Two hundred tons of hydrogen is good for up to 54x10^18 joules. A ten square kilometer jumpsail (bigger than any described in the canon) at 10 AU from the Sun can collect 1x10^15 joules in the week taken to charge the core.
You missed the point of what I was saying. Let's try this again. I'll repost that remark so that we don't have to look at it again.
No, it won't, because FASA physics is in play. It takes approximately 10 burn-days to charge a jump. Again taking the Merchant as an example, this is nearly 200 tons of fuel. The sail, meanwhile, is only 46 tons. There's no false economy here - the sails are a more efficient and cost effective solution.
To expand on that, I'm not aware of a problem with your math and I'm not challenging the actual science here. The problem is with your starting assumption that the formulas you're using accurately describe Reality According To FASA. They don't. This is why I said that this time around the problem isn't the fact that BattleTech economics are apparently made to make sense to Cthulhu, it's the fact that this is one of those times and places FASA's physics simply doesn't line up with reality. In this one case there's compelling economic logic to this decision.
If you open StratOps and turn to page 88, you'll see the following: "Each power plant-based charging attempt of the drive requires 10 burn-days of fuel[,]" which is modified by margins of success or failure on a control roll on that same page. It may go as low as 4 burn-days or if you truly screw up, it can go a lot higher, but 10 burn-days is a good quick and dirty reference. Taking the
Merchant as an example, we have a 120k ton hull according to TRO3057R page 92. The sail is 46 tons (30 + 120k/7.5k) per formula on StratOps page 149. The tons per burn-day figure is 19.75 for a hull this size (StratOps page 147), thus we have 197.5 tons, which I rounded up to 200 tons last time. Fuel costs 15k C-Bills per ton according to StratOps page 179, although if you have a whole bunch of units sitting around doing nothing during a maintenance cycle but cracking water, you can supply it for free but then you've got to ship it out, which requires more gas, and thus more fission, fusion, or solar engines to do the cracking, and thus more deferred maintenance. From the JumpShip owner/operator's point of view, that's probably academic. I will note that in this area there may be FASAnomics involved since that figure may be either cheaper or more expensive than it needs to be. Those of you who are interested may want to calculate the cost of sufficient .25 ton TR D fusion plants using the support vehicle rules to make hydrogen from water, their maintenance costs, and the incidental cost of shipping it out.
That standard 10 burn-days I mentioned costs 2.9625 million C-Bills. Anyway, fuel use may be somewhat lower but at best you can never expend less than 79 tons of fuel (1.185 million C-Bills) on this operation, and even being generous and saying they routinely manage a margin of success of 8 on the aforementioned control roll, you're expending at least 158 tons (2.37 million C-Bills) charging the core. This is without any questions about station-keeping. You've got to replace that about once a week and given where JumpShips usually jump in, doing it involves burning a lot of gas in and of itself. The cost of the sail, including the final cost modifier, is 2.875 million C-Bills. If you don't do better, it's cheaper in
one week to just pay the money out for the sail. It'll pay for itself on two jumps with an MOS of 8 on each one and in three jumps with the absolute minimum figure. In this particular case, the cost of just paying the sail is lower. They're using it because the physics is screwed up and, over time, it's cheaper than buying and shipping hydrogen
Whether the energy efficiency here has anything to do with the energy efficiency of this same hydrogen when used for thrust is another interesting question I'll leave for someone else.