In this episode some of the Enterprise's officers are stranded on a class M planet after their shuttle crashes. The fuel system is exhausted and they must find a new energy source in order to lift off and attempt to escape the planets vicious inhabitants.
Scotty settles on using the hand phasers as a power source. A close-up view of the shuttles weapons unit shows five hand phasers on board. With this new energy source the shuttle is able to lift off, despite some interference from the locals, and make orbit. Dialogue indicates that they then expected to use the engines continually for a further forty five minutes before exhausting them.
We have few details on this shuttle craft, but on hearing that it has been lost Kirk specifically states that it is 24 feet (7.31 metres) long. The scale diagram in the original series technical manual is not considered canon, but is a close match to the shuttle actually seen on screen in every detail; looking at this and canon sources, we can see that the shuttle craft is 4.4 metres wide and 2.5 metres high.
Given these dimensions, a shuttle can be expected to mass somewhere around 5,000 kg - compared to an average family car which masses somewhere between 600 and 1,200 kg. Assuming that the planet in question is a close match to Earth - indicated by the class M designation - orbital velocity would be approximately 7,600 m/s. The kinetic energy required to orbit the shuttle craft is thus :
K.E. = 0.5 x m x v2
= 0.5 x 5000 x 76002
= 1.444 x 1011 Joules
Which means that each phaser contributed a total of 2.888 x 1010 Joules.
Now in fact, the total energy of a phaser would be more than this. At least three of the weapons where fired during the episode, two of them firing sustained bursts of several seconds. Breaking free of the natives attack on takeoff would also require some energy. Running the engines for three quarters of an hour once orbit is achieved is another difficult proposition - it indicates that the shuttle actually orbited within the upper atmosphere, rather than out in space. How much energy this would require is open to question, as is the energy required to maintain the shuttles artificial gravity field, which operated throughout the flight.
Conservatively, I would guestimate that the total energy involved in this situation would be double the simple orbital energy figure. This would give a typical type 2 hand phaser a capacity of approximately 57.76 GigaJoules.
The endurance of a phaser is another unknown. We know the weapons can indeed run out of power; the best reference is in the episode 'The Omega Glory'. Captain Tracey talked of a battle with the Yangs in which they sacrificed hundreds just to draw his forces out. He says the Yangs then attacked in massive numbers, overwhelming his men. He tells Kirk that during the battle his forces exhausted four of their phasers, and nearly drained his own completely, in the process of killing 'thousands' of Yangs. Another section of dialogue indicates that many Yang bodies remained after the battle; this demonstrates that the phasers where not used at a particularly high setting, as these settings leave no remains.
In order to make the power figure for the phaser output as conservative as possible, I will assume that some 9,999 Yangs where killed by the five phasers - any more and Tracey would more correctly have said 'tens of thousands'. This would indicate that each phaser contributed some 2,499 deaths (rounded down to the nearest integer). The original series Technical Manual does not indicate any form of detailed control of the duration of firing; one must simply depress the trigger more quickly for a faster burst. The Khoms where not a very well disciplined force, and so we can expect longer than necessary bursts - one second of firing per death seems reasonable to me. This indicates that a phaser would be able to fire for a total of almost 2,500 seconds at this moderate power level - nearly 42 minutes. This gives us a figure for a medium kill setting of the TOS Type 2 phaser of 23 MegaWatts.
We can only guestimate the ratio of maximum power to levels which would kill yet leave a body. The TNG Tech manual gives an energy index for setting 16 (maximum) of 1.55 x 106, while setting 6 (the lowest setting which seems to guarantee a kill) has an energy index of 2,700 - a ratio of over 570 to 1. If this held true for TOS then the maximum phaser power output could reach as high as 13,000 MegaWatts. This would exhaust the power pack in 4.5 seconds of continual firing at maximum power.
There are factors which could push these numbers in either direction. As we are already assuming that the Khoms are not well versed in the use of a phaser, they are also likely to have missed many, or even most, of their targets - though it is presumably quite easy to aim a phaser which would have no recoil and a very visible beam. Even so, it is likely that misses were common and this would lower the energy of each shot. On the other hand, the assumption that 9,999 Yangs where killed is probably over-estimated by a good deal. Also, it may well be that Tracey had many more phasers at his command but was overrun before more than four where used up; certainly he had the opportunity to bring as many as he wished down from his ship. This would increase the power of the phaser, as each one would use its energy on a smaller number of targets.
The only number we can be sure of for the TOS type 2 phaser is that it's energy capacity is above 28.8 GigaJoules, and probably in the 60 GigaJoule range.
Using various guestimates, I would say that a low-level kill setting output of about 23 MegaWatts is a pretty solid number.
The maximum output is almost certainly lower than 13,000 MW, since surely some advances must have been made in the century or so between the this episode and the TNG era. A figure in the region of 500 - 1,000 MegaWatts seems likely. This would empty the energy cell in approximately one to two minutes.
This contrasts strongly with the TNG Tech Manuals assertion that the Type X phaser array, the most powerful in service on starships in the early TNG era, has a total output of only 1,020 MegaWatts. It is simply more proof, if proof where needed, that this figure is ludicrously low and seriously out of line with the bulk of the evidence.
Using the commonly stated conversion that a one megaton explosion releases 4 x 1015 Joules, then an overloading type 2 phaser would have an explosive effect equal to that of approximately 15 tons of TNT. This is certainly in line with the level of panic we see when phasers are set to overload in episodes such as 'The Conscience of the King', but personally I regard it as a somewhat shaky number - I doubt that the megaton-energy conversion is quite that linear over such a wide range!