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?
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?