Table Of ContentUpper limits on the probability of an interstellar civilization arising in the local Solar
neighborhood
Daniel Cartin∗
Naval Academy Preparatory School, 440 Meyerkord Avenue, Newport, Rhode Island 02841-1519
(Dated: January 26, 2015)
At this point in time, there is very little empirical evidence on the likelihood of a space-faring
species originating in the biosphere of a habitable world. However, there is a tension between the
expectationthatsuchaprobabilityisrelativelyhigh(givenourownoriginsonEarth),andthelack
of any basis for believing the Solar System has ever been visited by an extraterrestrial colonization
effort. Thispaperseekstoplaceupperlimitsontheprobabilityofaninterstellarcivilizationarising
onahabitableplanetinitsstellarsystem,usingapercolationmodeltosimulatetheprogressofsuch
5
ahypotheticalcivilization’scolonizationeffortsinthelocalSolarneighborhood. Tobeasrealisticas
1
possible, the actual physical positions and characteristics of all stars within 40 parsecs of the Solar
0
System are used as possible colony sites in the percolation process. If an interstellar civilization is
2
very likely to have such colonization programs, and they can travel over large distances, then the
n upperboundonthelikelihoodofsuchaspeciesarisingperhabitableworldisontheorderof10−3;
a
on the other hand, if civilizations are not prone to colonize their neighbors, or do not travel very
J
far, then the upper limiting probability is much larger, even of order one.
2
2
I. INTRODUCTION included in the model developed by Cartin [4]. There a
]
h percolation model was also used, but instead of a fixed
p Afterdecadesofhumanexplorationofspace,itappears lattice, the observed positions of all known star systems
- thattherehavebeennovisitorstotheSolarSystemfrom within 40 parsecs (pc) of the Solar System were taken as
p
o other star systems. This is in sharp contrast to the (per- colonization sites.
p hapsnaive)viewthat,withhundredsofbillionsofstarsin The main focus of Ref. 4 was the following question:
. theMilkyWaygalaxy,thelikelihoodoffindinghabitable supposing each star system is the point of origin for a
s
c planets around many of these stars, and the possibility space-faring civilization with a given sociological drive
i of life arising on a good proportion of these planets – and technological prowess, how many could reach the So-
s
y as evidenced by the seeming rapidity of life developing lar System with their colonization process? To codify
h on the young Earth – there should be a large number this, the two parameters of the model were a probabil-
p of technological civilizations, and at least one of these ity p for a given settled system to colonize a particular
[
shouldhavedevelopedspacecraftcapableofreachingthe neighboringstellarsystem(“sociologicaldrive”),andthe
1 Solar System. The variance between this supposition, maximum travel distance Dmax that its spacecraft can
v and the lack of evidence for any visitations, is known as travel, which thus determines adjacency (“technological
4 the Fermi paradox [2, 6]. Its resolution has vexed many prowess”). The Monte Carlo simulations of these per-
0
mindsoverthedecades,althoughatthispoint,empirical colation processes allowed the calculation of the number
9
data sufficient to address the question is lacking. of systems originating colonization programs that could
5
This has not stopped the development of theoretical reach the Solar System as a function φ(p,D ). In
0 max
. models whose intent is to provide insight into the Fermi other words, φ(p,Dmax) is a correlation function reflect-
1
paradox. An as example appropriate to our discussion, ing whether the Solar System and any other given star
0
Landis[8]originallydevelopedamodelofinterstellarcol- system are in the same cluster of systems colonized by a
5
1 onizationprocessesasasimplepercolationmodel, where given interstellar civilization.
: eachsettledsystemhasagivenprobabilitypofcolonizing This paper asks a different question, namely: for this
v
all of its neighboring systems; since the model was sim- calculated function φ(p,D ) and the evident lack of
i max
X ulated on a cubic lattice, this fixed the number of such visitors to the Solar System, what are the upper limits
r neighborstobesixforeverysinglestarsystem. Agener- onthelikelihoodofspacefaringlifeoriginatingonagiven
a alizationofthismodelwasgivenbyHairandHedman[5], habitableplanet,asafunctionofpandD ? Thenum-
max
where the cubic lattice was still used, but colonization ber of interstellar civilizations reaching the Solar System
along diagonal connections was allowed – so each system mustbethenumberofstellaroriginsitesforcolonization
had up to 26 neighboring systems – and p was the prob- efforts could possibly reach us, multiplied by the proba-
ability for each neighboring system (rather than all) to bilityofastar-faringcivilizationarisingateachstarsys-
becolonizedbyaneighbor. However,neithermodelused tem. Fromtheself-evidentlackofvisitors–meaningthe
the actual “geography” of star systems; this aspect was numberofcolonizingeffortsreachingtheSolarSystemis
less than one – the probability of a species willing and
able to execute a colonization program must be simply
the inverse of the probability of such civilizations evolv-
∗ [email protected] ing in a given star system. Section II develops this idea
2
further, by using an equation for the expected number of civilizations visiting the Solar System is given by1
of interstellar civilizations based on a break-down of the
(cid:88)
variousrelevantfactorsinamannersimilartotheDrake (cid:104)N(cid:105)= N φ (p,D ) (1)
k k max
equation, and solving for the probability of a spacefar-
k
ingspeciesarisingonagivenhabitableworldintermsof
current constraints on such exoplanets. As with Ref. 4, We can parametrize N much like the Drake equation to
k
all types of stellar systems are lumped into three broad get
categories–brightsinglestars,dimsinglestars,andmul-
(cid:88)
tiple star systems. Specific forms of φ (p,D ) are ob- (cid:104)N(cid:105)= f n f f φ (p,D ) (2)
α max p,k h,k (cid:96),k S,k k max
tained by fitting the results of the simulations in Ref. 4
k
to a modified power law, where α runs over the three
system types. Using these fitting functions, an example where the parameters fp,nh,f(cid:96), and fS, for each stellar
case is shown for a choice of habitable planet occurrence system k, are defined in Table I.
rates around various star system types, and the maxi-
mum possible probability of a spacefaring species arising Symbol Definition
within the local Solar neighborhood are calculated as a f Fraction of stars with planets
p
function of p and Dmax. These results are summarized nh Number of those planets within
and discussed in Section III. the habitable zone of their stellar system
f Fraction of planets in habitable zone that
(cid:96)
develop life
f Fraction of life-bearing planets developing
S
a star-faring civilization
TABLE I. Definitions of parameters used in equation (2) for
II. CONSTRAINTS ON LOCAL theexpectednumber(cid:104)N(cid:105)ofinterstellarcivilizationsreaching
INTERSTELLAR CIVILIZATIONS the Solar System. The total probability Nk of a space-faring
civilizationarisinginagivenstarsystemistheproductN =
k
f n f f .
p h (cid:96) S
As mentioned in Section I, this work begins with
an equation based on the Drake equation, which Since Ref. 4 only considers those star systems within
parametrizes the number of extraterrestrial civilizations 40 pc, we limit our discussion to that volume to find the
in the Milky Way galaxy from a combination of astro- expected number of civilizations reaching the Solar Sys-
physical,biological,technologicalandsociologicalinputs, tem. Theoretically, at least, the astrophysical properties
with an eye towards estimating the total number of civi- ofeachstellarsystemwithinthatvolume–i.e. thenum-
lizations which are available for radio or other communi- ber of planets in each system, and the number of those
cations. A similar idea is useful in estimating the possi- within the habitable zone for that system – could be di-
bility of direct contact with an alien civilization. Specif- rectly measured, or at least estimated based on the age,
ically, such visits are only possible if (1) a potentially spectral type, and metallicity of each star present in the
space-faring civilization arises in another star system, system [10]. However, even those projects such as the
and (2) this civilization has the willingness and technol- Kepler spacecraft have garnered a huge amount of infor-
ogy to reach the Solar System, possibly after colonizing mation about exoplanets, the study of other worlds is
one or more star systems along the way. It is now pos- still young, and only broad outlines can be seen at the
sible to make educated guesses on the likelihood of the moment. Thus, to match current knowledge, the sum
first point. The result of this computation is a variable (2) for the expected number of visiting civilizations is
N ∈ [0,1], for a star system k, where N denotes the divided into three pieces, according to whether each sys-
k k
probability of an interstellar civilization arising in star tem consists of a single star of spectral class K or earlier
systemk. Notethatweusetheterm“system”here,since (“bright”), a single star of class M or other non-stellar
we include the possibility of space-faring species origi- object (“dim”), or a system of multiple stars (“multi-
nating within a multiple star system. The second is an ple”).
appropriate setting for the percolation model developed In addition, there are two parameters more in the as-
earlier in Ref. 4. The results calculated in that paper trobiologicalrealmthantheastronomical,namelyf and
(cid:96)
implicitly give a correlation function between the Solar f . The fraction f of habitable zone worlds where life
S (cid:96)
System and all other systems in question, e.g. the prob-
ability that the Solar System and another given system
arebothwithinthesamecolonizationcluster,basedona
choice for colonization probability p and travel distance 1 Equations (2) and (3) of Brin [2] are similar in spirit to the
D . Forastellarsystemk,thiscorrelationprobability equationspresentedhere,butusedifferentparameterstodefine
max
is denoted φ (p,D ). Then the expected number (cid:104)N(cid:105) themodelspace.
k max
3
evolves is unknown at the moment2, although this could
be constrained by potential near-term space missions,
such as those craft currently exploring Mars, or possi-
ble missions for Europa, Titan and Enceladus. On the
other hand, the fraction f of inhabited worlds where
S
an interstellar civilization arises is completely unknown,
since as far as we know, the answer is zero. In light of
our ignorance of the processes behind f and f , it is
(cid:96) S
reasonable to keep the model as simple as possible; thus,
for the following the product f f is assumed to be the
(cid:96) S
sameforallstarsystemtypes,andtheremainingparam-
eters are taken to be identical for a given system type
α ∈ {b,d,m}. This means the sum (2) for the expected
number of visiting civilizations is given by
(cid:20) (cid:21)
(cid:88) (cid:88)
(cid:104)N(cid:105)=(f f ) (f n ) φ (p,D )
(cid:96) S p h α k max
α k∈α
FIG. 1. Contour plot of the maximum possible probabil-
ity f f of a star-faring civilization arising on an Earth-like
The results of Ref. 4 allows the functions (cid:96) S
planet in another star system, with spacecraft from that civ-
(cid:88) ilization reaching the Solar System. This graph uses the
(cid:104)N (p,D )(cid:105)= φ (p,D )
α max k max choice of Earth-like planets existing in other star systems
k∈α (f n ) = (0.0232,0.309,0.0174), for bright and dim stars,
p h α
and multiple star systems, respectively.
tobecalculatedforallsystemskwithin40pcoftheSolar
System with a particular system type α, so the previous
equation (2) for (cid:104)N(cid:105) will be used in the form
As a way of seeing the behavior of the number of sys-
(cid:88) tems that can reach the Solar System, we fit each of the
(cid:104)N(cid:105)=(f f ) (f n ) (cid:104)N (p,D )(cid:105) (3)
(cid:96) S p h α α max three functions (cid:104)N (cid:105) to a modified power law in p, given
α
α∈{b,d,m} by
civTilhiziasteioqnusattioonreafocrhtthheeeSxoplaerctSeydstneummibsenrootfhienltpefrusltealslairt (cid:104)Nα(p,Dmax)(cid:105)=Aαpc1,αDm2ax+c2,αDmax+c3,α (5)
stands,sinceitdependsonunknownprobabilitiesf and
(cid:96) ThisgivesusthecoefficientsgiveninTableII,wherethe
f . However, the only observable fact that can be used
S values of (cid:104)N (cid:105) are fit for p ∈ [0,1] and D ∈ [3.0,7.0]
inthisdiscussion–namely,thecompletelackofevidence α max
pc. ThelargeandnegativecoefficientofthelinearD
thattheSolarSystemhasbeenvisitedbyextraterrestrial max
termsintheexponentshowthat,asthemaximumtravel
spacecraft–doesservetoconstraintheproductf f . In
(cid:96) S distance increases, a smaller value of the colonization
otherwords, thefactthat(cid:104)N(cid:105)<1allowsthecalculation
probabilityisnecessaryinordertoachievethesameover-
of upper limits on the product f f , since this limit on
(cid:96) S all value of (cid:104)N (cid:105), for a given system type. However, the
the maximum value of the equation (2) for (cid:104)N(cid:105) implies α
positive quadratic coefficient shows that this tendency
levels off a bit as D increases. Both of these effects
(cid:20) (cid:21)−1 max
(f f )< (cid:88) (f n ) (cid:104)N (p,D )(cid:105) (4) are more pronounced for the bright stars and multiple
(cid:96) S p h α α max
star systems than they are for the dim stars.
α∈{b,d,m}
All quantities on the right-hand side of this inequality System type A c (pc−2) c (pc−1) c
1 2 3
are either probabilities now grounded in exoplanet sur- bright 2074 0.4905 -5.743 18.52
veys (f ) and climate models (n ) for these worlds, or
p h dim 1598 0.2351 -2.714 9.668
elsecalculatedfromthepercolationmodel((cid:104)N (cid:105))ofRef.
α multiple 1575 0.4896 -5.753 18.60
4, basedonassumedvaluesforthesociologicaldriveand
technological abilities of an interstellar civilization, i.e. TABLE II. Fitting coefficients for the functions
the colonization probability p and maximum travel dis- (cid:104)N (p,D )(cid:105) giving the number of stellar systems of
α max
tance D , respectively. a given type α∈{b,d,m} able to reach the Solar System for
max
a given choice of colonization probability p and maximum
travel distance D (in pc). The functional form of
max
(cid:104)N (p,D )(cid:105) is given in equation (5).
α max
2 As noted by Spiegel and Turner [9], current knowledge about
theevolutionoflife(specificallyonEarth)doeslittletolimitthe As a specific instance of the use of this model, values
valueoff(cid:96). fortheparametersarechosenbasedonageometricmean
4
ofresultsgivenintheliterature, citedbyWinnandFab- evolving within any star system is around 10−5. Rela-
rycky [10]. Specifically, the occurrence rates from Table tivechangesinthesizesof(f n ) forthethreedifferent
p h α
2 of that work are used, where the three listings for M system types α may change the shapes of the contours
stars are used for dim stars, the four entries for FGK giveninFigure1,buttheupperlimitvaluesforagivenp
are combined with the GK entry for bright stars, and andD willbealteredonlyforabsolutechangesinthe
max
theoccurrencerateforhabitableplanetsinmultiplestar order of magnitude for the number of habitable planets
systems is chosen as 75% of that for bright stars. This per system, regardless of system type.
gives the parameter choices of
(f n ) =[(f n ) ,(f n ) ,(f n ) ]
p h α p h bright p h dim p h multiple III. CONCLUSIONS
=(0.0232,0.309,0.0174)
The maximum possible values of f f are then calcu- In this paper, the model of interstellar colonization as
(cid:96) S
lated, and a contour plot showing the results is given apercolationprocess,originallydevelopedbyLandisand
in Figure 1. We now discuss the results shown in this appliedtothephysicalpositionsofallstarsystemswithin
figure; these comments relate specifically to the choice 40pc,isusedtodevelopupperlimitsontheprobabilityof
of (f n ) used here, but the broad outline remains the aninterstellarcivilizationarisingonanEarth-likeplanet,
p h α
same regardless of what values are chosen. andspacecraftofthatcivilizationreachingtheSolarSys-
tem. Themodelwasparametrizedinamannersimilarto
• Interstellar civilizations capable of making longer the Drake equation, with coefficients giving the fraction
journeys between star systems (higher D ), and f of stars with planets, and the number of those plan-
max p
more likely to begin these journeys (larger p), have ets n within the habitable zone of each stellar system,
h
ahigherchanceofreachingtheSolarSystem. Thus, thefraction ofhabitable planetsf thatactually develop
(cid:96)
the empirical fact that visiting spacecraft of this life, and finally the fraction of those biospheres f which
S
type are not observed places stronger limits on originate a space-faring civilization. At this point in our
f f , and so the upper limit of this product de- knowledge, there are now empirical constraints on the
(cid:96) S
creases in size as we move towards the upper-right first two parameters, but the latter are more or less free.
hand corner of Figure 1. The philosophy of this paper is to use a simple model in
order to place some upper limits on those values dealing
• Specifically,themaximumrangeoftraveldistances
with the origin of life and interstellar civilizations, since
for interstellar colonizers studied in this research
there is currently no observational data on these issues.
was chosen to be 7.0 pc. If such a civilization al-
Our ignorance of many of the processes involved in the
ways colonized its neighbors (p = 1), then the up-
evolutionofatechnologicalspecies, andthelikelihoodof
perlimitontheprobabilityofsuchaspeciesarising
that species choosing to develop interstellar settlements
within a biosphere less than 40 pc away from the
leads to the choice of the simple model given here, with
Solar System is f f ≤0.00171. In other words, at
(cid:96) S as few parameters as possible to give a reasonable look
most only one of out every 585 habitable planets
at the question at hand.
within the local Solar neighborhood could be the
However, in some sense we have just moved our igno-
cradle of an interstellar civilization in order not to
rance–insteadofleavingf andf open, thisworkputs
have evidence of their spacecraft in our Solar Sys- (cid:96) S
upper limits on those values at the expense of two new
tem.
coefficients, the probability p of a space-faring species
• There is a relatively large regime of colonization to colonize one of its neighboring star systems, and the
probabilitiespandmaximumtraveldistancesDmax maximum distance Dmax that can be considered ”neigh-
where interstellar civilizations would never reach boring”. Yet this shift in focus may be helpful, in that it
theSolarSystem,eveniftechnologicalcivilizations puts a different light on questions relating to the Fermi
capable of star travel arose on every single Earth- paradox. Indeed, as shown elsewhere [3], a maximum
like planet in the Solar neighborhood. The reach travel distance Dmax of about 3 pc is sufficient to allow
of these interstellar civilizations could be quite siz- effective interstellar travel, i.e. actual travel paths are
able, as seen in Ref. 4. However, the practical ef- lessthantwicetheshortestdistancebetweenendpoints,
fecthereisthatthereisaregioninp−D space withthebenefitofpassingthroughintermediatestarsys-
max
where (cid:104)N(cid:105)<1 regardless of the value of f f . tems, allowing for replenishment and refit. As for the
(cid:96) S
colonization probability p, as yet humanity itself has not
Recall that the product f f deals with the chances taken this step, but it is for each of us to evaluate the
(cid:96) S
of an interstellar civilization arising on a given habitable likelihoodofthishappeningwhentechnologyhasreached
planet, where we have estimated values for the expected the point where this question is a viable one.
numberofsuchplanetsperstellarsystem(asdividedbe- The percolation model presented here, however, only
tweenbrightanddimstars,andmultiplesystems). Since includesnodesrepresentingallstarsystemswith40pcof
the latter is on the order of 1-10%, then the most strin- theSolarSystem, andthusdoesnotaccountforthepos-
gent limit on the probability of a space-faring species sibility of interstellar civilizations arising outside of that
5
volume. Indeed,itisperfectlyconceivablethatagalaxy- two adjacent systems, remains a problem to be studied
widecolonizationeffortmayhavestartedfromanypoint in future simulations. However, having a relatively low
within the Milky Way, as long as the colonization prob- value of p would imply that the upper limits on f f
c (cid:96) S
ability p is large enough, leading to much smaller upper would be very small, or else a galactic civilization would
bounds on f f . To truly study this issue, the perco- havearisenandeasilyreachedoutSolarSystem,contrary
(cid:96) S
lation model would also have to be extended onto the to observation.
galacticscale. Thestellarnumberdensity–theexpected Another important caveat for the work here is that
numberofstarsystemsfoundinsideacubicparsecofvol- none of the relative motion of star systems in the Solar
ume, for example – is not constant over the Milky Way, neighborhood is taken into account. This motion leads
butvariesaccordingtobothdistanceawayfromthecen- to changes in positioning over periods of only 103−104
ter of the galaxy, as well as scale height away from the yr [1]. The Monte Carlo simulations used to model the
galactic plane [7]. Thus, the study of percolation clus- percolation process in Ref. 4 are “timeless”, in the sense
ters using a physically realistic distribution of simulated that all systems are static, and the speed of colonizing
stellar systems would not correspond to previous mod- spacecraft is not considered. It is likely this time scale
els from statistical mechanics. Said in a different way, is roughly the same as the process for a newly settled
the critical probability p , i.e. the probability to have a star system from an interstellar civilization to built its
c
galaxy-spanning cluster, needed as a function of D own colonizing spacecraft, and then this craft reaching a
max
is currently unknown. Thus, the size of the largest clus- new system, so the dynamical changing of available star
ter,correspondingtotheinterstellarcivilizationwiththe systems for a star-faring species would definitely affect
greatest reach, as a function of the colonization proba- theresultspresentedhere;thisalsoisanissueforfurther
bility p and the maximum travel distance D between work.
max
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