Since I don’t understand what y’all are talking about (and calling me a physics neophyte is giving me far too much credit), I’ll agree.
Nice work, Stranger.
This is mostly true; however you’d have to account for whether the the final path of the spacecraft has some component that points back into the planet’s previous orbit, or points ahead toward its future path. I the case of the former, the spacecraft would actually slow down, whereas in the latter it’ll speed up (with respect to the planet, and of course the Sun; the instantaneous velocity with respect to an Earth-bound observer would depend additionally on the motion of the Earth.)
If the planet were not moving–somehow, just sitting stationary in the sky–then the final speed of the spacecraft (in a parabolic orbit) would be the same as the approach speed at the same distance. Since this never happens in real life, it’s not a consideration. You will always gain or lose some speed, though I have to admit to not having looked at the gains. For the most part, when a spacecraft is performing a swingby maneuver it only stays in the sphere of influence of the dominant body for minutes, and I wouldn’t expect a massive (order of magnitude) change in speed unless the swingby fell very deeply into the gravitational well. The change in velocity (direction), however, can be very dramatic.
I would have gone into more detail on the topic, including an example problem, but the article was getting overly long as it was, and Ed trimmed out a few bits that he felt were too technical for the audience.
Using onboard propellant lets you add the impulse (“delta-V”) of your engine to that from the gravitational attraction of the planet. Done at the right time, i.e. close to periapsis (closest approach) can allow you to make dramatic changes to your orbit and/or resultant speed beyond what you’d be able to do with a unpowered (free) orbit. If you fire back along your course, you’ll fall into a shallower orbit, allowing you to stay in the planet’s influence for longer and warp the parabola around. You won’t get any more energy directly from the burn than you would from using the motor in free space (energy must remain balanced), but you’ll be able to utilize the swingby target for longer, sapping away at its momentum in order to give your craft a greater path change or more speed. So it can be said to amplify the effect of your swingby manuever. These maneuvers are very tricky, though; if not done at precisely the right time, they can send the spacecraft veering wildly off-course with no hope of recovery.
Stranger
I have to disagree with you here - in the example of Cassini (as shown in the Wikipedia article link I presented,) the delta-V required decreased from 17.5 km/sec to 2 km/sec because of the 4 gravity assists (with engine burns while deep in the some of the gravity wells). I would say that is very close to an order of magnitude effect - 17.5 to 2!
I have to disagree with you here also. When the engine burn is along the direction of motion while deep in the gravity well, the final speed of the spacecraft far from the planet will be significantly larger than it would have been without the burn. This does not violate conservation of energy since the burned propellant was left deep in the gravity well. Thus the potential energy that the “to-be-burned” propellant had before it fell into the well can be converted into kinetic energy of the final spacecraft after it has climbed out of the well. Energy is balanced. Again this is a big contribution to the Cassini spacecraft’s nearly order of magnitude decrease in fuel requirements.
Again, for a better explanation of both of these points please read the Wikipedia article: Gravity assist - Wikipedia
Ponder on that statement for a bit. Why must energy remain balanced?
FrankH has it right, but let me attack the explanation from a different flank:
Reaction engines (rockets and jets) produce thrust that is largely constant regardless of vehicle speed. Thus the power produced is proportional to the speed of the vehicle at the time that the engine is burned. The same engine, burning the same amount of fuel, is much more powerful at perigee than it would be in free space. A more powerful engine, burning for the same duration creates more energy.
The *energy doesn’t balance *as compared to a free-space burn.
This is essentially the same issue that makes low bypass jet engines a poor power source for low speed aircraft. To obtain reasonable efficiency, the vehicle speed needs to be a large fraction of the exhaust speed. Passenger jets don’t accelerate gently at the start of their takeoff roll because the pilot is making allowances for passenger comfort. The initial acceleration is low because the engine efficiency is horrible at low speed.
Bah, missed the edit window.
Energy DOES balance if you include the energy imparted to the exhaust as well as that given to the vehicle. Efficiency is increased because the exhaust ends up moving slower.
More on the amount of speed gained: Under ideal circumstances, and assuming no burn through the entire procedure, a spacecraft could gain speed in a slingshot maneuver of up to twice the planet’s speed. This is independant of the spacecraft’s initial speed, but does depend very critically on the spacecraft’s exact direction of motion. In actual practice, the gain in speed from a single slingshot will generally be less than this, but I’m a theory guy, and I don’t know how much gain is typical (or if the assumption of a no-burn slingshot is at all realistic).
Also, I’ll add that the planning of orbits to take advantage of flybys and such is something of a black art. Computers are very good at taking a rough orbit plan and polishing it up into a local maximum, but deciding on that rough orbit plan (“We’ll use the Moon to slingshot to Venus, then swing around for an Earth flyby, then back to Venus, then Mars, then onto Saturn”) is the sort of thing that computers aren’t very good at, and the few humans who are good at it, can’t explain how they do it.
That depends on whether we’re talking of velocity or speed. Certainly, the changes in velocity are very significant; being able to warp around a planet and change your vector by 10 or 15 degrees is a massive savings in propellant. But, as can be seen from this chart relating probe speed in relation to the Sun (derived from the Horizons Ephemeris System) the actual change in speed from the maneuver itself is small, about 1-4 km/s. (The spacecraft gradually gains speed as it approaches each body, but I don’t regard this isn’t a part of the swingby maneuver itself, because speed picked up on approach will be lost in leaving the body’s SOI.) Cassini’s Saturn Orbital Injection and insystem manuevers actually have a greater change in speed than any of the on-route planetary swingbys, though this comes from a slow initial speed and being deep in the sphere of influence of a massive gas giant; you’ll notice that the following drop in speed is almost as dramatic.
Guys, go back and read the previous statement. You get the same amount of momentum directly from firing your engine whether you do so firing deep inside a gravity well or out in space well away from any gravity source. The speed of the propellant mass after it has left the nozzle is irrelevant to this; the only momentum change you get from the rocket depends on the instantaneous impulse from the exhaust speed and mass of exhausted propellant, dv=v[sub]e[/sub]*ln((m+m[sub]e[/sub])/m) (assuming ideal efficiency in your engine). Gravity has nothing to do with this. It’s true that you’re leaving a little bit of mass behind at the planet’s orbital speed, and that helps you in the same way that tossing ballast overboard helps a balloon rise (metaphorically speaking) but the overall change in mass is small and basically irrelevant.
However, by firing your engine while deep in the gravity well of a large body, you can dwell within the strong influence and thus transfer more of the planet’s orbital speed. This is the only mechanism by which you gain additional energy or change momentum beyond what you get from the rocket motor itself. Nature keeps her books balanced with regard to celestical mechanics.
Stranger
On the first missions where swingbys were used (Pioneer 11, the Voyagers) only small adjustments were made to keep the craft on-course for the planned rendezvous, with no accelerating burn made, so no-burn approaches are quite realistic. And while you’re correct that the momentum change could be twice the orbital speed of the planet (assuming that the probe is coming straight into the planet at its orbital speed and an elastic [del]collision[/del]momentum transfer), in practice you’d never be able to keep it in position long enough for the planet’s gravity to reverse the course of the probe and bring it to the planet’s orbital speed. Swingby approaches last only a few minutes–with the critical area being on the order of a few seconds–and the spacecraft itself will only stay within the planet’s sphere of influence for a few score of hours.
There are a lot of different parameters and, of course, demands from scientists to fulfill their scientific objectives (and the physical limits of the craft itself, which probably shouldn’t be flying through Jupiter’s massive Van Allen belts or the Jovian-Io flux tube, regardless of how ballistically optimized the course is). It is a very fuzzy, if not quite wicked, problem, and solving it relies on the experience and ingenuity of mission navigators. There are some rules of thumb that can be more-or-less automatically applied to generate a few discrete families of solutions, but in the end it does (at least for now) require human judgement to discern which profiles are best.
Stranger
Despite including this quote in an earlier post I neglected to address it. The reason jet engines are inefficient at low speeds is because the oxidizer and main component of the propellant–that is, air–is moving very slowly, and at essentially ambient pressure; the engine has to do all the work at sucking the air in and pressurizing. At higher velocities the air becomes compressed and you get a higher mass flow rate, and the efficiency continues to increase until you get to mechanical and/or fluid dynamics restrictions (i.e. the engine can’t spin fast enough or the backpressure of the intake overtakes the increased pressure from velocity), at which point you’d like to convert your engine to a ramjet.
This has nothing to do with a rocket motor in vacuum, as the rocket has to carry its own oxidizer (in the case of a chemical rocket) and propellant. There is no increase in rocket efficiency at any speed, and exhaust velocity of a rocket in vacuum is strictly a matter of engine design, and the impulse and energy are strictly a matter of the mass and nozzle velocity of propellant exhausted.
Stranger
Impulse is forcetime. Energy is forcedistance. Changing the speed ipso facto changes the relationship between time and distance.
<bolding mine>
Assume the same nozzle, same mass of propellent burned, same nozzle velocity. Which is constant when speed is varied? If energy is constant, then impulse will vary with speed . If impulse is constant, then energy will vary with speed.
10 N*sec acting at an average velocity of 10 km/s increases the vehicle energy 1/10 as much as if the same impulse had acted at Vavg=100km/s.
Energy=force * distance (still, I hope!) How much kenetic energy is imparted to a rocket engine on a stationary test stand?
If you consider locking down the engine an artificial constraint, (or you want to put the energy into the planet the test stand is mounted to) then set the engine in motion, and let it fly free: Consider a retro (at start of burn) rocket firing to produce a 180 degree change in vehicle velocity vector, with equal speed before and after the burn. Isn’t the vehicle’s KE the same after the burn as before? (neglecting mass of fuel) Average velocity during the burn is zero, and average effiency for that burn is also zero. (actually negative if you consider the mass loss due to fuel burned)
Exhaust velocity and the change in momentum imparted is only relative to the vehicle, or more specifically the nozzle, regardless of how fast the craft is moving prior to thrusting (assuming that you’re carrying your propellant with you, and exclusing relativistic effects which are not an issue here). In the case of a rocket on a test stand, the impulse is imparted to the rotational momentum of the Earth, making it spin faster (or slower, or sideways, depending on the orientation). It’s not a noticable change, because the Earth is so big in comparison to a puny rocket like a Saturn V or an SRB, but the energy and momentum are conserved.
Conservation of momentum and conservation of energy apply to systems, not individual objects within a system; in your example, you have to consider the propellant as well as the spacecraft, in which case the total momentum of the system still sums to the orginal amount with the exhausted propellant and momentum required for the balance, and the change in energy can find its complement in the kinetic energy imparted upon the propellant.
In the case of our spacecraft swinging by the planet, the objects comprising the system (bringing with them their initial momenta, and kinetic and graviational potential energies) are the spacecraft, its propellant, the planet, and the Sun. The balance energy balance between the Sun and the planet is a given. The probe loses momentum and kinetic energy as it travels from the Sun, and gains it as it heads toward another massive body, like the planet, per basic Newtonian ballistics. Aside from this, the only way it can effect a change in momentum is to either shoot out some propellant–gaining momentum per the aformentioned Tsiolkovsky, which makes no regard to the initial velocity of the craft, or to acquire (or lose) additional momentum from the planet. The planet can’t somehow amply the effect of the rocket regardless of how close it is without giving away some momentum of its own, which is what occurs during a swingby, and the ability to do so enhanced by the judicious firing of the motor to give the craft a greater opportunity to gain a larger change in momentum/velocity/direction.
There are no free lunches in Newtonian mechanics, although free coffee and cookies can often be found at physics colloquia, which was often the prime attraction for attending lectures about “Anomalous Photoluminescence Behavior From Amorphous Ge-Si-Ca Structures In A Synthisized Argon Matrix.” Anyway, both energy (the sum of kinetic and gravitational potential) and momentum are conserved in the system. The only change you will get is either by firing your engine–getting exactly the amount of momentum change described–or leeching some off from another body, in the case of a swingby. You don’t get “extra” momentum outside of those from anywhere.
Stranger
Stranger, you are right that the change in momentum for a given engine burn is the same no matter what your initial speed. Thus you always get the same delta-V in all cases. But you are missing the point that the kinetic energy is proportional to the velocity squared so the absolute change in the kinetic energy will be larger if the inital speed is larger. There is no free lunch and the only way the craft can get a larger absolute increase in kinetic energy is if the burned fuel in the exhaust gets a larger absolute decrease in it’s kinetic energy.
Consider a craft in a very elliptical orbit around a planet. Let’s say that it only has enough fuel, or equivalently, enough delta-V to go into a circular orbit if it fires the engine when it is at the apogee. If you perform that same burn at the perigee, where the craft is traveling much faster and is much deeper in the potential gravity well of the planet, then the absolute change in the kinetic energy will be much larger and the craft will at least go to a much higher elliptical orbit, or in fact may be able to escape from the gravity well entirely.
Don’t worry, total energy is still conserved. The extra energy for the craft came from the fact that the burned fuel was left deeper in the potential well of the planet. In the case of the burn at apogee, the fuel was not very deep in the potential energy well of the planet, but when it was burned at perigee, the fuel was very deep in the potential well and thus the craft could get enought of an extra “kick” to perhaps escape the planet entirely.
So there is no free lunch, but you can make the fuel pay for the lunch by giving it a larger potential energy deficit. Clear?
Two observations:
[ul]
[li]No one has made an “It ain’t rocket science” joke yet. (On accounta it is rocket science.)[/li][li]I might be the only one who thinks that Jovian-Io flux tube would be a good name for a rock band. Or perhaps one of my Fantasy Baseball teams.[/li][/ul]
RR
This is true, but ultimately the extra energy a spacecraft making a gravity-assist manuever picks up is from the orbital energy of the planet, which is decreased by having to haul around the additional mass of the propellant like an annoying kid sister. The mass fraction of the propellant expended is not that large relative to the spacecraft itself, though and the larger amount of momentum transfer comes not from firing the engine but staying as closely within the sphere of influence of the planet for as long as possible, getting “free” energy by being dragged along with the planet.
Stranger
This should be saved for posterity!!! One of the best quotes ever on this board. 
Yes, it is clear that the biggest effect is probably from the purely passive gravity assist due to encounter with the moving planet. But I think you are still not getting the point. If the engine is fired along the direction of motion of the craft when it is at its perigee, the craft will speed up and will thus spend less time under the “sphere of influence of the plane.” So it is not all about spending the maximum time near the planet.
I put “probably” in italics because I would still like to see some numerical examples. Could your JPL contact give you any numeric insight into any of these issues?
In particular, you claim that current spacecrafts never get any where near the “twice the planet’s speed” limit that Chronos claims is the maximum amount of speed that can be gained from an un-powered slingshot maneuver. So what fraction of this factor of two maximum is actually achieved currently? 10%? 1%? 0.01%?
I would also love to see a comparison of the amount of fuel needed to do a burn deep in a gravity well to acheive a certain effect as compared to the amount of fuel needed to get the equivalent effect with a midcourse burn that followed an un-powered slingshot.
But by moving faster, it is also able to make a closer approach and still escape, which increases the gravitational influence of the planet on the spacecraft. It’s not just how long it spends in within the SOI but how close it gets, and this is not a simple calculation, particularly when the planet itself has a significant motion.
There are no simple answers to either of these questions; the amount of gain you might get from the planet depends upon the relative speeds of the planet and the spacecraft; how closely and at what angle the latter passes with respect to the path of the former; how much impulse from, for how long, and at what point the motor is fired; how strong the gravitational field is; and what the change in angle is. It’s not like comparing the gas mileage of a car on flat ground compared to efficiency at a certain incline; there are a large number of factors, and because this is a classic 3-body problem (albeit one for which the complexity can be somewhat reduced by assuming that the spacecraft is essentially massless in comparison to the other bodies and/or using the “patched conic” method with assumptions) there is no explicit solution.
This isn’t a difficult problem–at least as a first order approximation–to solve using numerical integration and readily available commerical solvers like Matlab, Scilab, or Mathematica, but even a crude estimate of a specific case is highly dependent upon the boundary and initial conditions, so attempting to make any general blanket estimates is wrong-headed. In general, we can say that if you pass on the sunward side that your speed (and kinetic energy) will decrease, and if you pass on the leeward side your speed will increase, but beyond that the magnitude of effect is affected by the particulars of the maneuver. I can confidentally say that gaining twice the orbital speed of the planet–for any body in the Solar system–is essentially impossible (this would require capturing an object, bringing it up to just below planetary escape speed at the apoapsis, then slinging it forward to escape on the return, a feat that even the magnificient Jupiter’s and its colossal gravity field is not capable of) but the limit of what can be gained depends upon the above parameters.
If you take a look at the graph of speed wrt the Sun versus time in the Wikipedia article you linked to you can see what speed changes the Cassini spacecraft experienced, but I make no claims of the “typicality” of such; each maneuver is its own unique problem, which is why these guys slave and worry over their calculations and continue to develop and refine tools to help them plot such trajectories.
Stranger
I said: "If the engine is fired along the direction of motion of the craft when it is at its perigee, the craft will speed up and will thus spend less time under the “sphere of influence of the plane.”
Stranger, the engine burn makes NO difference regarding how close the craft can get to the planet and still escape. Even without an engine burn, the craft can get as close to the planet as it needs to get. The only thing that would prevent the craft from escaping would be if it actually hit the planet or the atmosphere of the planet. The burn makes no difference unless it happens to make the craft hit the planet.
I agree, that it is complicated, that is why I was hoping your “rocket scientist” contact at JPL could shed some numeric light on the answers to the questions.
I just now took a very close look at that graph again and by doing a hand extrapolation of the speed of Cassini before the encounter with Jupiter to just after the encounter with Jupiter, I would estimate that Cassini achieved a 15% to 20% speed increase from this encounter alone. And since Cassini was coming up from behind Jupiter and was thus travelling faster than Jupiter, it probably picked up about 20+% of Jupiter’s orbital speed in the encounter. So I would consider that to be very significant fraction of the planet’s speed, but I agree it is not a factor of 2.
With the other 3 encounters it is much harder to guess at what speed increase was achieved - especially since there were obvious significant engine burns (vertical straight lines) at each of these encounters. Jupiter was the only close encounter without an (obvious) engine burn. There was only1 other small burn that occurred far from any planet. Thus the fact that the great majority of the fuel was burned deep in the gravity wells (as seen by the large delta-Vs) and the fact that this convoluted orbit only took a total delta-V of 2 km/sec compared to a delta-V requirement of 15.7 km/sec with no gravity assists, leads me to believe that the usefulness of an engine burn deep in a gravity well is very significant, but I can’t put any number on the deep gravity well burn effect alone since it is mixed up with the 4 gravity assists.
I’m going to say this one more time; there is no general answer to this question. How much “gain” a spacecraft will get from a gravity-assist depends on a large array of parameters. I’m not going to bug mission planners at JPL–who were generous enough with their time to answer my questions and review the submitted draft of the Staff Report–and who have better things to do with their time with a query to which they would, I’m confident, come back with essentially the same answer.
In any case, it doesn’t require a “rocket scientist” to run a simulation and answer your question for a specific case; for a simple iterative approximation, it doesn’t even require a working knowledge of calculus, just the basic laws of Newtonian mechanics, analytic geometry, and simple programming that could be done in any language more sophisticated than Logo or an number of commercial or open source numerical solvers. The first reference in the Report–Prussing’s Orbital Mechanics–will give you everything you need to know to do this problem, and specifically Section 5.4 (Solution of the Rocket Equation with External Forces, pg 87-8), Section 6.2 (The Impulsive Thrust Approximation, pg 99-102), and Chapter 7 (Interplanetary Mission Analysis, pg 120-136) will give you specific equations upon which to base a simulation. I’d be happy, via private message, to help you with any questions you have about setting up such a sim, but I’m not going to do it for you as it is out of scope for the article, which was to present a general overview of the hows and whys of using a gravitational assist.
Stranger
I apologize for not making myself clearer. I was not asking for a general answer. I was asking for a specific answer on a realistic route. For an example, take the route of the Cassini spacecraft and turn specific things off and on, one at a time, and measure the effect on the speed of Cassini once it leaves the sphere of influence of the planet.
For a very concrete example, take the first Cassini encounter with Venus where a rocket burn was done deep in the gravity well. What I would like to know is the following:
[ul]
[li]1. What was the delta-V of the burn deep in the gravity well?[/li][li]2. What was the speed of the spacecraft after it left the sphere of influence of Venus?[/li][li]3. Now assume there was no engine burn deep in the gravity well and compute what the speed of the spacecraft would have been at approxiamtely the same distance from the sphere of influence of the planet.[/li][li]4. Finally assume that Venus wasn’t there at all - what would the speed of the craft be at approximately the same distance from the Sun as was measured in 2. or 3. above.[/li][/ul]
The difference between the two speeds measured in 2. and 3. would then give me an approximation for the size of the delta-V burn that would be required in free space if the original delta-V burn deep in the gravity well had not performed. I know the actual route of the craft would be different without the burn in the gravity well so there is not an easy way of defining exactly the same point in the route to define as the place where the craft leaves the sphere of influence of Venus, but just do the best we can.
Now if I had to guess I would guess that the delta-V difference that would be measured by the procedure outlined above would be on the order of a 100% or larger effect. In other words the difference in the speed of the craft in case 2 and 3. would be at least twice as large as the size of the delta-V burn performed deep in the planet gravity well in case 1. I get the impression you think it might be only a 10% or less effect, but I apologize for attempting to put words in your mouth.
That would answer my question about the effect of an engine burn deep in a gravity well.
If you now compare the speeds measured in 3. and 4. above and compare that difference to the speed of Venus, you would get an answer to my second question about what fraction of the planet’s speed can be acquired by a spacecraft in a passive gravity assist maneuver.
I can get a very approximate answer to this question in one case: From my eyeball measurement of the Cassini speed graph, I believe that Cassini did acquire about 20% of the speed of Jupiter during it’s encounter. This admittedly, is a lot less than the theoretical maximum of 200% of the speed of Jupiter. To really nail this down, I would like to know what are the equivalent numbers for the other 3 encounters Cassini had with Venus and Earth. I cannot measure the effect from the graph by eye – to really do it right you would have to perform the procedure outlined above for each of the encounters.
The reason I want to look at the other encounters is because I suspect that the encounters with Venus and Earth would be making a much bigger than 20% effect - simply because the overall speed changes are so much more dramatic as can be seen in the graph. But I cannot tell from the graph alone since delta-V burns are also present at each of these encounters.
Now again, I would have thought you would have been surprised by the 20% Jupiter effect that I measure just from the graph. Would you have guessed a smaller number? My impression is that you would have, but I again apologize for attempting to put words in your mouth.
I guess this discussion should just wind down now since neither of us is going to convince the other with just words - numbers are needed. So shall we just agree to disagree on the magnitude of these effects until numeric information becomes available?