This flag means math

With that unfortunate name of "Retarded Time", Thales Azevedo and Anderson Pelluso revisit Pierre Bouguer's Pirate Problem from AD 1732. This, we're told, founded pursuit-analysis as a branch of mathematics - I suppose Lambert's Problem is a Keplerian subset.

There's a pursuer P, anthropomorphed as a pirate (or privateer); and a target, the merchant. The merchant travels in a straight line. Assume P receives information on the merchant's position accurately and immediately. For Cartesian purposes this is best modeled in deep space where the merchant is beelining between star-systems without propellant (we'll get back to this). By whatever means, the pirate vessel immediately accelerates at the merchant to hit its chosen speed - faster than the merchant's. For whatever reason, the pirate holds that speed constant - but its velocity-vector always points to the merchant. In our coöordinates, the pirate starts (0,0) and sets off once the merchant hits the X-axis. Mathematically the merchant starts (m₀,0) going (δx_m, δy_m). Draw the graph of the pirate's trajectory y = f(x).

Unlike Lambert, Bouguer's problem was soluble nonalgorithmically which solution Bouguer derived himself. It is always soluble, in fact, if only the pirate is faster than the merchant (although he'll want to get there sooner than later).

Bouguer was talking the high Caribbean seas 'pon de flat eart', so wunna can correct the merchant's path for a geodesic 'pon de globe if wunna have to. In our normal space, the merchant's geodesic should be taken as between planets, probably Hohmann but maybe cycler. And of course any pirate who knew an unsuspecting merchant's destination would consider kicking off his own geodesic, to intercept without trimming sails / blowing propellant (hence Lambert). Bouguer didn't overthink this, but if you're interested Paul Nahin comes recommended.

In space Azevedo and Pelluso note the obvious problem, even interplanetary: communication is not instantaneous. (In the oceans one might consider a U-boat hunting the merchant via sonar.) Here the pirate does not point to where the merchant is at time t, but where the merchant was at earlier time t_r - hence, that time is retarded. Jokes about the mental state of pirates who can't relativity, aside (maybe they lack a bead on the parallax).

A-&-P point out that the retarded-time concept is already used - in electrodynamics. But electrodynamics be overly hard. A-&-P considered other relativistic solutions already published especially C. Hoenselaers' "Chasing relativistic rabbits" doi 10.1007/BF02107933. A-&-P deemed this inordinately hard as well, since although the light is going at 300000 km/s, the ships don't have to be. We can handle the setting and the motive, next post. A-&-P suggest this pirate-problem as an entertaining and gentle introduction to retarded-time, for the university freshmen, before they go on to using this in the higher courses.