> But they only have 1 machine, and so their chance of finding a valid hash _must_ be half.
No, Bitcoin's network is currently set up to guarantee that a new hash is found in roughly 10 minutes. It's not about how many computers you have in relation to the earth, you need there to be few enough computers on earth and structural enough changes to Bitcoin's technology that it takes the current network on Earth more than 16 minutes to find a hash.
Adding more computers to your orbital array will not change the hash rate on Earth, the network is set up to take the computing power into account and keep the hash rate at about 10 minutes.
Go back to my footrace analogy. What you're currently trying to do is to beat me in a race by getting a progressively faster and faster car, or by introducing 2 cars into the race instead of just 1. None of that matters unless you can get your cars onto the track in less than 10 minutes, because in 10 minutes I will be done with the race and have moved on to the next hash.
You're not pitting your computer network against the earth, you're just pitting it against the latency. You need the on earth hash rate to be slower than 16 minutes, which is not how Bitcoin is currently set up.
> and if that tipping point is greater than 50% of the available hashing power, then you might as well just forge instead of doing it for real...
To be fair, a 51% attack might actually be a feasible idea for your satellite network if you have enough computers there, but it would also come with the fun side effect of making all of your coins useless and crashing the bitcoin bubble.
I suppose the other thing you could maybe do is get so many computers in orbit accepted into the network that you actually drive up the hash difficulty until all of the computers on earth can't solve the problem in 10 minutes; purely because the built-in hashing requirements of Bitcoin become so high that your network's computing power is necessary to finish the transaction in a reasonable time at all. I vaguely suspect you'd need at least close to 50% of the network to do that, but I might be wrong, I haven't done the math.
The downside of that scenario is that you've now raised the hash rate for Bitcoin to 26 minutes, because the 10 minute hash rate is going to stay the same regardless of how many computers are on the network, even if 90% of them are orbiting the sun. But to be fair again, a 26 minute hash rate only matters if you're trying to build a working currency, which pretty much nobody in Bitcoin land seems to care about at this point.