Imperial College

Colonizing the Galaxy

A computer simulation based approach to Fermi’s paradox

January 19, 2009

Final report

Matthieu Schaller

Colonizing the Galaxy

Our sun is one of 100 billion stars in our galaxy. Our galaxy is one of billions of galaxiespopulating the universe. It would be the height of presumption to think that we are the only

living things in that enormous immensity.- Werner Von Braun -

The picture on the title page is a photography of the spiral galaxy NGC 4414 taken bythe Hubble Space Telescope. This galaxy is supposed to look like our Milky Way and couldcontain several other intelligent civilizations.

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CONTENTS Colonizing the Galaxy

Contents

1 Introduction 3

2 Research 32.1 Colonizable Stellar Systems . . . . . . . . . . . . . . . . . . . . . . . 32.2 The Galactic Habitable Zone . . . . . . . . . . . . . . . . . . . . . . 42.3 Number of colonizable stars . . . . . . . . . . . . . . . . . . . . . . . 42.4 Probability for a star to have a habitable planet . . . . . . . . . . . . 52.5 Spaceships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 Crafting a ship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 Probability for a spaceship to reach his target . . . . . . . . . . . . . 62.8 Colonizing a planet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Computer Simulations 7

4 Results 94.1 Primary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Default Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Exploring the parameters . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Conclusion 17

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1 Introduction Colonizing the Galaxy

1 Introduction

The discovery of exoplanets in the last decade has given a new youth to an oldquestion: “Are we alone in the Universe ?”. Many attempts have been done inthe history of science to answer this question with arguments that were more orless serious. In the 1950s Enrico Fermi did a simple calculation and concluded thatextraterrestrial civilizations must have visited the Earth. Since no signs of alienshave been discovered, he asked his famous question: “Where are they ?”, which isknow has “The Fermi Paradox“.

His idea was to use a statistical argument. The number of stars in the galaxy(∼ 1011) is huge enough to assume that there must be at least one other intelligentcivilization mastering interstellar travels. Thus, this civilization must have begun tospread out of its stellar system. Arrived to this stage, the time needed to reach allthe stars of the galaxy is, as he computed, approximatively 10 million years, which ismuch less than the age of the Milky Way ( 10 billion years). Hence, this civilizationmust have reached the Earth and we should have seen signs of its presence.

The aim of this project is to do computer simulations based on actual physicaldata to check if the simple calculation done by Fermi is really accurate. The com-puter program will simulate the travels of interstellar ships from stars to stars1 andmeasure the time needed to colonize a random generated distribution of stars. Wewill then extrapolate the time needed to reach all stars of the Milky Way and hencediscuss Fermi’s result for different input parameters and see whether the parameterthat we chose are relevant or not.

2 Research

This part presents the research we have undertaken to collect the most actual andrealistic data. There are two kinds of information we need. Firstly, we need toconsider which stars are colonizable and how many such stars there are in the wholegalaxy. Secondly, we need to consider the different spaceships that a civilizationcould use to travel between stars and how this civilisation would colonize a planet.

2.1 Colonizable Stellar Systems

One of the most difficult point is to determine which stars are colonizable and whichone must be left out. All planets inside a given a solar system are not habitable.Some are too near from the central star and receive too much heat (and radiations)from the central sun (e.g. Mercury) and some others are too far away and thusreceive not enough energy (e.g. Pluto). Furthermore, planets far away from theirstars are supposed to be gas-planets rather than rocky planets.

The (human) habitable zone (HZ) of a star is the zone around a star in which theenergy received per unit surface per unit time is approximatively equal to the solarconstant (1360 W/m2). For all stars, there exists a shell around the star where theenergy received per unit surface equals the solar constant. But for big stars (morethan 1.5 M�), a planet doesn’t have time to evolve enough to get habitable, since

1Technically it would rather be from a planet around a given star to a planet around anotherstar.

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2.2 The Galactic Habitable Zone Colonizing the Galaxy

Figure 1: The Galactic Habitable Zone

big stars only live for a couple million years. One the other hand, small stars maynot have enough matter around them to form planets. We will then restrain the setof stars which can have planets to stars with stellar types F,G and K2.

2.2 The Galactic Habitable Zone

The concept of habitable zone can be extended to the whole Milky Way. Indeed,life is not possible everywhere in a galaxy. The stars that are in the bulge or inits neighborhood are surrounded by an important number of Supernovae and theradiations that they emit is too dangerous for human life and for the life as we knowit. In addition, the central galactic black hole also emits highly energetic particlesthat would kill all living species on a planet too near of it. On the other hand, thezone very far from the center nearly only contains hydrogen or helium. This impliesthat stars don’t have rocky planets around them. These two limiting factors haveled the exobiologist to the definition of a Galactic Habitable Zone (GHZ) which hasthe form of a torus. The dimensions of this torus is showed on the picture 1.

For the reasons explained before, it seems to be impossible to colonize starsoutside the GHZ. So we will only consider the stars inside the GHZ for our purposeand consider the galaxy as colonized when the GHZ is fully colonized.

2.3 Number of colonizable stars

The number of stars in the Milky Way is too large to do a simulation with the realmap of stars, even with the restrictions found in the previous sections. Hence, wemust use random distributions based on one parameter, the stellar density. Thisvalue is well known and can be obtained through three different ways: integrationof the luminosity function, analysis of the galaxy’s mass or by a statistical mea-surement in the Gliese-Catalog3. The commonly accepted value is approximativelyρ = 0.0015 low-mass stars per cubic light-year4 and we will use this value as a

2This is also the approach chosen by the NASA for their Kepler Mission, which will searchhabitable planets in the neighborhood of the sun.

3The catalog only contains stars around the Sun, which are obviously inside the GHZ. .4P.Kroupa, The distribution of low mass stars in the Galactic disc, Royal Astronomical Society,

Monthly Notices vol. 262, 1993

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2.4 Probability for a star to have a habitable planetColonizing the Galaxy

constant throughout our work.

2.4 Probability for a star to have a habitable planet

All stars that have a habitable zone don’t automatically have a planet. The simplestway to deal with this fact is to set a general probability Pp for a given star to havea planet inside its HZ and therefore to have a habitable planet. The value of Pp isin the center of a a huge debate and the literature gives value between 100% andone single habitable planet in the whole Galaxy (the Earth). The NASA proposesvalues between 20% and 60% as an assumption. Our simulations will explore thisparameter to check if it is an important element or if it doesn’t affect the colonizationtime at all.

2.5 Spaceships

The question of the kind of spaceship to use is central in answering the Fermiparadox. Technologies used to travel to the moon can obviously not be used totravel to another star. In his book, R.Zubrin5 proposes different kind of ships. Wehave selected two among them, the fusion rocket and the solar sail, because thesetechnologies seem to be affordable with the Humanity’s knowledge. The fusionrocket uses the energy released during the fusion of Deuterium ( 2H) and Helium-3( 3He) atoms. As these two elements can be produced with Lithium and Hydrogen,there are no problems to find fuel for these ships everywhere in the Milky Way.

Figure 2: One of the possible fusion reaction used to produce energy ( http://www.

swip.ac.cn/xsyd/jbzs/fusion1.htm)

Such a ship could theoretically achieve a speed of 0.1c which would set α-Centauriat only 43 years of travel. This is an acceptable time and could be done withoutproblems. The other solution we have kept is the solar sail. This kind of ship doesn’tneed any fuel to travel and could achieve a speed of 0.15c if it were pushed by astrong laser beam with a radius of 100 m. Building such a laser would take a longtime but it could be used to push more than one ship once it has been built.

Finally, we need to consider the acceleration and braking time. For the braking,the best solution would be to use a Magnetic Sail and to use the braking created bythe interstellar medium interacting with a strong magnetic field. The whole calcula-

5R. Zubrin, Entering Space: creating a spacefaring civilization (New York: Jeremy P.Tarcher/Putnam a member of Penguin Putman Inc., 1999)

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2.6 Crafting a ship Colonizing the Galaxy

tion leads to a braking time of approximatively twenty years for both vehicles6. Thisseems to be really consequent but if there were no magnetic brake, the ship wouldhave to use some fuel to brake and could then achieve only half of the speed it couldreach in our case, which would lead to a longer travel time in general. However, thedistance covered during the braking is relatively small (the brake is a function of thespeed) and can be neglected in comparison with the rest of the travel. On the otherhand, the acceleration time is very short (∼ 2 months) and can also be neglected. Ifwe add the fact that there are no relativistic effects here because the speed involvedare too far from c to be affected by the contraction of length, we can deduce thatthe total travel time is simply the distance divided by the speed plus the constantbraking time.

2.6 Crafting a ship

Crafting an interstellar ship, even if all technologies have been developed, will prob-ably not be easy and will take a certain amount of time. For example, the Solar Sailrequires a 300 km× 300 km wide sail to reach the speed of 0.15c. This can only bedone in space and thus requires a long time to be completely done. For the Fusionrocket, the main problem is the gathering of tritium and deuterium. In both cases,we had to choose different times, which we called crafting times to take into accountthese facts.

2.7 Probability for a spaceship to reach his target

The history of space travel has shown many times that traveling through space isdifficult and that this difficulty increases with the travel time. It is hard to estimatewhich facts are important and what could cause a ship to ”crash“ before it reachesits target. Hence, the best way to handle with this is to use a probability. We defineP to be the probability for a given ship to reach its target. As this probabilitydepends on the time of travel, we have decided to use an exponential decay as afunction of the travel time to represent this probability. The exponential decay isdriven by a parameter γ defined such that P (100) = Pa and P (0) = 1, where Pais a parameter of our simulation. This seems to be an easy way to modelize thedifficulty to travel over centuries.

2.8 Colonizing a planet

The same kind of problems arise when a ship reaches a planet. There are plenty ofelements that could prevent a colony to survive on the planet. We could think at alack of water or no solid ground to build for examples. The best way to representour ignorance of the exoplanets we are going to colonize is again to use a probabil-ity. We have defined Ps as the probability that a given colony survives on a givenplanet. This parameter is very hard to estimate and can vary a lot if we considerthat the colonies will depend on their environment or not. If a colony dies that doesnot imply that this stellar system is definitely uncolonizable. One could think forexample that the next attempt will bring more specialized material to compensatethe mangles. In other word, a planet is never uncolonizable. This might be a re-stricting factor but one could think that in many years the humanity will be able to

6The whole calculation is done in R. Zubrin, Islands in the Sky: Bold New Ideas for ColonizingSpace (New York: Wiley, 1996)

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3 Computer Simulations Colonizing the Galaxy

Figure 3: The probability for a ship to reach its target as a function of the traveltime

survive in any conditions.

After being installed, a colony that survives may not be able to produce enoughenergy or to gather enough material to build new interstellar ships. This impliesthat the colonization of other systems would not be possible from this new startingpoint. Again, the number of planets that wouldn’t offer the possibility to produceships is best represented as a probability. Hence, we have defined Pd as the prob-ability that a given colony (that has survived) will be able to produce interstellarships. This parameter is very important because if a planet cannot offer the materialneeded, it won’t be possible to continue and this planet will just be an end point ofthe colonization process.

Before being able to send ships to other systems, a freshly arrived colony needssome time to settle in and to develop itself. We have represented this settling timethrough the parameter tsettle. The value of this parameter is also quite hard toestimate because it depends on what is exactly going to be sent through space. Ifthere are only a few persons, it would take a huge amount of time for them topopulate the planet enough in order to produce enough energy for another spacetravel. On the other hand if the ship contains a lot of material and some robots thatcould recreate human beings from DNA and quickly build power supplies, this timecould be much lower.

3 Computer Simulations

We have chosen to use the C++ programming language to write our program. Thecomplete source code and its documentation can be found on my website7.

Simulating the colonization of the whole Milky Way is impossible because of7http://www.matthieu-schaller.no-ip.fr (C++ projects section)

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3 Computer Simulations Colonizing the Galaxy

its size. Solely the number of stars is already too big to save each of them on acomputer. Thus, we have chosen to simulate the colonization of a sphere with aradius of about 300 l.y. Such a sphere contains approximatively 100′000 stars. Thissize is big enough to be acceptable for our work and small enough for our algorithmto terminate in less than one hour. Such a sphere is also easier to deal with thana torus8, because there are less border conditions. A sphere of 300 l.y. easily fitsinside the GHZ-torus and thus, our simulation would only represent the start of thecolonization, when the civilization has only conquered the nearest planets. We willthen extrapolate the results to the whole Milky Way9.The main informations wewant to collect are the number of colonized stars and the distance to the furthestcolonized star as a function of time. Obviously, the sphere containing all the colo-nized stars will grow as would a spherical wave in the air. As the GHZ is a torus,the colonizing-wave will behave as the sound does in a circular pipe. If the radius ofthe pipe is big enough, the wave will just propagate as it would in a straight pipe.The same should apply for our colonizing-wave and this is an argument to show thatone can extrapolate results from a small sphere to the whole GHZ without loss ofgenerality. Hence, the time required to cover the galaxy would be the time neededfor the furthest colonized planet to be at a distance of π · radius(GHZ). This givesus a lower bound for the value we are looking for. We can get an upper boundby adding to this value the amount of time needed to colonize a sphere of radius2000 l.y. (the radius of the GHZ torus) from a unique central starting point. It is anupper bound because it would take less time to colonize a sphere if there are morethan one starting points, which is the case when the colonizing-wave has reachedthe opposite of the torus. With these two values one can get an estimation of thecolonizing time for the GHZ.

As there are many different parameters, which can individually take a lot ofvalues, it was not possible to explore all combinations. We have decided to use a”standard“ simulation and then to change only one parameter for each simulationwe do in order to explore a little bit each of the possible directions. We can thendeduce what effect the combination of many changes will have on the time of colo-nization.

We have done researches for two different kind of ships. Hence, we have imple-mented a choice algorithm that will decide which ship is the best for every traveltaking in count the crafting and traveling times. We have chosen a set of values asthe standard parameters. They are shown in the table 4.

Description Symbol ValueProbability for a given star to have a planet Pp 0.5Probability that a colony survives on a given planet Ps 0.85Probability that a given colony develops itself Pd 0.9Probability for a ship to reach its target after a 100 years trip Pa 0.95Settling time tsettle 1000

Figure 4: Standard values for the main parameters8The actual shape of the GHZ.9We will first need to prove that the results can effectively be transposed from a relatively small

sphere to the whole GHZ.

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4 Results Colonizing the Galaxy

here are some other parameters that we could have changed, for example thecrafting times, but as these times are really small in comparison to the settlingtime, we have chosen not to change them. This can also be explained by the factthat all these times could have been represented as only one big parameter, insteadof 3 or 4 different times, which are too specific for the kind of simulation we aredoing. The values we used for these parameters are 12 years of crafting for thefusion rocket and 42 years for the Solar Sail. These values have been evaluated bylooking at how much time this would take nowadays to craft these ships and collectthe energy. But as this is not a really good approximation, one could just take thisas a constant added to the settling time10.

4 Results

4.1 Primary Results

he first interesting result we got is the fact that only one kind of ships was used, thesolar sail. The following pictures shows a small simulation running. We used thecolor code displayed on the table 5.

Color DescriptionWhite dots Not yet colonized starsGreen dots Colonized stars with a colony that develop itselfOrange dots Colonized stars with a colony that doesn’t develop itselfPink dots Colonized stars that have stopped sending shipsYellow lines Solar SailsRed lines Fusion rockets

Figure 5: Color-code used in our simulations

The following picture (figure 6) shows a screenshot of our program.As we can see, there are no fusion rockets and this is the case for all simulations,

apart maybe for the one or two ships that are sent at the beginning, probablybecause of the required crafting time of the laser. The other thing we rapidly cameon was the fact that our simulations were very slow. A simulation with a hundredthousands stars took several hours to terminate and it would not have been possibleto run all the simulations we wanted to do with big enough systems.

Thus, we decided to change our ”algorithm“ to take into account the fact thatsome colonized stars are too far away from non-colonized stars to send ships witha reasonable probability of success. We decided that stars far away from the un-colonized stars will stop seeking for available targets and just stay inactive. Theseinactive stars are painted in pink on the previous picture and are located in thecenter of the sphere of colonized stars (as expected). We also restrained the zone inwhich a star seeks for possible targets to a cube with an edge of a few light yearsaround it. This second condition may change a little bit the results if a star is faraway from all the other11 but this is very rare (less than 0.1% chance to occur) andnever happened to us.

10It is a little bit more complicated, because a given star can send more than one ship. But thiskind of approximation could de done without affecting the overall result.

11In which case, no ship would be sent to colonize it.

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4.2 Default Simulation Colonizing the Galaxy

Figure 6: A screenshot of the program

These two modifications allowed us to run bigger simulations in a shorter timeand hence to check precisely if it was reasonable to transpose results from smallsimulations to bigger ones.

4.2 Default Simulation

The first big simulation we did was the one with the default parameters, that aredisplayed in the previous table. The following figure shows a plot of the distance tothe furthest colonized star as a function of time.

Figure 7: Distance to the furthest colonized star as a function of time

The blue curve corresponds to the measured values and the red line is a linearregression of the values. As one can easily see, the distance grows as an affine map

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4.2 Default Simulation Colonizing the Galaxy

(which would be linear if we removed the first 1000 years in which nothing happens).This is a very interesting point because it allows us to transpose the result to biggersimulations very easily. As we know the slope of the curve, we can compute thetime needed to reach a star that is at any distance from the starting point. Thisapplies to spherical distributions of stars but as discussed before, the GHZ torus isbig enough to allow us to use spheres as an approximation. By applying the formulashowed on the plot, for a distance d = π · radius(GHZ), we get a colonization timeof:

t ∼=25000 · π

0.017∼= 4.6 · 106 years

This time is much smaller than the age of the Galaxy and hence shows that wecould confirm Fermi’s paradox, but we need to go further to explore more in detailsthe parameters. And this time is not the time needed to colonize the whole GHZbut just the time to reach a star which is exactly on the opposite of the Earth.

An interesting thing on the previous plot is the fact that the data seem to growas a stair function with each step being approximately 1000 years long. This isthe settling time we have chosen. We can deduce from this plot, that this is animportant limiting factor. These steps appear because all the colonizable stars ina given region have been reached and it is not possible to go further until all thenew colonized planets have developed themselves, which in our case takes 1000 years.

Another important parameter is the number of colonized stars as a function oftime. The following plot (figure 8) shows the data obtained.

Figure 8: Number of colonized stars as a function of time

Again, there are small oscillations with a period of 1000 years, which have thesame origin than before. But the global behavior of the curve is as expected, it growsquicker and quicker. A log-log plot from the same curve gives a slope of 3.4, whichimplies that the number of colonized stars grows as t3.4. One could have expecteda curve in t3, but the difference can be explained if we consider the following curve.

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4.2 Default Simulation Colonizing the Galaxy

If we draw sphere with a radius equal to the distance to the furthest colonized starand then compute the proportion of colonized stars inside this sphere, we get thecurve displayed on figure 9.

Figure 9: Proportion of colonized stars inside the sphere as a function of time

As one can see (with the help of the red line), the curve slowly grows. Thisimplies that as the sphere grows, the ratio of colonized stars inside the sphere alsogrows and thus, the expansion rate is bigger than 3 as the previous plot showed.

To verify if our simulations are correct, we can check if the different probabilitieswe have set as entry parameters remain constants during time.

Figure 10: The two probabilities as a function of time

These two curves show that, apart from the beginning where the system hasto start, the behavior of the simulation is exactly as expected and the parametersconverge to the input parameter. Thus, we can consider our simulation program asgood enough to give reliable values.

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4.3 Exploring the parameters Colonizing the Galaxy

The colonization time calculated before is the time at which the ”colonizing-wave“ has reached the opposite of the torus but there at this time, there are still alot of free stars available. As discussed before, we need to evaluate the time needed tofill a sphere of radius 2000 l.y. The simulation of such a system is very long becauseof the number of stars it contains but after more than one day of CPU-time, we gota time of 1.6 ·105 years. This represents less than 5% of the lower bound. Hence, wecan take the value 1.05 · lower bound as an upper bound for all our simulations. Thismay be wrong for some parameters but as the simulation of such a sphere needs avery long time, it is an approximation that we have to do. Hence, the value for thetime needed to colonize the Galaxy with the default parameter lies between 4.6 and4.9 million years.

4.3 Exploring the parameters

The different parameters we have chosen were not totally randomly picked up butnearly. So we need to see if they modify the overall result or if they don’t play asignificant role in this problem. As discussed before, the only relevant we want tocompute is the time needed to colonize the galaxy. The colonization density is nothelpful because of the geometry of the GHZ which is not the same than the sphereswe use in our simulations. The rest of the discussion is then an exploration of thedifferent input parameters to see which influence deeply the colonization time andwhich just let it stay around 5 · 106 years.

As we already noticed in the first simulation, the settling time seems to be animportant factor. Hence, we decided to let it vary between 100 and 10000 years. Wethen represented the lower and upper bound for the colonizing time as a function oftsettle.

Figure 11: The time needed to colonize the Milky Way as a function of tsettle.

As one can easily see, the colonizing time is a linear function of the settlingtime. This was expected after the results in the standard simulation where we sawthat there were a lot of breaks in the growth of tsettle years. Hence, as tsettle grows,the length of the breaks grows with it and thus the overall length is longer. This

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4.3 Exploring the parameters Colonizing the Galaxy

parameter plays a very important role. We chose tsettle = 1000 because it seemed tobe a reasonable time for building everything that is needed to send new ships, butthis is also quite a long time if we compare it with the evolution in our history sincethe 10th century. We could easily divide this value by two to get an as reasonablevalue with a final colonization time nearly divided by two. The important fact isthat with a reasonable settling time, the colonization time of the whole GHZ remainsunder 107 years.

Another parameter which can vary a lot is the speed a ship could achieve. Asexplained before, nearly all the travels in our simulation were done with the solar sailat a speed of 0.15c. Hence, it could be interesting to change the ship traveling speedand to use only one kind of ship to see the influence. We have taken different val-ues for the speed from Zubrin’s book12. They are summarized in the following table.

Rocket Achievable speedChemical Rocket 0.00058cFission Rocket 0.02cFusion Rocket ( 2H + 2H) 0.077cFusion Rocket ( 2H + 3He) 0.1cSolar Sail 0.15cAnti-matter Rocket 0.33cSpeed of light c

Figure 12: Speeds of different interstellar ships

Chemical rockets correspond to the traditional rockets used to travel inside oursolar system. Doing the same process than with tsettle, we got the following plot(figure 13).

Figure 13: The time needed to colonize the Milky Way as a function of the travellingspeed

12R. Zubrin, Entering Space: creating a spacefaring civilization (New York: Jeremy P.Tarcher/Putnam a member of Penguin Putman Inc., 1999)

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4.3 Exploring the parameters Colonizing the Galaxy

The first important thing to see is that there is something wrong with chemicalrockets (The dots that are at zero on the left). They are so slow that no ship canreach its target because of the probability Pa. After a very long time, some starsmay have been colonized but the colonization process stops rapidly. This occurredevery time we tried this simulation and implies that colonizing the galaxy with achemical rocket and all other parameters being the one from the standard simulationwould not be possible.The other interesting fact is that the curve seems to reach a constant value when thespeed of travel increases. There is only a small difference in the colonizing time be-tween 0.15c and c. Hence, the evolution in technology is not absolutely necessary tofully colonize the galaxy. Solar Sails that one could be able to build with technologiesnot much more developed than ours would be enough to fulfill the colonization. Thisis quite surprising as one could thing that this is the most important parameter, butas seen before, the settling time is more important. This is also interesting becauseit implies that a civilization does not need to be much more developed than ours tobe able to colonize the Milky Way. Hence, the number of civilizations able to colo-nize the entire Galaxy is bigger than if it were only possible with ships that travelat the speed of light or which would use antimatter. Fermi’s statistical argumentis in a certain manner reinforced or at least not totally wrong. One can comparethis to the Earth’s civilization: If discovering the wheel were the only technologicalrequirement to conquer the world, there would have been more ”competitors“ thanif nuclear fusion is the only solution to dominate the other civilizations.

One of the main unknown in this problem is the number of stars that have ahabitable planet. We modelised this as the ratio Pp. We have again undertaken thesame process and got the plot displayed on figure 14.

Figure 14: The time needed to colonize the Milky Way as a function of Pp

Again, we have a different behavior for small values than for bigger. If 20% ofall stars have a habitable planet or if they all have a habitable planet, there will notbe a huge difference in the colonizing time. More planets will imply shorter travelswith a greater chance of success but more stars to colonize. And we would only see

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4.3 Exploring the parameters Colonizing the Galaxy

a difference of a factor 2 or 3. On the other hand, if less than 10% of all the starscan be colonized, the colonizing time increases dramatically. It is again because ofthe too huge distance between the stars that would cause most of the ships not toreach their targets. The plot does not show any value for Pp < 0.1 because thesimulations were too long to terminate in a reasonable time.

If the ships were faster, this problem would arise at a smaller Pp but would notdisappear. Hence, this parameter is very important because it is not something thata civilization can change by improving its technology, it is a part of the Milky Way’sgeography. As we don’t know the true value of this parameter we cannot really chosea value and see what would be the real colonization time. But if it is more than10%, there wouldn’t be an important difference in the final time between all possiblevalues of Pp.

Another element that can be pointed at on this plot is the small irregularityaround Pp = 0.6. This is probably a small fluctuation because we did only one”measure“ for each Pp and not a large amount before taking the mean value. If wewere looking for a precise value, we would have had to perform a set of simulationswith the same parameters and then compute the mean value and the variance. Butas we only want an order of magnitude, doing one simulation is sufficient13. Thisexplains the small fluctuations that appear on some plots.

The last parameters that one can explore are the different probabilities Ps, Pd

and Pa which we set to represent all the possible problems that could occur duringthe colonization. The effects of these three parameters are nearly the same. If theyare not too small, they don’t affect very much the colonization time. They justmultiply it by a factor 2 or 3. But if the parameter is very low, we get, as one couldexpect, a very long colonization time. There is noting surprising here, everythingbehaves exactly as one could thing it will do. Here is for example a plot of Ps (figure15):

Figure 15: The time needed to colonize the Milky Way as a function of Ps

13One could argue that, this is wrong and that we got a special case in the tail of the distributionbut the porbability to have so many simulations in the tail of the distribution at an order ofmagnitude away from the ”true“ mean value is ridiculously small.

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5 Conclusion Colonizing the Galaxy

There are again some fluctuations, but one can easily see that if Ps > 0.2 there areno relevant differences in the colonizing times we obtained. The same observationsalso hold for the two other probabilities. We can conclude that these elements don’tplay an important role, if they are not too low.

5 Conclusion

The goals of the project have clearly been reached. We have been able to producea simulation program which could compute the colonization time of the galaxyaccording to different parameters, such as the travel speed or the number of starswhich bill be colonized. After having set some physical restrictions (GHZ, speedof travel, crafting times,...), we have launched a set of simulations to explore thedifferent free input parameters that were left over.

We found that the colonization time for the whole GHZ would be approximatively5 · 106 years and the different parameters would not change the order of magnitudeof this value, apart from the cases were we set the different probabilities to lownumbers or the speed to a small fraction of c. Surprisingly, the travel speed was notthe most important constraint on the final time. Going faster than 0.1c would notincrease a lot the total time but would need technological solutions that we cannotafford nowadays. The parameter that slows down the most the colonization time isthe so called settling time. The overall colonization time is approximatively a linearfunction of this time and thus it would be a real boost if the settling time weremuch shorter than the one we took as a reference. All the other parameters did notplay an important role and didn’t affect importantly the time needed to colonize thewhole galaxy.

Even with the worst parameters, the time required to reach all the stars insidethe GHZ is approximatively two orders of magnitude smaller than the age of theMilky Way. This confirms the calculation done by Fermi and we can reinforce hisquestion by saying ”Where are they ? They are late ! “

Matthieu Schallermatthieu.schaller08@imperial.ac.uk

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LIST OF FIGURES Colonizing the Galaxy

List of Figures

1 The Galactic Habitable Zone . . . . . . . . . . . . . . . . . . . . . . 42 One of the possible fusion reaction used to produce energy ( http:

//www.swip.ac.cn/xsyd/jbzs/fusion1.htm) . . . . . . . . . . . . . . . . . . . 53 The probability for a ship to reach its target as a function of the travel

time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Standard values for the main parameters . . . . . . . . . . . . . . . . 85 Color-code used in our simulations . . . . . . . . . . . . . . . . . . . 96 A screenshot of the program . . . . . . . . . . . . . . . . . . . . . . . 107 Distance to the furthest colonized star as a function of time . . . . . 108 Number of colonized stars as a function of time . . . . . . . . . . . . 119 Proportion of colonized stars inside the sphere as a function of time 1210 The two probabilities as a function of time . . . . . . . . . . . . . . . 1211 The time needed to colonize the Milky Way as a function of tsettle. . 1312 Speeds of different interstellar ships . . . . . . . . . . . . . . . . . . . 1413 The time needed to colonize the Milky Way as a function of the

travelling speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414 The time needed to colonize the Milky Way as a function of Pp . . . 1515 The time needed to colonize the Milky Way as a function of Ps . . . 16

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