The effect is surely in line with both the proximity of the 'slingshotter' and 'slingshottee' (proportional to the inverse square of the centre-to-centre distance?) and
the amount of time at that proximity (proportional?1
An extremely energetic pass-by has less 'nearby time', and so I'm not sure if sending such a projectile at a more significant (relative/sun-relative?) velocity is going to help, if it passes the point of greatest influence2
so much quicker. To overcome that, you need to aim your projectile closer
. And then you have the problem of atmosphere-skimming, because Jupiter isn't a point-mass that can be passed arbitrarily close.
At the very least, you need to consider the trade-offs that come from the Jovian aerobraking manoeuvre itself, when you do that. Which seems to send us into the territory of Could we speed up Earth's rotation
, and of course
sending in projectiles to directly impart velocity upon the planet (each below the limit of any significant 'exfoliation' of the atmosphere... another problem!) and changes the scenario. So we'd ideally want to avoid that and restrict our flybys to only so close as to stay beyond all but the most tenuous atmosphere. Or at least configure the probe to be more aerodynamic (c.f. "Night Glider" orientation modes, and other variations on this, for the ISS solar panels) so it remains an insignificant aspect of the calculations.
So (and, even assuming I've been on the money so far, I'll let someone else
do the exact maths) there's probably a maximum fly-by speed that would actually be useful in the momentum-bleeding of Jupiter, for any given fly-by path by any given mass/cross-section of object. Which sounds like a complicated optimum to work out, if you're not just going to send a Pinto-sized probe at near-'c' velocity straight in against Jupiter and count it as a success when the average accumulated
momentum of the resulting rapidly (and, in parts, relativistically) expanding debris cloud is calculated as now having effectively zero orbital velocity... 1
If 'held' in position, but as there's an approach/pass/withdraw curve, it probably ends up being an integration of the changing centre-to-centre (multiplied by the effective of the operating angle at dt
?) force, as each body's trajectory changes the exact interaction.2
The trajectory line-segment where all points have above-mean mutual gravitational attraction, compared with the whole path?