I I need some support, please, for a gravity assist analysis

[PeroK]

If the Earth's frame of reference were inertial, it would be the case, but it is not.

The Earth is in a free fall orbit of the Sun. You don't get much more inertial than that.

Honestly, you're missing the basics of vector kinematics and analysis in more than one (inertial) reference frame.

 

[LesRhorer]

And because they both accelerate due to the Sun's gravity at nearly exactly the same rate, you can analyze their interaction locally in a free falling frame fixed to their common center of mass. It's an approximation, but so is ignoring the fact that our entire solar system orbits the center of the galaxy.

Nope, not even a little. Not even close. The orbital motion of the barycenter of the planet / asteroid system is crucial, especially since the net effect of the planet's gravity on the asteroid's speed is dead zero. Once again, the planet's effect on the speed of the asteroid at any given displacement from the planet is absolutely zero. A free body cannot under any circumstances change the orbital energy of any object under its influence. If we have a large object in free space and a smaller object is heading toward it, the inbound speed of the smaller object at 100,000 kilometers will be perfectly equal to the outbound speed at 100,000 kilometers. For a given pair of masses, the energy of the system is ONLY dependent upon their separation. Add in a third extremely massive object, and suddenly the energy of the first object has effectively nothing to do with the second object, as opposed to it being only due to the second object before the appearance of the third object. It is the same with the momentum. The momentum of the barycenter of the two body system in the frame of reference of the first body is essentia;ly only due to the movement of the second body relative to the barycenter. Plop in a giant third object, and the barycenter is suddenly nowhere near either of the first two bodies. The momentum of the first object relative to the barycenter suddenly jumps from an infinitesimal value to some truly giant value depending on the actual mass of the first and third objects and their separation. The net acceleration of the second body jumps from zero to three times the orbital velocity of the first object. Adding the initial and final vectors together, we get a change in speed of twice the orbital velocity of the first body.

No matter how one looks at it, a number like 10 -20 Km/sec in a three body system is huge compared to zero in a two body system. In addition, the one and only inertial reference frame for a gravitational system is the barycenter of the system. In the case of a solar system like ours, said barycenter is very near the center of the Sun. Trying to place a frame of reference near the center of a planet for any trajectory that spans a significant fraction of the radius of the solar system, or even just a few planetary diameters, inevitably results in a non-inertial reference frame, and cannot be accurately matched to real-world mechanics without some seriously complicated math. Newton's law of gravitation considering only the mass of the planet and a small vehicle won't cut it, unless one can live with being up to an order of magnitude off and in completely the wrong direction.

 

[A.T.]

.... a number like 10 -20 Km/sec in a three body system is huge compared to zero in a two body system.

We are moving around the galactic center at 230 km/s. Why is it fine to ignore that?

 

[Filip Larsen]

I have only skimmed the posts in this thread so I might be repeating something already mentioned, but the hyperbolic fly-by of a planet by a probe (the "elastic collision" mentioned above) must be seen from the perspective of the Sun in order to appreciate why it is considered an orbital maneuver relative to the Sun (some actually forgets this), and the fly-by "maneuver" can be conceptually be understood as a rotation of the probes velocity vector relative to the Sun, where the rotation angle is a function by the planet mass and the probes closest distance from it.

Math-wise the whole thing can be modeled fine by a sequence of three two-body problems (Sun orbit before, the hyperbolic fly-by, and Sun orbit after) that are then patched together.

 

[PeroK]

We are moving around the galactic center at 230 km/s, which is even bigger. Why is it fine to ignore that?

Just to add that, AFAIK, the Sun has an almost negligible effect on the orbit of the Moon about the Earth.

We can take the Sun's varying gravity into account for anything and everything. The question is at what point it's necessary for accuracy.

In any case, the concept of gravity assist does not depend on the Sun's varying gravity. Which seems to be what the OP refuses to accept.

 

[PeroK]

Math-wise the whole thing can be modeled fine by a sequence of three two-body problems (Sun orbit before, the hyperbolic fly-by, and Sun orbit after) that are then patched together.

Precisely!

 

[Juanda]

This is a pretty interesting thread.

First of all, how are we really defining "Gravity-assisted maneuvers"? As someone said near the beginning of the thread, the important thing is that the math is right so that the orbit of the spacecraft works as intended. Definitions of the concepts are irrelevant to the physical results.

@Drakkith offered a case in #17 involving only 2 bodies and he obtained a transfer of energy from the Earth to the asteroid. The total energy of that system is shared between the kinetic energy of the two bodies (Earth and the asteroid) and the gravitational energy of the field (similar to a spring connecting them).
When the asteroid reaches the same distance after the flyby the gravitational energy is the same since it is only dependent on the distance. However, the asteroid's kinetic energy is greater after the flyby. Therefore, some of the kinetic energy from Earth was transferred to the asteroid (it could be a spacecraft so the whole maneuverability makes more sense).

How does this fail to be a gravity-assisted maneuver? Posts #24 argues against it and #25 defends it again. So far, I consider this to be a valid gravity-assisted maneuver involving only two bodies.

@LesRhorer, if the people from that other forum are like you mentioned, is taking part in it really worth it? Discussions to see who is right are pointless more often than not. Online discussions are even more so.
That attitude differs a lot from what I usually found in this forum by the way. It's not as much about proving who is right but proving what's right and trying to understand the topics that come up in the process. I didn't encounter celestial mechanics in a while and I'm having a blast.

 

[LesRhorer]

We are moving around the galactic center at 230 km/s, which is even bigger. Why is it fine to ignore that?

The math surrounding the gravity assist requires the small object to be optimally traveling at escape velocity for the primary. It also needs to be traveling at a speed similar to the orbital velocity of the primary. One can see why this needs to be the case, because the gravitational influence of the giant body must exceed the gravitational influence of the primary for a significant portion of the trip. Otherwise we would say the object is "captured" by the primary, and the giant body's influence is comparatively small throughout its entire trajectory. At that point, a 2 body analysis can be close enough, but then again we would not be talking about an interplanetary trip.

As you say, the solar system's orbital velocity s around 230 Km/s. Any trajectory with an escape velocity that high would intersect the Sun, but at much lower velocities the influence of the galaxy is not going to be felt until well beyond the solar system. I have not done the math, but if our Sun were a neutron star, it might be possible to get a 200 Km/s gravity assist from it. Otherwise, we can consider the planets to only be circling the Sun - approximately.

The math of celestial mechanics is not in any way simple, and even for very restricted problems like a planetary gravity assist the results can be seemingly non-intuitive or even wrong. No matter what, one is always dealing with rotational systems and one is almost always dealing with multiple gravitational bodies. Like in this case. It is under every analysis a three body problem requiring contributions from a giant object, an extremely large object, and a tiny object, all under only local gravitational influence (which means a limited range of velocities). If either Voyager had approached Jupiter too slowly (which implies they never came from too very far away) they would have been captured by Jupiter, or perhaps even crashed. Had they been going too fast, then the amount of speed increase would have been small compared to the initial speed and been of little worth.

 

[A.T.]

I have not done the math, but if our Sun were a neutron star, it might be possible to get a 200 Km/s gravity assist from it.

The Sun being a neutron star is irrelevant for this possibility.

all under only local gravitational influence (which means a limited range of velocities).

The range of velocities is irrelevant for the local approximation. The local approximation merely ignores the gradient of the Sun's gravity. That's is the only reason it might not be accurate enough for some applications. All your arguments based on velocity make no sense.

 

[Drakkith]

In these cases, however, we can ignore the effect of all the planets on the Sun, since it is so huge, and similarly ignore the effect on each planet on every other since they are relatively much smaller and so far from each other, and totally ignore the effect of a tiny spacecraft on everything. For intra-planetary travel, we can never ignore the Sun and wind up with anything like an accurate prediction. Thus: three bodies.

Irrelevant. We are discussing gravity assists in general, not just the specific case of a gravity assist inside a star system with multiple bodies. And we certainly aren't discussing long range orbital trajectories, just the short-range interactions between bodies during a gravity assist.

The question of whether or not a gravity assist can take place with only two bodies necessarily requires us to examine a case with only two interacting bodies. As my simulation demonstrated it is entirely possible for a gravity assist to occur in a two body system. If you want to argue that this is unrealistic, that's fine, but then the very question is moot as the universe does not contain any two-body systems, only approximations of them.

So what do you actually want to know? If you want to know about a gravity assist in a two body system then define what that actually means, because as far as I can tell it only means that energy is exchanged between two bodies relative to some inertial frame of reference, such that they end up with differing speeds and trajectories after the interaction than before.

 

[weirdoguy]

they are unwilling to even listen (or heaven forbid admit they are wrong)

Are you willing to admit you are wrong?

 

[LesRhorer]

This is a pretty interesting thread.

Thanks! I always enjoy spawning a spirited, yet polite, debate.

First of all, how are we really defining "Gravity-assisted maneuvers"? As someone said near the beginning of the thread, the important thing is that the math is right so that the orbit of the spacecraft works as intended. Definitions of the concepts are irrelevant to the physical results.

True, mostly. It is very easy to accidentally make an error in physics that causes one to apply the math incorrectly and wind up with an answer that is physically incorrect.

@Drakkith offered a case in #17 involving only 2 bodies and he obtained a transfer of energy from the Earth to the asteroid. The total energy of that system is shared between the kinetic energy of the two bodies (Earth and the asteroid) and the gravitational energy of the field (similar to a spring connecting them).
When the asteroid reaches the same distance after the flyby the gravitational energy is the same since it is only dependent on the distance. However, the asteroid's kinetic energy is greater after the flyby. Therefore, some of the kinetic energy from Earth was transferred to the asteroid (it could be a spacecraft so the whole maneuverability makes more sense).

That is specious for a couple of reasons. First of all, it is a bit misleading to say the energy is "shared" between the Earth and the asteroid. While true, in the two body analysis, virtually all the energy is "owned" by the small body. Why? Because the only proper motion of the system is that of the barycenter. Since we restricted the mass of the two bodies so that the mass of the asteroid is essentially insignificant, the barycenter is in essence a fixed point at the very center of the Earth, and the Earth's velocity relative to that point is basically zero. Thus effectively all the energy is invested in the asteroid, and none in the Earth. That single flaw by itself invalidates the entire analysis, because there is a vast amount of gravitationally moderated energy and a vast amount of gravitationally moderated momentum involved with the Earth's path around the Sun. Furthermore, the proper motion of the Earth / asteroid system is not around its own barycenter, but rather around the Earth / asteroid / Sun system, which is effectively the center of the Sun.

Secondly, the asteroid's energy is not greater after the flyby in the frame of reference of the Earth / asteroid system. It is precisely the same as it was before. In this case, however, he chose to look at the system from a frame of reference offset from the barycenter of the system. From that frame of reference, the asteroid started out with zero energy, and wound up with considerably more. This is the very same mistake being made by the guy in the other forum. There is nothing wrong mathematically (or physically, for that matter) with choosing a different frame of reference, as long as it is inertial. In this case, as I have shown, it is actually not, but if Earth were in deep space it would be. The point here is changing the frame of reference changes the apparent energy of the system, but it does not transfer any energy to the system or to any component of it. The total change in velocity is always the same for both frames of reference. One can see this in various ways, but the simplest, really is given by the fact any acceleration is only produced by a force on a massive object. In this case, that force is dependent only upon the separation of the Earth and the asteroid and their masses. None of those are frame dependent. Neither is the change in energy of the system (as opposed to one part of the system relative to an external reference frame) which for every two body system is exactly zero. (See above.) To properly account for any increase or decrease in the energy of the Earth / asteroid system, one must first account for the Earth's kinetic energy WRT the external frame of reference. All this can be done, but it is one whole heck of a lot easier to simply understand the acceleration due to gravity at any point along the trajectory of a 2 body system is precisely the same for any other point with the same displacement. I is an orbit, folks, and unperterbed orbits involving isotropic spherical bodies are always, ALWAYS, ALWAYS perfectly symmetrical. (Not that real asteroids are anything like isotropic spheres, but that is far beyond the scope of this discussion.) The fact one can change the relative velocity of one side of a circular orbit to be zero and the other side to be twice the velocity relative to the barycenter is irrelevant. It doesn't boost anything.

How does this fail to be a gravity-assisted maneuver? Posts #24 argues against it and #25 defends it again. So far, I consider this to be a valid gravity-assisted maneuver involving only two bodies.

If one wishes to change only the direction of a spacecraft's motion but not its speed, nor the inclination of its orbit, then one can do so by sending it around a large object in deep space. That is a simple 2 body problem. If one wished to call it a "gravity assist", then that is OK (ish), too,m because all definitions are arbitrary. It s not what NASA did with Voyager or many other spacecraft, however, andone must then come up with a different term to describe what NASA did. As you yourself pointed out, it is ot the definitions that matter, but the physical results. One cannot obtain the results NASA did with Voyager, Cassini, etc. if only two bodies are involved.

An orbital flyby of Mercury results in a huge increase in speed, (over 40 Km/s) despite the fact the planet is only about 5 times as massive as our moon. A flyby of Neptune only results in a fairly small overall acceleration (about 5 Km/s), despite it having more than 300 times the mass of Mercury. That really should tell one all one needs to know anout whether the Sun's influence in a maneuver like that performed by the Voyager spacecraft, Cassini, and New Horizons. The Messenger spacecraft used assists at Earth, Venus, and on three separate flybys of Mercury in braking maneuvers to slow itself down. These maneuvers all resulted in actual changes in speed and direction compared to the Sun. They are not an artifact of changing the reference frame.

@LesRhorer, if the people from that other forum are like you mentioned, is taking part in it really worth it?

Most of the people there are not like those two. A few are, but most are quite reasonable, if nonetheless untrained in physics. As to whether I feel it is worth it... well it is to me. I hate inaccuracy - especially scientific inaccuracy with a seething passion. I have done so for nearly sixty years. It is not liable to change. I also despise charlatans, sand I am fairly sure this guy is one. He is also an insufferable jerk. He even admits to that.

Discussions to see who is right are pointless more often than not.

Not with me. I make plenty of mistakes, and I am always more than happy to be corrected, whenever I am in fact wrong. If I am ultimately wro9ng in this case, I absolutely want to know about it. I am pretty sure I am not, since not only do I have a fairly decent understanding of the physics involved, but every reputable reference I can find agrees with me. That includes online publications from NASA, Scientific American, and even Wikipedia, not that the latter is by any means always correct. Neil deGrasse Tyson also agrees, although he also does not bat 1000. I have personally caught him in more than a couple of errors and others have caught him also.

Online discussions are even more so.

In a forum whose principle thrust is debunking pseudo-science? Perhaps not so much?

That attitude differs a lot from what I usually found in this forum by the way. It's not as much about proving who is right but proving what's right and trying to understand the topics that come up in the process.

Absolutely. Obviously, it stings a bit to find out one was mistaken, but I certainly hope my ego is more than strong enough to take a little jab once in a while. Certainly I will never call anyone here names, which this guy did the moment he "met" me.

I didn't encounter celestial mechanics in a while and I'm having a blast.

I am definitely glad to hear you say it. Certainly I consider it well worth my time, even if I am in some way badly mistaken and even if I do not eventually get the assistance I would like. In my case, I earned all this almost 40 years ago, but I still come across it from time to time. Of course that may beg the question of how much I may have forgotten in the nearly four decades since.

 

[jbriggs444]

First of all, how are we really defining "Gravity-assisted maneuvers"?

True, mostly. It is very easy to accidentally make an error in physics that causes one to apply the math incorrectly and wind up with an answer that is physically incorrect.

I note the lack of response to the request for a definition. I have seen multiple walls of text from @LesRhorer. But no definition is forthcoming.

I stand by the recommendation in #2. We ought not engage. This member is not participating in good faith.

 

[A.T.]

The total change in velocity is always the same for both frames of reference.

And that's exactly why we can use the local frame orbiting with the Earth to compute the change in velocity. As explained in the link posted by @Filip Larsen:

https://en.wikipedia.org/wiki/Patched_conic_approximation

I hate inaccuracy

All of physics is approximation. Whether it's appropriate depends on the application.

 

[Filip Larsen]

if our Sun were a neutron star, it might be possible to get a 200 Km/s gravity assist from it.

It doesn't really make sense to talk about using the Sun alone for gravity assist maneuvers relative to our galaxy if the probe starts out in a bound orbit around the Sun (which all human made probes do). To be considered a gravity assist maneuver the probe would have to make a hyperbolic⁽¹⁾ close fly-by of some massive object that is not the primary relative to which you'd want to gain or loose orbital energy.

However, you are correct that the more dense the gravity assist mass is the closer the fly-by can be made and thus the greater angle the velocity vector can be turned. In the (unrealistic and degenerate) limit of a point-like mass the assist maneuver could turn the velocity vector almost 180 deg relative to the primary.

(1) "Hyperbolic fly-by" is a bit of a pleonasm, since a fly-by kind of imply being on a hyperbolic trajectory.

 

[Dale]

First of all, I hesitate to speak of it as a collision. In ordinary usage, a collision tends to connote a very brief, perhaps virtually instantaneous interaction, not something that spans weeks, months, or even years.

You can hesitate, but as far as the math goes it works.

A collision is an interaction between two bodies where we don’t care about the details of the interaction itself, but only the “before and after” states. For an elastic collision it must conserve momentum and kinetic energy. Any external force must be negligible during the collision.

So a gravitational assist meets these specifications, usually better than a car collision or a billiard ball collision. It is both valid and standard practice for several decades.

the fact is in the frame of reference of the barycenter of the two smaller bodies, the interaction is not elastic. Object V either picks up or drops a rather large amount of energy.

This is not true. Please show your calculation for this specific claim so we can help you with the math.

Obviously, in a mechanical collision there is not usually any transverse displacement (displacement perpendicular to the initial velocity vector) and there is never any force exerted on the components after the event is completed. Consequently, there is also no acceleration of any component once the event is complete.

Obvious counterexamples are automobile collisions and billiard ball collisions. Friction, gravity, and the normal forces all act on automobiles and billiard balls before, during, and after their collisions.

The question is not whether external forces exist, nor even if they are small. The question is whether they are separable. Is the total force on each object the sum of two way forces and does the “external” force change significantly during the detailed part of the collision?

This part of the discussion is done. Treating a gravitational assist as an elastic collision is well established in the professional scientific literature:

https://doi.org/10.1119/1.3072898
https://doi.org/10.1119/1.1621032
https://doi.org/10.1007/s10569-007-9114-5
https://doi.org/10.1088/0143-0807/15/4/002

No matter how one slices, dices, smashes, or purees it, a gravitational assist (as opposed to a simple orbit) requires at a minimum three bodies

Please see the above references. Particularly the last one where it explicitly acknowledges the three body problem while still treating the gravitational assist as an elastic collision.

Further claims that the gravitational assist cannot be treated as an elastic collision should be supported with explicit references from the peer reviewed professional scientific literature.

 

[LesRhorer]

I note the lack of response to the request for a definition.

I did respond.

I have seen multiple walls of text from @LesRhorer. But no definition is forthcoming.

Untrue. Although I mentioned it previously, I will clarify again right now.The only reference from any knowledgeable sources I have ever seen is the method used by NASA for Voyager, Casini, etc. to increase the speed of a vehicle relative to some very massive body by the amount equal to the orbital speed of a body of intermediate size. This is often termed the "Slingshot Effect". The effect can be used not only to reduce the amount of fuel required to reach escape velocity (or slow down drastically without using any fuel). It can also be used to inflect the trajectory of a spacecraft away the original plane of the trajectory, as was done with Voyager I. I certainly thought all that was perfectly clear, but in light of your post I must suppose not.

Furthermore, no interaction between two bodies is ever capable of achieving the aforementioned results. All two body interactions result in perfectly symmetrical trajectories. Thus, simple changes in the direction of a craft along the plane of its initial trajectory with no net increase in the energy of the craft with respect to the barycenter of the two body system qualifies.

If you insist on using some other term which meets these criteria, then please provide it and I will accede to your request. Otherwise, let's move on. This is neither fundamental to the discussion nor a reasonable use of my time.

I stand by the recommendation in #2. We ought not engage. This member is not participating in good faith.

I consider that to be a very serious accusation. I am compelled to demand you support it. Having done so, I will immediately amend whatever actions of mine in any way lack good faith. Failing that, you need to stop it.

 

[Juanda]

Post #42 makes sense about orbits being symmetric when only two bodies are involved.

The CoM of the two bodies won't move since there are no external forces involved. Let's use that as a reference frame then.

The proposed experiment by @Drakkith in #17 necessesarly will look like this if the velocities are below the scape velocity.

Where the green line is to show the position of the bodies after the flyby. I feel like the kinetic energies of the bodies before (black line) and after (green line) should be exactly the same.

If the velocities are high enough (as I believe is the case in proposed simulation in #17 by @Drakkith), the orbits will no longer be elipses and they won't close at their far ends. (I wish I could explain it better but I can't find any diagrams). Still, the same reasoning applies:

Where the green line is to show the position of the bodies after the flyby. I feel like the kinetic energies of the bodies before (black line) and after (green line) should be exactly the same.

@Drakkith do you agree with this? Maybe there is something wrong with the simulation? The results you provided no longer make sense to me. I assume it's being solved numerically. Maybe the acquaricy is being an issue.

 

[jbriggs444]

The CoM of the two bodies won't move since there are no external forces involved. Let's use that as a reference frame then.

I suspect that this is key. Picking a reference frame. We can all agree that in the case of a two body situation viewed from the reference frame of the combined center of mass, both objects emerge with the same speed that they entered, albeit in new directions.

Viewed from another reference frame, one object can gain speed at the expense of the other.

To be picky, the CoM of the two bodies will not accelerate. It can "move" if we choose to adopt a reference frame where it is not at rest.

 

[Dale]

no interaction between two bodies is ever capable of achieving the aforementioned results. All two body interactions result in perfectly symmetrical trajectories.

This is not correct. The aforementioned results can indeed result from a standard elastic collision with only two bodies in any frame of reference other than the center of mass frame. Only the center of mass frame has the symmetry you mentioned.

The laws of physics do not require us to use the CoM frame.

 

[LesRhorer]

It doesn't really make sense to talk about using the Sun alone for gravity assist maneuvers relative to our galaxy if the probe starts out in a bound orbit around the Sun (which all human made probes do).

True. This would probably have to be some vehicle coming in from deep space. As you very clearly are aware, the required parameters for a gravity assist are rather narrow. Instituting an actual hyperbolic orbit around a neutron star from only a few million Km away would be tricky, indeed. From a few parsecs, not so much.

To be considered a gravity assist maneuver the probe would have to make a hyperbolic⁽¹⁾ close fly-by of some massive object that is not the primary relative to which you'd want to gain or loose orbital energy.

Yes, that is precisely my point, although I labeled the flown-by object as the Primary and the distant object as the third body. I realize this is not traditional,but it made it easier to transition from a two body system in free space to a similar system in orbit about the additional body, labeled by you (and most folks) the Primary. It of course makes no fundamental difference what we label them.

However, you are correct that the more dense the gravity assist mass is the closer the fly-by can be made and thus the greater angle the velocity vector can be turned. In the (unrealistic and degenerate) limit of a point-like mass the assist maneuver could turn the velocity vector almost 180 deg relative to the primary.

Precisely.

(1) "Hyperbolic fly-by" is a bit of a pleonasm, since a fly-by kind of imply being on a hyperbolic trajectory.

A salient point, upon which I touched earlier. Circular and even somewhat extreme elliptical orbits about the middle-sized objects are too near the planet to be affected by the distant giant in anything more than a modest perturbation of the orbit. The craft must come from and leave into regions where the giant body dominates. Only in that case will the usually rather modest centripetal force from the giant object make the sort of difference we see with Voyager, etc. My entire point is in order to see any such modification to what would otherwise be an actual hyperbola, it requires the presence of an additional body.

 

[Dale]

My entire point is in order to see any such modification to what would otherwise be an actual hyperbola, it requires the presence of an additional body.

This is simply not true. You do not need an additional body to consider a hyperbolic trajectory from a different frame than the CoM frame.

 

[A.T.]

The craft must come from and leave into regions where the giant body dominates. Only in that case will the usually rather modest centripetal force from the giant object make the sort of difference we see with Voyager, etc.

What matters is the gradient of the bigger body gravity, not its magnitude. Even in regions where the Sun's gravity is far greater than planet's gravity, you can still use the planet's local free falling frame, if the variation of the Sun's gravity is sufficiently small in that region.

 

[Juanda]

This is simply not true. You do not need an additional body to consider a hyperbolic trajectory from a different frame than the CoM frame.

I also think you can achieve hyperbolic trajectories with only two bodies if the incoming speeds are high enough. However, before and after flyby conditions/states would be the same, right?

So, in that case, you would not be really stealing energy from one body to pass it to the other in the way it was done during the Voyager missions to increase the speed of the probes or other similar missions.

Do you think the reasoning is correct? In post #48 I claimed a similar thing to what I'm posting here although I was missing the word "hyperbolic" at that time.

 

[jbriggs444]

you would not be really stealing energy from one body to pass it to the other

Kinetic energy is not an invariant. It is relative to a reference frame. Many people use the term "really" to refer to invariants only.

The same interaction can be described in frames where energy goes from one body to the other, from the other body to the one or is not transferred at all.

 

[LesRhorer]

This is not correct. The aforementioned results can indeed result from a standard elastic collision with only two bodies in any frame of reference other than the center of mass frame. Only the center of mass frame has the symmetry you mentioned.

No, it can't. No matter what, an interaction between two solid bodies can never result in the system gaining energy. That would be an absolute violation of conservation of both energy and momentum. It also cannot ever lose momentum, and can only "lose" energy if something internal to the system can convert kinetic energy to some other form, like heat, Changing to a different frame of reference never has any effect whatsoever to the internal energy of any system.

The laws of physics do not require us to use the CoM frame.

Not exactly true. As long as the CoM of the system is not accelerating, we can choose any FoR we like. If it is accelerating, however, it is not the case. A major point here, however, is that an accelerating CoM cannot ever happen with only two objects. One must absolutely infer a third body, and a barycenter for the larger system that is not at all coincident with the CoM of the two bodies. The inference is really quite straightforward, and absolutely concrete.

It is quite another thing to deal with an external frame of reference to a genuinely two body system. As a simple example, let's take a solar system consisting only of one main sequence star and one roughly Earth-sized planet. The planet has a perfectly circular orbit with an orbital velocity of 10 Km/s. 'Pretty ordinary.The kinetic energy of the system relative to its CoM is essentially totally invested in the planet, and it is perfectly constant. A few trillion Km away, along floats a space probe at 10 Km/sec towards the star. Measuring the velocity of the planet, the probe sees it alternately slowing momentarily to a dead stop and then speeding up to a peak of 20 Km/s. The measurements are perfectly correct, but it does not mean the planet is somehow loosing energy and then gaining it back. It is never gaining or losing energy at all. We can always choose an external FoR such that any orbit is non-symmetrical relative to the external FOR, In fact, ANY FoR whose velocity is not the same as the CoM of the system will see the orbit as skewed. It does not mean the two body system is inelastic. In a gravity assist, the two smaller bodies ARE ineslatic in the FoR of the planetary barycenter. The entire three body system is elastic WRT the barycenter of the entire three body system.

 

[weirdoguy]

No, it can't.

Yes it can, and this discussion is getting pointless. You are the one who is wrong and unwilling to admit it.

 

[Juanda]

Kinetic energy is not an invariant. It is relative to a reference frame. Many people use the term "really" to refer to invariants only.

The same interaction can be described in frames where energy goes from one body to the other, from the other body to the one or is not transferred at all.

Total kinetic energy depends on the frame of reference. That's for sure. Work however remains constant. Or at least that's how I recall it from the last time I had to derive it for a similar conversation regarding elastic collisions.

If two bodies interact gravitationally with each other, the orbits will be either elliptical or hyperbolic depending of the initial velocities of the bodies. We previously agreed the reference frame at the CoM between the two seems to be very useful so I will keep using that one during the discussion.

Before (black line) and after (green) flyby the kinetic energies of the bodies are the same. Do we agree on this? If they are the same, no work has been transferred. No matter the reference frame chosen, that will remain true. Therefore, to "steal" energy from one body and give it to the other, having only 2 bodies interacting with each other is not enough for that to be a possibility.

 

[Juanda]

Yes it can, and this discussion is getting pointless. You are the one who is wrong and unwilling to admit it.

C´mon, don't be harsh. I actually think we are making progress.

Again, I think it is more convenient to focus the conversation on WHAT's right instead of WHO's right.
It's a long thread already so I'll try an unofficial summary of it to help us get back on track. OP, correct me if this doesn't summarize the content discussed so far.

We're trying to figure out if it's possible to pass energy from one body to the other during a flyby. For example:
State 1: 5h before flyby.
State 2: Nearest point in the orbits.
State 3: 5h after flyby.

@LesRhorer claims that it is necessary to have a 3rd body if you want to have any energy transfer between the bodies. Otherwise, incoming and outgoing velocities will be the same. Velocities are dependent on the FoR chosen but the point is that there is no work or energy transfer between them which does not depend on the FoR. I do agree with this.
On the other hand, other people claim that no 3rd body is necessary to accomplish the extra kick.

Only one of those things (2 bodies NOT OK - 2 bodies OK) can be true so it's just a matter of time before we figure out which one it is.

I am no moderator or anything like that but I find the topic especially interesting and I wouldn't like the thread to derail into something nasty looking.

 

[PeroK]

Total kinetic energy depends on the frame of reference. That's for sure. Work however remains constant. Or at least that's how I recall it from the last time I had to derive it for a similar conversation regarding elastic collisions.

If two bodies interact gravitationally with each other, the orbits will be either elliptical or hyperbolic depending of the initial velocities of the bodies. We previously agreed the reference frame at the CoM between the two seems to be very useful so I will keep using that one during the discussion.

View attachment 329334

Before (black line) and after (green) flyby the kinetic energies of the bodies are the same. Do we agree on this? If they are the same, no work has been transferred. No matter the reference frame chosen, that will remain true. Therefore, to "steal" energy from one body and give it to the other, having only 2 bodies interacting with each other is not enough for that to be a possibility.

No. That's completely wrong. Energy is frame dependent. That's of fundamental importance.

Simply consider the rest frame of either body. In that frame a body must gain energy in a collision. And in the rest frame of the other body it must lose energy.

Also, don't confuse total mechanical energy conservation with invariance. The former means no change over time in a given reference frame. Invariance means the same in all frames. Total loss of energy in an inelastic collision is invariant.