Super eccentric looping orbits.

[Seegull]

I am imagining a stable orbit pattern something like this.
The large object might be a plant (as labelled) or a star.
The small object might be a space craft (as labelled) or a natural satellite.

The orbiting object makes an almost complete orbit at highspeed near the closest point and then makes a long comet like loop far out into space.

Does orbital mechanics permit something like this?

It would be o.k. for my purposes (perhaps even desirable) if the long loops orientation slowly precesses over multiple cycles.

I vaguely remember seeing a potential energy simulator many years ago that suggested such things were possible. I can find no trace of it now.

Both answers in principle and possibly mathematical solutions welcome.

 

[statickid]

Am I wrong thinking this just an exaggerated Hohmann rendezvous? :shrug:

One major difference between the scales is that if it is a planet then there will be a significant gravitational influence when the bodies meet up. Also it will be more of a flyby either way.

[edit:]

Ooops sorry I just looked at your drawing again, I was confused by the lines.
There is no way I can imagine this being possible without an enormous ΔV budget. Maybe my imagination is lacking.

 

[Linguofreak]

I am imagining a stable orbit pattern something like this. View attachment 15247
The large object might be a plant (as labelled) or a star.
The small object might be a space craft (as labelled) or a natural satellite.

The orbiting object makes an almost complete orbit at highspeed near the closest point and then makes a long comet like loop far out into space.

Does orbital mechanics permit something like this?

It would be o.k. for my purposes (perhaps even desirable) if the long loops orientation slowly precesses over multiple cycles.

I vaguely remember seeing a potential energy simulator many years ago that suggested such things were possible. I can find no trace of it now.

Both answers in principle and possibly mathematical solutions welcome.

For Newtonian gravity it is not possible, all orbits are exactly conic sections (ignoring perturbations from other bodies).

In General Relativity such orbits actually exist, but anything as extreme as you have illustrated will only happen in close proximity to a black hole. One of the early successes of General Relativity is that it was able to explain extra precession in Mercury's orbit that could not be explained by the influence of the other planets. That extra precession is a much less extreme example of this.

---------- Post added at 02:46 ---------- Previous post was at 01:43 ----------

To be a bit more specific, any inverse square force, such as Newtonian gravity, will have orbits that are exact conic sections. General Relativity is inverse square in the limit of infinite distance from the central body, but the exponent gets higher at short distances, so that at progressively smaller radii GR is inverse cube, inverse quartic, and, at the Schwarzschild radius, the strength of gravity is infinite.

This causes orbiting bodies to dwell near perihelion longer than they normally would, for multiple orbits in extreme cases, which causes the apsides to precess.

The objects in the solar system are too big, relative to their masses, for this effect to be significant, but it is an observable (but small) deviation from the inverse square case for Mercury.

 

[Seegull]

This is getting interesting

So the effects of 'Relativistic precision' are apparent in the Mercury=>Sol orbit because:
A. They are close to each other
&
B. Sol is much bigger than Mercury.

You mention massive singularities so I am guessing that the more extreme the mass ratio the stronger the effect.

So, if we changed from Sol to a much heavier star (perhaps 100 solar masses) and a planet that was a little heavier than mercury (between 10 to 20 mercury masses) the effects of Relativistic precession would be even more apparent, leading to a slightly 'spiragraphic' orbit.

Mix in some extreme elipticality in the orbit and you get what I was thinking of. Maybe not as extreme as in my sketch. That was only to over-emphasise what I was getting at.

Thanks very much for explaining this to me so clearly (unless I've just completely mis-understood what you wanted to explain. In which case, please try and rescue me from my own ignorance).

 

[boogabooga]

The orbiting object makes an almost complete orbit at highspeed near the closest point and then makes a long comet like loop far out into space.

Does orbital mechanics permit something like this?

Yes, but you have to burn a lot of propellant. For example, the Apollos made an earth orbit or two before launching into an elliptical orbit to the moon.

I won't happen on its own in Newtonian physics.

 

[Linguofreak]

So the effects of 'Relativistic precision' are apparent in the Mercury=>Sol orbit because:
A. They are close to each other
&
B. Sol is much bigger than Mercury.

Not quite. The effects of relativistic precession are apparent because the ratio between the distance between them and Sol's Schwarzschild radius is relatively small. Mercury's mass doesn't have much to do with it, though I think if Mercury were similar in mass to Sol it would add somewhat to the effect.

Now, Sol's physical radius is much larger than its Schwarzschild radius, so the effects of GR are small even for orbits that nearly hit the sun at perihelion. If you squeezed Sol down to 3 km radius, however, it would become a black hole, and then you could constrict a horizon-grazing trajectory that would come in from infinity, loop around the black hole an arbitrary number of times, and then escape back out to infinity.

The Schwarzschild radius is directly proportional to mass, at 3 km / solar mass.

 

[Seegull]

The effects of relativistic precession are apparent because the ratio between the distance between them and Sol's Schwarzschild radius is relatively small.

The Schwarzschild radius is directly proportional to mass, at 3 km / solar mass.

Ah! Got it now, the pivotal characteristic of the system is really the density of the objects (especially the large one). So for my spiragraph orbit to work nicely, I need a (lets assume star) that is far denser than Sol (at a similar mass) or of a similar density but much heavier (the radius of the star would increase at a rate proportional to the cube root of the mass but the Schwarzschild radius grows directly proportional to the mass, i.e. much more rapidly).

Now I just have to research if any real stars would fit my purpose....

Is there a rule of thumb about how much precession a certain ratio of Schwarzschilderihelion radius might cause?

 

[Linguofreak]

Now I just have to research if any real stars would fit my purpose....

Only neutron stars (depending on how much precession you need) and stellar-mass black holes (for any amount of precession), but those will have extreme tidal forces near the object (so any spacecraft/planet/solid object in general will be torn apart on such trajectories)

For a supermassive black hole, you can get arbitrarily large amounts of precession for arbitrarily low tidal forces, depending on mass).

Is there a rule of thumb about how much precession a certain ratio of Schwarzschilderihelion radius might cause?

Not just a rule of thumb, for a given eccentricity there will be an exact formula, though I don't know it off the top of my head.