Adam Hibberd
In the UK Spring of 2017, I derived the theory for solving interplanetary trajectories, which enabled me to develop a powerful software tool for optimising hight thrust spacecraft missions, a tool which I called Optimum Interplanetary Trajectory Software (OITS).
For those of you fascinated by mathematics, in particular mathematical formulae, the two equations which are pivotal to the functionality of OITS are as follows:
Where α is the angle between the approach velocity (hyperbolic excess speed VA) with respect to the planet and the departure velocity (hyperbolic excess speed VD') with respect to the planet and also observe that α + π= θ + θ'. In this context θ and θ', are respectively the angle made between the arrival asymptote with the periapsis point at the planet's centre and the angle between the departure asymptote and the periapsis point at the planet's centre.
The second of these equations expresses the derivative of the left hand side of the first with respect to θ, thus enabling the solution of the first via a Newton iteration method.
These two equations together allow the solution of the encounter hyperbolae for each planet visited on the route to the destination, and note in the case of Project Lyra, this was of course the first interstellar object to be discovered 'Oumuamua.
If you have the motivation, go to all the documentation for OITS on github here, where you will also find diagrams illustrating the definitions of the above parameters, α, VA, VD', θ and θ'.
However I understand that not everyone has a technical bent, but for these people there is help at hand!
Simply go to my guide for laypersons, which was published in Principium, the online quarterly magazine for the Initiative for Interstellar Studies, i4is.
The issue in question was number 29, published November 2019, and to save you the bother of searching for it on this website, here's a link which will make things easier for you and will take you right there.