A probabilistic analysis of the Fermi paradox in terms of the Drake formula: the role of the L factor
MMNRAS , 1–9 (2015) Preprint 23 September 2020 Compiled using MNRAS L A TEX style file v3.0
A probabilistic analysis of the Fermi paradox in terms ofthe Drake formula: the role of the L factor
N. Prantzos (cid:63)
Institut d’Astrophysique de Paris and Sorbonne Universit´e, 98bis Bd Arago, 75014 Paris, France
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
In evaluating the number of technological civilizations N in the Galaxy through theDrake formula, emphasis is mostly put on the astrophysical and biotechnological fac-tors describing the emergence of a civilization and much less on its the lifetime, whichis intimately related to its demise. It is argued here that this factor is in fact the mostimportant regarding the practical implications of the Drake formula, because it deter-mines the maximal extent of the ”sphere of influence” of any technological civilization.The Fermi paradox is studied in the terms of a simplified version of the Drake formula,through Monte Carlo simulations of N civilizations expanding in the Galaxy duringtheir space faring lifetime L. In the framework of that scheme, the probability of ”di-rect contact” is determined as the fraction of the Galactic volume occupied collectivelyby the ”spheres of influence” of N civilizations. The results of the analysis are used todetermine regions in the parameter space where the Fermi paradox holds. It is arguedthat in a large region of the diagram the corresponding parameters suggest rather a”weak” Fermi paradox. Future research may reveal whether a ”strong” paradox holdsin some part of the parameter space. Finally, it is argued that the value of N is notbound by N=1 from below, contrary to what is usually assumed, but it may have astatistical interpretation.
Key words:
General:extraterrestrial intelligence – Galaxy: disk
Most of the effort in establishing the chances for the ex-istence of life elsewhere in the universe has been put onevaluating the various astronomical and biological factorsthat may lead to the emergence of life on the surface ofterrestrial-type exoplanets around other stars in the MilkyWay. Twenty five years after the discovery of the first exo-planet, we have now started to have sufficient data to allowus to think that such objects are relatively common in theGalaxy (Petigura et al. 2018). However, we have not yet anyevidence about the existence of other lifeforms, even at themicroscopic level, elsewhere in the Universe.Metrodorus, disciple of Epicurus, seems to have beenthe first to formulate the main argument for the existence oflife forms and intelligent beings beyond Earth, in the thirdcentury BC: ”to consider the Earth as the only populatedworld in infinite space is as absurd as to assert that in anentire field sown with millet only one grain will grow”. Theidea of an infinite space, populated by an infinite number ofatoms and their combinations, was a key ingredient of the (cid:63)
E-mail: [email protected] atomistic philosophy of Leucipus, Democritus, and Epicurus(Furley 1987).Today, proponents of extraterrestrial intelligence (ETI)invoke essentially the same argument, although the conceptof infinity is not used any more (because it is difficult tohandle and it may lead to paradoxes, e.g., in an infiniteuniverse, everything, including ourselves, could exist in aninfinite number of copies). But the number of stars in ourGalaxy, ∼
100 billion, is considered by some - mostly as-tronomers - to be large enough as to make Metrodorus’ ar-gument applicable to the Milky Way. Others, however - evo-lutionary biologists, in particular - are not impressed by thatnumber and remain skeptical concerning ETI (e.g. Simpson1964; Mayr 1992).In the past sixty years or so, the debate on ETI waslargely shaped by the ”Drake equation” and the ”Fermi para-dox”. The former - which should rather be called ”Drakeformula” - was proposed by Frank Drake (Drake 1961) andbecame ever since the key quantitative tool to evaluate theprobabilities for radio-communication with extraterrestrialintelligence (CETI). The latter - also known as ”Fermi’squestion” - was formulated a decade earlier by Enrico Fermi,but it remained virtually unknown until 1975, when itwas independently re-discovered twice (Hart 1975; Viewing c (cid:13) a r X i v : . [ phy s i c s . pop - ph ] S e p N. Prantzos where are they?” ) it opposes ahealthy skepticism to the optimistic views on ETI: if thereare many of them, why don’t we see any evidence of theirpresence on Earth or in its neighborhood? (see Cirkovic2018, for a recent comprehensive overview of the subject ).It is not known what kind of calculations - if any - Fermidid to evaluate the chances of the Earth being visited byextraterrestrial civilizations, during a lunch-time discussionwith colleagues at Los Alamos in 1950 (see Jones 1985, foran account of that discussion). Several attempts have beenmade to quantify Fermi’s question through numerical sim-ulations, based on various assumptions (Newman & Sagan1981; Jones 1981; Fogg 1987; Landis 1998; Bjørk 2007; Hair& Hedman 2013; Zackrisson et al. 2015; Carroll-Nellenbacket al. 2019). These studies reached different conclusions onthe paradoxicality of Fermi’s question, both from the quan-titative and the qualitative point of view, e.g. the timescalesrequired for colonization of the Galaxy (Webb 2015; Ash-worth 2014); however, none of those works makes explicituse of the Drake formula.In this work, a new framework is introduced for thestudy of Fermi’s question, using a simplified form of Drake’sformula (Sec. 2) to evaluate the number N of technologi-cal civilizations. In this framework, the possibility of N < L plays a key role for thestudy of Fermi’s question, because the volume of the ”sphereof influence” of communicating civilizations (either throughdirect contact or through ELM signals) depends on L foran isotropic expansion. This dependence is illustrated byMonte Carlo simulations performed for various values of therelevant parameters N, L and υ (the typical speed of theexpansion front of the civilizations), in a disk galaxy withthe dimensions of the Milky Way (Sec. 4). The main re-sult is that the probability of contact can be defined as thefraction of the Galactic volume occupied by the common vol-ume of the spheres of influence of the ensemble of galacticcivilisations. This original presentation allows one to displayin terms of the Drake formula and in a compact form, thethree physical explanations mostly discussed as solutions tothe Fermi paradox: civilizations are rare, too short-lived, orunable to expand at sufficiently high speed in the Galaxy(Sec. 5). On the occasion of a now famous meeting -the first one onthe Search for Extraterrestrial Intelligence (SETI) - that heorganized in Green Bank, Virginia, Drake tried to evalu-ate the number of radio-communicating civilizations in ourGalaxy (Drake 1961). In its original formulation, the Drakeequation reads N = R ∗ f p n e f l f i f t L (1)where R* is the rate of star formation in the Galaxy (i.e.,number of stars formed per unit time), f p is the fraction ofstars with planetary systems, n e is the average number ofplanets around each star, f l is the fraction of planets wherelife developed, f i is the fraction of planets where intelligent life developed, and f T is the fraction of planets with tech-nological civilizations. Obviously, N and L are intimatelyconnected: if N is the number of radio-communicating civ-ilizations - as in the original formulation by Drake - thenL is the average duration of the radio-communication phaseof such civilizations (and not their total lifetime, as some-times incorrectly stated). On the other hand, if N is meantto be the number of technological or space-faring civiliza-tions, then L represents the duration of the correspondingphase (see Prantzos 2013, for additional comments on theDrake formula ).The Drake formula obviously corresponds to the equi-librium solution of an equation similar to the equation ofradioactivity for the decay rate D of a number N of radioac-tive nuclei: D = dN/dt = - N/L, where L is the lifetimeof those nuclei. In the steady state, where the productionrate P is equal to the decay rate D, one has N = P L. In asimilar vein, the product of all the terms of the Drake for-mula, except L , can be interpreted as the production rate Pof technological (radio-communicating or space-faring) civi-lizations in the Galaxy .On the basis of this analogy, it was suggested (Prantzos2013) that the practical implications of the Drake formulawould be more clearly evaluated if its original seven termswere condensed to only three. The aim was twofold: (1) toillustrate quantitatively some implications of the number Nfor SETI and CETI, and (2) to use exactly the same frame-work for a quantitative assessment of the Fermi paradox.The Drake equation is now written as N = R ASTRO f BIOTEC L (2)where R ASTRO = R ∗ f p n e (3)represents the production rate of habitable planets, and f BIOTEC = f l f i f t (4)represents the product of all chemical, biological and socio-logical factors leading to the development of a technologicalcivilization.Obviously, while the product R ASTRO f BIOTEC repre-sents the ”production term” of technological civilizations,the lifetime L represents the ”destruction term” in the steadystate situation described by the Drake formula.From the three terms of the modified Drake equation(Eq. 2) R ASTRO is the only one reasonably well studied atpresent and expected to be well constrained in the fore-seeable future. The first of its terms, R ∗ , is already con-strained by observations in the Milky Way to be 4 stars/yr.The present day star formation (SF) rate is 1.9 M (cid:12) /yr(Chomiuk & Povich 2011) and there are 2 stars per M (cid:12) ina normal stellar initial mass function (IMF) like the one ofKroupa (2002). However, its average past value was proba-bly higher by a factor of 2, and we shall adopt the value of4 M (cid:12) /yr which corresponds to an average star productionrate of < R > ∼ − ; this average SF rate reproduceswell the stellar mass of 4 10 M (cid:12) or the ∼ stars of The same formula is obtained within a different framework, theLittle’s law (Little 1961), well known in probability theory andstatistics. MNRAS , 1–9 (2015) he impact of the L factor on the Fermi paradox the Milky Way if assumed to hold for the age of the GalaxyA ∼
10 Gy.We shall assume that only 10% of those stars are appro-priate for harboring habitable planets, because their masshas to be smaller than 1.1 M (cid:12) , i.e. they have to be suffi-ciently long-lived (with main sequence lifetimes larger than ∼ (cid:12) , to pos-sess circumstellar habitable zones outside the tidally lockedregion (Selsis 2007). This leaves aside the most numerousclass of stars, namely the low mass red dwarfs: their intenseand time-varying activity (e.g. Paudel et al. 2019, and ref-erences therein) and the limitations on their circumstellarhabitable zone (Haqq-Misra et al. 2018; Schwieterman et al.2019) largely balance the effect of their larger number withrespect to solar-type stars. In any case, considering themwould increase the planet numbers by factor of 10-20 butthis would change little the conclusions, given the extremelylarge uncertainties of the other factors of the Drake equation,as illustrated in a recent analysis (Wandel & Gale 2019).A recent analysis of the Kepler DR25 and Gaia DR2data (Hsu et al. 2019) finds that the statistics currently avail-able on extra-solar planets around solar-type (FGK) starspoint to 20% of the surveyed stars possessing planets withsizes 1-1.75 Earth radii and orbital periods of 237-500 days.This fraction may be considered to describe the product f p n e in the Drake equation. It is, admittedly, a rather optimisticestimate, its only merit being that it imposes a plausible up-per limit on the fraction of such solar-type stars. Combinedwith the aforementioned formation rate of 0.7-1.1 M (cid:12) stars,it leads to R ASTRO ∼ f BIOTEC is not constrained from below. Ithas a maximum value of f BIOTEC , Max = 1 (correspondingto f l = f i = f t =1), which is rather implausibly optimisticbut constitutes a useful upper limit. Because of that, thenumber of N technological civilizations at steady state isbound from above by the value N MAX = R ASTRO L = 0.1 L (with L expressed in yr) in the Drake formula.The re-formulated Drake equation (Eq. 2) appears in agraphical form, in Fig. 1, after fixing R ASTRO =0.1 y − andplotting N as a function of L . A slightly different form of thisdiagram ( f BIOTEC vs L ) appeared in Prantzos (2013), butthis one is more straightforward and illustrates the situationin a clearer way. In this log-log diagram, values of N vs L for a given f BIOTEC are represented by straight quasi-diagonal lines. Taking into account that f BIOTEC ≤
1, thesolid line at f BIOTEC =1 bounds N from above, i.e. thereare no civilizations on the upper left part of the diagram inthe steady state situation. Same values of N are obtainedfor different combinations of f BIOTEC and L , but we arguethat the dependence on L is more important regarding itsimplications.On the right axis of the figure appear, on a differ-ent scale, the typical distances between such civilizations.Indeed, as noted in Prantzos (2013), the Drake formulawas meant to evaluate the chances of establishing radio-communication with ETI in the Galaxy, but in fact it saysvery little on the probability of contacting them: numberN alone is totally insufficient to evaluate that probability. Figure 1.
Number of civilizations N versus their average life-time L, assuming R ASTRO = 0.1/yr. Straight lines correspondto biotechnological factors f BIOTEC =1 (solid), 10 − , 10 − and10 − (dotted). Typical distances D (in light-years l.y.) betweencivilizations for each N are obtained from the discussion of Sec. 2and are indicated on the right axis. The bottom (yellow shaded)region corresponds to N <
1. The statistical significance of thatregion (illustrated by the point N = 0.2) is discussed in Sec. 3.
From the point of view of interstellar communication or di-rect contact it is a completely different matter to have 10civilizations inside the whole Galaxy or inside a globularcluster. For that purpose, one more step has to be taken, toconnect the lifetime L to the typical distances between suchcivilizations, in order to account for the finite speed of elec-tromagnetic (ELM) signals (or transport); those distancesdepend on N and on the dimensions of the system, in thiscase the Milky Way.To evaluate typical distances between Galactic civiliza-tions it may be assumed that, to a first approximation, theGalactic disk is described by a cylinder of radius R G =12kpc ( ∼
40 000 l.y.) and height h G =1 kpc ( ∼ . By equating the volume of the Galactic cylinderV G = π R h G with the sum of N volumes of spheres of aver-age radius r occupied by each civilization , one obtains theaverage distance between two civilizations as D = 2 r = 2 (cid:18) G R N (cid:19) / ( D < h G ) (5)In the case of a small number of civilizations (say N < > h G and the appropriate expression isthen D = 2 r = 2 R G / (cid:112) ( N ) ( D > h G ) (6) A better approximation would be to consider the exponentialprofile of the stellar disk of the Milky Way, but the conclusionsdepend little on such assumptions (factors of order unity) andmuch more on the unknown factors of the Drake formula.MNRAS000
40 000 l.y.) and height h G =1 kpc ( ∼ . By equating the volume of the Galactic cylinderV G = π R h G with the sum of N volumes of spheres of aver-age radius r occupied by each civilization , one obtains theaverage distance between two civilizations as D = 2 r = 2 (cid:18) G R N (cid:19) / ( D < h G ) (5)In the case of a small number of civilizations (say N < > h G and the appropriate expression isthen D = 2 r = 2 R G / (cid:112) ( N ) ( D > h G ) (6) A better approximation would be to consider the exponentialprofile of the stellar disk of the Milky Way, but the conclusionsdepend little on such assumptions (factors of order unity) andmuch more on the unknown factors of the Drake formula.MNRAS000 , 1–9 (2015)
N. Prantzos
It is interesting to notice that even for a hundred thou-sand civilizations, typical distances are of several hundredl.y., making contact - either by radio-signals or space probes- difficult.Notice that in the simplified picture presented here, it isassumed that all civilizations have similar values of L , thatis, the dispersion ∆ L is ∆ L (cid:28) L . This need not be the case.Statistical treatments, considering ∆ L ∼ L , and canonicaldistributions have been studied (Maccone 2010; Glade et al.2012). However, in any case, the unknown mean value of L plays a more important role than the equally unknown formof its distribution. The region below the line N = 1 is not necessarily void. Itcorresponds to values of N < L : f = L /T.In Fig. 2, this is illustrated by assuming arbitrarily that L = 2 My and T = 10 My. Four civilizations appear within40 million years, and last for 2 My each. Their summed life-time is 8 My, that is, they exist for 8/40 = 0.2 of that timespan. For an external observer, the probability of findinga technological civilization anytime in the Galaxy is then0.2 and this number may be considered as the number N ofcivilizations of lifetime L in the Galaxy at a given momentassuming steady state.The case N < Figure 2.
Statistical interpretation of the case N=0.2, illustratedin the bottom panel of Fig. 1. L/N is the typical timescale ofappearance of civilisations of lifetime L in the Galaxy.
A simple method to evaluate the conditions under which theFermi paradox holds in the framework of the Drake formulawas suggested in Prantzos (2013): the paradox is assumed tohold when the N civilizations, expanding in the Galaxy at afraction β of the speed of light c, are able to cover collectivelywithin their ”spheres of influence” the whole volume of theGalaxy during their lifetime L .The adopted scheme for galaxy colonization is known asthe ”coral model”and was suggested in 2007 by J. Benett andS. Shostak (Bennet & Shostak 2017, latest edition) and fur-ther developed (including a statistical description) in Mac-cone (2010). It assumes that once a civilization masters thetechniques of interstellar travel, it starts a thorough colo-nization/exploration of its neighborhood for its whole life-time L . Colonization proceeds in a directed way, i.e. it con-cerns only stars harboring nearby habitable planets, whichare detected before the launching of the spaceships. Shipsare sent to new stars not from the mother planet but fromthe colonized planets in the colonization front and they arelaunched after some time interval Dt following colony foun-dation. This gives enough time to the colonizers to install onthe new planet and prepare the next colonizing mission. No-tice that υ is the effective velocity of the colonization frontand not the velocity of the interstellar ships, which has to belarger than υ (to compensate for the time required for instal-lation and preparation of new missions, but also for the factthat the new missions would not always head ”outwards” butalso ”sideways” for a full exploration of the neighborhood).Notice also that the lifetime L corresponds to a single civi-lization and includes all its offspring colonies; in other terms,the colonies do not count as different civilizations, since theydo not originate in an independent way.Fig. 3 provides illustrations of this scheme, through sev-eral Monte Carlo simulations for various combinations of thevalues of the involved parameters. It is assumed that N civ-ilizations emerge at random places in the Milky way disk and at random times during the last period of duration L .In other terms, at any time a flat distribution of ages is ob-tained. The spatial distribution of N takes into account thestellar surface density profile of the Galactic disk ( ∝ exp(-R/3 kpc)), i.e. their surface density is larger in the innerdisk. After their emergence as a technological species, civi-lizations are assumed to start expanding in the Galaxy filling A simulation along those lines was recently performed inGrimaldi et al. (2018) to explore probabilities of SETI throughradio-signal research. MNRAS , 1–9 (2015) he impact of the L factor on the Fermi paradox Figure 3.
Monte Carlo simulations of N civilizations of lifetime L in steady state, appearing in the Galaxy at random places and timesand expanding spherically during time t < L at speed υ /c (where c is the speed of light), for various values of N (=2, 10, 100, and 1000from bottom to top) and L (=10 , 10 , 10 , 10 and 5 10 y, from left to right). The values of υ /c are given in the bottom left of eachpanel. The Sun is located by the blue symbol at 8 kpc from the Galactic center and the solar circle is indicated by the blue dashedcurve. The four-arm spiral pattern and the bar of the Galaxy are (approximately) indicated by grey curves. The size of the filled circlesindicates the projected aerea of the ”sphere of influence” of each civilization covered during its age t
Monte Carlo simulations of N civilizations of lifetime L in steady state, appearing in the Galaxy at random places and timesand expanding spherically during time t < L at speed υ /c (where c is the speed of light), for various values of N (=2, 10, 100, and 1000from bottom to top) and L (=10 , 10 , 10 , 10 and 5 10 y, from left to right). The values of υ /c are given in the bottom left of eachpanel. The Sun is located by the blue symbol at 8 kpc from the Galactic center and the solar circle is indicated by the blue dashedcurve. The four-arm spiral pattern and the bar of the Galaxy are (approximately) indicated by grey curves. The size of the filled circlesindicates the projected aerea of the ”sphere of influence” of each civilization covered during its age t
N. Prantzos civilizations are common, they are unlikely to fully occupythe Galaxy, because at some point of their expansion, theirmutual interactions could reduce the pace of colonization,leaving some portions of the Galaxy unoccupied for periodsof the order of L . They base their arguments on a mathe-matical analysis drawing from the ideas of theoretical ecol-ogy and they suggest that the Earth may be found in suchan unoccupied region, thus providing another explanationof the Fermi paradox. Landis (1998) performed simulationsbased on percolation theory to simulate the expansion of civ-ilizations throughout the Galaxy. Depending on the adoptedparameters of his model, he found that large unoccupied re-gions may be found within colonized volumes, thus explain-ing the Fermi paradox. Recently, Carroll-Nellenback et al.(2019) adopted a multi-parameter scheme - taking also intoaccount the natural motion of stars within the Galaxy - toshow that clusters of continuously occupied systems, as wellas quasi-void regions could co-exist in the Milky way.All the aforementioned works and many others (see e.g.the Introduction in Carroll-Nellenback et al. 2019) adoptseveral extraneous parameters, beyond the factors of theDrake formula, and those parameters play a crucial role inthe final outcome of the corresponding studies. We thinkthat for the first time a framework using explicitly and ex-clusively the parameters of the Drake formula is adopted;the addition of one more parameter, the speed of the expan-sion front, is mandatory in order to place the discussion inthe context of the Milky Way taking into account its size. Our scheme provides a quantitative answer to Fermiˆa ˘A´Zsquestion in terms of probabilities: assuming that there areat present N civilizations in the Galaxy, appearing randomlyin space and time and exploring their neighborhood for time L at speed υ , the probability that our solar system is ”cur-rently” (i.e. in the last L ) within at least one of their ”vol-umes of influence” V i can be defined as the fraction F ofthe Galactic volume V G occupied by the ensemble of thosevolumes: F ( N, L, υ ) = (cid:80) Ni =1 V i V G (7)For large N, the sum tends to N < V > , where the averagevolume is < V > = 1 L (cid:90) L V ( L ) dL = 1 L π (cid:90) L ( υt ) dt = π υL ) for ( υL < h G )= 1 L π h G (cid:90) L ( υt ) dt = π h G υL ) for ( υL > h G )(8) Figure 4.
Results of a systematic investigation of the N vs L plane for υ /c=0.001, assuming a formation rate of habitable plan-ets R ASTRO =0.1 yr − in the Milky Way. Blue diagonal lines indi-cate the number N as function of L and correspond to f BIOTEC =1 (maximum value, solid line), 10 − , 10 − and 10 − (dottedblue lines). The probability of the N civilizations covering theGalactic volume with their ˆA´nspheres of influence ˆA˙z is colourcoded. Probabilities of 10 − , 10 − , 10 − and 1 are indicated bythe green thin curves. In the yellow region to the right of thefigure, the probability is ≥
1. To its left, the probability of con-tact decreases with L − , to become negligible in the cyan shadedregion. The six models of Fig. 3 which adopt υ /c=0.001 appearwith the corresponding letters (see the text). This leads to F ( N, L, υ ) =
N < V > V G = 13 N ( υ L ) h G R for ( υL < h G )= 13 N ( υ L ) R G2 for ( υL > h G ) (9)In fact, the dependence of the probability on L is evenstronger than it appears from Eq. 9, because N is not anindependent variable, it depends on L according to Eq. 2where the independent factor is f BIOTEC rather than N. Re-placing N in Eq. 9 through its expression of Eq. 2 leads toa dependence of the probability on L or L : F ( f BIOTEC , L, υ ) = 13 R ASTRO f BIOTEC L ( υ L ) h G R ( υL < h G )= 13 R ASTRO f BIOTEC L ( υ L ) R G2 ( υL > h G )(10)Thus, although f BIOTEC and L enter the Drake formulalinearly, the latter has a much greater impact on the Fermiparadox than the former: a variation in f BIOTEC by fourorders of magnitude can be compensated by a variation of L MNRAS , 1–9 (2015) he impact of the L factor on the Fermi paradox by a factor of ten. Despite that, we shall keep our subsequentdiscussion in terms of the (N, L ) variables, which are moreintuitive.An illustration of our quantitative evaluation for thecase υ /c=0.001 is provided in Fig. 4 for R ASTRO =0.1 y − .The solid line at f BIOTEC =1 bounds N from above, i.e. thereare no civilizations on the upper left part of the diagram inthe steady state situation. The yellow shaded region to theright corresponds to F ≥
1, indicating full coverage of theGalaxy and even overlapping spheres of influence . Thisregion indicates the parameter space whereas Fermiˆa ˘A´Zsquestion does not admit a physical answer: they should behere but we don’t see them, so where are they ? However,the values required for the various parameters are ratherhigh. If space-faring technology is trivially developed in theGalaxy ( f BIOTEC ∼ civilizations should co-exist (up-per right part) each one expanding for more a million yearsat one thousandth of the speed of light in order to fullycover the Galactic volume and thus render the observed ab-sence of contact with them truly problematic. The situationwould be equally problematic in the case of rare technology( f BIOTEC ∼ − ), requiring a dozen civilizations to expandat the same speed but for about 10 years (lower right part).Regions to the left of the yellow region correspond toprogressively lower probabilities, as indicated by the threesolid green curves at F =1, 10 − , 10 − and 10 − , where civ-ilizations are too few or too short-lived to collectively col-onize the whole Galaxy. The importance of the factor L isclearly seen: a reduction of L by a factor of 10 decreases theprobability of covering the Galactic volume by a factor of100-1000, making contact rapidly improbable. On the otherhand, a decrease of f BIOTEC by a factor of 100 can be easilycompensated by an increase of L by a factor of 5-7, as canbe seen in the case of each of the green probability curves. Inthe cyan shaded region to the left of the curve F =10 − theprobability is practically zero. Fermi’s question can be easilyunderstood in physical terms in that region, without invok-ing any ”sociological” reasons (e.g. unwillingness to exploreor contact us, cosmic ”zoo” etc.)Each of the lettered points in Fig. 4 indicate the corre-sponding models in Fig. 3, which adopt υ /c=0.001. Again,as discussed before, the role of L is clearly and quantita-tively illustrated: the probability P decreases by a factor of10 when L decreases by 100 (going from Model S to Q, forthe same N=1000), but it decreases more slowly, by a factor100 when N decreases also by 100 (going from S to D, forthe same L =10 ).Simulations for different values of υ /c, from 0.9 to0.0001, are summarized in Fig. 5: the shaded area to theright of each curve, identified by the corresponding υ /c, isthe region of F ≥ No attempt is made here to evaluate the impact of overlap-ping spheres of influence on the outcome of the calculation, i.e.would the contact between civilizations stop the expansion of oneof them (or both) or, on the contrary, would it accelerate thatexpansion?
Figure 5.
The Fermi paradox presented in the N vs. L plane,in terms of the Drake formula and for R ASTRO =0.1 yr − inthe Milky Way. It is assumed that the colonization front expandswith average speed υ /c=0.9, 0.1, 0.01, 0.001 and 0.0001, as in-dicated by the dotted curves. The Fermi paradox holds to theright part of each speed curve, i.e. the probability of a collectivecolonization of the Galaxy by N civilizations is =1 in that re-gion. In the region in the bottom left part the Fermi question canbe understood in physical terms, since the probability of collec-tive occupation of the Galaxy is practically zero (”No paradox”).To its right, probabilities are formally high, but conditions arerather too ”optimal” in some cases (thus a ”Weak paradox”). Itremains to be seen whether there are regions of the parameterspace where conditions are ”reasonable”, allowing one to qualifyFermi’s question as a ”Strong paradox” (see the text). paradox” is satisfied. It is essentially a simple ”geometrical”criterion which can be expressed as N ( υ L ) ≥ G R for ( υL < h G ) N ( υ L ) ≥ G2 for ( υL > h G ) (11)for the case of the ”coral model” adopted here. To the left ofeach curve, the probability of contact rapidly decreases (theinequalities of Eq. 11 are inversed) and there is no Fermiparadox for the corresponding value of υ /c (as in Fig. 4).The case of υ /c=0.9 is shown only for illustration purposes,as it is improbable that civilisations may expand at suchhigh speed (and even at υ /c=0.1), because the spaceshipsshould travel even faster than that.In Prantzos (2013) it is argued that if N is ”small” (arbi-trarily taken to be N < υ /c > f BIOTEC
MNRAS000
MNRAS000 , 1–9 (2015)
N. Prantzos must certainly be lower than its maximal possible valueof 1. For illustration purposes, we have put the value f BIOTEC =10 − in Fig. 5, but the actual value may turnout to be much smaller. Future observations may reveal thepresence of life elsewhere (Kopparapu et al. 2019) probingthus the value of f l , the first of the three factors of the term f BIOTEC in Eq. 4. On the other hand, evolutionary biologistsargue for values of the intelligence factor f i much lower than1 (see Lineweaver 2009, and references therein), in whichcase the region of the ”strong paradox” would shrink muchmore.For all those reasons, it is argued that in a fairly largeregion of the diagram (orange shadowed in Fig. 5) Fermi’squestion constitutes only a ”weak paradox”. It remains to beseen whether there are regions to the right of the diagramwhere Fermi’s question can be considered as really mean-ingful (”strong paradox”), i.e. : if there are so many, live solong and expand at not an unreasonable speed, why arenˆa ˘A ´Ztthey here? From Fig. 5 it appears that, for ”realistic” expansionspeeds ( υ /c < L ∼ f BIOTEC , L and υ ), accounting alsofor the astrophysical setting i.e. the size of the Galaxy. Butthis ”quantification” obviously cannot help with other im-portant issues related to the Fermi paradox, e.g. ” to assertthe presence of extraterrestrials on Earth in the recent or dis-tant past, what kind of tracers should we seek for? ”(see e.g.Schmidt & Frank 2019, and references therein). The lack ofa convincing answer to that question obviously limits theutility of Fig. 5. In this work the Fermi paradox is analyzed in terms of a sim-plified version of the Drake formula - originally suggested inPrantzos (2013) - and the role of the civilization lifetime L is emphasized. Several novelties are introduced in the dis-cussion.a) The condensed form of Drake’s formula (N =R ASTRO f BIOTEC L ) is presented graphically in the plane Nvs L , assuming that the ”astronomical” factor R ASTRO canbe determined from present-day and forthcoming observa-tions; based on current understanding, R ASTRO =0.1 yr − is adopted throughout this work. The plane is covered bythe different values assumed for the ”biotechnological” fac-tor f BIOTEC and N is bound from above by the value N=0.1 L (where L is expressed in yr), since f BIOTEC , Max =1.b) The possibility of N < L of a civilization issmaller than the typical timescale T of the emergence of two successive civilizations in the Galaxy, i.e. N= L /T. With thisinterpretation, the Drake formula covers the case of civiliza-tions being ”alone in space” (within the Milky Way), but not”alone in time”.c) It is argued that, in the steady state situation ex-pressed by Drake’s formula, the lifetime L plays a key role,even larger than f BIOTEC , despite the fact that both fac-tors enter the formula in a linear way. The reason is thatthe volume of the ”sphere of influence” of communicatingcivilizations (either through direct contact or through ELMsignals) depends on L for an isotropic expansion, at least inthe framework of the ”coral model” for Galactic colonizationadopted here.d) This dependence is illustrated by Monte Carlo simu-lations performed for various values of the parameters N, L and υ (the typical speed of the expansion front of the civi-lizations), in a disk galaxy with the dimensions of the MilkyWay.e) A quantitative criterion is proposed to evaluate thechances of contact: the probability of contact is the fractionof the Galactic volume occupied by N ”volumes of influence”during the last period of duration L . Eq. 9 and 10 revealthe strong influence of the factor L on the discussion of theFermi paradox.f) This criterion allows one to define in the plane N vs L regions where the probability of contact is high or low,for a given assumed value of the expansion speed υ (Fig. 4).For sufficiently large values of N and L , a probability is P ≥ if they are so numerous and expand for so long andsufficiently rapidly, then why are they not here? The caseP > N and L ) to the size of the Galaxythrough the expansion speed υ .h) This original presentation allows one to display quan-titatively in a single figure (Fig. 5) and in a compact form,the three physical explanations mostly discussed as solu-tions to the Fermi paradox(see Webb 2015, and referencestherein): rare civilizations (low N ), too short-lived (low L ,making them unable to arrive here, even if they are numer-ous and/or expand rapidly), or unable to expand at suffi-ciently high speed (low υ ).i) It is argued that in a large region of the dia-gram the corresponding parameters suggest rather a ”weak”Fermi paradox. Future research may reveal whether a”strong”paradox holds in some part of the parameter space.In any case, it appears that for ”realistic” expansion speeds( υ /c < ∼ The number N of Galactic civilisations is obtained bythe Drake formula and reflects a steady-state.
In principle, adifferent framework may be conceived, in which the ”produc-tion rate” of civilisations is not connected to the birth rate ofhabitable planets or does not correspond to a steady state:for instance, they may appear in ”waves” in time and/or inspace following some ”special” event, e.g. the passage of a
MNRAS , 1–9 (2015) he impact of the L factor on the Fermi paradox spiral wave or some other perturbation. Such considerationswould make the analysis considerably - and unnecessarily,at this stage - more complex.b) Civilisations appear randomly in the Galactic volume (the MC simulations of Fig. 3 take into account the radialstellar density profile, but the analytical criteria of Eq. 11 donot); the situation would be obviously different if, for somereason, some Galactic places are systematically favoured.c)
The expansion front expands as in the ”coral model” and its radius increases as r exp ∝ L , leading to a strong de-pendence of the ”volume of influence” on L ( V exp ∝ L ). Inother models, like the diffusion model adopted in e.g. New-man & Sagan (1981), the radius increases as r exp ∝ √ DL -where D is the diffusion coefficient - and the dependence on L is weaker ( V exp ∝ L / ). Although it is trivial to calculatesuch models in the adopted framework (replacing the appro-priate quantities in Eq. 8 to 11), it is hard - and certainlynot intuitive - to decide about the values of the diffusioncoefficient D and even about the physical meaning of suchmodels.For several decades, the Drake formula played an im-portant role in the search for extraterrestrial life, providinga framework to formulate our current understanding abouta very complex phenomenon such as the development of lifeand intelligence in the astrophysical setting of the MilkyWay (Prantzos 2000). In this study, we show that it can alsobe used to constrain quantitatively the ”physical” answersto Fermi’s question. Forthcoming developments in variousfields related to astrobiology, space sciences, communicationtheory, big data analysis etc. are expected to enrich furtherour understanding of this topic (e.g. Cabrol 2016, and refer-ences therein). ACKNOWLEDGEMENTS
I am grateful to my colleague Gary Mamon at IAP for in-valuable help with the graphics of this work (and many oth-ers).
REFERENCES
Ashworth S., 2014, Journal of the British Interplanetary Society,67, 224Bennet J., Shostak S., 2017, Life in the Universe, 4th edition.Monograph, PearsonBjørk R., 2007, International Journal of Astrobiology, 6, 89Cabrol N. A., 2016, Astrobiology, 16, 661Carroll-Nellenback J., Frank A., Wright J., Scharf C., 2019, AJ,158, 117Chomiuk L., Povich M. S., 2011, AJ, 142, 197Cirkovic M. M., 2018, The Great Silence: Science and Philosophyof Fermi’s Paradox. Oxford University PressCrawford I. A., 2018, Direct Exoplanet Investigation Using In-terstellar Space Probes. p. 167, doi:10.1007/978-3-319-55333-7 167Drake F. D., 1961, Physics Today, 14, 40Fogg M. J., 1987, Icarus, 69, 370Furley D., 1987, The Greek Cosmologists: vol. 1: The Formationof the Atomic Theory and its Earliest Critics. Cambridge:Cambridge University PressGlade N., Ballet P., Bastien O., 2012, International Journal ofAstrobiology, 11, 103 Grimaldi C., Marcy G. W., Tellis N. K., Drake F., 2018, PASP,130, 054101Hair T. W., Hedman A. D., 2013, International Journal of Astro-biology, 12, 45Haqq-Misra J., Kopparapu R. K., Wolf E. T., 2018, InternationalJournal of Astrobiology, 17, 77Hart M. H., 1975, QJRAS, 16, 128Hsu D. C., Ford E. B., Ragozzine D., Ashby K., 2019, AJ, 158,109Jones E. M., 1981, Icarus, 46, 328Jones E. M., 1985, Technical report, Where is everybody? anaccount of Fermi’s questionKopparapu R. k., Wolf E. T., Meadows V. S., 2019, arXiv e-prints,p. arXiv:1911.04441Kroupa P., 2002, Science, 295, 82Landis G. A., 1998, Journal of the British Interplanetary Society,51, 163Lawton J., May R., 1995, Extinction rates. Monograph, OxfordUniversity PressLineweaver C. H., 2009, Geochimica et Cosmochimica Acta Sup-plement, 73, A769Little J. D. C., 1961, Operations Research, 9, 383Maccone C., 2010, Journal of the British Interplanetary Society,63, 222Mayr E., 1992, Naturwissenschaftliche Rundschau,, 45, 264Newman W. I., Sagan C., 1981, Icarus, 46, 293Ostriker J. P., Turner E. L., 1986, Journal of the British Inter-planetary Society, 39, 141Paudel R. R., Gizis J. E., Mullan D. J., Schmidt S. J., BurgasserA. J., Williams P. K. G., Youngblood A., Stassun K. G., 2019,MNRAS, 486, 1438Petigura E. A., et al., 2018, AJ, 155, 89Prantzos N., 2000, Our Cosmic Future: Humanity’s fate in theUniverse. Monograph, Cambridge University PressPrantzos N., 2013, International Journal of Astrobiology, 12, 246Schmidt G. A., Frank A., 2019, International Journal of Astrobi-ology, 18, 142Schwieterman E. W., Reinhard C. T., Olson S. L., Harman C. E.,Lyons T. W., 2019, ApJ, 878, 19Selsis F., 2007, Habitability: the Point of View of an Astronomer.p. 199, doi:10.1007/978-3-540-33693-8 7Simpson G. G., 1964, Science, 143, 769Viewing D., 1975, Journal of the British Interplanetary Society,28, 735Wandel A., Gale J., 2019, arXiv e-prints, p. arXiv:1907.11098Webb S., 2015, If the universe is teeming with aliens...where iseverybody?. Springer InternationalZackrisson E., Calissendorff P., Asadi S., Nyholm A., 2015, ApJ,810, 23This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000