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Posted by: Anton Martynov Sep 1 2015, 05:18 AM

I was involved in a casual conversation recently about the exploration of bodies in Solar System, in particular about fly-by approach vs. orbiter approach. In particular, I was saying that the main challenge with orbiter missions is that either you have to decelerate once you reach your target (requires tons of fuel on board), or you have to travel along the trajectory that would take an unreasonable amount of time to reach the target. As an example of the latter, I said that you could launch a spacecraft into a trajectory that would be a part of an elliptical orbit around the Sun with the perihelion around Earth and the aphelion around the destination. That way, when the spacecraft reaches its target, its speed relative to the planet will be slow enough for it to be "picked up" by the planet's gravity, and it will start orbiting the planet.

The other person in the conversation pointed out that it's been shown that when two bodies pass each other and influence each other gravitationally, it's not possible for them to start orbiting each other (or, in the case of one object being much more massive than the other (planet vs. spacecraft), simply "one orbiting another"). He said that either the more massive object will simply alter the trajectory of the passing smaller object, but not capture it, or the smaller object will crash into the bigger one.

And this is something that, to me, "intuitively" shouldn't be right, but I don't have enough expertise to prove that it's wrong. My counter-arguments are:

- Some of the natural satellites in the Solar System are believed to be captured by the planet as they were passing by (true, these are mainly hypotheses, but people wouldn't make such hypotheses if this wouldn't be possible?).

- If the object is passing by the planet at the speed less than what is required to enter the orbit, then it will crash down onto the planet. If the object is passing at the speed greater than the escape velocity, then it will continue flying without being captured by the planet. Surely if the object's speed is between these two values, it has to start orbiting the planet? (Not necessarily in a perfect circular orbit of course).

As further proof for the second point, I calculated the elliptical orbit with the perihelion at Earth (1 AU) and aphelion at Uranus (19.2 AU). Sure, it would take 16 years to get to Uranus, but the required takeoff speed would be 41.1 km/s (relative to Earth, that would be 11.3 km/s, so just barely above Earth's escape velocity, so we're good here), and the spacecraft's speed when it arrives to Uranus would be 2.1 km/s relative to Sun, and -4.6 km/s relative to Uranus. This 4.6 km/s speed happens to be the speed of a circular orbit at 266000 km from the center of Uranus.

So, in my understanding, we can launch the spacecraft from Earth at 11.3 km/s into the elliptical orbit, and then some 16 years later it will pass by Uranus and be captured by it. If we make some small course corrections along the way so that it passes 266000 km from Uranus, it will even be a circular orbit.

... but, like I said, maybe I'm missing something that won't allow the spacecraft to be captured by the planet's gravity? My whole point was that in this scenario you don't have to use fuel for anything else other than takeoff. Of course, some course corrections would be inevitable, but at least you won't have to try to decelerate from New Horizons-like speeds.

Posted by: Herobrine Sep 1 2015, 06:20 AM

I'll leave it to the more educated minds on the forum to give you a firm answer, but it's been my understanding that the 2-body problem doesn't allow for a capture, given ideal point-/spherical masses in a vacuum, but that the 3-body problem does. So, it's supposedly possible for capture to occur when there are three masses interacting in the right way. I don't know about other possible capture mechanisms; I imagine there are some.

Posted by: Explorer1 Sep 1 2015, 06:36 AM

I'm pretty sure the above is correct; in a simple two body situation it is not possible to get 'in-between' escape velocity and a collision. It will either be a hyperbolic/parabolic 'orbit' or an impact trajectory. One cannot get an ellipse without intervention, either by firing thrusters or a flyby of something; that's how all those natural moons got captured; they were in the right positions to exchange a bit of momentum and slow something down to remain in the SOI, but even then it doesn't always work.
Compare Triton with Shoemaker-Levy 9; both were captured, but one lasted a single orbit of a few years around Jupiter before impact and the other has lasted for many millions of orbits around Neptune.
Sometimes visuals are better than words (select generate proto disc): http://www.nowykurier.com/toys/gravity/gravity.html Notice how often moons show up around the 'planets', and how often they are unstable.
Maybe Cassini could have been captured with a Titan flyby alone, but it was obviously a much easier engineering feat to just fire the engines for capture and use Titan later for adjustments

I had the same question a long time ago, but a few hours of Kerbal Space Program will also suffice to demonstrate this.

Posted by: Herobrine Sep 1 2015, 07:19 AM

Oh, gosh, this reminds me of something I forgot I'd even done.
If you want to see the N-body problem solved for millions of particles of equal mass simultaneously, and you have the Unity web player or are willing to install it, and you have a good graphics card, you might enjoy the available at http://herobrinesarmy.com/gravity

It's all calculated and rendered using GPU compute, so GPU power will make a big difference in performance here. If I recall correctly, it's scroll wheel to zoom and moving the mouse to turn, and that's it. The more you zoom in, the brighter the particles are rendered.

Edit: The more I look at this, the more I'm pretty sure this is actually one of the ones that cheats by lumping mass into "galaxies" and treating the individual particles as massless. I made so many different versions of this, years ago, that it's hard to remember which one this is. Either way, it's N-body, but it's likely the only massed bodies are the individual galaxy centers.

Posted by: tanjent Sep 1 2015, 07:46 AM

In a two body interaction, in vacuum with point masses, etc., can't we say that what happens after closest approach has to be symmetric with what happened before closest approach? The relative approach velocity of the smaller object should accelerate until closest approach and then decelerate with an identical time/distance profile, so if the smaller object wasn't in orbit before the approach, it won't end up in orbit afterwards either. Caveat: there may be some relativistic effects to consider for the accretion disks "orbiting" black holes, but in a planetary-scale Newtonian universe, symmetry should rule out getting into orbit.

But these ideas about multiple-body collisions are fascinating. Under what circumstances is it possible to attain orbit around Saturn with a gravity assist from Titan? Or around Pluto with a gravity assist from Charon? With any body A of a given mass, there must be a limit to the amount of delta-v a spacecraft B can borrow relative to a third body C, without B colliding with A. And if B can be a piece of rock or ice, then collisions are permissible too, like in Triton's case. I'm sure that solutions like these have not been overlooked by the writers of mission proposals, so they must be impossible for the mentioned cases, but impossible by how much?

Posted by: Anton Martynov Sep 1 2015, 10:24 AM

I remember having made a "gravity simulator" like this one a few years ago as well. It wasn't a web app, but a standalone program instead (all calculations were on CPU); I left it pretty underdeveloped, but I remember running into the precision problem pretty quickly. With a fixed time step, very elongated elliptical orbits (and in general any orbits that have a high velocity at some point) become extremely inprecise. When the planet "curves" around the sun at a very high speed, it's very important to have a smaller time step there, otherwise the orbit becomes very approximate.

In this simulation, I can see the same problem with the time step being too big in some cases. Look at elongated elliptical orbits and examine parts of them that are near the massive star - you'll see that the curve is a noticeably polygonal chain. In my old implementation that I mentioned above, elongated elliptical orbits exhibited the same instability as in this implementation, but the smaller (= more precise) the time step was, the more stable the orbits were.

I don't want to jump to conclusions, and please don't take my words as a criticism of the implementation you linked to: I certainly don't say that it is bad (and it's definitely better in so many aspects than mine was), but I just wanted to point out that at least some of the instability / "rotation" of elongated orbits is because of not having enough precision during close approaches.

I remember that I wanted to implement individual time steps for every object as an attempt to fix this, but never really got around to doing it. The idea was to choose the time step for an object based on its current velocity (faster object = smaller time step needed) instead of having a fixed time step for all of them.

In fact, even when I used to input "perfect" parameters for objects (as in, spawn a planet at these coordinates with this speed, which should have given the planet this or that type of "perfect", circular or elliptical, orbit), lack of precision was slowly but surely accumumating and resulting in the orbit deteriorating over time.

---

To everyone: If we "somehow" spawn an object at distance N from the planet with the speed that an object going in a circular orbit at this distance should have, would that object get into this orbit? To extend this question, what if a spacecraft flies at this particular speed into this particular point of space (with the help of some small course corrections beforehand), will it get into this orbit? And if not, why? I understand that the general consensus says "no, it won't start orbiting the planet", but I want to understand what I'm missing (what will prevent this, so to say).

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I agree with this approach, but I'm not sure if we can extrapolate orbits (not just some particular types of orbits, but orbits in general) into the past. Is it possible that for some location and speed of the object, there are multiple variants of how this object got here and got this speed?

Also as a counter-argument for this particular "symmetry" argument: in the frame of reference of the planet, "before" and "after" closest approach situations are not exactly the same. The Sun - the thing that got the spacecraft into the necessary point in space - is somewhere else now in this frame of reference. (Of course, with the Sun being there and influencing stuff, it's no longer a two-body problem).

Posted by: Floyd Sep 1 2015, 11:38 AM

I have not done orbital calculations, but was a physics major for the first two years of college--so have a feel but not a solid answer. The problem I see with your thought experiment is you can drop a spacecraft into the proper position and velocity for a circular orbit of a 3rd body, but when you start working backwards towards the joint orbit from earth you will find you would have had to fire rockets to slow the probe to get there. If you think of an elliptical orbit of earth pointing directly away from the sun out 2 AU, and look at the second focus spot and calculate how massive an object you would need to capture a spacecraft at the spacecraft's velocity at it furthest point out, then drop the 3rd body into place when our spacecraft is at the point of its orbit closest to earth (and also sun), and then follow what happens. As it goes in its orbit, it will speed up as it approaches the 3rd object and not go into orbit because it is going too fast. As you play with increasing the mass of the 3rd body to try and capture it, you get closer and closer to orbital capture....but just as you think you are about to succeed...SPLAT... litho-braking... Regardless of where you drop the 3rd object or how massive it is, you will always be (sped up to be) going too fast to orbit, but crash landing can be arranged.

Posted by: siravan Sep 1 2015, 01:43 PM

You cannot have capture in a pure two-body problem. This is simply a result of the conservation of energy. Using the usual conventions, the total energy (i.e. the sum of kinetic and potential energy) in a bound state is negative, but for two bodies that start from infinity with non-zero velocities, the total energy is positive.

In your example, the relative velocity between the s/c and Uranus is calculated assuming no gravitational pull on the s/c by Uranus. In fact, as s/c gets closer to Uranus, its velocity increases such that in the closest approach, it is above the orbital velocity a that distance.

Now, the situation is different in a three-body problem. Here, one body can be ejected and carries away the extra kinetic energy and allows the two other bodies to bind.

Posted by: HSchirmer Sep 1 2015, 01:45 PM

Yes, you can move around the solar system in the manner you describe. For three reasons-

First, your hypothetical has 3 bodies, Earth, spacecraft, Uranus, so IIRC, it must have an orbital solution.

Second, real planets have- moons, atmospheres, exospheres and tidal effects (energy dissipation mechanisms)
that can create capture orbits, and eventually circularize elliptical orbits.

Third, what you are describing would actually be a chaotic orbit, not a classical elliptical orbit and a circular orbit.
I think the technical term you are looking for is "low energy transfer".
You're essentially "gravity surfing" along unstable orbits between the planets.
Nasa had a discussion about this technique, updated to include Lagrange points, and modern ability compute chaotic effects.



Somebody came up with a cool term - https://en.m.wikipedia.org/wiki/Interplanetary_Transport_Network to describe the web of low delta-v routes that run among the planets.

Posted by: Herobrine Sep 1 2015, 02:26 PM

A leapfrog integrator can substantially improve the accuracy and stability of gravitation simulations without increasing the computational load very much. The ability to keep millions of (massless) particles in fairly stable orbits around the galaxy centers in that simulation I linked, despite a large time step, is due to the use of leapfrog integration.

Yes. In fact, it already is in orbit; it has been spawned in orbit.

If, by small course corrections, you could get it to pass through that point at that speed, then yes, it would orbit. However, as Floyd explains, you can't really ever get in that situation without artificially imparting a large delta-v on the craft en route and/or upon arrival. The "small course correction" in this scenario is actually an orbit insertion burn and wouldn't be very small. What you could do, potentially, is use the atmosphere of your target (if it has one), or even that of one of its moons, to burn off enough delta-v to make it more of a "small course correction" needed, either as a full aerocapture, with a burn to raise periapsis out of the atmosphere, an incomplete aerocapture that requires a small burn to finish the job, or if the atmosphere is too thin, multiple passes over many solar orbits. That last option sounds like a trajectory planning nightmare to me, but it may still be possible if the stars align and you have enough time; someone may be able to correct me if there are reasons you could never get multiple passes to work.

Posted by: Explorer1 Sep 1 2015, 05:25 PM

In the meantime, there's a boat load of these gravity simulator programs on any Google search; they're certainly great time wasters, while also being educational.

Posted by: djellison Sep 1 2015, 06:50 PM

Yes - obviously.




The part you're missing is that by the time you are at that point, the spacecraft has been falling thru the planets gravity well to a speed greater than that of the orbital velocity you want, and so you're doing a flyby.

Not to put too fine a point on it - but if orbital capture were this easy.....everyone would be doing it. Galileo's orbit insertion was 630m/sec - Cassini's was 626m/sec - MESSENGER, after putting itself thru many gravity assists still needed 800m/sec+ for it's orbit insertion burn. A Uranus orbiter is expected to need around 800m/sec. Mars orbiters often have burn of 1 km/sec to go into orbit.

Another way of thinking of the problem..... say you are in an orbit with a Perihelion of 1AU and Aphelion of, say, Saturn orbit. When you are aphelion, you are travelling significantly slower than Saturn. Assuming you time it right and you arrive at that point as Saturn does - Saturn is now catching you up - so when you enter it's gravity well you are now falling towards Saturn...thus, in the solar frame of reference...you're slowing down. Your orbit around the sun is now LESS Saturn like than when you arrived.

Posted by: HSchirmer Sep 1 2015, 08:15 PM

Good way to put it.

There is an asterisk to that though. There. is a distinction between orbital capture, and a quasi-stable orbit.
Getting a stable circular orbit around an outer planet it energy intensive.
But, there are lots of energy cheap unstable orbits around interesting destinations.
Orbits that are only stable for a century or so, then the space craft gets ejected from the system.
Generally, not a problem for this situation.

Actually, that could be a blessing in disguise.
The US military recently asked for ideas about how to use decomissioned spy satellites.
As I understand it, US intelligence agencies had a bunch of Hubble sized telescopes in orbit, but pointed DOWN to keep an eye on things.
Next generation drones are cheaper / better / less predictable than spy satellites, thus triggering the US orbital telescope garage sale.

I suggest taking one or two of the decomissioned keyhole (IIRC) spy satellites, drag them down to ISS, polish them up,
drop in a state of the art CPU, add a giant antenna, top off the volatiles, bolt on some sunshields and a Dawn style ion engine.
Then give them a swift kick out to L2 Earth-Sun and then start pushing them through the "Interplanetary Transportation Network"
While it is on the way to Mars, the "mobile Hubble" could do the usual "stars/galaxy/cluster" style astronomy.
Then, when it gets close to something local and interesting, say asteroid or comet, it observes that for several months.
Then back to stars and galaxies.
Next, a tour through the Jovian system, orbit for several months or a few years, get the Galilean worlds, then out.
Stars and planets for a few years, then Saturn.
Repeat for Uranus, and Neptune, and Pluto.

Posted by: ZLD Sep 1 2015, 08:35 PM

"As I understand it, US intelligence agencies had a bunch of Hubble sized telescopes in orbit, but pointed DOWN to keep an eye on things."

I love the idea and I think it would be a great use for them but if they are the ones I recall, those Keyhole satellites http://www.space.com/16000-spy-satellites-space-telescopes-nasa.html. According to that article, they also are basically just the bare telescope without any instrumentation so they are also unfinished. As for the currently orbiting Keyhole satellites, I think some are still in use while others are derelict already.

Posted by: HSchirmer Sep 1 2015, 09:15 PM

Yep, those are the telescopes.
Darn, I thought, (eh hoped) there were more than two of them....

Wow, curious where would Hubble be if IT had been set on the "low delta-v" tour when it was launched?
What if 20+ years ago, Hubble had been pushed out to the outer solar system when it was launched?

Wow, aAfter thinking about it, I really like the idea of cold war weapons being converted to grand tour telescopes.
Our swords-into-plowshares telescopes tour of the solar system.
Old spy satellites and nuclear icbms get reconfigure them into space telescopes visiting the planets.
Spy satellites provide the optics.
Warhead material becomes RTGs to power them.
ICBMs repurposed to lift telescopes out of earth orbit, ion engines send them out for exploration.

Posted by: djellison Sep 1 2015, 10:58 PM

Indeed. The examples I cited ( MRO, Cassini, Galileo, MESSENGER, Uranus orbiter ) almost all end in exceptionally eccentric orbits - they are pretty much the minimum delta V you could possibly use whilst remaining gravitationally bound to the spacecraft. MESSENGER wasn't hugely excentric - but far from circular. It's already highly energy intensive to get into ANY orbit - let alone a circular one.

in the case of Cassini and Galileo, multiple gravity assists from moons were used to manipulate the orbit thereafter.

In the case of MRO - it's aerobraking that makes up the difference.


The spare NRO hardware equates to enough hardware for one telescope. It's to be used as the basis for the proposed WFIRST mission. Getting a spacecraft that massive ( >11 tons ) into some sort of orbital tour of Mars, asteroids and the outer solar system would require extraordinary amounts of Delta V that would pretty much require next gen solar-electric prop ( which obviously isn't going to work beyond Jovian like distances )

Posted by: HSchirmer Sep 2 2015, 02:19 AM

Correct. Except when it isn't.

The beauty of chaotic orbits that require little to no delta-v is that 0 x anything = 0
Zero x 350kg craft = zero thrust.
Zero x 11 ton craft = zero thrust.

http://www2.esm.vt.edu/~sdross/talks/ross-csulb-2003.pdf
http://forum.nasaspaceflight.com/index.php?topic=26217.0

Idea is, old spy satellite and icbm components could be reporposed into a group of space telescopes.
If we aren't using them to spy on each other, might as well send them out to get pictures of other planets and moons.

Take a spy satellite telescope, add a Dawn style ion engine powered by a RTG, and a big antenna to transmit back to earth.
Send it out to Earth's L2 point, then let it traverse the roughly zero-delta-v chaotic orbit pathways from earth's L2 out to the gas giants and beyond.
While it is in-between planets, telescope could look at stars and galaxies just like hubble.
But when it closes in on planetary sytems, we get great images of planets, moons etc.
That takes lots of time, but, if the telescope does science on the way out, who cares as long as our kids and grand-kids get great photos?

Posted by: HSchirmer Sep 2 2015, 02:56 AM

Cool map, with classical transfer orbit delta-v

http://www.dailymail.co.uk/sciencetech/article-2679107/Mind-interplanetary-gap-Space-subway-map-reveals-travel-planets-moons-solar-Tube.html

Cooler map, with chaotic minimal delta-v
http://www.gg.caltech.edu/~mwl/publications/papers/IPSAndOrigins.pdf

Posted by: djellison Sep 2 2015, 03:11 PM

You say zero thrust - then talk about an RTG powered Dawn Thruster.

Dawn's thrusters require approx 10x the output of a typical RTG. Also - assuming you could apply every ounce of Delta V that Dawn has generated and apply it to a Hubble sized payload, the total Delta-V it would impart would drop from approx 10km/sec to 1.4 or so.

Exploring the Earth-Moon region with little Delta-V is easy. It's been done time and again. That Scientific American article is somewhat disingenuous to talk about the '300 million miles' that Genesis traveled....when it actually just went to L1 and back, via L2... in an geocentric frame, a few million miles, not hundreds of millions. It also negates the fact that getting from LEO to L1 requires >3.5km/sec in the first place.

Getting out to Mars, Jupiter, Saturn and beyond....I see no reference to the zero Delta-V trajectories in either of the papers you cite.

I'd be delighted to see a realistic trajectory that gets 11 tons from LEO or even L1 to Mars, Jupiter and beyond with zero delta V in a time frame realistic for the lifetime of a spacecraft. Say, 10-20 years.