poly: RE: Seeking StarFlight Functional Forms

-----Original Message-----
From: Robin Hanson [SMTP:hanson@econ.Berkeley.EDU]
Sent: Thursday, February 05, 1998 2:51 PM
To: Forrest Bishop; carl feynman
Cc: polymath@cco.caltech.edu
Subject: Seeking StarFlight Functional Forms

Hi. I'm working on a paper, tentatively titled "Burning the
Cosmic Commons: Evolutionary Strategies for Interstellar
Colonization," building on Carl's insight that we can derive
colonization behavior from the selection effect of what it
takes to stay on the frontier.
  
Beat me to it. Oh well, I'll just have to think of another publishable idea. Serves me right for procrastinating.
  
Part of the paper proves some general results independent
of specific functional forms, but I also want to include
computer simulations of specific cases. For such specific
cases I need to pick specific functional forms, which I'd like
your advice on choosing.

The basic functions in the model are these:
R(s) = the resources available after growing at an oasis for
       time s. R(0) = 1. R'(s) > 0
  
Let me suggest a sigmoid form for this. It's the usual form for growth proportional to the product of capital and virgin resources:

R(s) = r exp(alpha s) / (1 + exp(alpha s))

Here, r is the limiting amount of resources, and alpha is the growth rate. Lots and lots of things grow this way.
  
1/A(x*b(v,h)) = the fraction of probes that survive traveling
       distance x at velocity v, with hardness h. A(0) = 1.
       A'(x) > 0.
  
How about using Gompertz's law of mortality? The mortality rate as a function of age rises exponentially. The free parameters are the base rate and the doubling time. It's a generic feature of mortality in most species.

I was looking on the web for a closed-form equation for the resulting A, which I couldn't find, when I came across http://ariadne.if.uff.br/~tjpp/racco/racco.html, which as far as I know provides the first mechanistic explanation for why Gomertz's law is so general. I spent some time trying to come up with a model for it a few years ago; it's nice to know someone has done the job.
 
C(v,h) = the resource cost to make one probe with v,h.
 
There may well be economies of scale in the construction and launching of probes. Many of the good starflight proposals I have seen involve the construction of a reusable launcher.

Q(P) = the fraction of probes which land at open oases, given
       a fraction P of them are occupied.
  
Wouldn't this depend on where the probe was relative to the frontier?

--CarlF
Received on Thu Feb 5 23:02:14 1998