"Masks"
In this episode we see Enterprise-D melting comet.
To make our comet boil, we need to do following things:
1) Raise temperature of ice to melting point (0°C or 273 K)
2) Melt ice ( E = 333.55 kJ/kg or 333 550 J/kg)
3) Raise temperature of water to boiling point (373 K)
4) Make water boil (2257 kJ/kg or 2 257 000 J/kg)
Heat capacity:
ice at -10°C : 1.938 J/(cm^3 x K)
water at 25 ° C : 1.938 J/(cm^3 x K)
water at 100 ° C : 4.2160 J/(cm^3 x K)
After ice/water have reached sufficient temperature to melt/boil (0°C for melting, 100°C for boiling), you need to add additional energy for melting/boiling to actually occur. To melt ice, that energy is 333 550 J/kg; to boil water, you need 2 257 000 J/kg. Density of water at 0°C is 999.8395 kg/m^3; density at 100°C is 958.4 kg/m^3.
Mass of comet is around 3.868474 x 10e13 kg.
So for step one we need 1.1394x10e19 J.
Step two requires 1.29x10e19 J.
Step three requires 8.17836 x 10e18 J.
Step four requires 8.73 x 10e19 J.
So energy required in total is 1.197838 x 10e20 J. Gigaton is 4.184 x 10e18 J. So total energy delivered is around 28.629 Gt over course of 10 seconds, on 10% power. Actually, it would be more like 25 Gt due to object inside, or 2.5 Gt per second at 10% power. NDF, if applied, might lower actual DET value somwhat (to around 0.25 Gt per second at 10% power).
Other examples
In "Who Watches the Watchers" Riker considered a 4.2 Gigawatt generator sufficient to power "a small phaser bank". In "The Mind's Eye" Data says a Type 3 phaser rifle was using energy at a rate of "one point oh five megajoules per second". In "A Matter of Time" Geordi cautions that it will be difficult to control the ship's phasers to within 0.06 Terawatts (60 Gigawatts). That would seem to confirm that shipboard phasers are in Terawatt range at least, possibly even in Petawatt range given that we're talking about 24th century.
Also, here we have very interesting theory about phaser output, by Graham Kennedy.
"So 5 GW hand phasers are reasonable, and even 73 GW is not impossible. "
This is my conclusion for Type-2 hand phasers output. If we reduce that to 50 MW, Type 10 would come as 5 PW. If my conclusion is correct, then Type-10 phaser would have output of 500 PW, or 0.119 Gt per second. If my Masks calculations are good, output would rise to 1.046 x 10e20 W or 10.46 exawatts.
During TNG, Enterprise D drilled hole 50 meters wide at minimum and 2800 km deep in 14 seconds. Phasers were specially modified to prevent tecton instabilities and earthquakes which would be caused by normal phaser beam. That is 5.5 billion cubic meters of rock melted. Granite has density of 2.7 g/cm^3 average. That gives us 1.48 x 10e16 grams of granite melted. Granite has heat capacity of 0.79 (kJ/kg K). Granite melts at 1215 to 1260 ° C. So delta t=1200 K. Energy is 1.4 x 10e19 J over 14 seconds or 1 x 10e18 J per second. That is 2.39 gigatons per second.
Conclusion: Phasers are 2.39 Gt per second minimum
Also, term phaser originally comes from photon maser, which suggests that phaser is primarly particle weapon using photons (hand phasers were described as particle weapons on several occasions), with NDF/chain reaction as optional extra (althought we don't know that for certain).
Generations incident
D 12 was able to destroy 1 058 968 052 to 2 117 936 103 cubic centimeters of Enterprise-D's hull with each shot, depending if only outer hull was damaged or inner too. Actual value is between these two. I will take that shots heated hull 15 000 K. So we have 4.76 x 10e18 to 1.143 x 10e19 joules per shot for our outdated D-12. That is 1.138 to 2.73 gigatons per shot.
Note: Special thanks to Starfleet Jedi forum for providing screenshot showing size of comet in relation to size of object buried iside (episode Masks).
22nd century phaser cannons
Designed as a starship-based version of the hand-held phase pistol, the phase cannon was rated for a maximum power output of 500 gigajoules (given occasional confusion of basic physical terms by Star Trek writers - which only worsened over time - it is quite possible that actual output is 500 GW, and not 500 GJ - for that, however, we have no confirmation. Still, given that shots from these cannons generally last for about 1 second, it is essentially same).
UPDATE: Output of warp core is calculated at 5.84x10e24 W. If phasers use 10% of core output, then we are talking about 5.84x10e23 W or 139.579 teratons per second as high end.
On other note, 4.2 GW is enough to power small phaser bank, presumably consisting of single emitter. I will take that to be shuttlepod emitter. Now, GCS emitters are certainly larger than those on shuttlepod, and I will take them as being 8 times more powerful.
Emitters are in two rows. 12.82 meters long segment has 9 emitters in one row.
Array is incomplete ellipse, with long axis being 321 and short one 214 meters. Array itself is 7/8 of length of ellipse, which gives array length of 749.875 meters. So total count of emitters is 526 in one row, and 1052 total. So total energy output of GCS main array is 35 347 GW high and 4 418 GW low end (low end is taking GCS emitters as being equal in power to those on shuttlepod - GCS certainly does not use "small" emitters, but it is useful as lower end).
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