Stable/unstable orbits in N-body system - how does that work?

[michal.don]

Hi,

could anyone explain/link me to an explanation, preferrably a simplified one, what determines whether an orbit is stable or unstable in real solar systems/N-body systems generally? I heard a lot about bodies in unstable orbits, but I can't get my head around how that works.

Thanks,

Michal.don

[Green Baron]

 

I don't think there is a simple explanation. Afaik only the 2-body problem has an exact solution.

A special case, the restricted 3-body-problem, (1 mass=0, movement in a plane) can be approximated. I am not a mathematician and can only accept what i read.

For my understanding the general n-body problem quickly gets chaotic. Simulations can only approximate the general behaviour.

Gravitation perturbations in solar system can be approximated, e.g. to predict a position of a planet at the time of arrival of a spacecraft. But orbits are in principle never really stable, algorithms are based on optimizations and simplifications and spacecrafts can correct the flight path.

Hope that's generally not totally wrong :-)

Edit: under 3-body considerations, bodies that enter resonant orbits are considered "stable".

 

Edited February 1, 2017 by Green Baron

[ElWanderer]

Wikipedia has a Stability of the Solar System page: https://en.wikipedia.org/wiki/Stability_of_the_Solar_System

Confusingly, it also has a page on "Orbital Stability" that has nothing to with orbits, but is related to mathematical functions.

[michal.don]

Thanks, both these pages are quite informative, but I'm afraid the math is too advanced for me to properly understand

Michal.don

[p1t1o]

This is how I understand the n-body problem:

Its not that n-body systems are impossible to predict, it is that the errors build up so rapidly that they essentially become impossible to predict beyond a certain length of time. This is because where each mass will be at time X is dependent on where each other mass will at all other times leading up to X, so not only does the sheer volume of calculations necessary very quickly become arbitrarily large, even tiny errors in first assumptions (such as "what is the initial position+velocity of planet A") also very quickly become unmanageable.

Our solar system, for example, I think can be predicted to decent accuracy for several thousand years (not sure of the exact figure, but it is some finite number of years, large in human terms, small on the cosmic scale) at some point the errors become so large that they encompass the whole system, so when you try and predict where they will be beyond a certain length of time, all you can say is that you are not really sure, but somewhere in this "zone".

Yes you can say, well the likelihood of Jupiter being ejected from the system to beyond pluto is very, very low, but you would be unable, mathematically, to show that it couldn't happen.

***

As for stability, I think the general definition goes somewhere along the lines of: 

"A system is stable if it returns to its original conditions after receiving a minor perturbation. A system is unstable if minor perturbations result in departure from original conditions." 

In this context, the "system" could be any "system", not just an orbital one.

A ball sitting in a bowl is a good example of a stable system.

A ball balanced carefully on another ball is a good example of an unstable one.

***

Combining those two principles, you find that with n-body systems, figuring out how stable it is is non-trivial. There are no pointers or characteristics you can look at and say "This n-body system is stable."

The only thing you can do is collect as much data as possible and do as much hard maths as possible to figure out how long it will take before your predictions lose quality, and even then there is uncertainty because all of your measurements/observations will have their own error bars.

Of course you can imagine many systems where it will be obviously unstable, and others which are obviously more stable, but not only are these hard to come by in nature, the exact nature of the stability of the system will still elude you.

[michal.don]

Ok, let me rephrase the original question a bit - I think I "understand" (more like I have an idea) that the calculations are pretty difficult and that the system itself is not a stable one because all the bodies affect each other. But I'm curious how this fact affects behaviour of planets and their moons, for example in our solar system. I heard that the orbit of the Moon is getting lower and lower, and in distant future, it might get so close to Earth it will be torn apart. Which force lowers the orbit? Is it gravity of the Sun? Certain moons are believed to be captured astroids. How does a celestial body get captured into orbit of another one (without Tylo gravity assist  )? Is there a "simple" explanation for these things? Generally, these, that work in another way than Sphere-of-influence and on-rails orbits we know so well from KSP?

Thanks,

Michal.don

[tomf]

The moon is getting further away from Earth. It's gravity raises a tidal bulge on Earth, but because Earth's rotation is faster than the moon's period the bulge is dragged slightly in front of the moon. This accelerates the moon, dragging it into a higher orbit. It is currently moving away at 3.78cm/year. The friction also slows the earth's rotation, until eventually the earth will rotate at exactly the same speed as the moon orbits and the process will stop.

I'm pretty sure that Capturing moons is only possible with the assistance of assisting bodies.https://en.wikipedia.org/wiki/Asteroid_capture

Edited February 1, 2017 by tomf

[kerbiloid]

For example, an asteroid revolves around the Sun close to the orbit of a planet, and their relative speed is just about, say, 1 km/s.
While the asteroid passes far from the planet, it's all right.
But when the asteroid approaches close to the planet (say, tens thousands kilometers), it's relative speed (that 1 km/s) is lower than escape speed relative to planet. And suddenly it's inside the planet sphere of influence with "elliptical" speed. It can nothing more, except continue revolve around the planet.

Edited February 1, 2017 by kerbiloid

[Green Baron]

Yes, the moon is getting away due to conservation of momentum as @p1t1o explained. And the days are getting longer in the same turn. It won't leave earths orbit and as far as i know it will not force earth in a bound rotation during the "lifetime" of the solar system. Not sure, something around tens of billions of years if nothing comes in between :-)

 

[Steel]

5 hours ago, michal.don said:

... I heard that the orbit of the Moon is getting lower and lower, and in distant future, it might get so close to Earth it will be torn apart. Which force lowers the orbit? Is it gravity of the Sun? Certain moons are believed to be captured astroids. How does a celestial body get captured into orbit of another one (without Tylo gravity assist  )? Is there a "simple" explanation for these things? Generally, these, that work in another way than Sphere-of-influence and on-rails orbits we know so well from KSP?

Thanks,

Michal.don

So in the cases above (with the exception, as noted by others above, that the moon is in fact getting further away, due to conservation of angular momentum) the gravitational influence of some other body in the system has caused a perturbation to the the orbit of the object and cause effects that are impossible in a 2-body simulation like KSP (i.e capture of a body). Most of the time this will be Jupiter or the Sun as these are the dominant masses in the system but it could quite equally be another close-by planet or another moon of a planet that causes the capture of an object.

[kerbiloid]

Probably that theme about Moonocalypse relates to a fresh article.
Though, the astronomer predicts Moon-Earth collision 65 BLN years later (i.e. ten times greater than to the Sun-Earth burning), which makes to think that PR works.

[Green Baron]

"predicts" is the right word in this context :-)

 

[magnemoe]

19 hours ago, p1t1o said:

This is how I understand the n-body problem:

Its not that n-body systems are impossible to predict, it is that the errors build up so rapidly that they essentially become impossible to predict beyond a certain length of time. This is because where each mass will be at time X is dependent on where each other mass will at all other times leading up to X, so not only does the sheer volume of calculations necessary very quickly become arbitrarily large, even tiny errors in first assumptions (such as "what is the initial position+velocity of planet A") also very quickly become unmanageable.

Our solar system, for example, I think can be predicted to decent accuracy for several thousand years (not sure of the exact figure, but it is some finite number of years, large in human terms, small on the cosmic scale) at some point the errors become so large that they encompass the whole system, so when you try and predict where they will be beyond a certain length of time, all you can say is that you are not really sure, but somewhere in this "zone".

Yes you can say, well the likelihood of Jupiter being ejected from the system to beyond pluto is very, very low, but you would be unable, mathematically, to show that it couldn't happen.

***

As for stability, I think the general definition goes somewhere along the lines of: 

"A system is stable if it returns to its original conditions after receiving a minor perturbation. A system is unstable if minor perturbations result in departure from original conditions." 

In this context, the "system" could be any "system", not just an orbital one.

A ball sitting in a bowl is a good example of a stable system.

A ball balanced carefully on another ball is a good example of an unstable one.

***

Combining those two principles, you find that with n-body systems, figuring out how stable it is is non-trivial. There are no pointers or characteristics you can look at and say "This n-body system is stable."

The only thing you can do is collect as much data as possible and do as much hard maths as possible to figure out how long it will take before your predictions lose quality, and even then there is uncertainty because all of your measurements/observations will have their own error bars.

Of course you can imagine many systems where it will be obviously unstable, and others which are obviously more stable, but not only are these hard to come by in nature, the exact nature of the stability of the system will still elude you.

It depend a lot on the main forces acting on an body, Moon interact mostly with Earth making it easy to predict. Halley comet's orbit can not be predicted much more than 600 years into the future.
Problem is that it crosses the orbit of Jupiter and Saturn, tiny changes affect the high Ap a lot who changes time for Pe and this timing is critical for the next orbital calculation giving an error who grows fast. 
Same issue with an space probe like Hugens in an elliptical orbit around Saturn with flyby of many moons making long term predictions as in 100 orbits pretty impossible. 

You also have orbits where you can predict issues in the future like then Earth get an temporary moon, an tiny asteroid or more common spent stages for Moon missions end up in solar orbits before getting captured by earth / moon again do some orbits before getting expelled by Moon, this is also seen with asteroids in KSP, they enter Mun SOI trajectory changes so they enter Kerbin orbit until they again enter Mun SOI and get expelled. 

[kerbiloid]

Orbital movement is a harmonic oscillation. Like a cord on a harp. Every planet - another cord.
Earth - 31.7 nHz, Pluto 1.3 nHz (about -29th octave).
Every of them oscillates, causing others to oscilate in time.
And every cord in this harp is distorted and mistuned. 
N-body problem is: can you ever get the unison on a distorted harp with several tens of mistuned cords.

 

Spoiler

 

[YNM]

In every particle condition, you need a phase space : position vector, velocity vector, and acceleration vector. The problem with N-body problem is one : Lack of a significant stationary point, which complicate the predictability of said phase spaces.

In 2-body problems, it is true that there's no need to be a stationary point. But, arbitrarily, you can take one of the bodies as that.

In limited 3-body problems, you can take both objects to be very close to not moving wrt to each other.

In higher-body problems, you'll have a hard time picking a stationary point in the system. The reason why over time they tend towards deviation is actually thanks to the way we do it today - by some approach along Euler method, because we need it to be well-documented for each body. Now, even while using Lagrangian or Hamiltonian mechanics isn't out of question, remember that they're not deterministic in nature - you can tell whether something will happen or not, but how, you have no idea. It's like ideal gasses - you can't tell where each particle really is (apart in simulations).

[Steel]

3 hours ago, YNM said:

In every particle condition, you need a phase space : position vector, velocity vector, and acceleration vector. The problem with N-body problem is one : Lack of a significant stationary point, which complicate the predictability of said phase spaces.

In 2-body problems, it is true that there's no need to be a stationary point. But, arbitrarily, you can take one of the bodies as that.

In limited 3-body problems, you can take both objects to be very close to not moving wrt to each other.

In higher-body problems, you'll have a hard time picking a stationary point in the system. The reason why over time they tend towards deviation is actually thanks to the way we do it today - by some approach along Euler method, because we need it to be well-documented for each body. Now, even while using Lagrangian or Hamiltonian mechanics isn't out of question, remember that they're not deterministic in nature - you can tell whether something will happen or not, but how, you have no idea. It's like ideal gasses - you can't tell where each particle really is (apart in simulations).

While this may well be a good answer, I'm not sure describing stationary points in phase space is going to help OP to get a simple and intuitive understanding of why many orbit in n-body systems are unstable

[YNM]

I suppose the bigger points is that the system is not deterministic. You can have a pretty overall picture of it every time but you can't tell where each particle will end up in.

[pincushionman]

The solar system orbits aren't "stable," they're "stable enough."

[WinkAllKerb'']

2 minutes ago, pincushionman said:

The solar system orbits aren't "stable," they're "stable enough."

that's the point , autonomous on various layer and recalculation on the flow along the trip to adjust ^^

Edited February 4, 2017 by WinkAllKerb''

[YNM]

21 hours ago, pincushionman said:

The solar system orbits aren't "stable," they're "stable enough."

Doesn't stop any "doomsday-size" meteors to come every "dooms"day

[pincushionman]

1 hour ago, YNM said:

Doesn't stop any "doomsday-size" meteors to come every "dooms"day

I should have clarified - I meant planets, dwarf planets, and large moons. On the other hand, it's well-established that SSSBs are unstable over the long term, and the really small stuff (as in doomsday-size) isn't even stable in the short term. Partially because of the chaos effects, but also because other forces like solar wind, drag,  and the Yarkovsky effect become non-negligible on those scales

[natsirt721]

Kepler says that orbiting bodies move on ellipses, but that's really just an approximation.  In an n-body system, every body pulls on every other body, not just the sun and its satellites.  This creates disturbances, irregularities in the otherwise elliptical orbits of the satellites.  Sometimes these irregularities have positive feedback - they tend towards configurations that are 'unstable' (eg; bodies ejected from a system or dramatic changes of orbit).  Sometimes they have negative feedback - they tend towards configurations that are 'stable' (eg; orbital resonance between bodies).  Whether a system is 'stable' or not depends on, well, everything.  Gravity, radiation pressure, tidal forces, non-spherical bodies, the list goes on.  Sometimes resonant systems are also unstable, and sometimes otherwise unstable configurations are actually periodically stable.  There's no good way to tell just by looking at a system, and therein lies the problem.  I think p1t1o had it spot on when they said:

On 2/1/2017 at 9:33 AM, p1t1o said:

...the exact nature of the stability of the system will still elude you.