The Athra is a capital ship like Enterprise and Voyager are capital ships. They go on long range missions to survey the dimensional sea and hopefully solve Lost Logia shaped problems with no support from Midchilda until they cruise all the way back.
Hm.
Classically, international treaties have defined "capital ships" as the largest and most formidable warship type then in existence; the notable exception was aircraft carriers not being regarded as capital ships until 1942, by which point Pearl Harbor and Midway had done a good job of proving the point that they were very effective, commensurate with or superior to a battleship of roughly comparable tonnage.
The fact that a ship goes on long range independent solo missions to survey the dimensional sea is not,
in itself, proof of capital ship status. Indeed, most navies would assign that responsibility to a relatively smaller class of specialist ship. There are good reasons to do this. First, it makes the survey ships cheaper, so you can afford to operate more of them and cover more territory on a fixed budget and fixed number of personnel. Second, optimizing a ship for extreme range, endurance, and mechanical reliability will typically come at the cost of having less room to cram military hardware aboard, because there are always tradeoffs in ship design. A ship optimized to win battles against hard-hitting opposition will not look quite like a ship optimized to explore strange new worlds (compare
HMS Victory to
HMS Beagle; this pattern has persisted for a long time). Third, survey ships will of necessity be very far away from any central location or base, and so cannot be recalled quickly in an emergency. The TSAB would presumably not want to be
completely unable to get its most expensive and largest class of warships concentrated to fight a pitched battle against some single threat to their heartland, creating a pressure to differentiate between survey ships and battleships.
...
Starfleet, of course, handles things a bit differently, but Starfleet is very, very
mildly military and likes to put its best foot forward in first contact situations by having its deep-space explorer ships like the
Enterprise-nil (TOS) and
Enterprise-D (TNG) be relatively well-founded and luxurious ships.
Especially by the TNG era. Most navies in the same position would have made the big
Galaxy-class starships more warlike and heavily equipped for combat (in case of the need to fight a major battle). For deep space exploration roles, they'd rely on something a lot smaller... appropriately something like the
Intrepid-class ships such as
Voyager, which is much smaller than the
Enterprise-D, has a much smaller crew complement, and as a happy side effect of being smaller, can do things like land on a planetary surface which the
Enterprise-D cannot.
Voyager is definitely not a "capital ship" in the normal sense. Note that it was never designed as a 'flagship' or 'go solve enormous problems' ship by itself. The part where the ship wasn't really supposed to be well suited for getting stranded in the Delta Quadrant and having to do that kind of thing by itself for years at a time was in fact kind of the point, as I recall.
...
Now, the ability to "solve Lost Logia problems without help"
does sound more like a capital ship mission... But then, Lost Logia problems are insanely diverse. Sometimes, having a good archaeologist or linguist on board would matter a lot more than having biggatons. And given how important Lost Logia problem can be, I'd expect
any TSAB ship to be under orders to try to examine
any Lost Logia problem and either try to solve it, scream for help, or both, regardless of that ship's tonnage or nominal fitness for purpose. If the ship can't handle it, well, that just moves their response towards the "scream for help" end of the spectrum; it doesn't mean they don't have a response at all.
...
So to sum up, I'd say that whether the
Arthra is a "capital ship" has little to do with whether it does long independent cruises (many navies had light ships classed as 'sloops' or 'cruisers' or 'frigates' for that role), or whether it tries to address Lost Logia problems it encounters, and everything to do with whether by TSAB standards it's a big chonky fighty ship.