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

Single star in corotating frame. This simulation only take 5 hours on 120 cores. It is a 48AU^3 box and the base grid is 60^3. I use 3 level of AMR. So the finest grid is 0.05AU^3.

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The orbital period of 6 AU simulation is 12 years, so almost 3 orbits has been done.

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The density contour of 3 AU separation simulation shows that something very similar to common envelop forms. The mass transfer mode between the two stars is between RLOF and WRLOF. I believe it is closer to the RLOF. The mass is flowing out through the L2 point. The orbital period of 3 AU simulation is 4.96 years. So more than 3 orbits has been done. The result has some similarity to this paper ​Sawada, Matsuda & Hachisu(1986). In that paper, they call the bridge that connects the two stars the elephant trunk.

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I tested two sets of data.

model 1 has extremely large mass loss rate - 1.5\times10^{-4}M_{\odot}/year and model 2 has 3\times10^{-5}M_{\odot}/year. In previous model, we can just achieve 1.5\times10^{-7}M_{\odot}/year because I may set the abseff a little higher (around 0.45?) and the resolution is poor. Also, we have changed the CFL number. Current CFL number is more appropriate.

Below is the mass loss rate in 3D single AGB star simulation.

We may use model 2 in our simulation. If the mass loss rate is still thought to be too high. I can lower the abseff to 0.45 and surface density to 8\times10^{-9}g/cc.

I can control the initial condition to avoid the strong shock now.

I can also control the variables outside the boundary cylinder to avoid in falling material and shocks.

Below are the simulation from model 1:

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Below are the simulation from model 2:

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Below is model 3:

I found that the resolution of radiation field is very important.

I have run binary simulation with low radiation field resolution and high resolution. The low resolution one has Bondi-Hoyle accretion but the high resolution one has Wind Roche-lobe accretion. The difference comes from the radiation force on the gas. The lower resolution one has higher radiation force on gas thus drive a faster wind. The difference in radiation force is tiny and the terminal velocity should not be too much different (which I will go back and check the single 3D AGB star wind).

Below is the low resolution run. It is faster and more stable than the high resolution run. The image show the side view of the density, velocity and Mach number contour. The result is pretty much in line with previous research on BH accretion ​Shima et al. (1985)​Ruffert (1996). More specifically, it resemble the result of low gamma and high Mach BH accretion. This is very reasonable because the cooling in our model is in fact equivalent to decreasing the gamma.

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The temperature of outer region (near boundary) increases unphysically. This can be the inappropriate boundary condition in the co-rotating frame.

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Below is a snapshot of high resolution result. This run also has problem. But it shows that there is a bridge which resemble Roche-lobe accretion. However, the position of this bridge is not at the L1 point (not even close, probably 0.8~1.0 AU away). This may be a sign of wind Roche-lobe overflow.

Actually, when go from Roche-lobe to BH wind accretion. There should be a separation that the bridge appear and disappear as the AGB star pulsate. That state can distinguish the RLOF and BH accretion and be a sign of WROLF.

Attached to this post.

1. I am half way to calculate a new EOS. It will consider the balance between: electron, hydrogen atom, hydrogen ion, hydrogen molecule, helium atom, HeII and HeIII.

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2. I updated the model 2 and model 4 in ​summary

It turns out that model 4 will have unphysical result.