})\sim 10^{3}$-$10^{4}$, or somewhat less for 2D
random beams of Fig. 4(a), which is essentially the number of habitable
planets being formed during a timespan of order $t_{G}\sim 10^{5}$ yr.

Such an upper limit of $N_{G}$ entails a corresponding lower bound on the
average distance $a_{E}$ between the emitters. Indeed, since within our
working assumption $N_{G}$ corresponds to the number of emitters releasing the
emissions, their number density can be expressed as $\rho_{E}=N_{G}/\pi
R_{G}^{2}$. This allows us to find from $\pi a_{E}^{2}\rho_{E}\sim 1$ that
$a_{E}\sim R_{G}/\sqrt{N_{G}}$, thereby implying that the lower bound on
$a_{E}$ is of the order $10^{3}$ ly for $\bar{L}\lesssim t_{G}$. Following the
same reasoning, we see that the typical relative distance between emitters
estimated by the Drake equation, $a_{D}\sim R_{G}/\sqrt{N_{D}}$, scales for
$\bar{L}\lesssim t_{G}$ as $a_{D}\sim a_{E}\sqrt{t_{G}/\bar{L}}$. The
difference between $a_{G}$ and $a_{D}$ stems from the fact that the Drake’s
number gives the average population of active emitters, which are only a
fraction of all $N_{G}$ emitters whose signals are present in the Galaxy. For
example, while assuming $\Gamma_{i}\sim\Gamma_{P}$ and $\bar{L}\lesssim 10$ yr
gives $N_{G}\sim 10^{3}$ and $a_{E}\sim 10^{3}$ ly, the Drake equation yields
$N_{D}\lesssim 1$ and a value of $a_{D}$ comparable to or larger than the
diameter of the Galaxy, meaning that in this case out of $\sim 10^{3}$
galactic emissions essentially none comes from currently active emitters.

As we have seen in the previous section, in addition to $N_{D}$ the longevity
$\bar{L}$ constraints also the number of the emission processes crossing our
planet, which is further affected by the directionality of the signals (Fig.
2). Assuming therefore a large number of galactic emissions does not
automatically imply similarly large values of $\bar{k}_{i}$. For example, even
taking $\Gamma_{i}\sim 0.1$ yr-1 (that is, $N_{G}^{i}\simeq 10^{4}$ for
$\bar{L}_{i}<t_{G}$), the expected maximum value of $\bar{k}_{i}$ ranges from
$\sim 10^{3}$ for $\bar{L}_{i}=10^{4}$ yr down to $\sim 0.1$ for
$\bar{L}_{i}=1$ yr in the case of isotropic emissions (Fig. 3) and 2D
lighthouses [Fig. 5(a)]. Within the same range of signal longevities, the
upper bound on $\bar{k}_{i}$ of beams in 2D and rotating beacons in 3D drops
to only $\sim 10^{-5}$-$10^{-1}$, Figs. 4(a) and 5(b).

A special situation is represented by a collection of beamed signals with axis
orientations distributed in the 3D space [Fig. 4(b)]. In this case,
$N_{G}^{\textrm{rb}}$ becomes independent of the signal longevity only when
$\bar{L}_{\textrm{rb}}\lesssim\langle\alpha_{0}\rangle t_{G}$, which for
$\langle\alpha_{0}\rangle\sim 2\arcmin$ represents lifetimes smaller than
$\sim 10$ yr. In this limit, $N_{G}^{\textrm{rb}}\lesssim 1$ for
$\Gamma_{\textrm{rb}}\sim 0.1$ yr-1 and the resulting $\bar{k}_{\textrm{rb}}$
is upper bounded by a negligible $\sim 5\times 10^{-8}$. Values of $N_{G}$ of
the order of $10^{5}$ can nevertheless be reached for 3D beams lasting at
least $1$ Myr, but even in this case $\bar{k}_{\textrm{rb}}$ is only $\sim
2\times 10^{-3}$.

Let us pause one moment to consider the implications of assuming each
habitable planet being the potential source of at most one emission process.
As shown above, this hypothesis entails an upper bound of $N_{G}^{i}$ of the
order $10^{3}$-$10^{4}$, implying therefore the possibility of technogenesis
arising on each habitable planet during the last $\sim 10^{5}$ years. This
exceeds by far the most optimistic stances, as such an assumption would imply
not only a non-zero probability that abiogenesis is ubiquitous in the Milky
Way, but also that intelligence and technology are inevitable outcomes of the
evolutionary path of life on each inhabited planet. As long as a one-to-one
correspondence between emission processes and planets is maintained, an upper
bound on $\Gamma_{i}$ (and so on $N_{G}^{i}$) should be more reasonably placed
to much lower values than $\Gamma_{P}$, leading to $\max(N_{G}^{i})\ll
10^{3}$-$10^{4}$ and to correspondingly small values of $\bar{k}_{i}$.

Our model, however, does not distinguish whether the emission events have
occurred once or multiple times within $t_{G}$ on a given planet, nor does it
rule out the possibility of the emitters far outnumbering the planets in which
technology arose, as for example self-replicating robotized lighthouses
swarming in the free space. Within such scenarios, $\Gamma_{i}$ could thus be
larger than the rate of emergence of technological civilizations capable of
releasing technosignatures and perhaps even comparable to, or in excess of,
$\Gamma_{P}$. The plausibility of a galactic population of $\sim 10^{4}$ short
lived (i.e., $\bar{L}_{i}\lesssim t_{G}$) emissions should however be weighed
against the requirement that all these emissions must have been released
during the last $\sim 10^{5}$ years in order to fill the galaxy.

Of course, it is still possible to have significantly large values of
$N_{G}^{i}$ and $\bar{k}_{i}$ even for relatively low birthrates if the mean
longevity is so long to prevail over the small values of $\Gamma_{i}$. For
example, signals emitted from isotropic sources or 2D lighthouses with a rate
as small as $\sim 10^{-5}$ yr-1 would bring values of $N_{G}^{i}$ and
$\bar{k}$ larger $\sim 100$ if their longevities exceeded $\sim 10$ Myr.
Similar values of $N_{G}^{i}$ are obtained for 2D beams and 3D lighthouses
with $\Gamma_{i}\sim 10^{-5}$ yr-1 and $\bar{L}_{i}\sim 10$ Myr, but the
reduced solid angle for $\langle\alpha_{0}\rangle\sim 2\arcmin$ makes
$\bar{k}_{i}$ as small as $\sim 10^{-2}$ (which drops to $\sim 10^{-8}$ in the
case of 3D random beams).

So far, we have discussed each type of emission processes separately to study
the effect of $\Gamma_{i}$, $\bar{L}_{i}$ and of the signal directionality on
$N_{G}^{i}$ and $\bar{k}_{i}$. However, in the most general case, different
types of processes may be present simultaneously in the Galaxy and the
contribution of each $i$-process to the total $N_{G}$ and $\bar{k}$ depends on
the respective occurrence frequency. To see this, we note that the different
expressions of $N_{G}^{i}$ and $\bar{k}_{i}$ given in Table 2 have the form
$N_{G}^{i}=\Gamma_{i}(\bar{L}_{i}+u_{i}t_{G})$ and
$\bar{k}_{i}=v_{i}\Gamma_{i}\bar{L}_{i}$, where $u_{i}$ and $v_{i}$ are the
dimensionless factors taking account the geometry and the directionality of
the signals. Owing to the assumed statistical independence of the emitters,
the quantities $N_{D}$, $N_{G}$, and $\bar{k}$ are simple linear combinations
of the different types of processes. We can thus write:

$\displaystyle N_{G}$
$\displaystyle=\sum_{i}\Gamma_{i}(\bar{L}_{i}+u_{i}t_{G})=\Gamma\bar{L}+\Gamma
t_{G}\sum_{i}x_{i}u_{i},$ (23) $\displaystyle\bar{k}$
$\displaystyle=\sum_{i}v_{i}\Gamma_{i}\bar{L}_{i}=\Gamma\sum_{i}x_{i}v_{i}\bar{L}_{i},$
(24)

where, as done in Sec.2.1, $x_{i}=\Gamma_{i}/\Gamma$ and
$\bar{L}=\sum_{i}x_{i}\bar{L}_{i}$. Considerations similar to those discussed
in the previous section apply therefore also to the more general case. In
particular, as seen from Eq. (23), as long as the total signal longevity
$\bar{L}$ is smaller than $\sim t_{G}\sim 10^{5}$ yr, the total number of
processes intersecting the Galaxy results to be proportional to $\Gamma
t_{G}\sim(10^{5}\,\textrm{yr})\times\Gamma$, regardless of $\bar{L}$.
Speculations about the abundance of short-lived ($\bar{L}\lesssim 10^{5}$ yr)
emissions in the Galaxy can thus be framed in terms of possible upper bounds
on the total birthrate $\Gamma$.

From Eq. (24) we see that the contribution of each type of emission to the
total number of processes crossing Earth strongly depends on the relative
abundance of signal types and their longevities. As shown in Figs. 3-5, the
contribution to $\bar{k}$ of isotropic processes and lighthouses in 2D would
likely dominate over other types of emissions of similar birthrates. For
example, assuming that the fraction of rotating beacons sweeping the galactic
plane is comparable to that of 3D beamed emissions, $x_{\textrm{lh}}\sim
x_{\textrm{rb}}$, the two would contribute equally to $\bar{k}$ only if the
mean longevity of the 3D beams is about $16/\langle\alpha_{0}^{2}\rangle$
times larger than that of the rotating beacons. For beam apertures of
$2\arcmin$ this corresponds to a factor $\sim 10^{7}$, so for a given fraction
of 2D lighthouses lasting in average $10$ years an equal amount of 3D beams
requires a longevity of $\sim 100$ Myr to contribute equally to $\bar{k}$.

As a last consideration, we note that the total birthrate $\Gamma$ in Eqs.
(23) and (24) can be eliminated using the Drake number $N_{D}=\Gamma\bar{L}$,
yielding:

$\displaystyle N_{G}$
$\displaystyle=N_{D}\left(1+\dfrac{t_{G}}{\bar{L}}\sum_{i}x_{i}u_{i}\right),$
(25) $\displaystyle\bar{k}$
$\displaystyle=N_{D}\sum_{i}x_{i}v_{i}\dfrac{\bar{L}_{i}}{\bar{L}},$ (26)

allowing us to translate in terms of $N_{G}$ and $\bar{k}$ the rich literature
devoted to the Drake equation. By further eliminating $N_{D}$ from (25) and
(26) we get

$\frac{\bar{k}}{N_{G}}=\dfrac{\sum_{i}x_{i}v_{i}\bar{L}_{i}}{\bar{L}+t_{G}\sum_{i}x_{i}u_{i}},$
(27)

which expresses the fraction of galactic signals crossing Earth in terms of
the remaining unknown temporal variables: the longevities. We note that Eq. 27
generalizes a similar formula derived for the case of isotropic signals in
Grimaldi et al. (2018) and Grimaldi & Marcy (2018). The two formulas are
however not fully equivalent because in those works $\bar{k}$ was put in
relation to the number of emission processes released during the last $t_{G}$
years rather than using the number $N_{G}$ of emissions physically
intersecting the Galaxy.

##  4 Conclusions

In this paper, we have introduced other statistical quantities than the Drake
number $N_{D}$ to characterize the population of EM technosignatures in the
Milky Way. We have considered the average number of EM emissions present in
the Galaxy, $N_{G}$, and the average number $\bar{k}$ of emissions
intersecting the Earth (or any other site in the Galaxy). Unlike $N_{D}$,
$\bar{k}$ and $N_{G}$ provide measures of the number of emission processes
that are not necessarily released by currently active emitters, but that can
be potentially detected on Earth ($\bar{k}$) or that still occupy physically
the Galaxy ($N_{G}$). In order to study how these indicators are affected by
the signal directionality we have considered in addition to the case of
isotropic emission processes also strongly anisotropic ones like narrow beams
pointing in random directions and rotating beacons.

Under the assumption that the emission birthrates did not change during the
recent history of the Galaxy, we have shown that $\bar{k}=N_{D}$ only for
isotropic processes and for emissions originating from rotating beacons
sweeping the galactic disk. In all the other cases considered (beamed signals
directed randomly and lighthouses with tilted rotation axis) $\bar{k}$ can be
orders of magnitudes smaller than the Drake number, showing that $N_{D}$ may
largely overestimate the possible occurrence of signals that can be remotely
detected.

We have further discussed at length $N_{G}$ as the proper indicator of the
galactic abundance of technosignatures. We have shown that $N_{G}$, leaving
aside the special case of narrow beams directed uniformly in 3D space, is only
marginally affected by the signal directionality. A central result of the
present study is that $N_{G}$ becomes independent of the signal longevity if
this is shorter than about $10^{5}$ ly, yielding therefore a measure of the
abundance of galactic technosignatures that depends only on the emission
birthrate.

## Acknowledgements

The author thanks Amedeo Balbi and Geoffrey W. Marcy for fruitful discussions.

## Data availability

The data underlying this article are available in the article.

## References

  * Anchordoqui et al. (2018) Anchordoqui, L. A., Weber, L. A., Soriano, J. F., 2017, Proc. 35th Interna- tional Cosmic Ray Conference PoS(ICRC2017). International School for Advanced Studies, Trieste, Italy, p. 254
  * Balbi (2018) Balbi, A. 2018, Astrobiology 18, 54
  * Behroozi & Peebles (2015) Behroozi, P., Peebles, M. S., 2015, MNRAS, 454, 1811
  * Ćirković (2004) Ćirković, M. M., 2004, Astrobiology, 4, 225.
  * Cordes et al. (1997) Cordes, J. M., Lazio, J. W., Sagan, C., 1997, ApJ, 487, 782
  * Drake (1961) Drake, F. D., 1961, Physics Today, 14, 40.
  * Drake (1965) Drake, F. D., 1965, The radio search for intelligent extraterrestrial life. In Current Aspects of Exobiology, Pergamon Press, New York, p. 323
  * Dressing & Charbonneau (2013) Dressing, C. D., & Charbonneau, D. 2013, ApJ, 767, 95
  * Dyson (1960) Dyson, F. J., 1960, Science, 131, 1667.
  * Enriquez et al. (2017) Enriquez, J. E., Siemion, A., Foster, G., et al. 2017, ApJ, 849, 104
  * Flodin (2019) Flodin, M., 2019, Determining upper limits on galactic ETI civilizations transmitting continuous beacon signals in the radio spectrum (Dissertation). Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-266824
  * Forgan (2009) Forgan, D., H., 2009, Int. J. Astrobiol., 8, 121.
  * Forgan (2014) Forgan, D. 2014, JBIS, 67, 232
  * Forgan et al. (2019) Forgan, D. et al., 2019, Int. J. Astrobiol., 18, 336
  * Frank & Sullivan III (2016) Frank, A., Sullivan III, W. T., 2016, Astrobiology, 16, 359.
  * Glade et al. (2016) Glade, N., Ballet, P., Bastien, O., 2016, Int. J. Astrobiol., 11, 103.
  * Gobat & Hong (2016) Gobat, R., Hong, S. E., 2016, å, 592, A96
  * Gray (2020) Gray, R. H. 2020, Int. J. astrobiol. 1-9; doi.org/10.1017/S1473550420000038
  * Grimaldi (2017) Grimaldi, C. 2017, Sci. Rep., 7, 46273
  * Grimaldi et al. (2018) Grimaldi, C., Marcy, G. W., Tellis, N. K., Drake, F., 2018, PASP, 130, 054101.
  * Grimaldi & Marcy (2018) Grimaldi, C., Marcy, G. W., 2018, Proc. Natl. Acad. Sci. U.S.A., 115, E9755.
  * Howard et al. (2004) Howard, A. W., Horowitz, P., Wilkinson, D. P., et al. 2004, ApJ, 613, 1270
  * Isaacson et al. (2017) Isaacson, H., Siemion, A. P. V., Marcy, G. W.,Lebofsky, M. et al. 2017, PASP, 129, 054501
  * Kardashev (1964) Kardashev, N. S., 1964, Soviet Astronomy, 8, 217.
  * Kiang et al. (2018) Kiang, N. Y. et al., 2018, Astrobiology, 18, 619.
  * Maccone (2010) Maccone, C., 2010, Acta Astronaut., 67, 1366.
  * Petigura et al. (2013) Petigura, E. A., Howard, A. W., & Marcy, G.W. 2013, Proc. Natl. Acad. Sci. U.S.A., 110, 19273
  * Prantzos (2013) Prantzos N., 2013, Int. J. Astrobiol., 12, 246.
  * Prantzos (2019) Prantzos, N., 2019, MNRAS, 493, 3464.
  * Price et al. (2019) Price, D., C., et al., 2019, AJ, 159, 86.
  * Seager (2018) Seager, S., 2018, Int. J. Astrobiol., 17, 294.
  * Tarter (2001) Tarter, J. 2001, Annu. Rev. Astron. Astrophys. 35, 511.
  * Tellis & Marcy (2017) Tellis, N. K., Marcy, G. W. 2017, AJ, 153, 251
  * Wolfe et al. (1981) Wolfe, J. H., Billingham, J., Edelson, R. E., et al. 1981, in Proc. Life in the Universe, NASA Conf. Publication 2156, ed. J. Billingham (Cambridge, MA: MIT Press), 391
  * Wright et al. (2018) Wright, J. T., Kanodia, S., Lubar, E., 2018, AJ, 156, 260.
  * Zackrisson et al. (2016) Zackrisson, E., Calissendorff, P., Gonzalez, J., et al., 2016, AJ833, 214.
  * Zink & Hansen (2019) Zink, J. K., Hansen, B. M. S., 2019, MNRAS, 487, 246.

##  Appendix A anisotropic emissions

####  A.0.1 $\bar{k}$ for random beams

Let us consider an emitter located at $\mathbf{r}$ transmitting since a time
$t$ before present a conical beam of aperture $\alpha_{0}$ (Forgan, 2014;
Grimaldi, 2017). During the entire lifetime $L$ of the emission, the beam axis
is held oriented along the direction of a unit vector $\hat{n}$. The region of
space filled by the radiation is the intersection between a cone of apex at
$\mathbf{r}$ and a spherical shell centred on the cone apex with outer radius
$ct$ and thickness $cL$. As done for the isotropic case, we neglect the
internal structure of this region arising in the case of an intermittent beam.

The angular sector formed by the overlap of the conical beam with the galactic
plane (grey region in Fig. 6) subtends the angle

$\beta=\left\\{\begin{array}[]{ll}2\arccos\\!\left[\dfrac{\cos(\alpha_{0}/2)}{\sin(\theta)}\right],&|\theta-\pi/2|\leq\alpha_{0}/2\\\\[8.0pt]
0,&\textrm{otherwise}\end{array}\right.$ (28)

where $\theta\in[0,\pi]$ is the angle formed by $\hat{n}$ with the $z$-axis.
From this construction, we se that the beam will cross the Earth if
$\mathbf{r}_{\textrm{E}}$ is located within the angular sector, that is, if
Eq. (9) is satisfied and $|\phi|\leq\beta/2$, where $\phi$ is the angle formed
by $\mathbf{r}_{\textrm{E}}-\mathbf{r}$ and the projection of $\hat{n}$ on the
$x$-$y$ plane, Fig. 6.

Figure 6: Schematic illustration of a beamed emission of duration $L$ and
aperture $\alpha_{0}$. The Earth and the emitter positions are denoted
respectively by the vectors $\mathbf{r}$ and $\mathbf{r}_{\textrm{E}}$. The
beam axis of the lighthouse is oriented along the direction $\hat{n}$ and
forms an angle $\theta$ with the $z$-axis. $\phi$ is the angle formed by the
direction of $\mathbf{r}-\mathbf{r}_{\textrm{E}}$ and the projection of
$\hat{n}$ over the galactic disk.The grey region denotes the overlap area
between the galactic disk and the beam.

By adopting a constant birthrate of beamed signals with random orientations of
$\hat{n}$, the integration over $t$ under the condition (9) yields the factor
$L$, as in Eq. (11). Introducing the random beam (rb) emission rate
$\Gamma_{\textrm{rb}}$ and the corresponding average longevity
$\bar{L}_{\textrm{rb}}$, the mean number of beamed signals crossing Earth
reduces therefore to:

$\bar{k}_{\textrm{rb}}(\alpha_{0})=\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}}\int\\!d\hat{n}\,g(\hat{n})\Theta(\beta/2-|\phi|),$
(29)

where $d\hat{n}=d\phi d\theta\sin\theta$, $g(\hat{n})$ is the PDF of the
direction of $\hat{n}$, and $\Theta(x)=1$ for $x\geq 0$ and $\Theta(x)=0$ or
$x<0$ is the unit step function.

In the case in which the beams are oriented uniformly in three dimensions
(3D), the PDF of $\hat{n}$ is $g(\hat{n})=1/4\pi$ and using Eq. (28) the
integration over $\hat{n}$ yields $\frac{1}{2}[1-\cos(\alpha_{0}/2)]$, which
is simply the fractional solid angle subtended by the beam (Grimaldi, 2017).
For beam directions distributed uniformly over the two-dimensional (2D)
galactic plane, $g(\hat{n})$ is a Dirac-delta function peaked at
$\theta=\pi/2$, $g(\hat{n})=\delta(\theta-\pi/2)/2\pi$, and the orientational
average reduces simply to $\alpha_{0}/(2\pi)$. For random 3D and 2D beam
orientations we obtain therefore:

$\bar{k}_{\textrm{rb}}(\alpha_{0})=\left\\{\begin{array}[]{ll}\dfrac{1-\cos(\alpha_{0}/2)}{2}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}},&\textrm{3D
random beams},\\\\[8.0pt]
\dfrac{\alpha_{0}}{2\pi}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}},&\textrm{2D
randombeams}.\end{array}\right.$ (30)

Under the assumption that the beams have angular apertures distributed over
small values of $\alpha_{0}$, Eq. (30) reduces to:

$\bar{k}_{\textrm{rb}}=\langle\bar{k}_{\textrm{rb}}(\alpha_{0})\rangle\simeq\left\\{\begin{array}[]{ll}\dfrac{\langle\alpha_{0}^{2}\rangle}{16}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}},&\textrm{3D
random beams},\\\\[8.0pt]
\dfrac{\langle\alpha_{0}\rangle}{2\pi}\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}},&\textrm{2D
random beams},\end{array}\right.$ (31)

where $\langle\cdots\rangle$ denotes an average over the $\alpha_{0}$ values.

####  A.0.2 $\bar{k}$ for lighthouses

Figure 7: Graphical representation of the effective volume encompassing the
regions swept during a time duration $L$ by a lighthouse (or rotating beacon)
of beam aperture $\alpha_{0}$. The Earth and the emitter positions are denoted
respectively by the vectors $\mathbf{r}$ and $\mathbf{r}_{\textrm{E}}$. The
spin axis of the lighthouse is oriented along the unit vector $\hat{n}$ and
forms an angle $\theta$ with the $z$-axis. The grey region denotes the overlap
area between the galactic disk and the effective volume.

We take a lighthouse (lh) located at $\mathbf{r}$ that started transmitting at
a time $t$ in the past and for a duration $L$ a conical beam of angular
aperture $\alpha_{0}$. The effective volume encompassing the region physically
filled by the EM radiation is formed by the overlap between the volume swept
by a cone (of aperture $\alpha_{0}$) rotating about the spin axis $\hat{n}$
with apex at $\mathbf{r}$ and a spherical shell concentric to $\mathbf{r}$ of
outer radius $ct$ and thickness $cL$ (see Fig. 7). The overlap of the
effective volume with the galactic plane forms two annular sectors (shown by
grey areas in Fig. 7), symmetric with respect to the line of nodes formed by
the intersection of the rotation plane with the galactic disk, and subtending
the angle $\alpha$ given by

$\alpha=\left\\{\begin{array}[]{ll}\pi,&0\leq\theta\leq\alpha_{0}/2,\\\\[8.0pt]
2\arcsin\\!\left[\dfrac{\sin(\alpha_{0}/2)}{\sin(\theta)}\right],&\alpha_{0}/2<\theta\leq\pi/2,\end{array}\right.$
(32)

where now $\theta\in[0,\pi/2]$ is the angle formed by $\hat{n}$ with the $z$
direction. As done for the case of random beamed signals, the intersection of
the angular sector with the Earth is given by the condition (9) and
$|\phi|\leq\alpha/2$, where $\phi$ is the angle formed by
$\mathbf{r}_{\textrm{E}}-\mathbf{r}$ and the line of nodes (see Fig. 7). For a
constant birthrate of the lighthouses, the average number of randomly
distributed spiralling beams crossing Earth is thus given by:

$\bar{k}_{\textrm{lh}}(\alpha_{0})=\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}}\int\\!d\hat{n}\,g(\hat{n})\Theta(\alpha/2-|\phi|),$
(33)

If $\hat{n}$ is distributed uniformly over the unit sphere (3D case), using
Eq. (32) and $g(\hat{n})=1/4\pi$, the integral over $\theta$ reduces exactly
to $\sin(\alpha_{0}/2)$ for $\alpha_{0}\leq\pi$ and $1$ otherwise. In the case
the spin axis is perpendicular to the galactic plane (2D case), $g(\theta)$ is
a Dirac-delta peak at $\theta=0$, so that the angular average in Eq. (33)
yields $1$. Expanding Eq. (33) for small beam apertures,
$\bar{k}_{\textrm{lh}}=\langle\bar{k}_{\textrm{lh}}(\alpha_{0})\rangle$ reads:

$\bar{k}_{\textrm{lh}}=\left\\{\begin{array}[]{ll}\dfrac{\displaystyle\langle\alpha_{0}\rangle}{\displaystyle
2}\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}},&\textrm{3D lighthouses},\\\
\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}},&\textrm{2D
lighthouses}.\end{array}\right.$ (34)

Figure 8: Top view of the galactic disk of radius $R_{G}$ with superimposed
areas covered by emissions of different geometries originated by a source
located at $(r,0)$. (a): The intersection of a beam with the $x$-$y$ plane
forms an annular sector (orange region) subtending an angle $\beta$, Eq. (28),
and an inner arc at distance $c(t-L)$ from the emitter. $\ell(r,\varphi)$ is
the distance from the emitter to the edge of the Galaxy along the direction of
the beam axis projected on the $x$-$y$ plane. A narrow beam intersects the
galactic disk when $c(t-L)\leq\ell(r,\varphi)$. (b): The two annular sectors
of angle $\alpha$, Eq. (32), denote the overlap area between the $x$-$y$ plane
and the effective volume spanned by a rotating lighthouse with spin axis
tilted with respect to the $z$-axis (see Fig. 7). The intersection of the
galactic disk with the rotation plane forms two segments of length
$\ell(r,\varphi)$ and $\ell(r,\varphi+\pi)$. The longest of them must be
larger than $c(t-L)$ in order for the effective volume to intersect the
galactic disk.

####  A.0.3 $N_{G}$ for random beams

In deriving the average number of random beams intersecting the Galaxy,
$N_{G}^{\textrm{rb}}$, we shall retain only the contributions at the lowest
order in the beam aperture $\alpha_{0}$, which simplifies considerably the
calculation. We take an emitter to be located along the $x$-axis,
$\mathbf{r}=(r,0)$, with the beam axis directed along $\hat{n}$ forming an
azimuthal angle $\varphi$ with $\mathbf{r}$, Fig. 8(a). The projection of the
beam axis on the $x$-$y$ plane defines the distance

$\ell(r,\varphi)=\sqrt{R_{G}^{2}-r^{2}\sin(\varphi)^{2}}-r\cos(\varphi)$ (35)

measured from the emitter position to the edge of the Galaxy (i.e. the circle
of radius $R_{G}$). As seen from Eq. (28), the intersection of the beam with
the $x$-$y$ plane is non-null only if the polar angle of $\hat{n}$ is such
that $|\theta-\pi/2|\leq\alpha_{0}/2$. Since $\alpha_{0}\ll 1$, $\hat{n}$ lies
approximately on the $x$-$y$ plane and a beam emitted at time $t$ for a
duration $L$ will intersect the galactic disk only when $c(t-L)$ is smaller
than $\ell(r,\varphi)$, as shown in Fig. 8(a). Conversely, if
$|\theta-\pi/2|>\alpha_{0}/2$ the only beams that intersect the galactic disk
are those that are still being transmitted at the present time, that is, those
such that $t<L$. After integration over $t$, $N_{G}^{\textrm{rb}}$ at the
lowest order in $\alpha_{0}$ is therefore given by:

$\displaystyle N_{G}^{\textrm{rb}}$
$\displaystyle=\int\\!dL\rho_{\textrm{rb}}(L)\int\\!d\mathbf{r}\gamma_{\textrm{rb}}(\mathbf{r})\int\\!d\hat{n}\,g(\hat{n})$
$\displaystyle\times\left[L+\Theta\left(\frac{\alpha_{0}}{2}-\left|\theta-\frac{\pi}{2}\right|\right)\frac{\ell(r,\varphi)}{c}\right]$
$\displaystyle=\Gamma_{\textrm{rb}}\bar{L}_{\textrm{rb}}+t_{G}\frac{\eta}{\pi}\int\\!d\mathbf{r}\gamma_{\textrm{rb}}(\mathbf{r})E(r/R_{G}),$
(36)

where $\eta=\alpha_{0}/2$ or $\eta=1$ if the direction of $\hat{n}$ is
distributed uniformly in 3D space or in the $x$-$y$ plane, and
$E(x)=\int_{0}^{\pi/2}\\!d\varphi\,\sqrt{1-x^{2}\sin(\varphi)^{2}}$ is the
complete elliptic integral of the second kind. Using a birthrate that is
constant over the galactic disk, the integration over $\mathbf{r}$ yields:

$N_{G}^{\textrm{rb}}=\left\\{\begin{array}[]{ll}\Gamma_{\textrm{rb}}\left(\bar{L}_{\textrm{rb}}+\dfrac{\langle\alpha_{0}\rangle}{2}\dfrac{4}{3\pi}t_{G}\right),&\textrm{3D
random beams}\\\\[8.0pt]
\Gamma_{\textrm{rb}}\left(\bar{L}_{\textrm{rb}}+\dfrac{4}{3\pi}t_{G}\right),&\textrm{2D
random beams},\end{array}\right.$ (37)

from which we see that $N_{G}^{\textrm{rb}}$ of narrow beams depends on the
angular aperture only in the 3D case.

####  A.0.4 $N_{G}$ for lighthouses

In the case the spin axis of a rotating beacon is parallel to the $z$-axis,
the effective volume encompassing the radiation (Fig. 7) intersects the Galaxy
as long as $c(t-L)$ is smaller than the maximum distance of the emitter from
the galactic edge, in full equivalence with the isotropic case. Assuming a
spatially uniform birthrate, the number of rotating beacon signals
intersecting the Milky Way is thus
$N_{G}^{lh}=\Gamma_{\textrm{lh}}(\bar{L}_{\textrm{lh}}+\frac{5}{6}t_{G})$, as
in Eq. (18).

In the more general case in which $\hat{n}$ forms an angle $\theta$ with the
$z$-direction, at the lowest order in $\alpha_{0}$ it suffices to calculate
$N_{G}^{lh}$ by considering the intersection of the rotation plane with the
galactic disk, which forms an angle $\varphi$ with the $x$-axis. As shown in
Fig. 8(b), the emitter, located at $(r,0)$, cuts the intersection line in two
segments, generally of different lengths. The longest of these segments has
length $\ell_{\textrm{max}}(r,\varphi)=\ell(r,\varphi+\pi)$ for
$0\leq\varphi\leq\pi/2$ and $\ell_{\textrm{max}}(r,\varphi)=\ell(r,\varphi)$
for $\pi/2\leq\varphi\leq\pi$, where $\ell(r,\varphi)$ is given in Eq. (35).
Since a non-null intersection with the galactic disk is obtained by requiring
the inner edge of the effective volume, $c(t-L)$, to be smaller than
$\ell_{\textrm{max}}(r,\varphi)$, we obtain:

$\displaystyle N_{G}^{\textrm{lh}}$
$\displaystyle=\\!\int\\!\\!dL\rho_{\textrm{lh}}(L)\\!\int\\!\\!d\mathbf{r}\gamma_{\textrm{lh}}(\mathbf{r})\\!\int\\!d\hat{n}\,g(\hat{n})\\!\left[L+\frac{\ell_{\textrm{max}}(r,\varphi)}{c}\right]$
$\displaystyle=\Gamma_{\textrm{lh}}\bar{L}_{\textrm{lh}}+\frac{t_{G}}{\pi}\int\\!d\mathbf{r}\gamma_{\textrm{lh}}(\mathbf{r})[E(r/R_{G})+r/R_{G}],$
(38)

where we have assumed that the orientation of $\hat{n}$ is distributed
uniformly over 3D. For a spatially uniform $\gamma_{\textrm{lh}}(\mathbf{r})$
the integration over $\mathbf{r}$ yields $2\Gamma_{\textrm{lh}}$, so that for
the two cases examined ($\hat{n}$ random and $\hat{n}\parallel z$)
$N_{G}^{\textrm{lh}}$ reduces to:

$N_{G}^{\textrm{lh}}=\left\\{\begin{array}[]{ll}\Gamma_{\textrm{lh}}\left(\bar{L}_{\textrm{lh}}+\dfrac{2}{\pi}t_{G}\right),&\textrm{3D
lighthouses}\\\\[8.0pt]
\Gamma_{\textrm{lh}}\left(\bar{L}_{\textrm{lh}}+\dfrac{5}{6}t_{G}\right),&\textrm{2D
lighthouses}.\end{array}\right.$ (39)