Formal definition of orbit?

[h4tt3n]

Hello,

 

I'm looking for a way to determine wether one object is in a closed elliptic orbit around another based on mass and state vectors alone.

 

For instance, if an asteroid is passing close by Jupiter, how do you determine wether it has actually been caught up into a closed orbit (assuming mass and state vectors are known)? At all times it is orbiting the sun, but how do you determine wether it is also orbiting Juiter?

 

Cheers,

Mike

[D H]

Nothing truly follows an elliptical path. The planets perturb one another's orbits. Further afield, the nearby stars perturb the planets' orbits slightly. Even further afield, the Sun is orbiting the galaxy. There is a gravity gradient (very slight, but there) that keeps the planets' orbits from being true ellipses. Then there is the slight issue that Newtonian gravity is but an approximation of reality.

[h4tt3n]

Nothing truly follows an elliptical path. The planets perturb one another's orbits. Further afield, the nearby stars perturb the planets' orbits slightly. Even further afield, the Sun is orbiting the galaxy. There is a gravity gradient (very slight, but there) that keeps the planets' orbits from being true ellipses. Then there is the slight issue that Newtonian gravity is but an approximation of reality.

 

True, no object follows an *exact* ellipse, but this is really besides the topic. A given object is either in closed orbit around another given object or not, the exact shape of that orbit beeing of less importance.

 

But how do you determine wether it is or not?

[Radical Edward]

a quick and dirty guess might be that the sum of the kinetic and potential energies would be less than the potential energy of the object at infinity wrt the larger object.

[Mr Skeptic]

a quick and dirty guess might be that the sum of the kinetic and potential energies would be less than the potential energy of the object at infinity wrt the larger object.

 

I'd go with this. If the kinetic energy is more than the potential energy (with respect to your object to be orbited), there is no way it's staying in an orbit. If it is far less, then it's unlikely to be able to escape the orbit even with the influence of other bodies. Of course it could also then be on a collision course, although if you assume point masses there's never a collision. If the kinetic is just slightly less than needed to escape, other orbiting bodies might transfer energy to it and make it escape. Also small orbits of massive objects could tear apart an orbiting object due to tidal forces.

[Sisyphus]

Or to put it another way, it is captured if its velocity is less than escape velocity for that object.

 

http://en.wikipedia.org/wiki/Escape_velocity

 

That's assuming it doesn't collide. A collision at less than escape velocity is just an orbit that intersects with the surface of the orbitted object.

[h4tt3n]

I'll try this out. It does sound plausible, although I think there's a pitfall. Smaller objects can orbit more than one larger object simultaneously. The Moon orbits the Earth, but it also orbits the Sun. If we pick the Moon's state vectors as an example, the method should point out Earth as the parent object, not the Sun.

 

I'll run this through my simulation software and see if there's a clear pattern.

 

Cheers,

Mike

[D H]

True, no object follows an *exact* ellipse, but this is really besides the topic. A given object is either in closed orbit around another given object or not, the exact shape of that orbit beeing of less importance.

 

But how do you determine wether it is or not?

Ahh. Misuse of terminology. You mean bounded, not closed. A closed orbit is one that (eventually) repeats state. A bounded orbit is one that (somehow or another) stays within some finite bounds. In other words, no escape.

 

There is no true way to know whether an orbit is bounded in a multi-body gravity field. Escape velocity definitely doesn't cut it. A couple of good boundaries are Hill's sphere and Lagrange's gravitational sphere of influence:

 

[math]

\aligned

r_{\text{Hill}} \approx a\left(\frac m{3M}\right)^{1/3} &&&\text{Hill's sphere} \\

r_{\text{SOI}} \approx a\left(\frac m{2M}\right)^{2/5} &&&\text{Sphere of influence} \endaligned

[/math]

 

Here r is the radius of the orbit of some very small object (e.g. a captured asteroid) about an object of mass m (e.g., a planet), which is in turn is in orbit of radius a about an object of mass M (e.g., the Sun).

[h4tt3n]

H_D:

 

Thanks for clarifying the concepts, now I understand why you brought up the exact elliptic orbit. As for the equations, this is definitely in the right direction. The problem here is, that in order to calculate wether a smaller object is likely to be in orbit around a larger object, you *first* need to know wether this larger object is orbiting an even larger one (in order to calculate the semimajor axis).

 

This is not a problem when working with a known, stable system like our solar system, where all larger objects behave nicely, and where you know the sun will always be the larger, central body everything else orbits around. But I can't seem to implement these equations in a more general form, which returns valid Hill spheres for bodys moving around in a more chaotic system.

 

Cheers,

Mike

---

Merged post follows:

Consecutive posts merged

Radical Edward, Mr Skeptic & Sisyphus:

 

Yes, the smaller object is captured by the larger if it's total energy (kinetic + gravitational potential relative to larger object) is less than zero, or if it's speed is smaller than the escape velocity (which is the same thing, really). The only issue about this is that an object can have E < 0 relative to several other objects at the same time. The Lunar Orbiter probes were orbiting the moon, Earth, and Sun at the same time. How do we "sort" them hierarchically?

 

Cheers,

Mike

Edited March 10, 2010 by h4tt3n
Consecutive posts merged.

[D H]

Emphasis mine:

This is not a problem when working with a known, stable system like our solar system, where all larger objects behave nicely, and where you know the sun will always be the larger, central body everything else orbits around. But I can't seem to implement these equations in a more general form, which returns valid Hill spheres for bodys moving around in a more chaotic system.

Even our own solar system is, in the long haul, a chaotic system. Mercury might well collide with Venus in the far future, for example. You are essentially asking for a predictable outcome for what is inherently a chaotic system. Such a beast does not exist.

[h4tt3n]

Yes, I realise this and that, as a consequence, you can't define a border around any larger body within which any orbiting object will remain stably for all eternity. That's also not what I'm looking for. I'm simply asking for a method that, assuming known mass and state vectors for all bodies, in any given system will determine which - if any - larger body a smaller body of our choice is orbiting at that moment. No prediction of future eventualities neccesary.

 

I've implemented the Hill Sphere with some success, but as stated on the wikipedia page it's only an approximation, and the true region of stability over any length of time may be only half as wide. It seems like just a qualified guess.

 

The SOI is basically the same equation, except it results in a smaller sphere, but unfortunately it also requires knowledge of the semimajor axis of the larger object trajectory around an even larger body.

 

At least there are some mostly certain methods to determine that a smaller object is not in a bounded orbit around another one:

 

-total energy of smaller body wrt to larger body is positive

 

-smaller body's speed is larger than the escape velocity of the larger body

 

cheers,

Mike