Space 19: Solar Power Satellites, Take 3

[wrog]

After my previous forays into the question of How to do Solar Power Satellites Right, here and here, there seem to yet more ways to simplify the math and get a better sense of the design space. Very annoying.

In particular, that WTF/4A energy flux formula is not, in fact, the final word on simplifying and, it seems, we can have a station at L2 behind Mercury and still have a wide variety of orbits available. Which is good because I don't know how else we can do a fixed collection point.(No no no no no)

So, rewind. Let's try this again:
Doing Solar Power Right

The plan: Have a ring of satellites in a single, not-necessarily-circular orbit around the sun, all soaking up sunlight and using it to charge antimatter cells. How do we get this to produce some number of kg per day? What are our actual choices?

One would think the number and sizes of the individual satellites shouldn't matter a whole lot. To first order, we'll just have some total acreage of solar panels being crammed into the orbit. Multiply by solar power flux and we're done, right?

However, if we make the satellites too small, then we're making more of them to cover the same area, and if they're all vaguely square/circular - the sensible way to build them - and thus correspondingly less wide, they'll use up more linear space in the orbit and eventually start bumping into each other. Which we very much don't want.

Life will also be easier if our satellites are evenly spaced, whatever the hell "evenly spaced" means in a non-circular orbit where, at any given time, individual bodies in orbit will necessarily have all different velocities and constantly changing mutual distances.

The key observation is that if they're all in the same orbit, then they're all moving in lock step. Assuming that none of the satellites are big enough to be gravitationally messing with any of the others, we're back at the two-body problem that Newton solved 300+ years ago. They're gonna do what they're gonna do and Ellipses are Forever. If it takes one of them, say, 5 days to go from point A to point B, the same will be true of the next satellite following along. So if we make measurements of the km²/day passing point A, we must get the same numbers at point B 5 days later.

Which means that when we're first injecting satellites into this orbit, we want to be sure to inject them at a constant rate - call this our solar panel current, i.e., make it so that the number of km²/day passing the insertion point remains constant.

Once we do this, the current will then be constant and also be the same constant everywhere else along the orbit. Individual velocities can still vary, but anywhere that we have satellites moving faster, they'll have to be more spread out to keep the current the same.

Thus, the place where we have to worry most about stuff colliding will be at aphelion, that point farthest from the sun where everything is moving slowest and hence most bunched together. A panel density can be obtained from dividing the panel current (km²/day) by the aphelion velocity (km/day). This number (km²/km) is how many km² of panel you'll see in any snapshot of 1 km's worth of orbit - or, rather, in every km of an imaginary circular orbit in which everyone is going at (constant) aphelion velocity and spaced out exactly as how they arrived at aphelion.


… which will necessarily be a lower bound on the actual spacings you see in the real orbit. Since I prefer to think in terms of spacings rather than densities, I'll be working in terms of this inverse number instead. That is, we'll define the spacing to be:

h = (panel current) / (aphelion velocity) = 1/ (panel density)

which gives us a number (km/km²), i.e., the number of km of (imaginary circular) orbit you need to snapshot/grab in order to see/accumlate 1 km² of solar panel.

If each individual satellite has area A, then hA is how far apart the satellites will be at aphelion. And if we're comfortable with that being the distance of closest approach, we'll be good to go.
Getting into the ball park

We can now give the formula for this orbit's total power production:

power = W / (2 R h)

where R (km) is aphelion distance, h (km⁻¹) is the spacing, and W is total output of the sun as before (use whatever energy-per-time units you want).

And, yeah, that's it. Just like the WTF formula, this is an exact solution (*) and somehow has none of the other factors you'd expect to see. Inverse-square-law fields have all of these wacky hidden tricks and stuff that cancels unexpectedly. Gauss is probably laughing at me.

(*) or, at least, as exact as we get in what is actually an n-body problem (Yes! This!) and, given that we're in close to the sun, maybe also some General Relativity bullshit lurking (what causes Mercury's orbit to precess) as well. It's also possible I'll lose because I'm ignoring Mercury's gravity (Yes! This!). we'll probably want (small) engines on the satellites to correct perturbations.
So,… if, say,

1. we want aphelion to be at the L2 point behind Mercury (58,130,000 km) (We don't, but this is just as good an example number as any), that convenient place I keep wanting to put a station - which is annoyingly just outside the total solar eclipse zone by about 18,000 km, but maybe having a mere 84% of the sun blocked might be good enough, and also maybe that distance is short enough that we can do some space-elevator bullshit so that the inhabited part of the station(No no no no no) - assuming there even needs to be one - can be hanging down inside the total eclipse zone anyway - and
   
   
2. we want the spacing factor to be, e.g., 36.635 km⁻¹ (to pick a completely random number),
   
   

we can just turn the crank:
3.68×10¹⁴ (kg/day, total solar power)= 86,401 (kg/day) 2 × 5.813×10⁷ (km, aphelion) × 36.635 (km⁻¹, spacing)
… which is roughly the number we were getting before, except we're not counting satellites, computing periods, or velocities or anything.

And, also, you can now easily see that if you want 10 times this much power, one way to get there is to reduce the spacing by a factor of 10, to a mere 3.6 km/km². Which will work reasonably well if the individual satellites are 1 km² (and thus 3.6 km apart), but not so well if they're 1/5 that size (200m square ⇒ area is 0.04 km² ⇒ spacing is 3.6km⁻¹×0.04km² = 144m, which is going to lose badly),
… and even with 1km² satellites we're arguably close to the size limit for this orbit.

… which in general will be a diameter of 4/𝛑h for circular satellites, with square ones being able to get away with being 1/h on a side provided you can keep them from turning.
Getting the Actual Orbit

Note that this does not yet nail down the orbit. If we want to find out, say, how much stuff we have to actually build or how close it'll all be getting to the sun, that's more work. There's one more parameter to specify, which we choose depending on what we most care about:

• if we want to keep our satellites out of the solar corona, we probably care a lot about the perihelion (closest sun approach) distance, which we'll call r, for which the formula that matters is
  
  ε = (R − r) / (R + r)
  
  where ε is
  
  
• eccentricity, which we could just specify directly, if we have a number we like.
  
  This is the measure of how much the orbit is squunched, ε = 0 being zero squunch (circular orbit), and values approach 1 as the orbit gets narrower with satellites diving in closer and closer to the sun, the limit being ε = 1 which would normally be a parabolic escape-velocity orbit except those have infinite aphelion, so if we also specify a finite aphelion, that means we're in this degenerate case where the satellites are just being dropped from aphelion directly into the center of the sun and never heard from again.
  
  Suffice it to say, we really want ε < 1.
  
  
• semi-major axis, usually denoted a, this being the distance from the geometric center of the orbit (not where the sun is) to either perihelion or aphelion, since ellipses are symmetric that way.
  
  In case you were wondering:  R = a(1+ε) and r = a(1−ε).
  
  As it happens, there are lots of other numbers you can use as proxies for the semi-major axis (i.e., they're all conveying the same information), including
  • orbital period, T = 2𝛑a√a/GM, but only if you know the magic constant GM/4𝛑² = 2.509462183311675×10¹⁹ km³/day² (M being mass of the sun and G being the gravitation constant, but for this you don't need those numbers separately)
    
    
  • orbital energy, E = −GM/2a
    
    
  and so on.
  
  
• total satellite area = panel area A × number of satellites N.
  
  If h is how far you have to travel (along that imaginary circular orbit) to see 1 km² worth of panel, and hA is how much linear space one satellite is taking up at aphelion, then hAN is all of the space, the circumference of that imaginary circular orbit, how far you go to see all of the satellites. You can also get this number from the aphelion velocity and the orbital period.
  
  So calculating AN given ε is somewhat straightforward:
  
  hAN = vT = 2𝛑a√(1-ε)/(1+ε) = 2𝛑R√(1-ε)/(1+ε)³
  
  Calculating ε given AN is unavoidably mysterious. If I were sane, I'd just use Newton's Method, but since there actually is a formula for solving cubic polynomials (like the quadratic formula, but people tend not to know this one because it's hideous) we can do this: ε = s − k/3s − 1 where s = ∛k(1 + √1 + k/27) and k = (2𝛑R/hAN)² As for where that comes from, well,… you can ask. Maybe one of these days I'll do a page on Galois theory.
  
  and AN has its own proxies, notably
  • the aphelion velocity v = hAN/T = √GM(1−ε)/R
    
    
  • the aforementioned panel current = AN/T = v/h
    
    
  and so on.

And then we do the wall of numbers to show the range of possibilities you get where production is 86,400 kg/day, satellite spacing is 36.635 km⁻¹, and aphelion is at the Mercury L2 point our chosen distance: perihelion (km)AN(km²)εcurrent (km²/day)  1,000,000  665,0550.966176 20,724  2,173,7631,000,0000.927906 30,256  3,000,0031,190,8710.901848 35,303 13,764,3133,000,0000.617095 69,729 58,129,9909,969,649   3.2×10⁻⁸ (~circular)112,685
Radius of the sun is around 700,000 km so it's not really advisable to try for closer approach than a million km. And while I'd like to think the corona doesn't go past 3 million km, it probably will whenever the sun gets in a bad mood.

That last column is the area (km²) of solar panel that will have to get serviced per day at the L2 station, which you'll notice is Rather A Lot (if the satellites are 1 km² each, then you've got one arriving every few seconds). Yes, this whole operation, like everything else in space, will have to be highly, highly, automated. Surprise.
Not that it should be that difficult. I imagine 99.99999% of the time a satellite will just pass on through, toss its charged antimater cells into a receiving net, and then the station will have a mass driver firing the empties at just the right speed so that the outgoing satellites can pick them up easily. Since the relative velocities will range from 8 to 40-some-odd km/sec, that's probably going to be the only way to do this.

Somewhat more interesting is the repair scenario, where you're either having to send a ship out to catch up with a broken powersat, and either fix it in situ or haul it away somewhere, which will be really expensive because you're totally changing its orbit. However, being at Power Collection Central you'll have as much energy as you need. Somewhat cheaper would be catching it with a tether and using that to swing it out of the stream sufficiently fast so that the next powersat coming along a few seconds later won't crash into it. And wow, will that have to work right the first time.

I still get amused at shows like "The Expanse" where it's imagined that people are going to be out there in space suits doing these jobs with their bare hands. Not that there can't be a role for humans. Person-In-Charge sitting in the Gods-Eye-View office in the total eclipse zone, running some Really High Level Software to monitor things. Maybe.

Personally, I think I'd feel better if actual people were kept well away from these sorts of operations.

All of the above is, of course, for just one orbit. At some point, the power needs will get beyond what one orbit can provide, and then …

… we, of course, start a second one.

which shouldn't be that big a deal at that point. Give it a slightly different orbital plane and the only places the new satellites will have any chance of running into any of the old ones in the first orbit will be at aphelion or perihelion. If we shift both of those numbers by a kilometer or two, that will suffice. We can even have them managed from the same L2 station, (unless we want multiple stations for redundancy, which we will at some point,… probably a lot sooner than we'll need that second orbit).

And then we build a third. You can probably see where this is going.

(Next: Dyson Spheres)

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