As I see it, the question of determining the phaser power of a GCS is tricky for several reasons. Phasers are unusual weapons that have been known to do fairly unusual things, which complicates the question of yield. You'll hear talk about
nadion effects and the like - so before we do any meaningful analysis, we have to first address the common myth that phaser yield and phaser effect are completely separate.
On high settings, a hand phaser typically disintegrates a human. From
"The Galileo Seven", we know that seven TOS era hand phasers have enough juice to lift an extra 150 pounds clear off a planet to escape it before it blows up. That's 600 megajoules right there - if you have 100% energy efficiency in liftoff, which is simply not going to happen. Realistically, since shuttles use thrusters, this points to multi-gigajoule energy capacities. The fact that overloading phasers are a major hazard, the sustained long-term discharge of 1.05 megajoules per second in "The Mind's Eye," even the fact that Data uses micro-fusion cells are all quite compatible with this.
So then we want to look at phaser
effect. TOS era hand phasers, on high settings, tend to make people completely disappear. The energy required to vaporize someone is around 100 megajoules. Truly spectacular hand phaser shots in TNG have gigajoule-range energy effects, but TNG phasers should have much better batteries, too. From this and similar evidence, we can conclude that the
power used by a phaser and its
apparent effect (including "phaser" vaporizations) should be very close to each other in terms of energy.
That established, we can now look at the actual
yield of a GCS phaser bank. We have several ways of getting at this problem - four, in fact. To my eye, the highest figures also come out of the strongest methods.
Power Generation
From a conservation of energy standpoint, the fact that a GCS can exit a system very quickly (as in "Relics") from a cold start indicates that a GCS has a very high rate of peak power generation. Even warp-speed displacement within a gravity field (as in "Descent") requires hundreds, possibly thousands, of exawatts.
In terms of quotations, "Deja Q" gives us our only realistic specific power output for a GCS - 12.75 exawatts when the warp core
isn't being pushed. There are occasional references to terawatt-range power outputs, but since a terawatt isn't enough energy to get something the size of a GCS out of Earth orbit very quickly, we pretty much have to throw those out. There are also a few references to petawatt/exawatt range figures in Voyager (such as
the quotation being disputed here), but "Deja Q" is the important one for our purposes. To make a long story short, we have to conclude that GCS can pull hundreds of gigatons of energy per second through the warp core.
If we assume that 1-10% of maximum power can be routed to phasers - which seems reasonable - we conclude phasers should put out multiple gigatons per second.
Phaser effects on inert bodies
More directly, we have several events in which phasers are used to drill rock, such as "Inheritance" and "Legacy." (See
here for the most recent discussion we had on the topic here on SFJ).
Here is the (brief) discussion of the TNG phaser drilling incidents on the main website. These incidents involve disintegrating a lot of rock. The quick summary is that a gigaton of phaser energy a second is about enough to disintegrate one seventh of one cubic kilometer of rock per second - a rate of material disappearance that fits pretty reasonably for all four rock-related incidents in TNG, from the E-D's presumed ability to blast its way out of the asteroid in "The Pegasus" on up to the sustained drilling into the mantle of "Inheritance."
There's also a comet in "Masks." Here, we really aren't talking about as much energy, but the E-D is firing its phasers at an explicitly stated 10% rate. As Mike DiCenso indicated above, "Masks" involved only a short duration phaser sweep, and the comet should be several cubic kilometers. The estimate on the main website is a bit higher, since it assumes the comet was vaporized and then superheated, but this also suggests around gigaton per second firepower. Here's the thing with "Masks," though: While vaporizing the comet is a lower-energy event than the phaser drilling cases, it is unique in explicitly indicating the task was accomplished with 10% phaser power.
We also have one incident in TOS - "
The Paradise Syndrome" - in which CCS phasers are used to blast something made of rock. There are a number of problems with the incident, such as the fact that the rock in question is supposed to be half the size of Earth's moon, but isn't remotely near spherical, but the stated goal of fracturing an extinction-event asteroid, and the literal documentarian reading of what "actually" happened (melting a 70 km patch on a strangely non-symmetric large asteroid), point to high energy levels. Since GCS phasers should be stronger than CCS phasers, this actually points towards stronger than GT/sec phasers on a GCS, but it's in there.
These are fairly flexible incidents on the whole. If you're willing to wave a hand over "Inheritance" and say that it's not drilling nearly as deeply as the Okudagram suggests, and minimize "The Paradise Syndrome" by shrinking the asteroid to something that we wouldn't be surprised to be non-spherical, you can justify anything from hundreds of megatons per second to tens of gigatons per second.
Comparison with photon torpedoes/General firepower
There's a major problem with trying to compare phasers with photon torpedoes: There's no actual agreement on which should produce a greater fraction of total firepower.
We do know that total firepower of a CCS is enough to completely obliterate a planetary civilization, but we don't actually know how phasers and photon torpedoes compare. We know that they tend to think of photon torpedoes for blasting asteroids, for example ("Rise" and "Pegasus" both demonstrate that thinking), and we know that for the
NX-01, phase cannons were weaker than photonic torpedoes.
However, that doesn't tell us that phasers are weaker or stronger than photon torpedoes on the whole. Photon torpedoes almost certainly deliver a faster
pulse of energy, and a different
kind of energy; phasers do the whole "magic disintegration" number, while photon torpedoes blow things up conventionally.
Since photon torpedoes themselves should have a maximum yield falling around 100 MT - 1 GT, and can also be fired at substantially lower yields, with a GCS easily being able to fire an average of a torpedo a second, we could
suggest that phasers are 10 MT/s-10 GT/s - somewhere within an order of magnitude of the raw power of the torpedoes.
General Order 24 strongly suggests going no lower even if you're minded to be minimalist, since even the updated CCS doesn't carry all that many photon torpedoes (only enough to obliterate a hundred cities or so), meaning that the destruction of an advanced planetary civilization can be carried out largely with CCS phasers in a fairly short timeframe.
I'm of the
opinion phasers are the main weapon of most Federation ships, with torpedoes being used for special effect. This explains why something the size of a GCS only carries a couple hundred of them, to me, and why we don't see the use of torpedo-boat style warships; however, others carry the opposite opinion, and there's not really anything in the canon to make sure of either.
Hull damage
One of the most curious cases of calculation is trying to compute phaser yield from actual hull damage. We know that GCS hulls are mainly tritanium, and in "Star Trek: Generations," we get a unique close look at the damage done by a phaser strike when shields aren't helping - each bolt that land puts a gaping hole across several decks, as seen in screencaps (see
here for screenies) and indicated in dialogue. It's worth noting that with the shields useless, the phasers actually did a lot more damage than the photon torpedoes - photon torpedoes might be more useful for knocking down shields, but they did very little to the GCS hull.
Now we have to know how thick the tritanium cladding on the outer hull is. Let's say about 30 cm, for the moment. Each blast is stripping away several hundred square meters of tritanium hull and then damaging the fringes, so call it 100 cubic meters of tritanium phaser-"vaporized" per blast - it's peeling/blasting away a little more than that, but that seems to be about how much is actually disintegrated (the photon torpedoes do a little less hull damage, we might note).
That leaves us with one more piece of information: How tough is tritanium? This is the real unknown. This is what we do know, from "Obsession" and "Arsenal of Freedom." Tritanium is 21.4 times as hard as diamond. Tritanium can be heated to 12,000 degrees Celsius without showing any sign of melting. Even thin tritanium plates and bulkheads are basically invulnerable to UFP hand phasers, which regularly emit blasts strong enough to disintegrate humans in a beam about a square centimeter.
Now, we would expect about 50 gigajoules of phaser energy to disintegrate a cubic meter of iron. If tritanium is only ten times as hard to disintegrate as iron, as we might guess from the temperatures we've seen tritanium heated to, then we're looking at a pretty low end of about ten kilotons per blast.
On the high end, however, we've
heard Yar standing there with a hand phaser looking at a partially melted plate of tritanium and say that UFP weapons can't do that. That phaser that she's packing can put 100 megajoules of disintegration energy in a beam a centimeter wide, and yet can't even
melt - let alone disintegrate - a tritanium plate a few centimeters thick? If 100 MJ/cm^2 were sufficient to punch a hole through several centimeters of tritanium, disintegrating several hundred square meters of tritanium cladding several tens of centimeters thick would require close to a gigaton. And that's an outdated D12, not a GCS.
Really, though, since we don't know much about tritanium, we
can't rule out much on hull damage, but the way that phasers destroy things (efficient disintegration with minimal collateral damage) and the incredible durability of tritanium against hand phasers lines up neatly with the gigaton figures.
