Is it profitable to save fuel for the Oberth effect during a Jupiter gravity assist?

Probes to the outer Solar system use a Jupiter flyby for gravity assist, all six such missions thus far have. While flying near a gravity well one can enjoy the Oberth effect which multiplies the acceleration per unit propellant consumed.

Have any spacecrafts using a Jupiter gravity assist fired their (small) rockets in order to profit from the Oberth effect?

What does the trade off look like between using all fuel at Earth to get into the Jupiter trajectory, versus saving the upper stage and much of its fuel for a big Oberth bonus blast at perijove? New Horizons passed by Jupiter only 13 months after launch, and took another 101 months to reach Pluto. Would a slower fuel laden first leg of the trip with a big burn at Jupiter have shortened its total travel time to Pluto?

• One problem I imagine: carrying all that fuel out there, for months, with no guarantee the main engine will fire. Gravity for the assist will always be there, but if you can't perform the burn, you'll miss the destination!
– SF.
Apr 6, 2017 at 22:42
• These corrections usually can be (and are) done with RCS. Sometimes it is done with the main engine - but it's always a risk. example
– SF.
Apr 7, 2017 at 7:06
• Because RCS is engineered with entirely different priorities in mind than main engines. Main engine focus is on maximizing specific impulse and thrust, while RCS prioritizes very high restartability, precise (not necessarily high) thrust power and rapid ignition and extinguishing. It boils down to: RCS has a lousy specific impulse, so it can't provide the mighty boost. And an engine that combines reliability/restartability of RCS with specific impulse of main engine is... somewhere over the horizon.
– SF.
Apr 7, 2017 at 8:25
• IF the engine fires after a couple months of inactivity. If it used cryofuels, they will have boiled off. If it used pyrotechnic ignitors, they could have gone bad (like harpoons on Philae). If it used hypergolics, they might have damaged/corroded/gunked up the plumbing, blocking the valves. It takes some very special engineering to be able to fire an engine months or years after launch. (and RCS typically use monopropellant, which is pretty good for that application but has lousy performance.)
– SF.
Apr 7, 2017 at 10:19
• Soyuz stays at ISS as lifeboat, and uses hypergolics for the reentry burn. Juno also used hypergolics. Boil-off of cryofuels is a couple percent per DAY, so a year later you'll arrive entirely dry! Never mind you need a gnat's fart worth of delta-V to reenter.
– SF.
Apr 7, 2017 at 15:12

When it comes to the orbital mechanics of this scenario, one thing to note is that when it comes to oberth effect and raising the orbit around the Sun, it's the total velocity relative to the Sun that matters, i.e. the Earth orbits the Sun at 30km/s, and the probe orbits the Earth at 8km/s, giving 38km/s to work with, the bare minimum ejection burn to intercept Jupiter then adds 6.5km/s and the probe is up to 44.5km/s, now the question is are you better off burning on top of that 44.5km/s, or burning at Jupiter?

Jupiter orbits the Sun at 13km/s and the probe should swing by at about 60km/s if it gets as close to Jupiter as possible, potentially a combined velocity of 73km/s. The tricky part is that at closest approach to Jupiter the probe probably won't be travelling in exactly the right direction for a prograde burn to enjoy maximum oberth effect with respect to the Sun - at Earth precise timing can arrange this, but when you're swinging by one planet and need to actually encounter a third planet there are constraints on the first encounter which will result in the oberth effect not being maximized, unless you wait a long time for the planets to line up in exactly the right way and when it comes to long period inclined and eccentric orbits like that of Pluto, that's going to be a long wait.

So this is not enough to rule out the possibility that it could be more effective to do some of the burn in Jupiter's gravity well, it's just to say you get an awful lot of oberth effect already by virtue of doing the burn deeper in the Sun's gravity well. My hunch is that if all you care about is maximizing heliocentric velocity you'd be slightly better off doing the burn at Jupiter, but if you wish to actually encounter a third planet you'd tend to be equally well served by completing the burn at Earth.

And there are also the practical constraints: completing the burn at Earth uses only the one stage for the entire ejection burn. A Jupiter burn would require an additional solid stage or a reliably restartable engine and a no leaks guarantee. And if the burn is in any way messed up you've just lost your best option for cheap course corrections, which is fine-tuning the encounter with Jupiter.

• Doesn't the gun powder make much more noise when fired at 60 km/s (+13 km/s) in perijove, than it does at Earth's 30 km/s relative to the Sun? If we skip targeting any particular planet, and just wanna get away from here as fast as possible. There're plenty of KBOs in any direction of the ecliptic to fit any launch window. Didn't Ericke propose a Jupiter flyby to shed all of Earth's orbital velocity so that a space craft could crash right into the Sun? Couldn't a similar concept be used to instead DOUBLE the spacecraft's speed outwards? Apr 7, 2017 at 13:27
• @LocalFluff yeah, if you don't care where you're going and just want to go fast then you'll get more out of doing the burn at Jupiter. I tried it in an orbital simulator and expending 3000m/s of "extra dV" expended at the Earth ejection burn resulted in crossing Neptune's orbit in about 12 years, in contrast doing that burn at Jupiter crossed Neptune's orbit in about 10 years, so you would seem to get a good deal more heliocentric velocity, though would still be looking at mission times measured in decades. But it also has to compete with "sun diving" schemes which have much higher potential. Apr 7, 2017 at 14:38
• If we only had a retrograde giant planet nearby, it'd be a trajectory power plant. Apr 7, 2017 at 14:59

I can't give you a full tradeoff on whether it makes sense to save some delta-v for the gravity assist, but if you know how to accurately compute the gravity assist itself, then I think this will allow you compute the delta-v for Oberth and gravity assist together.

Adding an Oberth maneuver to a simple gravity assist requires breaking the flyby into two pieces, which affects the "turn angle", i.e., the amount the spacecraft's incoming hyperbolic asymptote is rotated counterclockwise to its outgoing asymptote.

The vanilla-flavored formula for the turn angle is:

$$\delta = 2 \cdot \arcsin{\frac{1}{e}}$$

where $$e$$ is the hyperbolic eccentricity, which is $$1+\frac{r_p v_{\infty}^2}{\mu}$$.

But when you apply the Oberth burn at periapse, you have to break the turn into two pieces:

$$\delta = \arcsin{\frac{1}{e_{In}}} + \arcsin{\frac{1}{e_{Out}}}$$

Note also that in the pure gravity assist version, $$v_{\infty,In}=v_{\infty,Out}$$, but you now have to calculate the new $$v_{\infty,Out}$$ with the Oberth delta-v added. That will then get you the new eccentricity ($$e_{Out}$$), so you can calculate the turn angle.

The turn angle will be shallower, which generally reduces the gravity-assist delta-v by a small amount, but the Oberth burn more than makes up for it.