Mathematics Department

Susquehanna University

Selinsgrove, PA, 17870

email: brakke@susqu.edu

Home page: http://www.susqu.edu//brakke

July 7, 1998

By "technological civilization" we mean one that has the ability and desire for interstellar travel. Such a civilization can easily colonize the entire galaxy in a few million years, an eye blink in galactic history. For supporting arguments on this, see (Zuckerman and Hart, 1995). A colonized galaxy would rather quickly have its stars surrounded by solar energy collectors, as beings made maximum use of all available energy. Such "Dyson spheres" (Dyson, 1960) would emit copious infrared radiation, and be easily recognizable from Earth with current telescopes. We do not see any Dyson spheres, so we conclude that the galaxy has not been colonized yet. Based on this, we can make some interesting calculations about how long civilizations take to develop, and how long it will be until the next one.

This HTML version omits the mathematical formulas. The
full paper is available in PDF format.

**Numerical results:**
The probability density distribution of the interval between the first and second
civilizations turns out to be approximately an exponential distribution, with the mean
intervals given in Table 1. Each row represents a different rate of star system formation,
in units of systems per mean event time. The columns are for different numbers of
key events needed for the evolution of a civilization. As an example of using the
table, suppose 10^{6} systems have formed in the 10^{10}
year history of the galaxy,
and the evolution of a civilization takes 32 key events with a mean time of 100 million
years each. Then the system formation rate is 10^{6}
systems per 10^{2} event times, or
a rate of 10^{4}. From the table, the mean time between the first two civilizations
is 0.766 mean event times, or 76.6 million years.

Rate, per event |
Events per evolution | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 4 | 8 | 16 | 32 | 64 | |

1e-03 | 999.95000 | 999.94999 | 999.94999 | 999.95014 | 999.97625 | 999.96739 | 1000.0030 |

1e-02 | 99.998568 | 100.00534 | 100.01646 | 100.03558 | 100.07382 | 100.14910 | 100.29638 |

1e-01 | 10.042342 | 10.083671 | 10.161497 | 10.306881 | 10.572211 | 11.040728 | 11.836042 |

1e+00 | 1.1677528 | 1.3007202 | 1.5138553 | 1.8408931 | 2.3237438 | 3.0135552 | 3.9773596 |

1e+01 | 0.2342562 | 0.3498452 | 0.5340489 | 0.8036302 | 1.1900764 | 1.7387205 | 2.5037201 |

1e+02 | 0.0626808 | 0.1289611 | 0.2547671 | 0.4696012 | 0.7819415 | 1.2297784 | 1.8655367 |

1e+03 | 0.0198163 | 0.0566541 | 0.1426260 | 0.3026855 | 0.5672170 | 0.9475604 | 1.4970819 |

1e+04 | 0.0062657 | 0.0256481 | 0.0842004 | 0.2104898 | 0.4281309 | 0.7660517 | 1.2528633 |

1e+05 | 0.0019790 | 0.0117674 | 0.0510466 | 0.1511569 | 0.3379296 | 0.6392454 | 1.0771975 |

1e+06 | 0.0006261 | 0.0054301 | 0.0314289 | 0.1108497 | 0.2726252 | 0.5405007 | 0.9438092 |

1e+07 | 0.0001980 | 0.0025150 | 0.0195294 | 0.0824666 | 0.2234091 | 0.4659647 | 0.8385665 |

1e+08 | 0.0000626 | 0.0011660 | 0.0122016 | 0.0619768 | 0.1852362 | 0.4067610 | 0.7532391 |

1e+09 | 0.0000197 | 0.0005409 | 0.0076559 | 0.0469203 | 0.1549792 | 0.3582943 | 0.6829621 |

1e+10 | 0.0000062 | 0.0002510 | 0.0048126 | 0.0357124 | 0.1305905 | 0.3179054 | 0.6231450 |

1e+11 | 0.0000019 | 0.0001164 | 0.0030295 | 0.0272888 | 0.1106688 | 0.2837601 | 0.5700860 |

1e+12 | 0.0000006 | 0.0000540 | 0.0019086 | 0.0209117 | 0.0942212 | 0.2545491 | 0.5254267 |

Here we use Bayesian analysis to get the posterior probability distribution of the mean time for the evolution of civilization (from system formation), given that we are first. The inputs to this model are the observed time of out solar system formation, our observed evolution time, an assumed number of key evolution events, and an assumed rate of system formation.

**Numerical results:**
Table 2 gives the median total evolution time.
This particular table is calculated for
parameters suitable for our solar system, 6x10^{9} years for the
creation of our solar system, and 4x10^{9} years for the evolution of
ourselves. Note that the system formation rate here is per year, not per mean event
time. The table divides into two regimes. At low rates of system
formation, that formation is the determining factor in the rate of
civilizations, and our time of evolution is typical. We just happened
to be very lucky that our system formed early, or at all. On the other
hand, at high rates of system formation, evolution becomes the determining
factor, and typical evolution times are much longer than ours. We just
got very lucky in key events happening quickly.

Rate, per year |
Events per evolution | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 4 | 8 | 16 | 32 | 64 | |

1e-12 | 5.79e+09 | 4.78e+09 | 4.36e+09 | 4.18e+09 | 4.09e+09 | 4.04e+09 | 4.02e+09 |

1e-11 | 5.96e+09 | 4.86e+09 | 4.41e+09 | 4.20e+09 | 4.10e+09 | 4.05e+09 | 4.03e+09 |

1e-10 | 8.07e+09 | 5.82e+09 | 4.88e+09 | 4.43e+09 | 4.20e+09 | 4.10e+09 | 4.05e+09 |

1e-09 | 6.99e+10 | 2.26e+10 | 1.28e+10 | 8.94e+09 | 6.32e+09 | 4.79e+09 | 4.32e+09 |

1e-08 | 7.26e+11 | 9.33e+10 | 3.35e+10 | 1.91e+10 | 1.40e+10 | 1.16e+10 | 1.03e+10 |

1e-07 | 7.21e+12 | 3.05e+11 | 6.67e+10 | 3.02e+10 | 1.93e+10 | 1.48e+10 | 1.25e+10 |

1e-06 | 7.20e+13 | 9.76e+11 | 1.25e+11 | 4.41e+10 | 2.47e+10 | 1.77e+10 | 1.45e+10 |

1e-05 | 7.20e+14 | 3.10e+12 | 2.28e+11 | 6.21e+10 | 3.08e+10 | 2.03e+10 | 1.61e+10 |

1e-04 | 7.20e+15 | 9.80e+12 | 4.11e+11 | 8.62e+10 | 3.75e+10 | 2.33e+10 | 1.77e+10 |

1e-03 | 7.20e+16 | 3.10e+13 | 7.38e+11 | 1.18e+11 | 4.52e+10 | 2.59e+10 | 1.88e+10 |

1e-02 | 7.20e+17 | 9.80e+13 | 1.32e+12 | 1.60e+11 | 5.40e+10 | 2.88e+10 | 1.97e+10 |

1e-01 | 7.20e+18 | 3.10e+14 | 2.35e+12 | 2.17e+11 | 6.38e+10 | 3.21e+10 | 2.15e+10 |

1e+00 | 7.20e+19 | 9.80e+14 | 4.19e+12 | 2.93e+11 | 7.53e+10 | 3.56e+10 | 2.36e+10 |

1e+01 | 7.20e+20 | 3.10e+15 | 7.45e+12 | 3.93e+11 | 8.89e+10 | 3.93e+10 | 2.38e+10 |

1e+02 | 7.20e+21 | 9.80e+15 | 1.33e+13 | 5.28e+11 | 1.04e+11 | 4.32e+10 | 2.60e+10 |

1e+03 | 7.20e+22 | 3.10e+16 | 2.36e+13 | 7.06e+11 | 1.21e+11 | 4.71e+10 | 2.82e+10 |

**Numerical results.** Given that we are first, and given our system formation
at 6x10^{9} years, and our evolution time of 4x10^{9} years, Table 3
presents the expected mean time until the next civilization. Note that the system
formation rate is per year, not per mean event time. For low system formation rates,
we basically just have to wait a long time until the next system forms; evolution
of civilization happens quickly after that. For higher system formation rates,
we just have to wait until somebody is as lucky as we were, which takes about
the same amount of time. In any case, the mean waiting time is at least 10 billion
years. This is much longer than the mean interval between civilizations in Table 1
since that calculation did not take as given the time of appearance of the
first civilization.

Rate, per year |
Events per evolution | ||||||
---|---|---|---|---|---|---|---|

1 | 2 | 4 | 8 | 16 | 32 | 64 | |

1e-12 | 1.04e+12 | 1.03e+12 | 1.02e+12 | 1.02e+12 | 1.02e+12 | 1.02e+12 | 1.02e+12 |

1e-11 | 1.24e+11 | 1.14e+11 | 1.12e+11 | 1.11e+11 | 1.11e+11 | 1.11e+11 | 1.11e+11 |

1e-10 | 2.71e+10 | 2.25e+10 | 2.08e+10 | 2.02e+10 | 2.01e+10 | 2.01e+10 | 2.01e+10 |

1e-09 | 2.00e+10 | 1.50e+10 | 1.28e+10 | 1.16e+10 | 1.11e+10 | 1.10e+10 | 1.10e+10 |

1e-08 | 2.02e+10 | 1.51e+10 | 1.26e+10 | 1.14e+10 | 1.08e+10 | 1.04e+10 | 1.02e+10 |

1e-07 | 2.01e+10 | 1.51e+10 | 1.26e+10 | 1.14e+10 | 1.07e+10 | 1.04e+10 | 1.02e+10 |

1e-06 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.07e+10 | 1.04e+10 | 1.03e+10 |

1e-05 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.07e+10 | 1.04e+10 | 1.03e+10 |

1e-04 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.07e+10 | 1.04e+10 | 1.03e+10 |

1e-03 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.07e+10 | 1.03e+10 | 1.03e+10 |

1e-02 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.07e+10 | 1.03e+10 | 1.01e+10 |

1e-01 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.07e+10 | 1.03e+10 | 1.02e+10 |

1e+00 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.06e+10 | 1.04e+10 | 1.03e+10 |

1e+01 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.07e+10 | 1.04e+10 | 1.01e+10 |

1e+02 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.06e+10 | 1.04e+10 | 1.02e+10 |

1e+03 | 2.01e+10 | 1.50e+10 | 1.25e+10 | 1.13e+10 | 1.06e+10 | 1.03e+10 | 1.04e+10 |

Dyson, F. J. (1960). Search for artificial stellar sources of infrared radiation.

Zuckerman, B. and Hart, M. H., eds. (1995). Extraterrestrials: Where are They? Cambridge University Press.

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