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Exo-civilization Planetary Feedback

Exo-civilization Planetary Feedback Equation Derivation

Begin with most general form.

We begin with three coupled equations for the interaction between a exo-civilization population , a resource and which they draw energy from ® and the state of the planetary environment (E).

\frac{dN}{dt} =B_N - D_N + B_{RN}

\frac{dR}{dt} = -D_{NR} + D_{ER}

\frac{dE}{dt} = H_E + D_{NE} + D_{RE}

In the equations above generally B is used as "birth" terms that increase the variable and D is used for "death" terms that decrease the variable (though the meaning depends on the case). Each terms may have dependencies on the other variables - the "phase space" variables (N,R,E) - and other parameters/constraints.

In the above,

B_N is the population natural birth rate.

D_N is the population death rate.

B_{RN} is the additional population birth rate gained from extraction/consumption of resource R.

D_{NR} is the extraction/consumption rate of resource R due to activity of N.

D_{ER} is the modification of extraction/consumption rate of resource R due to changes in planetary environmental state E.

H_E is measure of capacity of planetary environment to return to an pre-civilization equilibrium state E_o. D_{NE} is the forcing of the planetary environment from pre-civilization equilibrium state E_o due to non-resource based population activity.

D_{RE} is the forcing of the planetary environment from pre-civilization equilibrium state E_o due to resource based population activity.

Choosing forms for the terms.

All alpha terms represent either growth rates or measures of the effect of variable i on variable j.

B_N = \alpha_N N.

D_N = \alpha_N N \frac{N}{K(E)}

where the death rate is now dependent on an environmentally dependent carrying capacity K(E).

B_{RN} = \alpha_{RN} \beta_{NR} N R

where \alpha_{RN} is a measure of the additional birth rate gained from extraction/consumption of resource R and \beta_{NR} is the per captia extraction/consumption rate of resource R

D_{NR} = \beta_{NR} N R

D_{ER} = \beta_{NR} N R (\frac{E}{E_c})

where E_c is the critical environmental state beyond which (E>E_c) resource extraction is no longer possible.

H_E = \alpha_E E (1- \frac{E}{E_o})

D_{NE} = \alpha_{EN} N

D_{ER}= \alpha_{ER} \beta_{RN} R N

Finally for the carrying capacity we choose,

K(E) = \frac{E-E_c}{E_c-E_o}

Exo-civilization Planetary Feedback Equations

Our final equations are,

\frac{dN}{dt} =\alpha_N N ( 1 - \frac{N}{\frac{E-E_c}{E_c-E_o}}) - \alpha_{NR} \beta_{R} R N

\frac{dR}{dt} = \beta_{RN} N R (\frac{E}{E_c}-1)

\frac{dE}{dt} = \alpha_E E (1- \frac{E}{E_o}) + \alpha_{EN} N E + \alpha_{ER} \beta_{RN} R N

Binary Star Fall Back Disk

Binary Star Fall Back Disk Variable Set

Here are variables which go into the HAFBAD problem (Highly Abstracted Fall Back Disk)

Stars

Primary Mass (solar masses): M_p

Binary Mass ratio: q=M_s/M_p

Orbital Separation: a

Outflow in the form of a brief Shell

Velocity of Shell (< escape velocity but >> sound speed: V_s

Density of Shell: \rho_s

Temperature of Shell (with sound speed constraint: T_s

Rotation of Shell: \Omega_s

Duration of shell ejection \Delta t_s

Simulation

Resolution defined in terms of injection radius R_0: N = R_0/\delta x

Planetary Wind Problem

Problem Set Up

We have 3 conditions to initialize: initial ambient medium (not important); the conditions in the base of the Hot Jupter (HJ) atmosphere; the conditions in the stellar wind at the orbital radius (a) of the HJ.

Initial Ambient Medium

The ambient medium is just a place holder to allow the planetary wind to expand into.

\rho_{a0} = m_h \mu n_{0a}

T_{a0}

c = [\frac{(k_mT_{a0}}{(\mu m_h)} ]^{1/2}

HJ Ex0base

\rho_{p0} = m_h \mu n_{pa}

T_{p0}

Stellar Wind at orbital radius a

\rho_{p0} = m_h \mu n_{pa} =\frac{\dot{M_s}}{(4\pi V_s(a) a^2)}

T_{s0}

V_{s0}

Test Runs

04/04/15

Test Run A: standard. 200x400. 1 level of refinement. Tmax = 1.0

Test Run b: standard. 100x200. 0 level of refinement. Tmax = 1.0

• seeing weird behavior?

Test Run C: standard. 100x200. 0 level of refinement. Tmax = 4.9; 50 frames

• seeing weird behavior.
• Wind does not expand uniformly

Test Run D: standard. 100x200. 1 level of refinement. Tmax = 4.9; 50 frames

• no change from C

Test Run E: standard. 200x400. 1 level of refinement. Tmax = 4.9; 50 frames

• no change from C, just better resolution
• is it the \gamma=5/3 or something about timing? Almost looks like the wind is turning off.

Grid of Runs

Lets begin with all runs having

Hot Jupiter

M_{p} = M_j

R_{p} = R_j

n_{p} = m_h

T_{p0} = 10^4

Stellar Wind

V_{s} = 100 km/s

#	lambda	Solar Wind Mach #	n_s/n_p
1	1.01	0.8	10-4
2	1.01	0.8	10-4
3	1.3	0.8	10-4
4	1.3	0.8	10-4
5	1.01	5.0	10-4
6	1.01	5.0	10-4
7	1.01	5.0	10-6
8	1.01	5.0	10-3
9	1.3	5.0	10-4
10	1.01	0.8	10-3

Fourier Transforms

Fourier Transforms

The basic equation for the fourier transform.

\Xi[f] = F(k) = \int^{+\infty}_{-\infty} f(x) e^{(-i 2 \pi k x)} dx

with the inverse

\Xi[F]^{-1} = f(x) = \int^{+\infty}_{-\infty} F(k)) e^{(i 2 \pi k x)} dk

And of course all the transform really means is this sum of intergrals.

F(k) = \int^{+\infty}_{-\infty} f(x) cos(2\pi kx) dx - i\int^{+\infty}_{-\infty} f(x) sin(2\pi kx) dx

Here we want to solve the signup function which is runs from -1 to 1 with a step at x=a

f(x) = sgn(x-a)

Use the derivative property of \Xi[f]

\Xi[f ] = \frac{1}{2\pi i k} \Xi[ \frac{df}{dx} ]

Since derivative of f is the direct delta function

\frac{df}{dx} = \delta(x-a)

and

\Xi[ \delta(x-a) ] = e^{-i2\pi k a}

Thus we have

\Xi[f ] = F(k) = \frac{1}{2\pi i k} e^{-i2\pi k a}

The next step is to define the Energy Spectral Density (ESD)

We use Parcivals Thm which tell us "energy" under the curve f(x)

\int^{+\infty}_{-\infty} | f(x)|^2 dx = \int^{+\infty}_{-\infty} | F(k)|^2 dk

Thus the ESD which we write as E(k) is

E(k) = | F(k)|^2

which for the sgn function is

E(k) = (\frac{1}{2\pi i k} e^{-i2\pi k a})^* (\frac{1}{2\pi i k} e^{-i2\pi k a})

E(k) = \frac{1}{4\pi^2 k^2}

So for a step function

E(k) \propto k^{-2}

which is exactly what you get in compressible turbulence because every shock is a sgn function.

Description of Problem

Results

movie

Results

movie