How do you define orbital shape parameters?

[SamBridge]

In what circumstances does an orbit become a hyperbolic trajectory and vice-versa? How do you actually change x^2+y^2=1 to x^2-y^2=1?

Edited June 13, 2013 by SamBridge

[John Cuthber]

If the kinetic energy is more than the gravitational binding energy the orbit isn't going to be an ellipse.

[swansont]

It will change because of some perturbation.

[SamBridge]

So there's no chance it would change on it's own...?

[D H]

In what circumstances does an orbit become a hyperbolic trajectory and vice-versa? How do you actually change x^2+y^2=1 to x^2-y^2=1?

 

I'm not quite sure what you're asking here, so I'll try to interpret your question in multiple ways.

 

 

1. What laws of physics change to make an orbiting body obey x^2+y^2=1 on one hand versus x^2-y^2=1 on the other?

 

The laws of physics are one and the same. This apparent change is just a result of trying to describe the behavior kinematically in Cartesian coordinates. A perhaps better way to look at things is in terms of polar coordinates, where the kinematic description of the motion becomes [imath]r(1+e\cos\theta) = p[/imath]. Here, r and θ are the polar coordinates, e is the eccentricity (a constant), and p is the semi latus rectum (another constant). A circular trajectory results if e=0, elliptical for 0<e<1, parabolic for e=1, and hyperbolic for e>1.

 

 

2. What physical condition distinguishes elliptical orbits from hyperbolic trajectories?

 

Energy. With gravitational potential energy [imath]\Phi[/imath] defined such that [imath]\Phi®[/imath] goes to zero as r tends to infinity, mechanical energy is negative for elliptical and circular orbits, zero for parabolic trajectories, and positive for hyperbolic trajectories.

 

 

3. How can an orbit be changed from elliptical to hyperbolic?

 

Simple: Increase the energy (i.e., increase the velocity) so that the total mechanical energy becomes positive.

4. What makes comets have hyperbolic trajectories?

 

This question is a bit trickier. Some of those long-periodic comets truly are on a hyperbolic trajectory. We have one chance to see those comets. After that one passage close to the Sun they exit the solar system, never to be seen again. Other comets have a reported eccentricity that is just slightly over one, yet these comets may well be seen again (perhaps thousands of years later, or longer). So what gives?

The answer lies in how orbital elements are calculated. For an object inside Jupiter's orbit, orbital elements are calculated from that object's heliocentric Cartesian position and velocity using the mass of the Sun only as the central body. For an object outside Jupiter's orbit, orbital elements are calculated from that object's heliocentric Cartesian position and velocity using the combined mass of the Sun and Jupiter (plus that of other giant planets orbiting inside the object's radial distance) as the central body.

To understand why this convention is used, the first thing to realize is that the very concept of Keplerian orbital elements is not quite valid for objects in our solar system. Orbital elements are valid for two isolated bodies subject to Newtonian gravity. Our solar system is not a two body, Newtonian gravity system. Orbital elements are a useful fiction.

This apparently weird way of calculating orbital element works quite nicely for an object whose orbit lies entirely inside or outside of Jupiter's orbit. For objects inside Jupiter's orbit, the giant planets have very little impact on the object's period. The primary effect is to cause the object's orbit to precess. The orbital elements for these inner solar system objects are best calculated by ignoring the presence of the outer planets. For objects outside of Jupiter's orbit, the primary impact of Jupiter is to increase the object's mean motion. Thus it's best to include Jupiter's mass when calculating orbital elements for those more remote objects.

What this means is that a comet's reported eccentricity can go from slightly more than one to slightly less than one as the comet crosses Jupiter's orbit on its outbound leg. There's nothing physical going on here. It's just a consequence of how astronomers compute eccentricity.

Edited June 13, 2013 by D H

[SamBridge]

So in other words it would never change on it's own, you'd just have to add some energy, which makes sense because hyperbolic trajectories break the orbit.

[D H]

So in other words it would never change on it's own.

Not exactly. Think of a rocket. Changing the orbit -- thats exactly what rockets do.

 

Comets are rockets. Not very controlled ones, but rockets nonetheless.

[SamBridge]

Not exactly. Think of a rocket. Changing the orbit -- thats exactly what rockets do.

 

Comets are rockets. Not very controlled ones, but rockets nonetheless.

This would imply there is an instance where without any outside force, the inertia of an object would shift angles due to eccentricity and thus become hyperbolic, but what doesn't make sense to me is it takes force to accelerate past the velocity which is required for it to be hyperbolic, which doesn't happen with inertia alone. The only time I can think this happens is when a comet happens to pass so close to the sun that it accelerates enough to break free of orbit. Otherwise, I don't know what you're saying.

It would be interesting to see the math changing as well though, to see exactly when the plus sign switches to a minus sign.

[D H]

This would imply there is an instance where without any outside force, the inertia of an object would shift angles due to eccentricity and thus become hyperbolic, but what doesn't make sense to me is it takes force to accelerate past the velocity which is required for it to be hyperbolic, which doesn't happen with inertia alone.

 

That doesn't make a bit of sense, Sam. You appear to be thinking that there is some extra-special barrier that separates closed orbits (circles and ellipses) from open ones (parabolas and hyperbolas). There isn't.