So if I'm reading this right, then at warp six to collect one gram a day the scoop needs to be around 106 square meters... right?
So if that's correct, the field would need to be about 9,200km^2 to collect one gram per second.
Do I get a cookie yet?
Given the "maximum" expected levels of hydrogen, yeah, 106 m^2... which is basically about a 5.81 m radius circle (for 1g/day)
I would recommend using the "average" number, though... so at WF6, you'd be talking about 107,000 m^2, or a circle of radius 184.37 m, for 1g/day.
So, the amount you get per second is 1/86,400 of that, or the area you need for 1g/sec is 86,400 times that.
So, for 1g/sec at WF6, you need a circle of 9,226,569,600 m^2, or a radius of 54,193 m.
Divide that area by 1000^2, to get it in km. That's 9,227 km^2, or a radius of 54.2 km. So yeah, you pretty much have it straight.
Sorry, no cookies here, though... all out. Girlscouts skipped my place this year it seems....
My question is... why would you need 1g/sec?
Remember that a single gram of reaction mass produces a LOT of energy. It's pretty much nonsensical to assume you need that much mass for conversion.
It's also nonsensical to assume that you're going to "react" 100% of what you're collecting. A fair portion would be used to replenish lost mass (atmosphere bleed, etc) and, most likely, the majority will be diverted to be used as reaction-mass for newtonian propulsion devices (a little mass can go a long way, remember, if you can accelerate it sufficiently first, of course). Maybe... MAYBE... 20% would be converted in the m/am reaction process, I'd guess. (After all, that's still allowing you, under sustainable-operation conditions, to produce 188,324,000,000,000 joules of energy a day.)
Yeah, they could "retank" every so often, but I don't prefer that approach. I really like the advantage that m/am reaction for power generation provides... insofar as it makes the ship, potentially, self-sufficient. No need to stop at "Space Texaco" every few thousand parsecs...
So for me, I presume that the ship is able to collect significantly more matter than is required to power the ship's systems, under "normal" loading, indefinately.
Given the sheer power output available, and the limits of materials and energy as we understand them today... I really think that the ratio of "collected" to "consumed for power generation" is pretty large.
If I were creating the ship, myself, with as much "magic" technology as I wanted, I'd give it a 5 km^2 "projected circle" for the primary collection "cone, and I'd convert 20% in the m/am reaction (using about 10% of the power output from that, or 2% of the energy potential in the available mass supply, to drive the antimatter "flip" system). The rest, I'd divert into cryogenic slush tanks, to serve as thrust mass, fusion "auxiliary power" fuel, and even replenishment of lost mass (atmosphere bleed, etc).
Even "only" 20% of the output from that 5km^2 "scoop" would be a LOT of energy.