Adam Hibberd
Having solved the problem of catching up with and staying alongside Halley's comet as it orbits the Sun (go here), the natural follow-up problem is how do we return to Earth, from Halley's orbit, with a sample on-board?
So, I decided to look into the problem of a sample return mission from Halley, in particular assuming departure from Halley on May 6th 2060, when it is approaching its northerly most passage w.r.t the ecliptic, and also when it is quite close to its perihelion point.
There are a couple of reasons for choosing such a location to depart Halley's comet. First, if a departure after this point is attempted then the comet will be heading downwards w.r.t. the ecliptic plane (i.e. with negative Vz) and with a huge velocity extremely difficult to reverse. (Of course, if Vz is negative a demanding U-turn would be necessary, so as to return to the ecliptic plane where Earth resides.) Second, departure must be late enough so there is plenty of opportunity for the s/c to rendezvous with Halley before this and exploit its loiter-time at the comet (1-2 years) to pick up a promising sample and then prepare for departure.
That's fair enough but now there is a huge elephant in the room, the fact is that if we are WITH Halley in its elliptical orbit around the Sun, then that means around perihelion we are travelling with a huge velocity w.r.t the Sun (~55 km/s) and retrograde also. This is extremely problematic.
However I have found a way of dissipating all that kinetic energy on leaving Halley, so that by the time Earth is finally reached, the hyperbolic excess speed w.r.t the Earth is a fraction of the s/c's original speed (in fact it would be ~15 km/s relative to Earth, a reduction of 73%).
My colleague Marshall Eubanks has dubbed this idea 'interplanetary badminton' which I think is quite appropriate and explains what is happening exceedingly well. Look at the Figure below to clarify.
The strategy is my old DJGA approach (Double Jupiter Gravtiational Assist), whereby we arrive at Jupiter, conduct a passive gravity assist (PGA), swing over the ecliptic plane eventually arriving at the same planet, Jupiter, 6 years later. The task of the leg of the journey up to the and including the first PGA is to remove all the radial velocity away from Earth (look at the orange segment (1) in the above Figure) and convert it into circular motion (refer to segments (2) and (3)), albeit this circular motion is at Jupiter's orbit and STILL retrograde.
But it turns out this is OK, since when we arrive at Jupiter again, we use another PGA to (a) reverse the retrograde motion to prograde motion and (b) confine the motion of the s/c to the ecliptic once again.
Thus we have by these two PGAs, removed a considerable amount of kinetic energy of the s/c, enabling the possibility of aerobraking in Earth's atmosphere and ultimately returning the capsule with the sample safely back to Earth where scientists can examine the contents using all the powerful instruments available to them.
'Eureka!' we have achieved a sample return of Halley! - and all using my 'Optimum Interplanetary Trajectory Software' (OITS - with Intermediate Points) which has a track record of reliability when I used it for solving the problem of catching up with 1I/'Oumuamua (Project Lyra).