Something I remember noticing when reading 'The Expanse' novels was the very optimal paths between planets and regions. I don't remember any mention of having to travel around the sun to reach a planet currently on the far side of the system. Is my memory correct that the ships always take a direct path to their destinations?
The Epstein drive is a 'magically easy' source of thrust and delta-V. As such, things like a gravity assist or Hohmann transfer orbit are a thing in the Expanse universe, but navigators can use more power to avoid slower, more efficient solutions. I don't recall if maneuvering around the sun ever became an issue, but doing that with the gas giants in the outer system and their moons was mentioned around book 2.
There were off-hand comments of ships aiming to where a station or planet was going to be when they arrive, which acknowledges that it wasn't fast enough to go straight-line aim-for-the-planet.
'Slingshotting' is a very stupid and inefficient way to get from a-to-b in the Expanse universe. And like most things that are stupid and inefficient, there's no shortage of people willing to try it.
He’d figured out afterward that she’d brought him because Mila Sana, a horse-faced Martian girl a little younger than him, had a thing, and they all thought it was funny to watch the ugly inner girl padding around after the half-breed, but by then he didn’t care. He’d met Silvestari Campos and he’d heard of slingshotting.
Went like this: Some coyo put together a boat. Maybe it was salvage. Maybe it was fabbed. Probably at least some of it was stolen. Didn’t need to be much more than a torch drive, a crash couch, and enough air and water to get the job done. Then it was all about plotting the trajectory. Without an Epstein, torch drive burned pellets too fast to get anyone anywhere. At least not without help. The trick was to plot it so that the burn—and the best only ever used one burn—would put the ship through a gravity assist, suck up the velocity of a planet or moon, and head out as deep as the push would take them. Then figure out how to get back without getting dead. Whole thing got tracked by a double-encrypted black net as hard to break as anything that the Loca Greiga or Golden Bough had on offer. Maybe they ran it. It was illegal as hell, and somebody was taking the bets. Dangerous, which was the point. And then when you got back, everyone knew who you were. You could lounge around in the warehouse party and drink whatever you wanted and talk however you wanted and drape your hand on Evita Jung’s right tit and she wouldn’t even move it off.
In the Expanse, ships using the Epstein drive are often capable of continuous high thrust for months at a time. Thus many ships routinely don't plan paths that are "optimal" in terms of energy consumption. Rather they intend to be continuously thrusting at a decent fraction of 1g, to provide gravity for crew comfort. Thus they plan a trajectory that will allow them to reach their destination while accelerating the entire time (changing the direction of acceleration at some point so as to arrive at the destination at the desired speed as well as the desired position).
Such a trajectory is definitely not a straight line. It's a complex curved trajectory influenced by the Sun's gravity (and that of any planets you pass close enough to) as well as the acceleration from the ship's thrust over the course of the journey. But it's definitely much "straighter" than just coasting on an orbit.
On the scale of the entire solar system, the Sun is incredibly tiny. The odds of any desired "straightish" path needing to pass dangerously close to the Sun are actually pretty low, even for destinations that are on the other side of the solar system. And even if the shortest path would pass too close to the Sun, you don't really have to "go around it" in the sense of taking a large detour; just angle your initial path a handful of degrees off the direction you would have taken, and now you'll miss the sun by millions of kilometres. It probably makes the course calculations more complicated than just "thrust in one direction for half of the travel time, then turn around and thrust in the exact opposite direction for the remaining time", but that can be worked out. This deviation would turn a "straightish" line into a somewhat less straight line, and thus add a relatively small percentage to the travel time; not into a massive semi-circle that takes significantly longer to traverse. (Doing even a distant flyby of the Sun like this means countering all of the orbital energy of the object you departed from, and then regaining a comparable quantity of orbital energy at the other end, which is pretty massive amounts of energy; so maybe a more curved path going nowhere near the Sun is what you would get with up to 1g of acceleration anyway, and the need to detour would never come up)
So the fact that the trajectory is a little longer than "optimal" by keeping clear of a margin around the Sun just isn't that narratively significant. The fact that it's much longer because the planets are very far out of alignment might be more significant, but in general the Expanse books don't actually try to sketch out the actual planetary alignments (so far at least; I'm about halfway through book 3). They do mention journeys taking weeks or months, so I just always assumed that some of those journeys between points in the Belt were "cutting across" the inner system to a greater or lesser degree, and that was the reason they were taking many weeks.
It's also worth noting that most of the navigation (in the books I've read so far) takes place in the asteroid belt, or the outer planets of the solar system. Orbits in the asteroid belt take multiple years, so if two objects are relatively close together it's going to take many years before they're on the opposite side of the solar system (and vice versa). So over the time scale of a single book, these distances just form the geography that navigators are working with; they've been vaguely similar for a long time, and will stay vaguely similar for a long time, so the specifics wouldn't be hugely noteworthy.
As you go further out the orbits get even longer. The ring is said to be beyond Uranus; Uranus' orbit takes 84 years. The ring is essentially going to be "on the same side of the system" for decades.
At the start of Abbadon's Gate Holden was trying to take a job that would send him to the opposite side of the solar system from the gate, so the books do mention the concept of being on the far side of the system from some other destination. But generally it's the travel time the books talk about, not the orbital positions that cause the specific travel times.
I just stumbled across specifically needing to get somewhere on the opposite side of the sun.
I've read the entire series and Memory's Legion and am reading thru a second time. Much better on second read! I'm going to openly discuss what happens in book 5 Nemesis Games so stop now if you don't want spoiler.
Nemesis Games chapter 34
A Mars ship group is carrying Mars Prime Minister Smith to Luna for a big pow-wow between Avasarala, Fred Johnson and Smith. At this time Earth/Luna are opposite the Sun from Mars PM ship's present position. The PM ship's escort group is drawn away by a clever ploy and a group of 6 other Martian warships 'conveniently near enough' are called on fill in as the PM's escort. But these other ships are in fact manned by Marco Inaros' belters and are tasked with boarding and taking control of the Prime Minister's ship. By chance, also aboard with PM Smith are Alex Kamal and Bobbie Draper with the racing pinnace Razorback stowed in a bay.
At end of Chapter 34... With the PM's ship boarded, Alex, Draper and Smith are put onto the Razorback in a last ditch effort to get Draper and PM Smith to Luna. Alex discusses the fact that the 'best course' is inside orbit of Mercury but Razorback is rated to only 1/2 AU from Sun's coronal surface. He then considers using Venus for slingshot but Venus is not in a location to be helpful. However, Venus slingshot might be useful in getting Earth military ships out to help protect Razorback so Alex sets course to Venus.