2-body gravitational coupling

[hstarbuck]

I have been reading a bunch of posts on collisions and the like, and I got to thinking about the necessary conditions for one body to capture another (in space of course). Upon first glance (inward) it seems obvious that when one body is much more massive than another this would be more likely to happen and form a stable system. Here are a few questions that come to mind on this phenomenon:
1. What incoming trajectories and velocities would create likely captures for an object that is much smaller than future host? (e.g. orthogonal to orbital radius at appropriate tangential velocity) What types of combo's work?
2. Are these two initial conditions (direction and velocity) different when the bodies are closer in mass? (e.g. two planets or two stars)
3. Do we observe or hypothesize many captured binary stars?
4. What is the most likely scenario once objects get really close? Choices: a)a quick fling b)dance to disaster c) occasional date d)stable relationship e)other

I have an old n-body program that I developed in college that I could try to remember how to compile and run on a computer in my closet that has LINUX, but it seems too far removed at the present.

[Chris Peterson]

I have been reading a bunch of posts on collisions and the like, and I got to thinking about the necessary conditions for one body to capture another (in space of course). Upon first glance (inward) it seems obvious that when one body is much more massive than another this would be more likely to happen and form a stable system. Here are a few questions that come to mind on this phenomenon:
1. What incoming trajectories and velocities would create likely captures for an object that is much smaller than future host? (e.g. orthogonal to orbital radius at appropriate tangential velocity) What types of combo's work?

None. It doesn't matter what the relative masses of the two stars are, they can't capture each other and end up in a stable orbit. In order for that to happen, you need at least one other massive body. So it is technically only possible for a binary star to capture a third component. However, that would not produce a stable system, and one of the three would end up getting ejected.

2. Are these two initial conditions (direction and velocity) different when the bodies are closer in mass? (e.g. two planets or two stars)

No.

3. Do we observe or hypothesize many captured binary stars?

No. AFAIK there are no candidates for captured binaries. The possibility of a capture through a 3+ body system is understood, but is extremely unlikely in practice.

4. What is the most likely scenario once objects get really close? Choices: a)a quick fling b)dance to disaster c) occasional date d)stable relationship e)other

Unless the objects actually collide, they will swing around each other and depart on hyperbolic orbits (with respect to each other), as each will exceed the escape velocity of the other.

[hstarbuck]

Thanks Chris. Unexpected . I thought that as a much smaller body approached a much more massive body--so much mass difference that big body can be thought of as stationary--some initial conditions could be set up (from far away) so that as the smaller body on its curved (parabolic or elliptical?) approaches the more massive body, at the moment the incoming body is moving with velocity orthogonal to radial position vector and equal to magnitude necessary to orbit at that given radius (v = (Gm/r)^1/2), then the incoming body would fall into orbit. Assuming high above the atmosphere this orbit would be stable. However unlikely the scenario, what is wrong with this reasoning?

[Chris Peterson]

Thanks Chris. Unexpected . I thought that as a much smaller body approached a much more massive body--so much mass difference that big body can be thought of as stationary--some initial conditions could be set up (from far away) so that as the smaller body on its curved (parabolic or elliptical?) approaches the more massive body, at the moment the incoming body is moving with velocity orthogonal to radial position vector and equal to magnitude necessary to orbit at that given radius (v = (Gm/r)^1/2), then the incoming body would fall into orbit. Assuming high above the atmosphere this orbit would be stable. However unlikely the scenario, what is wrong with this reasoning?

"Stationary" is just a matter of reference frames. With a two body system, it is often convenient to consider the more massive body the stationary one, but that's just a mathematical convenience.

One object can't "fall" into orbit around another. As if falls towards the other, its orbit is already defined. That orbit will almost certainly be hyperbolic, meaning that the object will slingshot around the other. Simply getting closer doesn't change the orbital parameters at all. The problem with a two-body system is one of energy conservation. Consider two objects that are infinitely far apart. Gravity will pull them together, and after an infinite amount of time they will come together at exactly each other's escape velocity. That's the limiting case. In reality, two objects moving towards each other will have had some external energy start their motion. They will have more velocity than if they had come from infinity, and will come together with more than their escape velocities. To form a stable system, you need a third body to absorb some of the velocity of one of the others, so it can be captured.

[hstarbuck]

That explanation makes total sense. Thanks. Good ol cons. of E.

[Wayne]

No. AFAIK there are no candidates for captured binaries. The possibility of a capture through a 3+ body system is understood, but is extremely unlikely in practice.

While not stars, we believe Triton (Neptune) was captured exactly this way. Neptune has a distinct lack of larger moons and Triton was obviously not formed there. Put the two together and you have a bunch of Neptune's former moons lurking around somewhere.

[Chris Peterson]

While not stars, we believe Triton (Neptune) was captured exactly this way. Neptune has a distinct lack of larger moons and Triton was obviously not formed there. Put the two together and you have a bunch of Neptune's former moons lurking around somewhere.

Yes, within a planetary system there are probably lots of opportunities for three-body (or higher) interactions. It's a good bet that some (or most, or all) of the moons of gas giants are captured bodies. And that moons that used to belong to these planets have been jettisoned.

[wonderboy]

While not stars, we believe Triton (Neptune) was captured exactly this way. Neptune has a distinct lack of larger moons and Triton was obviously not formed there. Put the two together and you have a bunch of Neptune's former moons lurking around somewhere.

Yes, within a planetary system there are probably lots of opportunities for three-body (or higher) interactions. It's a good bet that some (or most, or all) of the moons of gas giants are captured bodies. And that moons that used to belong to these planets have been jettisoned.

Could they have been eaten by the larger gas giant as well?

I would assume so.

Also, look at halleys comet? Could this not have been a moon that was jettisoned during capture by a large body? it travels as if it has been slingshot around our solar system, if not, feel free to rubbish my claim.

Paul.

[dougettinger]

Yes, within a planetary system there are probably lots of opportunities for three-body (or higher) interactions. It's a good bet that some (or most, or all) of the moons of gas giants are captured bodies. And that moons that used to belong to these planets have been jettisoned.

If moons have been captured by the gas giants then what three bodies made this possible ? Are we referring to the gas giant and one or more of its moons along with the captured body or are we referring to two neighboring gas giants and the subject captured body ?

As the number of bodies is increased in a multi-body system, is the capture mode increasingly enhanced ? Could you treat the combination of the Sun, Jupiter, and Saturn as a 3-body system that could possibly capture a perturbed body from the Kuiper Belt or the Oort Cloud ?

Doug Ettinger
Pittsburgh, PA

[Chris Peterson]

If moons have been captured by the gas giants then what three bodies made this possible ?

One of the bodies is the moon itself, and another is the capturing gas giant. That leaves one other interaction required; by far the most likely would be the Sun, although any of the other gas giants could do it.

As the number of bodies is increased in a multi-body system, is the capture mode increasingly enhanced ? Could you treat the combination of the Sun, Jupiter, and Saturn as a 3-body system that could possibly capture a perturbed body from the Kuiper Belt or the Oort Cloud ?

Yes. Actually, the system you reference is a four-body system, since you need to include the body being captured. However, in the Solar System the only bodies that can practically be involved are the Sun and the four gas giants. Almost all interactions will require bodies entering on the plane of the Solar System, although it remains possible (but unlikely) for out-of-plane objects to be captured if their first interaction places them in a new orbit with a node on the system plane. (I'm currently at a scientific conference, Meteoroids 2010, and a speaker described precisely that mechanism for producing what appear to be interstellar meteors from Oort cloud bodies- the interaction with Jupiter or Saturn adds energy to an object which was previously on an elliptical orbit).

In terms of your question, Kuiper and Oort bodies are already gravitationally bound to the Sun, so a three body interaction isn't required to keep them in the system.

[dougettinger]

In terms of your question, Kuiper and Oort bodies are already gravitationally bound to the Sun, so a three body interaction isn't required to keep them in the system.

I am getting better with the quotes, but I still screw-up sometimes. Thanks for your help.

Any Kuiper body coming from 50 to 100 AU's or an Oort body coming from 30,000 to 50,000 AU's will have a very elongated elliptic orbit. For this body to be captured into a smaller inner orbit, don't you need a three or four body interaction. Some of these bodies are also well outside the invariable plane or residual disk?

Doug Ettinger
Pittsburgh,Pa

[Chris Peterson]

Any Kuiper body coming from 50 to 100 AU's or an Oort body coming from 30,000 to 50,000 AU's will have a very elongated elliptic orbit. For this body to be captured into a smaller inner orbit, don't you need a three or four body interaction. Some of these bodies are also well outside the invariable plane or residual disk?a

Yes, that's precisely what is observed. Without further perturbation, comets from the Kuiper or Oort clouds are usually in hyperbolic orbits, or occasionally elliptical orbits of very high eccentricity. But they can interact with a gas giant (in practice, this is almost always Jupiter) and shed enough velocity to end up in a much less eccentric orbit. That's the mechanism used to explain the existence of short period (Jupiter and Halley class) comets. To be clear, the three bodies involved are the comet, the Sun, and Jupiter. These comets are mainly near the ecliptic, because the cross section for a Jupiter encounter is much greater for such bodies.