Adam Hibberd
There has been a lot of discussion around what would be the best route to catch that most mysterious and outrageously quirky interstellar interloper 1I/'Oumuamua, and I have personally investigated all sorts of spacecraft mission scenarios; using a combination of gravity assists (where kinetic energy is gained by the spacecraft's encounter with a planet but no on-board propulsion is used) and Oberth Manoeuvres (where kinetic energy is gained by firing the propulsion system at the periapsis of the Sun or a planet). Should the route exploit a Solar Oberth, Jupiter Oberth or a passive gravity assist of Jupiter for example?
Well all this wonderful research seems to side-step possibly the most obvious route, i.e. the direct passage, that is without any form of solar or planetary encounter on the way.
What becomes apparent when one thinks about this is that there are many challenges the direct route option would impose on mission planners, sure there is simplicity and robustness in a direct mision, but what would be the sort of level of 'DeltaV' - and therefore precious rocket propellant - that would be required, particularly when we have no 'gravity assists' or 'Oberth manoeuvres' to help us?
'Oumuamua is travelling at 26.3 km/s w.r.t. the Sun as it heads into the deep emptiness of space, and so in order to race it down, we must make sure we generate MUCH more speed than this since the gravity of the Sun will slow the spacecraft down as it heads off in pursuit - so we need a higher surplus or excess speed (to be precise a 'heliocentric hyperbolic excess' - i.e. the extra speed acquired on exiting the Sun's sphere of influence) than that of 'Oumuamua.
Not only this but we also know that 'Oumuamua is in quite a high inclination orbital plane (around 122 deg) and so consequently the spacecraft must thrust significantly outside of the Earth's own orbital plane (the Ecliptic plane) to make sure we head off in the right direction - actually in a direction torwards the constellation Pegasus (which has quite a high declination in the Northern hemisphere).
For these aforementioned reasons we can totally rule out chemical rockets, but is there a more capable propulsion system which might be up to the task?
If you look at the colour contour map provided we begin to see the problems I have outlined previously which I gave in qualitative terms, instead in the cold logic of quantitative data. But at first glance to the intelligent layperson, the plot may be rather hard to decipher, so let me explain.
On the horizontal axis (x) we have the launch date of the direct mission to 'Oumuamua. And for each launch date we can go up vertically a certain distance (y) in a scale given by the vertical axis (on the left-hand side) to get the 'hyperbolic excess speed at Earth'. This is NOT the same as the heliocentric excess speed, rather it reflects the speed needed on exiting EARTH's sphere of influence to catch up with 'Oumuamua.
Finally the colour code of each (x, y) coordinate expresses the total flight duration needed to eventually intercept 'Oumuamua, the key to this colour code is provided in the colour bar on the right and shows that the deeper blue the colour the less time is needed and on the contrary, the brighter the colour the longer the time needed.
It so happens I did some research into using Nuclear Thermal Propulsion (NTP) for Project Lyra, using my extremely powerful software tool I developed known as 'Optimum Interplanetary Trajectory Software' (OITS). This research can be found on my ResearchGate profile here.
Rather than you resorting to a detail reading of this document I show on the provided plot above two horizontal lines indicating the best Earth hyperbolic excess speeds achievable by two launch vehicles, namely the NASA Space Launch System (SLS) and the extremely capable SpaceX Starship Expendable, each of which is equipped with a payload in the form of an NTP-propelled spacecraft. We find that for both of these possibilities, direct missions to 'Oumuamua JUST become feasible.
Note that the SLS + NTP can only provide viability up to around 2036, with flight durations of around 30 years or so. The Starship Expendable is triumphant however in that depending on launch year, it can deliver a payload to 'Oumuamua in a matter of 10 to 20 years from launch.