I was reading the XKCD What If column about Voyager and it explained that Voyage picked up speed by gravitational assists - near approach swings past Jupiter and Saturn. Munroe wrote “Gravity assists aren’t paradoxical magic. It’s just like bouncing a tennis ball off a passing truck.”
That metaphor doesn’t work for me. I can understand how swinging past a planet (which isn’t really the equivalent of bouncing off of something) can change the direction of travel. But how would it increase the speed of the satellite?
When Voyager was approaching Jupiter it was being pulled in by Jupiter’s gravity and that would have increased its speed. But once Voyager passed its closest approach point to Jupiter and began moving away from the planet, Jupiter’s gravity would have been pulling it back and slowing its speed. I figure any speed increase that was gained during the approach would have been lost during the departure.
As I understand it the velocity of the object relative to the assisting planet is the same on departure as it is on arrival but remember that the planet itself is moving around the Sun itself and as the planet “hangs on” to the object then some of that orbital velocity is added to the object as it hurtles off.
So measuring the velocity relative to the sun or another planet would indicate that the object is now travelling faster.
As for a tennis ball analogy. Imagine you were standing by a road and lobbed a tennis ball at 10 mph towards a truck travelling at 30 mph.
The driver sees the ball approaching at 40mph and leaving his windsheild at 40mph (appx).
Relative to you though the difference is 10mph to 40 mph. you have used the motion of the truck to increase the velocity of the ball relative to you and change its direction.
Incidentally, in both cases (truck and planet) the larger object is slowed down slightly by it’s interaction with the smaller object. No such thing as a free ride.
As I said, laymans terms there but that’s how I think about it. (and wiser heads will be along with better and more complete explanations I’m sure)
A slingshot off jupiter actually slows jupiter down a little,
vis a vis conservation laws , this means the probe must have accepted some of its energy… Thats how you know you “it must lose what it gained” theory must be wrong.. Jupiter slowed down a little, therefore it didnt get back all of what it lost, so the probe must not be giving back all that it gained…
Why ? because the probe is moving outward at a great speed, its acclerated toward jupiter, but its outward movement takes it away from jupiter still and it escapes with a higher velocity … you can be sure the sum of the parts will add up, gravity law, vectors, very complicated, you’ll just have to do the math yourself or trust that it adds up.
They dynamics are relatively complicated but both Novelty Bubble and Ilisder are essentially correct: both the change in direction and net change in speed experienced outside of the target planet’s sphere of influence is due to momentum transfer between the planet and the spacecraft. Since the spacecraft is so much less massive than even a small planet, the amount of momentum change is insignificant to the planet but can be dramatic to the spacecraft.
Several years ago I wrote a Mailbag article on the topic. The article has since disappeared into the aether, but the thread that introduced it is here: [THREAD=413642]Do spacecraft swing around planets to gain speed?[/THREAD] (Ignore the discussion about thrusting at periapsis, as it is not germaine to the essential question.)
Intuitively, I’d see it like this: the “being accelerated towards the planet” period is significantly longer than the “hurtling away from it” period, because in that second period the vessel is traveling more quickly. So it spends more time gaining momentum than losing it.
(Yes, gravity applies at any distance, but for simplicity’s sake, let’s imagine it only applies at, say, 1 million km. From entering that zone to being at minimum distance from the planet will take n time. Going from minimum distance to exiting that zone will take n - k time, where k is positive.
It must be true now because I’ve even used maths.)
Sorry, but this is not correct. The reason can be understood intuitively by looking at energy (which is a scalar value with no direction) rather than momentum (which is a vector quantity). That is, gravitational energy is conserved, and so an object that starts out at distance S with a potential energy U, makes a parabolic trajectory around the planet, and then flies away to a distance S will have the exact same potential energy state U at that distance as measured from a coordinate frame fixed to the planet. It will, of course, accelerate as it approaches the planet, but it will decelerate at the same rate as it recedes, and there will be no net energy gain or momentum transfer.
However, if the planet is in motion relative to some exterior reference frame–in the case of a planetary swingby, relative to the Sun[SUP]1[/SUP]–the spacecraft that enters the sphere of influence[SUP]2[/SUP] of the planet will be pulled along like a car on a train. When it swings around and then leaves the sphere of influence on a different vector, the resulting change in momentum of the spacecraft must be balanced by a complementary change in momentum of the planet, and will very subtly change the orbital elements of the planet (e.g. the eccentricity or inclination will change, the semi-major axis or longitude of the ascending node will shift, the mean anomaly or argument of periapsis will move), even if the energy state of the spacecraft and planet are exactly the same. (They won’t be for a single swingby, but that is neither here nor there for the purpose of this discussion.)
The duration that it remains in the sphere of influence of the planet either coming or going is not a factor in the initial and final state calculation, although obviously the longer it remains in the sphere of influence the more momentum transfer can occur between the planet and the spacecraft, and the greater a change in spacecraft trajectory can result. Although it may not seem like a spacecraft passing on a close trajectory (but not impacting) a planet is similar to a tennis ball bouncing off a truck, both inherently involve a transfer of momentum and a resulting change in velocity of both bodies, and in fact, the swingby is the closest physical interaction to the idealized “purely elastic” transfer insofar as gravitational energy is conserved and there are essentially no other losses.
Stranger
[SUP]1[/SUP]Technically, the most neutral place to put an inertial reference frame is the barycenter of the Solar System, but as the Sun is dominant it is sufficient as an approximation.
[SUP]2[/SUP]The sphere of influence is the point at which the gravitational acceleration due to the planet is a greater influence than that of the Sun or any other body.
What are the limits to gravitational assist? Is the delta-V imparted to the probe limited by the velocity of the planet, or can we get some sort of “leverage”?
Take the case of a ball suspended on a string and getting hit by a truck going 40 mph. Can the ball leave the windshield at faster than 40 mph? I believe so, due to elasticity: it spends some time absorbing energy from the truck which it then releases as it expands back to its original shape, and thus bounces away from the truck at a speed higher than 40. Total momentum is preserved, of course. Is there a limit to this “extra” boost? Seems to me there is, but I can’t put my finger on it, probably because I’ve bungled something in the setup already.
Actually, it sort of is. When a tennis ball bounces off a truck, none of the atoms in the tennis ball actually make contact with any of the atoms in the truck - the interaction is all about fields repelling each other.
The key differences with gravitational assists is that it’s a different kind of field at play (gravity), and the mode of interaction is attraction, rather than repulsion.
In general, effects like that will be time-symmetric: The profile of the energy being stored in the elasticity during the first part of the collision will be exactly mirrored by the release during the second part. And even if you made some sort of weird elastic for which it wasn’t symmetric, you’re still limited by the total energy available. The ball bouncing away from the truck at the same speed it approached means that you’re using all of the energy available to you, and not wasting any of it. Unless you have some other energy source (the ball was compressed before being thrown or something), you can’t do any better than that.
Yes. The absolute upper limit to the “extra” boost is dictated by conservation of momentum: after the collision, the combined center of mass of ball+truck will be moving at the exact same speed and direction as it was before the collision. The real limit of the ball’s rebound speed (measured relative to the truck) will depend on the degree of elasticity of the collision between the ball and the truck. In the real world, elasticity is never 100%; some kinetic energy always gets pissed away as heat as the ball and truck deform during the collision.
OK, but by that token, if all of Jupiter’s momentum is transferred to the probe (or, all of the truck’s momentum transferred to the ball), then the resulting velocity is extreme. I assume there has to be a lower limit than merely total momentum remaining constant.
Chronos: that doesn’t quite answer my question. Clearly, the energy stored during compression equals the energy released during expansion. Also, this transfers energy from the truck to the ball. But what’s the limit on the energy transferred this way? Obviously, the upper limit is the truck’s kinetic energy, mv^2, but I’m guessing there has to be a much lower one. (And of course, coefficient of restitution is not 1, so some energy is lost to heat, but let’s ignore that.)
Please bear with me, because I’ve forgotten enough high school physics to be confused by the difference between preserving energy and preserving momentum, unless there is no difference.
More to the point of my query: if we assume Jupiter is a point mass with no atmosphere, and the probe is also, what’s the maximum velocity of the probe after the assist? Can it really take all of Jupiter’s momentum, if executed perfectly?
The limit to the energy that can be transferred is none, in the center-of-mass reference frame (which will be almost the same as the truck’s reference frame, assuming it’s much more massive than the ball). In that frame, the ball approaches the truck at some speed, and it is knocked away from the truck at at most the same speed.
For a perfectly elastic collision, kinetic energy and momentum are both conserved. So you can write two equations with two unknowns (the two objects’ velocities after the collision) and solve.
So, if I’m understanding the explanations correctly, it’s like Rollerball. (How’s that for a metaphor?) A player on skates grabs a hold of the player on a motorcycle and is able to use the motorcycle’s movement to speed up their own movement.
On approach, the satellite approaches Jupiter from the general direction of the Sun or Earth, I.e. almost 90 degrees to the orbit, a radius line from inside the orbit.
Plan the satellite path so it swings around Jupiter and goes out in a tangent to Jupiter’s obit, in the same direction as Jupiter’s path around the sun.
The probe leaves Jupiter’s effective gravitational influence with the same energy it came in with. If it was going 7mi/sec towards Jupiter on the way in when it first got close to Jupiter’s gravitational field, it is going away from Jupiter at 7mi/sec as it leaves.
The difference is, instead of going radially toward from Jupiter, it has turned about 90 degrees and is now going tangent to Jupiter’s orbit away from Jupiter. From the Sun’s point of view, it is now going 7 mi/sec PLUS the orbital speed of Jupiter, about 13km/s.
Of course, going faster than Jupiter, it travels on n elliptical router outward away towards Saturn’s orbit. Time this right, and you can do the same trick with Saturn and pick up a bit more speed.
(Yes, a planet’s gravitational field goes on forever, but effectively, at a certain point it is pretty much negligible compared to the influence of the Sun or other planets. )
The key point is the speed you enter Jupiter’s gravitational field is the speed you leave - relative to Jupiter! Relative to the sun, you have added a component of Jupiter orbital speed depending on what angle/direction you leave Jupiter.
Not quite like rollerball, more like rollerball if you grabbed a trailing bungee cord and slingshot from side entry to passing the motorbike.
Image you, stationary on roller skates and a motorcycle driving past you at 20 mph.
I stand behind you and start you rolling at 5mph. The motorcycle drives past at right angles to your travel. Me, being a dab hand at Newtonian calculations have ensured my push is timed perfectly to allow you to grab the back of the bike as it passes, swing yourself round through, say 45 degrees and scoot off in a new direction at a higher speed (relative to me).
The motorcyclist won’t see any change in your velocity but will certainly have felt you “stealing” some of their energy.
This is the analogy I use myself: Imagine that two people threw a baseball at each other. The two balls approach each other at the same speed, collide, and bounce apart. But it’s a slightly offset collision, so one ball is bounced upwards and the other downwards.
Assume that the collision is perfectly elastic, so each ball is ejected at the same speed it had before the collision. So neither ball gains speed. But one is now moving upwards, while the other is moving downwards, so one ball has gained gravitational potential energy, and the other has lost it.