Space 14 (cont.): Solar power satellites redux
(as I was saying in my tech wish list,)
To put some actual numbers on this, if we have the satellites grazing the sun (1 million km out)
So, this was all based on this theory that I had that there'd be a point to putting solar power satellites in close to the sun, that having them dive through the corona for a few hours is what you want. (I've apparently been watching too much Stargate: Universe.)
Strangely enough, this turns out not to actually matter in the way that I thought it would.
More generally, you would think that the energy flux (energy per m² of solar panel) received by a satellite over the course of a single orbit or portion thereof would be really complicated. What with distance from the sun and velocity of the satellite constantly changing, there'd be this nasty integral to do, there being any number of questions about ellipses, e.g., how to calculate the perimeter, that don't have easy answers in terms of familiar functions.
But this isn't one of them.
Herewith I present the formula for total energy flux received for a satellite traversing some angular fraction F of its full orbit - doesn't matter which F, i.e., it's the same energy for any wedge having a given angle at the center, no matter whether it's pointed towards perihelion (closest approach), aphelion (farthest away) or any other direction - where we let
- W = total wattage of the sun and
- T = the orbital period
(so that WT is the total energy emitted by the sun over a full orbit). Get ready for it...
WTF 4A
(Yeah, okay, I'm 12 today. But this is kind of how I feel about it.)
In case you were wondering A is area of the orbit for which there are any number of formulae depending on which information we have, e.g., πab if we have semimajor and semiminor axes. Or πa²√(1-ε²) or πb²/√(1-ε²), if they give us eccentricity instead. And so it goes.
Just to sanity check: For a circular orbit radius r, area A = πr² and doing a full orbit we get (WT)/4πr², as one might expect from projecting the sun onto a sphere of radius r with the satellite always being on that sphere. But the funky thing is how this all works no matter what the shape of the orbit is.
Short Reason Why: Angular momentum L = r²dθ/dt is conserved, so when we integrate received power over time to get energy, a whole lot of constants move outside: ∫(W/4πr²)dt = (W/4πL)∫dθ. Also L is twice the area swept out per unit time i.e., L = 2A/T. The rest is setting θ = 2πF and shuffling letters around.
To be sure, we want to be cranking the flux up as high as we can, which means making the area A as small as possible. But you can do that just as easily with a circular orbit as with that weird, highly eccentric one I was using, and then the satellites wouldn't have to deal with corona storms and other nastiness.
The only disadvantage is losing the convenient dark place behind Mercury where service station can hide from getting blasted by the sun. But perhaps if this is 10,000 years from now, our tech may be to the point where building something like that at, say, 1/3 of a Mercury orbit radius without there being a planet to shield it won't be that big a deal (after all, the satellites themselves will have to stand up to all kinds of abuse).
Also, it really won't make much difference having one big-ass satellite vs. lots of little satellites - my original reason for lots of little satellites was so that there could always be one in the corona getting charged up at any given time (on the assumption that the corona was the only place worth bothering having things charge up because that's the only place we get the huge energy flux, which is what turns out to be wrong, i.e., we actually do pretty well using the whole orbit), - …
… other than the usual Economies of Scale vs. the Putting All of Our Eggs in One Basket issues that need way more info about what the technology is actually going to be than we have at present.
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