# Flyby puzzler; starting from Earth, how many times can you use Jupiter flybys in one century?

Is there any limit to how many times you can increase velocity by repeated sling shot manoeuvres? and its answers have got me thinking, and that's always dangerous.

Suppose you have a durable RTG-powered spacecraft launched from Earth, and your goal is to use Jupiter-only flybys as many times as you can in order to set some sort of record before going off to explore some place further.

We've got to put some constrains on what a "flyby" is otherwise you'd just go into a 5.2 AU heliocentric retrograde orbit and pick up 16 conjunctions per century and call them "distant perturbative flybys".

So to be a flyby, the heliocentric characteristic energy $$C_3$$ and/or the semimajor axis $$a$$ have to increase by some factor.

I'll call that factor 10% arbitrarily. If there are profound or compelling answers that have a slightly lower factor they won't be balked-at :-)

Flybys for primarily inclination changes should obtain a similar delta-v to those above in order to count.

Question: This is a flyby puzzler; starting from Earth, how many times can you use Jupiter flybys in one century?

• "use Jupiter-only flybys" - that is, don't use any fly-by of another object in between? Aug 5 '21 at 10:52
• @asdfex that's what I was thinking, yes; keeping the problem as mathematically simple as possible it already seems challenging enough, and as a circular restricted 3-body problem there may be some helpful math tools available as well as the Tisserand stuff.
– uhoh
Aug 5 '21 at 11:00
• I thought about something like ping-ponging with Mercury to realign your orbit with the new position of Jupiter. That might allow for one encounter every couple of years. (but I'm not going to do the math and write an answer) Aug 5 '21 at 11:19
• @asdfex that's a great idea! If this one goes well then I'll ask a new question and open it up to some kind of multi-body problem.
– uhoh
Aug 5 '21 at 11:21

As always with such puzzlers, one need to find something to exploit heavily :)

Usually when constructing multi-flyby trajectories ("ball of yarn"), one relies on keeping a consistent semi-major axis in order to repeatedly flyby the same object, only modifying the eccentricity. Increasing $$C_3$$ or increasing semi-major axis is required in this case, so the usual approach doesn't work. (Increasing $$C_3$$ is equivalent to increasing the semi-major axis).

Luckily, there's still something left to exploit:

Flybys for primarily inclination changes should obtain a similar delta-v to those above in order to count.

So as long as an inclination change provides a large enough delta-v, we can keep the semi-major axis constant!

One immediate way of utilizing this is a "bouncing" trajectory, having a large inclination change flyby of Jupiter twice every orbit. In a century, one can fit 17 such bounces (if the initial transfer is quick enough).

These flybys are hyperbolic trajectories with a perijove above Jupiter's north (or south) pole. Such flybys only mirror the z-component of the entry and exit vectors, preserving heliocentric $$C_3$$.

This orbit is known as a "reflected backflip", which is discussed in Lunar Cycler Orbits with Alternating Semi-Monthly Transfer Windows by Uphoff and Crouch.

They do however primarily present the "single backflip" trajectory, which itself should qualify for this puzzler, doing 4 flybys for every 3 orbital periods (equivalent to 12 bounces over a century):

Since this is significantly easier to analyse, I will do so here.

For the planar part to have an orbital period of exactly half that of Jupiter, it needs to have a semi-major axis $$\frac{1}{2^{2/3}}$$ as large. That's a velocity difference at encounter of 4670 m/s.

We can calculate turning angle by:

$$\theta = -2 \cdot asin\left(-\frac{1}{1 + \frac{r_P v^2}{\mu}}\right)$$

Which means the maximum turning angle for 4670 m/s at a perijove of Jupiter's radius is 162 degrees, comfortably more than the 90 degree change required to turn a planar velocity difference into a pure inclination difference.

• @uhoh Conservation of heliocentric C3 merely requires that the Jovian entry and exit vectors have the same velocity component along the axis of Jupiter's motion. Every 90 degree inclination flyby with a perijove above a pole of Jupiter is a counterexample to your claim. Aug 6 '21 at 17:40
• perhaps "90 degree inclination change flyby..." would make more sense Aug 6 '21 at 18:08
• @uhoh ""90 degree inclination flyby with a perijove above a pole of Jupiter" isn't even possible." This is clearly not true. Any orbit with a perijove over the pole has an inclination of 90 degrees. Aug 6 '21 at 18:24
• @uhoh "Lunar Cycler Orbits with Alternating Semi-Monthly Transfer Windows" to the resque! Aug 6 '21 at 19:27
• @uhoh Yeah, that would be optimal under the present rules. The degenerate near-rectlinar halo orbit would in patched conic terms be an elliptic orbit around Jupiter, dipping just outside the SOI. So the upper limit that way is 47 flybys, although the inclination change in that extreme case is too low. Aug 9 '21 at 1:52