Okay so, shit... Kat has a number of ships that could be considered flagships. The thing is, she chooses ships based on where they will be operating and what she will be doing. Furthermore, whatever ship she is on at the time is considered a 'flagship' by very definition. As a Vice-Admiral of I-RED, whatever ship she's on is the one carrying the VADM's flag... thus, it is a flagship.
So now we're left with the question of... since any ship she's on is technically a flagship... which one does she prefer to use? Well that's a complicated question, one that occupies far too much of my time when preparing for any IC RP situation. (OOC PvP ship selection is very straightforward)
There are multiple criteria that are weighed for a given situation. The first is deployment time. How long will Katrina be deployed in this vessel? This is important, as ship hulls do not have indefinite supplies for the crew, and many ships are designed without crew comforts in mind. While OOCly, ships have no operational limits, IC I tend to follow lore.
Frigate - For solo deployments with minimal or no crew. Often lasting an hour or so, but with the capability to stay longer due to the minimal crew supply requirements. Often used for situations where there is little prep-time and the ship must be launched and on site as soon as possible, without knowing how long the deployment will last.
Destroyer - For patrols lasting longer than one day but no more than a week. Since the ship is cramped on the inside and supplies are strictly rationed for the crew, destroyers cannot tolerate long term deployments without resupply. Used on occasions where arriving on site is needed quickly, but engagements with other capsuleers is not expected.
Cruiser - For deployments of varying types lasting longer than one week. Since cruisers are relatively comfortable and designed for long term deployments without resupply, these ships are the baseline for most deployments. Capable of keeping a crew comfortable well past Katrina's own time limits for being cooped up in the capsule. This is the level at which crew endurance surpasses her own.
Battlecruiser - For longer term mobile deployments where firepower and combat endurance is more important and a show of force is necessary. Battlecruisers are even more spacious than cruisers by their nature. Similar to destroyers in that their focus is on combat more than comfort, they happen to be far more capable of keeping the crew happy with multiple shift rotations. Since battlecruisers can spend years out in space without docking, there is no real need to resupply at all.
Battleship - For immobile deployments where Katrina's presence in one specific area is required with the ability to land troops or equipment to planets on a large scale. They are not used in situations where she will be expected to pack up and move quickly. Battleships are used as a small mobile base of operations, and come equipped with the crew endurance and extra personnel to fill that role nicely. While unable to supply other ships, it is almost completely self sufficient and provides a method for Katrina's operations to continue with minimal reliance on stationary structures.
Capitals - As expected, capitals are literally a base of operations. This includes the flexible nature of an Orca, the transport capacity of a freighter, or the pure combat reliability of a carrier. Rarely deployed, very expensive, but with phenomenal operational usefulness.Now it's worth noting that where she will be deployed often influences what racial variant she'll be using. This is due to logistical reasons as well as 'blending in'. It's easier to find cheaper and higher quality supplies or spare parts for a Tempest in Metropolis than it would be to find the same for a Raven.
Every ship has a use, and it really depends on what the situation is to determine which ship gets used. This is all roleplay though, and if I need something for OOC reasons, I'll pull it out of the bag and simply use it regardless of what IC restrictions it ought to have.
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