wiki:u/johannjc/galactic_colonization

Version 1 (modified by Jonathan, 7 years ago) ( diff )

Beginning with the continuity equation for settled systems

where the source term is due to probe launches as opposed to diffusion - and follows the logistic equation

Here can be thought of as the average thermal velocity and is a constant, so we can pull it out of the derivative…

We then normalize the density of settled systems by the density of total systems

which gives us

We then look for self-similar solutions of the form

where

Upon substituting that in to the continuity equation we arrive at

or

which gives a front thickness of where is the average speed and is the speed of the front (maximum speed from maxwellian).

Note - I also tried

where

Substituting that form for the solution into the continuity equation we get

but that is no longer self-similar

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