So if the idea is to ping-pong between two planets, each ping-pong in the series will give roughly the same acceleration, which is related to the orbital velocity of the body you are slingshotting, i.e. linear speed increase per ping-pong. However, since the time taken to travel between the two bodies and perform the slingshot will get shorter and shorter, the rate at which the speed increases over time will actually be hyperbolic, not linear.

For this to work you would need to leave each body in roughly the direction you arrived, and to do that you need to be travelling roughly at escape velocity (https://en.wikipedia.org/wiki/Escape_velocity).

Escape velocity is not a fixed value, but is inversely proportional to the square root of the approach distance, r. Therefore in theory there is no limit to how much you can accelerate between the two bodies, provided you can approach them close enough.

That's the theory. Let's try some numbers for the planets mentioned, Jupiter (orbital velocity 13 km/s) and Saturn (orbital velocity 9.7 km/s). Let's use Voyager 2 as our start conditions, travelling at around 10 km/s when we first reach Jupiter and slingshot. The first round trip takes 6 years. Since we'll lose some speed from Saturn but gain more from Jupiter, let's assume we gain 5 km/s overall, per ping-pong. The next round trip will take 4 years (15 km/s when we reach jupiter), then 3 years (20 km/s)... to reach 10 000 km/s would take 20 000 ping-pongs. However, you'd be going so fast at the end that the total duration of the round trip would now only take 2 days, and surprisingly despite the small linear increase per ping-pong the total time taken to accelerate to that speed would “only” be about 86 years, which is in line with the outbound trip time to Proxima. By this stage you'd need to be approaching Jupiter at a distance of 2.5 km and Saturn at a distance of 0.75 km!

Of course planets are not points, you cannot approach them arbitrarily close like this because real objects have size and you will crash into them. Therefore the maximum velocity you can pass the planet at, and still return in roughly the direction you came from is the surface escape velocity. This is a much more sedate 60 km/s on Jupiter, nowhere near fast enough for a trip to Proxima (not to mention the fact that this is the velocity you have accelerated to at the point of closest approach- you lose velocity again on the way out). In fact by far the biggest useful sling-shot effect would be from the solar system itself, travelling around the galactic centre at ~220 km/s. The maximum slingshot effect is achieved by approaching in the opposite direction of the orbit and exiting in the same direction as the orbit, which adds double the orbital velocity to the start velocity, over 400 km/s in this case. Of course that means your destination better be somewhere along the tangent, which may mean picking a different destination, and it would still take ~2500 years even to get to Proxima.

So if the idea is to ping-pong between two planets, each ping-pong in the series will give roughly the same acceleration, which is related to the orbital velocity of the body you are slingshotting.

For this to work you would need to leave each body in roughly the direction you arrived, and to do that you need to be travelling roughly at escape velocity (https://en.wikipedia.org/wiki/Escape_velocity). This is the same order of magnitude as the speed of the Voyager probes (actually slower than them) and makes the technique of no practical use for travelling to Proxima Centauri.

In fact by far the biggest useful sling-shot effect would be from the solar system itself, travelling around the galactic centre at ~220 km/s. The maximum slingshot effect is achieved by approaching in the opposite direction of the orbit and exiting in the same direction as the orbit, which adds double the orbital velocity to the start velocity, over 400 km/s in this case. Of course that means your destination better be somewhere along the tangent, which may mean picking a different destination, and it would still take ~2500 years even to get to Alpha Centauri.

So if the idea is to ping-pong between two planets, each ping-pong in the series will give roughly the same acceleration, which is related to the orbital velocity of the body you are slingshotting, i.e. linear speed increase per ping-pong. However, since the time taken to travel between the two bodies and perform the slingshot will get shorter and shorter, the rate at which the speed increases over time will actually be hyperbolic, not linear.

For this to work you would need to leave each body in roughly the direction you arrived, and to do that you need to be travelling roughly at escape velocity (https://en.wikipedia.org/wiki/Escape_velocity).

Escape velocity is not a fixed value, but is inversely proportional to the square root of the approach distance, r. Therefore in theory there is no limit to how much you can accelerate between the two bodies, provided you can approach them close enough.

That's the theory. Let's try some numbers for the planets mentioned, Jupiter (orbital velocity 13 km/s) and Saturn (orbital velocity 9.7 km/s). Let's use Voyager 2 as our start conditions, travelling at around 10 km/s when we first reach Jupiter and slingshot. The first round trip takes 6 years. Since we'll lose some speed from Saturn but gain more from Jupiter, let's assume we gain 5 km/s overall, per ping-pong. The next round trip will take 4 years (15 km/s when we reach jupiter), then 3 years (20 km/s)... to reach 100 000 km/s would take 20000 ping-pongs. However, you'd be going so fast at the end that the total duration of the round trip would now only take 5 hours, and surprisingly despite the small linear increase per ping-pong the total time taken to accelerate to that speed would “only” be about 114 years, which is in line with the outbound trip time. By this stage you'd need to be approaching Jupiter at a distance of 25 metres and Saturn at a distance of 7.5 m!

Of course planets are not points, you cannot approach them arbitrarily close like this because real objects have size and you will crash into them. Therefore the maximum velocity you can usefully approach at, and still return in roughly the direction you came from is the surface escape velocity. This is a much more sedate 60 km/s on Jupiter, nowhere near fast enough for a trip to Proxima. In fact by far the biggest useful sling-shot effect would be from the solar system itself, travelling around the galactic centre at ~220 km/s. The maximum slingshot effect is achieved by approaching in the opposite direction of the orbit and exiting in the same direction as the orbit, which adds double the orbital velocity to the start velocity, over 400 km/s in this case. Of course that means your destination better be somewhere along the tangent, which may mean picking a different destination, and it would still take ~2500 years even to get to Proxima.

So if the idea is to ping-pong between two planets, each ping-pong in the series will give roughly the same acceleration, which is related to the orbital velocity of the body you are slingshotting, i.e. linear speed increase per ping-pong. However, since the time taken to travel between the two bodies and perform the slingshot will get shorter and shorter, the rate at which the speed increases over time will actually be hyperbolic, not linear.

For this to work you would need to leave each body in roughly the direction you arrived, and to do that you need to be travelling roughly at escape velocity (https://en.wikipedia.org/wiki/Escape_velocity).

Escape velocity is not a fixed value, but is inversely proportional to the square root of the approach distance, r. Therefore in theory there is no limit to how much you can accelerate between the two bodies, provided you can approach them close enough.

That's the theory. Let's try some numbers for the planets mentioned, Jupiter (orbital velocity 13 km/s) and Saturn (orbital velocity 9.7 km/s). Let's use Voyager 2 as our start conditions, travelling at around 10 km/s when we first reach Jupiter and slingshot. The first round trip takes 6 years. Since we'll lose some speed from Saturn but gain more from Jupiter, let's assume we gain 5 km/s overall, per ping-pong. The next round trip will take 4 years (15 km/s when we reach jupiter), then 3 years (20 km/s)... to reach 10 000 km/s would take 20 000 ping-pongs. However, you'd be going so fast at the end that the total duration of the round trip would now only take 2 days, and surprisingly despite the small linear increase per ping-pong the total time taken to accelerate to that speed would “only” be about 86 years, which is in line with the outbound trip time to Proxima. By this stage you'd need to be approaching Jupiter at a distance of 2.5 km and Saturn at a distance of 0.75 km!

Of course planets are not points, you cannot approach them arbitrarily close like this because real objects have size and you will crash into them. Therefore the maximum velocity you can pass the planet at, and still return in roughly the direction you came from is the surface escape velocity. This is a much more sedate 60 km/s on Jupiter, nowhere near fast enough for a trip to Proxima (not to mention the fact that this is the velocity you have accelerated to at the point of closest approach- you lose velocity again on the way out). In fact by far the biggest useful sling-shot effect would be from the solar system itself, travelling around the galactic centre at ~220 km/s. The maximum slingshot effect is achieved by approaching in the opposite direction of the orbit and exiting in the same direction as the orbit, which adds double the orbital velocity to the start velocity, over 400 km/s in this case. Of course that means your destination better be somewhere along the tangent, which may mean picking a different destination, and it would still take ~2500 years even to get to Proxima.