Between aux power, pushing the engines, and a solar wind that seems to push the ship along, they take less than a day to get to the area (though the engines will definitely want some time in port to retune them before getting into any sort of combat, and the paint job seems to be a little singed with solar-flare particles).
From the detritus of the battle, it seems to be the case that the Starguard was, in fact, trying to target engines rather than going for hull breaches. Unfortunately, while this did result in disabled ships in some cases, the engines are not really the safest things on a ship to be taking damage, and this sometimes resulted in a lot of radiation damage.
Ace, Katya, Sharra, and Max are definitely snooping around with the Diversity looking for Vance if Max thinks he's out there.When the Hippocrates takes off, Max thinks that the path towards Vance is "off planet, towards space." As you head out to the remnants of the battle, the path fades and becomes less clear - once you're out there, there doesn't seem to be a path to a derelict, or another ship, that finds Vance.
Ace, Katya, Sharra, and Max head around in the Diversity looking for survivors and trying to get a sense of Vance from other places; Max does end up finding a few not-really-a-path paths that lead them to pick up several bubble-pods from the Paramount, a Blue marine carrier. The bubble-pods are individual emergency lifepods, highly insulated and with a short-term life support; the two marines are both badly singed from some sort of explosion, but not dying. They initially try to stick to "name, rank, and serial number" but are persuaded by Sharra to give a few more details - they're from Vance's marine squad. The Paramount was assigned to a boarding action, but got shot way before they were expecting to draw fire. The ship was breached, and one of the forward torpedo mounts exploded. Neither of them is sure where exactly Vance was, compared to the torpedo mount, though they assure you he would have had a personal bubble-pod too.
As you report in to the Starguard (you've been asked to keep them informed of whoever you pick up) you learn that several of the other ships on rescue duty have picked up Paramount personnel, but no sign of Vance.
Jim gets sent on dangerous solo missions in the fighter ("thread your way through this field of unexploded munitions to retrieve the escape pods").No mine fields hanging around, but there is one heavy cruiser which has had its launching tubes slagged just as it was launching torpedoes; sensor ops indicate that 1) the torpedoes are still live, and could in fact go off in the launching tubes if the ship is badly jostled, and 2) there's a group of survivors on board cut off from the escape pods and airlocks, back near the engine core, which is itself getting more and more unstable. Jim parks the fighter a short ways from the cruiser, leaps across to it, and makes his way across the outside of the ship to the engines. He makes his way around the outside of the plasma manifolds just as the engines start to completely destabilize. He cuts a couple of manifold jets loose with his utility knife, dumping the core to space. His hands, caught in the plasma exhaust, seem strangely numb and unresponsive through the space suit after this. The five crew members are led out *through* the engines (in space suits), and then towed back to the Hippocrates. Shortly after everyone is rescued, the cruiser explodes, in proper dramatic form.
All five, but especially Jim, are immediately leapt upon by medical personnel once they get back to the Hippocrates; Jim's hands have bone-deep burns, and seem to be somewhat stuck to the inside of his suit. Eww.
The other person who I have a specific request for is Twig. I want him to squeeze everything he can out of the life support systems, so we can carry as many people as possible.He takes that as a challenge (though Ciernan proves quite useful in that regard, much to Twig's chagrin). Between the two of them, they say you will run out of space before you run out of life support.
Oh, and we'll ask Ruehan to just check to ensure that none of the doctors we invite on board has bitey shields.None seem to.
As the day wears on, the Hippocrates, with its medical tech and high doctor carrying capacity, ends up being something of a clearinghouse for cases being triaged from smaller rescue ships. This means there's less first-round rescuing being done, and a lot more surgery.
Katya and Martan start working as a transfer team, helping get the wounded from a docked rescue vessel into the Hippocrates with as little jouncing and fuss as possible. Kith makes gallons of tea and spends a lot of time cycling between different cases, adding hit points here and calming hysterics there. Those with first aid or medicine are put to work in the surgeries; those without pass out med patches and packs and stabilizers and Emerald until the Hippocrates is out, and help make sandwiches after that.
Grandfather Cass arrives on ship (having taken a slower transport out to the area), and generates placebo-effect medpatches and tea once the supplies run out.
In the end, nearly everyone brought aboard is saveable with conventional medical technology, or is already dead by the time they are found. The one set of cases who are not are the five engineers from the Elysium. The Elysium was another Blue marine carrier ship, which took an engine shot to the starboard maneuvering thruster, basically locking it on extra-full-blast. The ship started spiraling out of control towards the jump gate; these five engineers went into the inferno of the engine room and got the engine under control again. However, they've all taken far beyond a fatal dose of radiation, and they have a life expectancy of about a week, tops. Kith can devote herself full time to fixing one before he dies, or two if she's lucky.
If you are looking for volunteers for Kaufman treatments, the above would likely qualify; other possibilities include a Blue soldier and Red rescue offficer (who saved the Blue guy) who have been blinded by radiation.