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Arriving at our solar system with interstellar speeds from an awkward angle and assuming we have a good enough heat shield, would it be possible to chain a series of aerobrakes with various planets with thick atmospheres to finally be captured by one of them (without getting roasted or fly into infinity in the process) thus saving fuel for landing? Roughly what would the optimal trajectory look like? What planet would you choose for the first pass?

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An aerobraking maneuver at interstellar speeds is really, really scary for sure.  I highly doubt that such a maneuver is feasible, but with all the aero and heating glitches going around, who knows what's impossible or not.  If anything, I'd expect flybys to work better, but even then since you're traveling at such an insane speed it's unlikely you'll get too much of a slowdown.

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1 hour ago, Entropian said:

An aerobraking maneuver at interstellar speeds is really, really scary

I totally agree; a gravity assist chain would be much more safe and effective. As long as the incoming speed isn't too crazy (the upper limit is somewhere around 20-25 km/s), a craft could get captured around the Sun with a Jupiter assist. After that, a series of alternating gravity assists between Jupiter and Saturn could slow the craft down enough relative to Saturn that it would be able safely aerobrake at Titan. From there, it can simply parachute down to Titan's surface.

This trajectory would take decades if not centuries, but is quite safe (with enough assists, the final aerobraking speed can be as low as ~3200 m/s), and requires only minimal fuel for correction maneuvers.

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Jupiter can throw probes out of the Solar system with a single gravity assist, so the reverse will also be true if the trajectory is just right. Saturn, Uranus and Neptune are also decent targets but Jupiter's sheer size means it'll give the biggest slingshots.

Aerobraking, on the other hand, would almost certainly not work due to the ludicrous speeds involved. I once sent a probe flying past Pluto at a relative speed of 30km/s; despite Pluto's negligible atmosphere the probe was instantly incinerated the moment it entered the atmosphere, and while there was no heatshield on that probe I can't see how any amount of shielding would protect against a sustained aerobraking pass in the atmosphere of one of the gas giants.

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