Asri-unix.907
net.space
utcsrgv!utzoo!decvax!ucbvax!C70:sri-unix!Wedekind.ES@PARC-MAXC
Fri Mar  5 10:11:24 1982
A nearest neighbor calculation
	Here's an interesting attempt to calculate the average distance 
between technical civilizations in our part of the galaxy. A bit long, but 
it's only algebra and even if you think it's balderdash you might enjoy 
deciding just what parts you disagree with. 

	It comes from "Time and the Space Traveller" by L. Marder (his 
discussion is based on S. von Hoerner's  Dec '61  article in Science).

	First, a couple of "typicality" assumptions:

	1) Life and intelligence develop everywhere by the same rules
	   of natural selection given the proper surroundings and time.

	2) The average intelligent civilization will take about as long
	   as ours did to reach our current level of science, technology
	   and desire for interstellar communication.

	Not certain, but plausible. Now, for the local part of our galaxy, 
name these avg quantities:

	T0 = time from the birth of a star to develop a technical civilization
	T  = age of oldest stars
	V0 = fraction of stars which are "favorable", i.e. possess planets 
		capable of supporting life
	V   = fraction of stars which at present have a tech civilization
	L   = longetivity of the civilization after the technical stage is 
		reached, i.e. until destruction or degeneration.

	Von Hoerner considers 5 cases of longetivity limitation:

	1) Complete destruction of life
	2) Destruction of higher forms only
	3) Physical or mental degeneration and decay
	4) Loss of interest in science and technology
	5) No limitation.

	Let P1,..,P5 and L1,..,L5 be the probability and longetivity of the
respective fates (so that L5 = T - T0). Then the average longetivity is

		L = P1*L1 + ... + P5*L5,

and if each favorable star supports only one civilization in its lifetime 
then a fraction L/T of them will do so at any given time,

		V = V0*L/T  

(this assumes that star formation is a fairly constant phenomenon, so that 
star ages are evenly distributed up to the maximum value T). Now if L2 
and L3 are short enough it's worth counting the possibility that a second
civilization develops on the same planet in cases 2) or 3). This argument 
applied recursively changes the right hand side of the last equation to

		V = V0*L*Q/T, 

with

		Q = 1 + (P2+P3) + (P2+P3)^2 + ...  = 1/(1-(P2+P3)).

	Finally, if we call 

	D = mean distance betwixt neighboring technical civilizations, &
	D0 = mean dist between neighboring stars of ALL kinds,

then 
		D = D0/V^(1/3),

that is, D depends inversely on V's cube root. Now von Hoerner tries to 
assign values which people won't laugh at. For the Pi and Li (notice P2):

      Case           Est. range for Li       Adopted Li       Adopted Pi

1) Complete              0 - 200                100                 .05
 destruction
2) Higher                0 - 50                  30                  .6
 life only
3) Degeneration      10^4 - 10^5            3 X 10^4             .15
4) Loss of int         10^3 - 10^5             10^4                 .2
5) No limit             >= (T - T0)           T - T0                0

so that

	L = avg longetivity of 6500 years, Q = recurrence factor of 4.

He also sets

	T = star lifetime = 10^10 years, and
	D0 = neighbor dist = 7.5 light yrs.


Finally let's estimate V0, the fraction of stars with planets capable of 
supporting life. According to assumption 1) we can base this estimate 
solely on the existence and physical characteristics (temperature, etc) of 
the planets & stars themselves - say, for argument's sake,

	V0 = .01.

(von Hoerner used .1 for V0; we're being more conservative). We get

	V = 2.6 X 10^-8,

or one star in 40 million having a technical civilization at any given 
time, and

	D = 2500 light years  

for the expected distance to the nearest. Big changes in the Pi or in V0 
don't change D too much, because of the cube root dependence of D 
on V .  If you're a real pessimist, for instance, and set

	V0 = .0001,

you get

	D = 12000 light years.

	(Woops - the assumed value for D0 is not valid for a sphere of 
12000 ly radius centered on Sol - well, you get the idea). In either of these 
cases seperate calculations suggest that the most likely "technical age" of 
the first civilation encountered is many thousands of years. So we'd better 
be polite!


						Jerry					

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