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This is supposed to be a test of the tag. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Speculative or unreferenced answers, as well as those not supported by strong scientific theory, are not welcome. Long, comprehensive answers are desirable, but length and quality aren't always correlated.


A young B-type star (with a mass of about 10 M$_{\odot}$) is surrounded by a debris disk extending from about 2 AU to 1000 AU away. The disk has a mass of about 300 Earth masses - enough to form quite a lot of planetesimals. There is also an outer cloud of icy, comet-like bodies extending from 750 AU to 5,000 AU away.

The star stands out from other Sun-like stars, though, because recently a strong stellar wind has developed, which comes with a mass loss rate of about 1.0 $\times$ 10-6 M$_{\odot}$ per year. The stellar wind may die down eventually, but for now it's having quite a strong effect on the star and its disk. It is thought that much of the inner debris disk will be blown away very quickly.

A nearby red dwarf of about 0.76 M$_{\odot}$ (not a flare star, fortunately) is passing through. At its nearest point, it comes extremely close - an astounding 1500 AU away from the B-type star!

There will absolutely be some accretion by the red dwarf. But how much? An advanced civilization (Type II on the Kardashev scale) is watching closely. They're considering the red dwarf as a sort of "rest station". If planetesimals form, they could be mined for raw materials, and any icy bodies could be an excellent source of water - which can be turned into hydrogen and helium.

How can they figure out just how much dust from the debris disk and icy bodies will be captured by the red dwarf? Can they then predict if planetesimals are likely to form (though subsequent planetary formation isn't necessary)?

That bit could be construed as pure science, which it may be. But there's another, much more important question that's absolutely related to world building: Can the Type II civilization do anything to influence accretion?

This is what I'd like answers to focus on.


I've done some reading, and the situation is plausible. Accretion rates can be computed, and whilst encounters between stars this close are extremely rare, they're still possible. I didn't list density parameters, but those can be found rather easily in the scientific literature. Taking all this into account, I'm looking for answers based in .

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ I'd normally try an answer, but the hard-science tag has frightened me away some wary forest creature. $\endgroup$ – Serban Tanasa Mar 19 '15 at 23:29
  • $\begingroup$ @SerbanTanasa I was afraid that might happen. It could either deter crappy answers (good) or deter good answers (bad). Our of curiosity, what would your answer have been (I'd put it in the latter group, judging from your past answers)? $\endgroup$ – HDE 226868 Mar 19 '15 at 23:31
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    $\begingroup$ A type II Kardashev would have more use from low-gravity materials in the disk than from gravity-bound crap that requires energy to haul out... (Runs off into the shrubbery) $\endgroup$ – Serban Tanasa Mar 19 '15 at 23:34
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    $\begingroup$ I feel like this question would be a better fit in Astronomy. You could also ask the "hard science" part in astronomy and ask the "worldbuilding" part here once the first has been answered. $\endgroup$ – Maxime Lucas Mar 20 '15 at 12:00
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    $\begingroup$ I'm not entirely sure that the question is answerable as written - specifically, I suspect that the capabilities of a Kardashev type-II civilisation by definition include abilities that would be 'speculative' based on our current scientific understanding. $\endgroup$ – Toby Y. Mar 21 '15 at 10:50
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Perhaps your best source here is the recent Ribas, Bouy, and Merín 2015 paper, distinguishing between primitive and processed disks by the mass of the stellar object.

disks by age and mass

The take home point seems to be that larger stars tend to process or blow away their disks relatively quickly.

The actual N-body dynamics you describe (two stars and a granular disk) are far too complex to be easily treated here, but this is the closest I could find.

Recent simulations of the ONC cluster indicate that at least 20% of the stars undergo encounters closer than 300AU during the first 3Myrs of the cluster development (Olczak et al. in prep.).

Naturally such encounters will influence the mass and angular momentum distribution in the disc. The total mass of - and the mass distribution within - the disc after an encounter are important, as this directly influences the likelihood of the formation of planetary systems.

The change of the angular momentum distribution due to an encounter is of relevance since it is still unclear how the disc loses enough angular momentum that accretion of matter onto the star is possible. Although it is unlikely that encounters are the dominant source of angular momentum loss, their contribution is probably not negligible.

The main discussed effect is in a disk-disk context and has to do with changes in the angular momentum of the star-transferred mass that does not fall into the capturing star, and its impact on planetary formation. If you're interested in the topic, the entire paper is definitely worth a read, as is the bibliography (esp Pfalzner).

Speculation about the abilities of a Type II Civ is not a clear-cut case of , although we can definitely impose lower and upper bounds on their ability to modify a stellar environment based on their relative energy budget.

A Type I civilization has a low budget of ($10^{15}$ watts), while a typical type II could spend $10^{26}$ watts, whereas a Type III culture at its peak would command the power of a hundred billion suns -- upwards of $10^{37}$ watts. The best discussion on this remains the Freitas book Xenology.

From the perspective of a mature Type II civ, stars are merely piles of valuable resources that have unfortunately caught fire and sunk into the fabric of spacetime. Protoplanetary disks have 2 advantages: a distinct lack of deep gravity wells and the fact that they are not on fire. Again, this is speculative, but given that their energy budgets place overcoming the binding energy of planetisimals easily within their reach, they would look to a low-gravity protoplanetary disk in a way similar to how a 6-year old would look to Halloween: lots of delicious candy just lying around for the plucking.

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  • $\begingroup$ I'm not going to accept the answer because I'm still looking for a different one, but this is definitely a bounty-worthy answer. $\endgroup$ – HDE 226868 Mar 28 '15 at 23:05
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I decided to start working on my own answer after I asked the question, so here's the result of a few days' work.

An excellent (and very recent) paper regarding accretion is Debnath (2015), which can be applicable at least for material gathered onto the surface of the red dwarf.

Debnath assumes a static1, spherically symmetric metric: $$ds^2=-A(r)dt^2+\frac{1}{B(r)}dr^2+r^2(d \theta^2+ \sin \theta d \phi^2) \tag{1}$$ which uses the (-,+,+,+) sign convention. For now, we can leave $A$ and $B$ undetermined functions of $r$. We have to treat the surrounding matter as a perfect fluid, with a stress-energy tensor of $$T_{\mu \nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu \nu} \tag{2}$$ with $\rho$ and $p$ being the density and pressure, respectively.2 $u_{\alpha}$ is the four-vector, with the condition that $u_{\alpha}u^{\alpha}=-1$. For this fluid, though, $$u^{\alpha}=(u^0,u^1,0,0)$$ We can then re-write the earlier condition as $$g_{00}u^0u^0+g_{11}u^1u^1=-1$$ Substituting in that $g_{00}=g_{tt}=A(t)$ and $g_{11}=g_{rr}=\frac{1}{B(t)}$, as well as assuming (for simplicity) that $u^1=u$, we get $$\left(u^0\right)^2=\frac{\left(u^1\right)^2+B}{AB} \to u_0=g_{00}u^0=\sqrt{\frac{A(u^2+B)}{B}}$$ We can also calculate $\sqrt{-g}=\sqrt{\frac{A}{B}}r^2 \sin \theta$.

The law of conservation of energy states that $u{_\mu}T_{; \nu}^{\mu \nu}=0$.3 Putting this together with $(2)$ gives $$u^{\mu} \rho_{, \mu}+(\rho+p)u_{; \mu}^{\mu}=0$$ Doing it out, we find that $$C=-ur^2M^{-2}\sqrt{\frac{A}{B}} \exp \left[\int_{\rho_{\infty}}^{\rho_R} \frac{1}{\rho + p(\rho)} d \rho\right] \tag{3}$$ where $\rho_R$ is the density at the radius of the red dwarf and $C$ is a constant (which will be used later).

The rate of change of mass of the black hole, $\dot{M}$ (the negative rate of change of the mass of the fluid) is expressed as4 $$\dot{M}=\int T_0^1dS \tag{4}$$ where $$dS=\sqrt{-g}d \theta d \phi$$ From $(2)$, we get $$\dot{M}=4 \pi CM^2(\rho + p) \tag{5}$$

Abramowicz & Fragile (2013) give a slightly different expression in place of $(4)$ (Equation 125): $$\dot{M} = \int \sqrt{-g} \rho u^r d \theta d \phi \tag{6}$$ and use $(5)$ for the energy flux. Both expressions are applied to jets on black hole accretion disks.

Working off of Debnath, the total mass transferred to the red dwarf's surface is $$M=\int_{t_0}^{t_f} \left[\int \sqrt{-g} \rho u^r d \theta d \phi \right] dt \tag{7}$$ where $t_0$ and $t_f$ are the initial and final times during which the red dwarf accretes material.

I haven't quite figured out the full Roche lobe calculations just yet, but I was able to find some of the major equations. Paczynski (1971) mentions that the radius of the Roche lobe of the red dwarf is $$r_1=\left[\frac{2}{3^{4/3}} \left(\frac{M_1}{M_1+M_2} \right)^{1/3} \right]A \tag{8}$$ where $M_1$ is the mass of the red dwarf and $M_2$ is the mass of the B-type star, and $A$ is the distance between them.

The issue is that this is typically applied in binary systems, while, presumably, the red dwarf is traveling at a speed greater than the B-type star's escape velocity. It is, therefore, not orbiting it. So I'm not sure if the formula is valid.

Let's say that the civilization places a planet-sized object into the disk, inserting it at an orbital velocity $V_0$. It would then undergo Bondi accretion, as shown in Bondi (1951).5 In that paper, he goes from the expressions derived by Hoyle & Littleton and Bondi & Hoyle to get the accretion rate of $$\dot{M} = 2 \pi (GM)^2 (v^2 + c_s^2)^{-3/2} \rho \tag{9}$$ where $v$ is relative to the fluid. Taking the limit as $v \to 0$ gives the approximation shown on Wikipedia, although it differs by a factor of 2.

We can't just use this, though, because there are other things to consider. First, all the gas and dust in the disk is orbiting at the same rate as this object, so $V_0 \neq v$. Second, the conditions change. For each orbit the object makes, the density of the matter in the path through the disk changes, because it has been swept up. Finally, the object may be severely affected by Stokes drag.

The density issue can be dealt with by simply assigning the object a number of orbits $n$ at time $t$, and saying that during each orbit, it accretes $x$ percent of the gas and dust in its way. Once this is known, an expression can be derived for the accretion during each orbit.

The Stokes drag is slightly more interesting. As shown in a derivation by Gavnholt et. al. (2004), the formula is $$D=6 \pi \mu U a \left(1+\frac{3Re}{8} \right) \tag{10}$$ where $U=v$, $a$ is the object's radius, $\mu$ is the viscosity and $Re$ is the Reynolds number. This means that $$\frac{dv}{dt} \propto v$$ Knowing that, and placing the object in a circular orbit such that $$F_g=F_c \to G\frac{M_sm_o}{r^2}=\frac{mv^2}{r}$$ where $M_s$ is the mass of the B-type star, $m_o$ is the mass of the object and $r$ is the distance between them. we can write $v$ as a function of time and then solve for $r$ as a function of $v$, eventually witnessing orbital decay. Also, if $\rho$ is a function of $r$, we can further complicate everything. This also goes for the accretion experienced by the red dwarf.

I feel bad about not doing any actual calculations (i.e. with actual numbers), so I'll discuss a special case here: A dust disk surrounding a spherically symmetric body.

In a dust solution, $p=0$, so our generic equation of state $p=p(\rho)$ goes to 0. Imposing spherically symmetry means that $A=B$. Accounting for all this turns $(3)$ into $$C=-r^2uM^{-2}\sqrt{\frac{1}{1}} \exp \left[ \int_{\rho_{\infty}}^{\rho_R} \frac{1}{\rho + 0} d \rho \right]$$ $$=-r^2uM^{-2} \exp \left[ \ln \frac{\rho_R}{\rho_{\infty}} \right]$$ $$=-r^2uM^{-2} \frac{\rho_R}{\rho_{\infty}} $$

Plugging this into $(5)$ gives us $$\dot{M}=-4 \pi r^2u \frac{\rho_R}{\rho_{\infty}} \rho$$ Just take a given $\rho$, pick a velocity, and solve for $\dot{M}$.

I still haven't put in any numbers, but it's at the point where you don't have to do much to find the result.

Accretion by planetary-mass objects in debris disk has been observed, such as in Epsilon Eridani's debris disk (Greaves et. al. (2005); explored also by Backman et. al. (2008)). A good overview of the process is given by Janson et. al. (2013), while it was simulated by Stark & Kuchner (2009) and Nesvold & Kuchner (2014). The only issue now is to establish whether or not a Type II civilization could build such an object.


Footnotes
1This means we have to neglect rotation, which could be a problem.
2Were we to assume vanishing pressure, as in a true dust solution, things could get simpler (and, perhaps, more interesting). For now, though, we'll treat it as a perfect fluid, and treat it as homogenous.
3I'm using the convention in which a comma indicates a partial derivative and a semicolon indicates a covariant derivative.
4Raising and lowering indices via the metric tensor.
5In "thin disk" scenarios, the red dwarf might not undergo spherical accretion.

References
Abramowicz, M. A. and Fragile, P. C. "Foundations of Black Hole Accretion Disk Theory" (2013)

Backman, D. et. al. "Epsilon Eridani’s Planetary Debris Disk: Structure and Dynamics based on Spitzer and CSO Observations" (2008)

Bondi, H. "On Spherically Symmetric Accretion" (1951)

Debnath, U. "Accretion and Evaporation of Modified Hayward Black Hole" (2015)

Gavnholt, J. et. al. "Calculations of the Flow Around a Sphere in a Fluid" (2004)

Janson, M. et. al. "The SEEDS Direct Imaging Survey for Planets and Scattered Dust Emission in Debris Disk Systems" (2013)

Nesvold, E. R. and Kuchner, M. J. "Gap Clearing by Planets in a Collisional Debris Disk" (2014)

Paczynski, B. "Evolutionary Processes in Close Binary Systems" (1971)

Stark, C. C. and Kuchner, M. J. "A New Algorithm for Self-Consistent 3-D Modeling of Collisions in Dusty Debris Disks" (2009)

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  • $\begingroup$ +1 Hmm, perhaps should have stayed in the bushes. Very well researched answer! $\endgroup$ – Serban Tanasa Mar 23 '15 at 0:41
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    $\begingroup$ That is a big research indeed. I would up vote if only I could understand. $\endgroup$ – Vincent Mar 27 '15 at 20:19
  • $\begingroup$ @Vincent I can't really simplify it at all. Even I haven't gone through all the derivations. Thank you, though, for not just blindly upvoting. I suspect that other people have done so on the answer; I don't want that kind of thing. $\endgroup$ – HDE 226868 Mar 27 '15 at 20:22
  • $\begingroup$ This is great, but you don't actually your own question about the Kardashev II civ, do you? $\endgroup$ – AecLetec Mar 27 '15 at 22:49
  • $\begingroup$ @AecLetec I was suggesting that the insert an object to accrete debris, thus reducing rates. $\endgroup$ – HDE 226868 Mar 28 '15 at 1:30

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