# Dynamics of test particles in the formation of the solar system

#### Linguofreak

##### Well-known member
I've got a bit of a weird question. I'm thinking up an FTL system for a Sci-Fi campaign in which a large number of FTL access points exist, but they barely interact with normal matter (or each other), so you have to survey for them (barely interact, that is, except that they are affected by gravity, and that, if you walk up to them with a big lump of handwavium and <insert technobabble here>, you end up at another FTL access point somewhere far away). If there are a lot where you happen to be searching, the survey finds one relatively quickly, if there aren't, it could take years, decades, or centuries, but there are enough that you'll probably eventually find one anywhere. I'm trying to devise a rule for determining the probability distribution of positions and orbital elements for FTL access points around a star.

Since they barely interact with normal matter except by gravity and magic/handwavium, and since, for the sake of placing as little load as possible on people's suspension of disbelief (which is always nice in sci-fi), we don't want the large number of them we'll be using to contribute enough mass that they'd have gravitational effects inconsistent with present observations, we'll treat them as massless test particles.

I'll start from the assumption that these particles are in general milling around the galaxy fairly uniformly, and are captured by stars as they form. The unbound population can be ignored, as given the characteristics of the setting and the nature of the FTL system, they're only briefly useful, relative to the amount of time needed to survey for them, if they're on a e >= 1 trajectory (or a bound trajectory that is sufficiently close to e=1).

To state it semi-formally:

First, assume a star system that isn't a star system yet: a cloud of gravitating, collisionful matter that has just fallen over the edge into Jeans instability. Next, assume that along with this, we have a gas of collisionless, massless test particles filling all of space with uniform denisty. The test particles have some thermal distribution of velocities, and no net flow in any direction relative to the matter cloud.

Fast forward until the star system is a star system: the star and any planets have reached more or less their final masses and positions.

Unless I'm badly mistaken, at this point, some of the test particles have become bound to the star. The tl;dr question, summarizing the rest of the post, is:

"What does this cloud of bound test particles end up looking like?"

I will henceforth refer to the bound test particles as the "test cloud", and the unbound particles as the "test gas".

I expect that the answer to many of the specific points below will be "it has to be simulated numerically", but it would be interesting to know what heuristics there might be, as well as whether anyone is aware of any research in which this has been simulated numerically.

This is almost like a "galaxy formation with dark matter" problem (which has been simulated), except assuming a gravitationally negligible density of dark matter (so that the normal matter dominates the gravitational dynamics). I would assume that the "negligible density of dark matter" scenario has not been widely simulated.

In more detail:

1. Ignoring the formation of any planets (i.e, assuming all the interacting matter ends up in the star), and assuming no departures from spherical symmetry, what is the general density and velocity profile of the test cloud?
1a. Does the density profile only depend on the radius (will it look the same at the end for every star), or will it in general strongly depend on the timing of the collapse and the density profile of interacting matter during the collapse?
1b. If the density profile is just a function of radius (or if it's well constrained enough to speak of in terms of variations from an average case), what does the density function look like?
1c. Using the radius at which the velocity for a circular orbit is equal to the average velocity of the particles in the test gas as our reference radius, how does the final density of the test cloud at the reference radius compare to the initial density of the test gas?
1d. What does the velocity profile in the test cloud look like at any given point? It will of course deviate somewhat from being thermal, as high-velocity particles will be part of the test gas, not the test cloud and thus not part of the population we are considering, but will it otherwise still look more-or-less isotropic and thermal? What would averages and deviations of semi-major axis (relative to the radial coordinate of the point being considered) and eccentricity look like, respectively, for an isotropic, thermal distribution vs. the case we are considering? (I leave out angular orbital elements as we're assuming spherical symmetry for the moment).
1e. Going out on a limb for potential analogies in other physics, could the density profile be expected to look somewhat like the probability density for the position of the electron in a ground-state hydrogen atom? It feels like a fairly wild guess that's not horribly likely to be right, but the elements in common are a inverse-square potential, spherical symmetry, and (I think related to spherical symmetry) zero net angular momentum (of course, in the test cloud it's always going to be zero because they're massless test particles, but if we gave each particle a mass the net angular momentum would still be zero).

2. Dropping the spherical symmetry assumption, but keeping axial symmetry, if we now assume that a "protoplanetary" disk forms while the star is forming , but remains smooth (hence, axial symmetry), and that all interacting matter either eventually drops into the star or is ejected to infinity (so that we have approximate spherical symmetry in the star once formed, but a situation that deviates far from spherical symmetry while it's forming), how does the gravity of the disk during the formation of the star affect the orbital elements of the test cloud (as compared to the spherical case)?
2a. Will the only effect be to cause the LAN's of the test particles to precess gyroscopically (which will be irrelevant since we're still assuming axial symmetry), or will we actually see changes in inclination (i.e, a depletion or enrichment of particles with high inclinations relative to the disk, with changes to the density profile above/below the disk)?
2b. Will there be any change to the distributions of SMA and Ecc?

3. Now lets drop all symmetry assumptions and actually let planets form. I'm familiar, qualitatively, with the effect that planets/moons tend to have on asteroid belts/rings when everything is already in roughly coplanar orbits that are reasonably close to circular. I'm not, however, well versed on the quantitative details (e.g, how much of the material ejected from the orbit of an object, or an orbit in a resonance with it, ends up captured by the object, vs. swept into the primary, ejected, or just moved into a different orbit), nor on how a more random distribution of orbital elements will be affected, nor what the lack of collisions (thus making escape the only way by which test particles are removed) will do.
3a. For a planet of a given mass as a fraction of the primary, is there a function that describes the "shape" of the gap it will carve in a circular, coplanar disk of test particles (in terms how much the density of test particles is depleted as a function of difference in SMA from the planet). What can we then say about extending that function to include all of the orbital elements.
3b. Would we expect a significant population of test particles with periaspes inside the star (assuming sufficiently massive planets tossing things around)?
3c. Planetary migration is, of course, a concern, so we're likely to see depletion in areas that there aren't necessarily planets in when all is said and done, but will there be any significant effects from, for example, the phase when the disk has started to clump, but actual planets have not yet formed?
3d. Are we likely to see significant contributions to the test cloud density from test gas captures through interactions with planets, and what will the orbital elements of captured particles look like?

#### Urwumpe

##### Not funny anymore
Donator

If they are massless in the sense of zero resting mass, they would only have impulse in our universe, they can only move at the speed of light - from our point of view. So unless there is a black hole, they can not be bound by gravity to a star at all.

You thus essentially describe a constant flux of lightspeed fast particles with some energy distribution running through our universe, which are only interacting with us by some special, unknown, rare field-like mechanism.

Of course, much more strange would be: No mass, no impulse, nothing, nada at all... such a particle would not even qualify for existence in our point of view. But it could still exist and interact with the universe.

---------- Post added at 11:16 ---------- Previous post was at 11:02 ----------

Or do you mean test particles in the sense of those used in CFD simulations, that follow the airflow, but don't change it?

In that case, the question should be: To which volumes or masses should this test particle be tied? Should it follow the spacetime? Or any other special mass?

Should they just be pulled by gravity of the star without pulling the star? In that case, you would have a cluster of orbits and a concentration of particles near the star (perihelion), some particles would just leave the solar system, others would enter it...

(Hey... it should be possible to render this situation...)

Do the particles interact with the gravity of planets as well? If the particles have no inertia, they should not accumulate at the L4 and L5 points of planets, but rather something more strange....

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#### Linguofreak

##### Well-known member

If they are massless in the sense of zero resting mass, they would only have impulse in our universe, they can only move at the speed of light - from our point of view. So unless there is a black hole, they can not be bound by gravity to a star at all.

Drat. I knew I was going to have to clarify this.

By "massless" I mean "having negligible gravitational influence, even in aggregate, for the purposes of the model". Also, unless you're dealing with the galactic center, over the distance scales I'm considering (whole solar systems), we can assume Newtonian physics. The volume in which relativity is relevant near any stellar mass black hole will be kilometers in radius, while I'm considering things on scales of AU.

"Massless" here does *not* mean "propagating at exactly the speed of light in all reference frames".

#### Urwumpe

##### Not funny anymore
Donator
So, I get this right when I summarize it as:

"Each particle represents a n-body trajectory though a solar system?" And in case of question 1: A "two-body Keplerian orbit"

#### Linguofreak

##### Well-known member
So, I get this right when I summarize it as:

"Each particle represents a n-body trajectory though a solar system?" And in case of question 1: A "two-body Keplerian orbit"

Yes, basically, though in the case of questions 1 and 2 we're basically starting with free trajectories at the beginning, doing n-body during the collapse, and we end up with a Keplerian situation at the end (a cloud of test particles around a massive, spherically symmetric object, thus effectively a point source), and we want to know the distribution of final Keplerian orbits for those trajectories in the initial set of free trajectories that were captured during the collapse.

For question 3, we are basically considering n-body trajectories, and are first looking for what the distribution of bound test particle trajectories comes out of the initial collapse phase, and then, once we have the planets at their final masses and on their more-or-less final orbits, we're wanting to know which test particle trajectories remain close to Keplerian trajectories in the long term, and which parts of the phase space will be swept clean by a planet with a given orbit and mass.

#### Urwumpe

##### Not funny anymore
Donator
Not sure if the collapse is really important there to have any long-term effects on the "particle" density. The youngest known star is about 1000 years old and the gas density remains largely constant for a longer period before the cloud reaches ignition temperature.

#### Linguofreak

##### Well-known member
Not sure if the collapse is really important there to have any long-term effects on the "particle" density. The youngest known star is about 1000 years old and the gas density remains largely constant for a longer period before the cloud reaches ignition temperature.

Well, at the beginning, when you just have a diffuse molecular cloud, there's not much of a potential drop between infinity and the center of the cloud, and for the test particles, you can basically treat the situation as flat space. As the cloud collapses, your gravity well gets deeper and steeper, and you're presumably going to start collecting test particles. What I'm not sure of is whether you can just have a fully formed star suddenly pop into existence (rather than forming through a collapse process) and still get the same distribution of test particles around the star.

As for the dynamics of the interacting matter during the collapse, for most stars, the star has accreted its full final mass before hydrogen fusion brings it into long-term equilibrium (though I'm not sure where planet formation generally is by that time), so for the purposes of questions 1 and 2, the collapse can be considered complete before hydrogen ignition. For very large stars the main sequence lifetime is often on the order of the timescale for the star to form, so it may still be accreting matter well into its main sequence lifetime. But before all of that, there is the phase where the cloud is contracting pretty much in free fall, and then the protostar phase where there is a hydrostatically supported core with matter falling on to it. During the initial collapse, and the part of the protostar phase where a large part of the mass of the cloud has not yet fallen onto the star (or been ejected), the situation can't just be modeled as a point source, but I'm not sure how relevant this is to the final state of our test cloud.

#### Urwumpe

##### Not funny anymore
Donator
Well, the question is: How much time should have passed since the formation of the star? The orbit period of Pluto is just 250 years, since the solar system stabilized, there had been millions of Pluto orbits to the current situation.

And so would the test particles. And since we can assume an even distribution outside the solar system and thus a constant influx into it (for now), the situation right after the star formation should quickly be "overwritten" by the effects of the planets and the flux to and from outside.

#### Linguofreak

##### Well-known member
Well, the question is: How much time should have passed since the formation of the star? The orbit period of Pluto is just 250 years, since the solar system stabilized, there had been millions of Pluto orbits to the current situation.

Except for very massive stars that wouldn't live that long, I think assuming a billion years as our timescale is probably in the right ballpark.

And so would the test particles. And since we can assume an even distribution outside the solar system and thus a constant influx into it (for now), the situation right after the star formation should quickly be "overwritten" by the effects of the planets and the flux to and from outside.

We can assume a constant flux of unbound test particles *through* the solar system, so the "dent" in the unbound population of test particles that the formation of the solar system would make would be quickly filled in, but it's not obvious to me that, after the system had formed, test particle captures on planets would make a significant contribution to the bound population. My reasoning is that the ejection rate of interacting matter from our own solar system today is quite low (all the material with a short ejection half-life is already gone), but the capture rate seems to be even lower (close encounters between planets and solar-system bodies are rare, but happen at a reasonable rate due to sheer numbers, but we've only ever seen two interstellar objects). But, on the other hand, the interacting matter has been concentrated in the plane of the ecliptic by collisions, and without collisions the test particles would probably have a more spherical distribution, which might minimize their contact with the planets, lowering the ejection rate.

My general assumption is this: the collapse phase will capture a fair number of test particles due to the potential well deepening as they travel through, leaving them with insufficient energy to escape. Once the star has reached its final mass and something close to its final radius, this process will have ended. As planets form, they will start ejecting and capturing test particles, but at a low rate in general, and a lower rate for captures than collisions. But if the collapse process doesn't build up a population of bound test particles that is denser than the unbound population flowing in and out from infinity, then planets will tend to encounter unbound particles more often than bound particles, so the rate of capture will exceed the rate of ejection until they do. I'm trying to get information on which scenario is likely to be the case.

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