This question is inspired by the one asked here:
and I note that (though I've heard of these before) there are, apparently, 3 important fundamental "zones of inescapability" with regard to a black hole that can be identified on the basis of physical principles and which can be defined by the Schwarzschild $r$ coordinate and its critical radius $r_S$: one of these is, of course, the event horizon, at $1\ r_S$, from which no escape at all is possible. The second one is the "photon sphere", here referred to as the zone at the "innermost bound circular orbit", at $1.5\ r_S$, at which a photon (particle of light) traveling at right angles (i.e. constant $r$-coordinate) can complete one circuit around the hole and within which only impossibly powerful rockets could get you out. Orbits above this point but below the "innermost stable circular orbit", out to $3\ r_S$, are possible but require a continuous application of (likely ungodly large) engine propulsion. Above this, more-or-less normal orbits are possible, though they precess more and more rapidly as the general-relativistic effect that is incipient with the orbits of the planets becomes fully mature in these environs.
Now, the trick, of course, is the qualifiers regarding propulsion: "impossibly powerful", "ungodly large" which basically are likely words for "it can't be done with realistic technology". Which, then, of course, is what begs this question - what can be done with such?
For my "realistic" scenario, a generous interpretation will be used which is "a matter-antimatter fueled spacecraft with a wet mass 100x its dry mass". The goal is for the craft to execute one of two manoeuvres:
- lower itself into an orbit around the hole, orbit a few times, and then take off again (so it must have fuel for both trips) and return to where it launched from, and
- establish a "slingshot" trajectory around the hole that takes it as close as possible before shooting it back out into space (this may be easier?).
I consider this "reasonable" because any travel to a black hole will already be an extreme problem in interstellar travel, much less the kinds of targets that will already be in use for this kind of proposal which will almost surely have to be supermassive black holes to be interesting and thus requiring travel at least to the center of the Galaxy (~200 000 Pm) if not intergalactically (distances of many Zm), with tera- or petaseconds of travel time (thousands to millions of years) at least, or something like wormholes if they are possible. Hence I presume the technology to make such a craft will be readily available to any civilization that is capable of pulling this feat off to begin with.
The reason we need supermassive black holes is tidal forces and ideally we want to get as close to the horizon as possible. Possible targets may be Sagittarius A* and M87* (the black hole that the picture was taken of), both supermassive black holes, the latter much more so still, at the centers of galaxies. However, M87* presents the additional challenge of an extremely active matter environment around it, which may foreclose that possibility even further, though even Sgr A* might be too much. Hence, failing that, we can just consider a hypothetical black hole of mass equal to either, that is suitably "quiet", if nothing else is possible. (or perhaps imagine that with M87*, it has entered a quiescent period by the time the ships arrive as compared to the time when it sent the images we saw)
What would be the closest approach possible, in both scenarios, while still having enough fuel to get away (or not being destroyed by other effects)? We should also assume that the craft starts out at great distance where gravity is hopefully unremarkable, but do not budget for more than 250 Ms shipboard total mission time (a bit less than 2893 days of 86.4 ks each), measured from time of release from mission departure station to time of return.