Changes between Version 1 and Version 2 of u/johannjc/galactic_colonization


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Timestamp:
10/05/17 23:06:46 (7 years ago)
Author:
Jonathan
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  • u/johannjc/galactic_colonization

    v1 v2  
    1 Beginning with the continuity equation for settled systems
     1Consider the diffusion equation for settled systems in the free-streaming limit (which I think should be appropriate)
    22
    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).
    44
    5 where the source term is due to probe launches as opposed to diffusion - and follows the logistic equation
     5We can then combine that with a logistic style source term that accounts for local probe activity
    66
    77$S=\frac{1}{T_p}\rho(1 - \frac{\rho}{\rho_{max}})$
    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...
    108
    119We then normalize the density of settled systems by the density of total systems
     
    2927$\frac{\partial \eta}{\partial \xi} = -\frac{\eta (1-\eta)}{T_p (\mathcal{V}-v)}$
    3028
    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
     29which 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).