Changes between Version 1 and Version 2 of u/johannjc/galactic_colonization
- Timestamp:
- 10/05/17 23:06:46 (7 years ago)
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u/johannjc/galactic_colonization
v1 v2 1 Beginning with the continuity equation for settled systems 1 Consider the diffusion equation for settled systems in the free-streaming limit (which I think should be appropriate) 2 2 3 $\frac{\partial \rho}{\partial t} = -\frac{\partial \rho v}{\partial x} + S$3 $\frac{\partial \rho}{\partial t} = k \frac{\partial ^2 \rho}{\partial x^2} \rightarrow \frac{\partial \rho}{\partial t} = v \frac{\partial \rho}{\partial x}$ where $v$ is the free streaming velocity roughly corresponding to the thermal velocity. (For flux-limited radiation diffusion, this is just the speed of light). 4 4 5 where the source term is due to probe launches as opposed to diffusion - and follows the logistic equation 5 We can then combine that with a logistic style source term that accounts for local probe activity 6 6 7 7 $S=\frac{1}{T_p}\rho(1 - \frac{\rho}{\rho_{max}})$ 8 9 Here $v$ can be thought of as the average thermal velocity and is a constant, so we can pull it out of the derivative...10 8 11 9 We then normalize the density of settled systems by the density of total systems … … 29 27 $\frac{\partial \eta}{\partial \xi} = -\frac{\eta (1-\eta)}{T_p (\mathcal{V}-v)}$ 30 28 31 which gives a front thickness of $T_p \left (\mathcal{V}-v \right)$ where $v$ is the average speed and $\mathcal{V}$ is the speed of the front (maximum speed from maxwellian). 32 33 34 35 Note - I also tried 36 37 $\eta(\xi)$ where $\xi = \frac{x}{\mathcal{V} t}$ 38 39 Substituting that form for the solution into the continuity equation we get 40 41 $\frac{\partial \eta}{\partial t} = -\frac{\partial \eta v}{\partial x} + \frac{1}{T_p} \eta (1-\eta)$ 42 43 $\xi \frac{\partial \eta}{\partial \xi} = \frac{v}{\mathcal{V}} \frac{\partial \eta}{\partial \xi}-\frac{t}{T_p} \eta \left (1 - \eta \right )$ 44 45 $\frac{\partial \eta}{\partial \xi} = \frac{t}{T_p} \frac{\eta \left ( 1 - \eta \right )}{\frac{v}{\mathcal{V}} - \xi}$ 46 47 but that is no longer self-similar 29 which is just a logistic curve with length scale $L=T_p \left (\mathcal{V}-v \right)$ where $v$ is the thermal speed and $\mathcal{V}$ is the speed of the front (maximum speed from maxwellian).