This episode features the discovery of a Dyson sphere by the Enterprise-D. During the course of the episode the Enterprise is pulled inside the sphere by a set of tractor beams. The following dialogue ensues :
Ensign : 'We've lost main power, auxiliary power down to 20%.'
Worf : 'We are being pulled inside.'
Ensign : 'Auxiliary power failing.'
Data 'The resonance frequency of the tractor beams is incompatible with our power systems. Warp and impulse engine relays have been overloaded. I am attempting to compensate.'
Ensign : 'The tractor beams have released us sir.'
Riker : 'Hold position here until we can get our bearing.'
Picard : 'Full sensor sweep Mister Data. Where are we?'
Data : 'Approximately 90 million km from the stars photosphere. I am reading a great deal of surface instability. It may be-'
Ensign : 'Sir. The inertial motion of the tractor beams is still carrying us forward. Impulse engines are offline and I can't stop our momentum. We're falling directly into the star.'
In a subsequent scene the E-D crew manage to force the ship into orbit with the manoeuvring thrusters.
Ensign : 'We're in orbit Captain. Our altitude is 150,000 km.'
Riker : 'I'll see about getting main power back on line.'
And later, when Data reports a large flare directly ahead of the ship :
Picard : 'Shields.'
Worf : 'Shields are up. But only at 23%.'
Data : 'The star has entered a period of increased activity. Sensors indicate that the solar flares will continue to grow. In 3 hrs our shields will no longer be sufficient to protect us sir.'
The Dyson sphere was initially stated to have a dimaeter of 200 million kilometres - approximately 67% that of Earth orbit around our own sun. Assuming for the moment that the sphere was sized so that the interior surface recieved approximately the same amount of energy as our own planet does from Sol, about 1,300 Wm-2, this would give us a total output for the star of 1.6 x 1026 Watts.
The precise meaning of the altitude reference is crucial - would the 150,000 km figure be measured from the centre, or from the surface of the sun?
Assuming the former, we can calculate the power intensity incident on the Enterprise-D as :
Ei = 1.6 x 1026 / (4 x pi x (1.5 x 108) 2)
= 1.6 x 1026 / 2.83 x 1017
= 5.66 x 108 Wm-2
The area of the shields is not precisely known; however, they are approximately ellipsoid and measure some 750 x 250 metres when seen from the side. The ship should thus intercept approximately 147,281.25 m2. Therefore the total power on the ships shields would be :
P = 5.66 x 108 x 147,281.25
= 8.336 x 1013 Watts
= 83.36 TeraWatts
So in three hours, the shields would need to absorb a total of :
E = 8.336 x 1013 x 3600 x 3
= 9 x 1017 Joules
Or about 900,000 TeraJoules. And this is from damaged shielding - full shields would presumably have vastly greater capacity than this.
Alternately, if the altitude is measured from the centre of the sun then this would add anything up to another 400,000 km or so to the true figure. This would bring the capacity of the low power shields down to 66,940 TeraJoules.
During the episode it was revealed that the sphere had been abandoned because the star within it had become unstable, periodically emitting large amounts of radiation and mass. Datas dialogue revealed that the star had begun to enter such a phase as the Enterprise went into orbit - 'Sensors indicate that the solar flares will continue to grow', is how he put it. Hence, it seems that the average power output of the star was going to increase considerably during the next three hours - a factor which could push the shield numbers up by almost any amount. We also do not know the initial conditions within the sphere - it may well have been that the natives preferred environment was a much brighter and hotter one than class M, or much darker and colder for that matter.
Put simply, this episode proves only one thing beyond any possible doubt : against wideband electromagnetic input, the badly damaged shields of the Enterprise-D can withstand an absolute minimum of tens of thousands of terajoules and quite possibly a good deal more.
Assuming that shield capacity is directly proportional to the strength figure Data quoted, then the maximum capacity for the shields would be :
E = (100/23) x 9 x 1017 Joules = 3,913,000 TeraJoules.
But of course, this is yet another loaded assumption because Data could have meant that the shields had 23% of their normal power running through them, or 23% of their maximum energy absorbtion rate, or... you get the idea.
During my own speculations about the strength of Starship shielding, I assigned the Galaxy class a shield capacity of 5,400,000 TJ; this is some 38% above the top figure generated by 'Relics', which seems well within the range which can be accounted for by increased solar activity. On the other hand, it is an eighteenfold increase over the lower end of the range, which is still certainly feasible but less likely.
Of course, 2369, while the shielding figures on my pages are for 2374/5. During this time Starfleet has encountered the Dominion, and apparently most or all starships and starbases have had their shields modified to resist Dominion attack; metaphasic shielding was also invented during this interval. Between them these three factors could easily account for a considerable increase in shielding levels on the Galaxy class.