My car can go about 400 miles before it needs gas.
How far or long can Star Trek ships go between 'fill-ups'? I seem to recall a couple times where the dilithium needed to be recrystallized, or they needed it or another element for some power system. I thought dilithium wasn't really the source of energy (e.g. gasoline) but just what kept the reaction running smoothly (oil).
It depends on the vessel and what they are "Filling Up" on. If it is just fuel then it depends on the amount of space on the ship is dedicated to storing deuterium and anti-deuterium. they both fuel to Warp drive, and the deuterium is used in the fusion reactors for the Impulse Drives and as a general power source. The nacelles gather ambient hydrogen and deuterium through the bussard collectors while in flight, so as long as the ship has power and is in flight they should be able to generate deuterium from the surrounding stellar material. And from there they could produce anti-deuterium using power from the fusion reactors. Of course there would be diminishing returns as the power needed to gather the material and convert it to anti-deuterium is greater than the amount of deuterium they could gather out side of mining a large source (nebula, ice asteroids, ect). Not only that there are consumables related to crew, food, water, air, etc. that effect how long between resupplies the ship will support.
But, continuing the metaphor, your car does not need just fuel to go, it has other consumables such as oil, coolant, transmission fluid, wiper fluid, break pads, etc. While they do run out or go bad it is at a much lower rate. Dilithium could be put in this category as the crystals eventually break, de-energize, or otherwise are overtaxed. Also, on the Intrepid class vessels they introduced gel-packs as a replacement for isolinear chips, though more robust (self repairing, non-rigid construction) and capable (more paralleled) they are much more difficult to manufacture, pretty much impossible on a vessel. And beyond that you have all the coils, conduits, plating, etc. that break down and need to be replaced in the course of normal operations, not to mention damage taken from hazards.
So there seems to be 3 things you could be talking about: Resupply - Topping off on fuel and consumables Minor Refit - The equivalent to an oil change and tune up, replacing dilithium, cycling new coolant, replacing gel-packs, minor repairs Major Refit - Overhauling, changing capabilities, upgrading, major repairs.
Depending on the class of vessel and mission you will have different intervals of resupply and refit. These are ballparks I am creating from memory and a hanfull of disagreeing web sites, as I don't have any tech manuals handy.
Although dilithium isn't technically the "fuel" (deuterium and anti-deuterium are), it is so critical to producing usable power, that due to the fact that it crystalizes in use means it eventually "runs out" (stops being able to regulate the matter/anti-matter reaction).
In Star Trek IV: The Voyage Home, Scotty and Spock find a way to re-crystalize dilithium, by the 24th century this could be achieved while it was still in the dilithium chamber.
In light of this, the limits of replenishment have moved to mainly to anti-deuterium reserves.
Depend voyager have a antimatter generator like galaxy class an then if they can find enough raw material that can use replicators for most of thing but have to find with bussard collector the gas elements to recreate deuterium for the core and fusion reactors of impulse engine ,but because space isen't empty they can cross other species that can share resource in exchange of something,for ships without aniltimater generator its must go to a star base to resupply with antimatter to fill it's pods ,than the autonomy of a ship depends on the speed of use of antimatter in relation of its reserve of it and the deuterium for the fusion reactor as emergency power for the system when the core go offline
It seems to have more to do with "whenever the writers need to use the 'we need to refuel' trope" than it has to do with either physics or even consistency within the canon. While us geeks tend to want to believe that our science fiction worlds are (or even should be) consistent within themselves, it's more common for writers to not care because they'd rather not bother with that kind of work, especially when you have deadlines for TV episodes. More often than not, they'd rather just work with cliches because they're easy shorthand.
For reference, modern nuclear submarines never need to be refuelled for the lifetime of the ship, which is typically about 25 years. In the Star Trek universe, the reactors on their ships run on matter-antimatter reactions, which is basically the most power-dense fuel physically possible. While I'm sure that warp travel requires truly astronomical amounts of power, a "mere" 100 kg of antimatter[1] would provide it for years on end. It's not a huge leap of faith to assume that Starfleet engineers would be capable of ensuring that at least a design-lifetime's supply antimatter goes along with every ship. It's not exactly mass-constrained (meaning the amount of mass you have to put on the ship exceeds the mass the ship can carry) like diesel fuel is.
No, I'd be more concerned about feeding the crew. Except for the fact that this is also taken care of by replicators, which are so essential to the Star Trek universe that they are single-handedly responsible for the need to have any kind of economy at all. My understanding is that these replicators also operate indefinitely, although not without maintenance.
[1] For what it's worth however, creating antimatter (it's not exactly something that occurs naturally, or if it does, it's extremely rare in the universe) in itself requires an astronomical amount of power. But if you can make enough to run starships back and forth at all, all it would take is extra time to amass enough to run them for their entire design lifetime.
Based on the tng technical manual mostly from the graph on page 55, the power vs speed vs fuel usage can be calculated.
warp field mechanics dictate that fields are made up of layers. The faster you go the more layers you need and the harder it becomes to maintain each one. (per layer (cochrane), Exp 3 for warp 1 and Exp 8 for warp 9, in millions of joules).
At warp 6, at 390 cochranes at 1 EXP 6.6 mega joules per cochrane is 1.55 Quadrillion Joules. At warp 9, at 1516 cochranes, at 1 EXP 8 per cochrane, is 151.6 Quadrillion Joules.
Assuming joules are watts, a ship travelling at warp 9 would burn 3.3kg of fuel a second (at 51% coil efficiency, Energy of 302 EXP 15, divided by C squared is 3.3 approx) and at warp 6 would burn 0.0172kg a second (at close to 100% coil efficiency).
Warp Energy physics also dictactes that the ratio of matter to antimatter changes with warp factor. 10:1 for warp 1, and 1:1 at warp 8 and above.
with a standard cruising speed of warp 6 for 3 years, a ship would go through about 1.63 million kilograms of fuel with about 1/5 being antimatter (a 4:1 matter-antimatter ratio for warp 6). with 326,000 kilos of antimatter and a ratio of 1:1 a ship would burn through this at warp 9 in just 54.9 hours or 2 days.
so 3 years down to 2 days, depending on how fast you are going, and what region of space you are in. (subspace instabilities cost more energy to travel around as the warp field is much harder to maintain). the antimatter generator is for emergency use only (usually if you've encountered an anomaly that has neutralized the anti-matter) and would only provide enough to limp to the nearest starbase, a few kilos at most.
This also explains that the klingon sleeper ship would have been travelling at a low warp factor. Travel times between close stars taking decades requiring stasis, instead of the few days it would take a tng-era ship.
As in real life, speed for a starship comes at the cost of fuel and fuel efficiency.
