wrog

Hyperbolic Geometry 2 — How to Get Seriously Lost

Mon, 14 Sep 2026 23:21:50 GMT

Hyperbolic Geometry, continued from part 1, in which we learned about critical distances and critical angles.

Quickie review:

Tilty Line is at some angle 𝛼.
Horizontal Line is some distance z up.
These meet (form a triangle) only if the distance isn't too large or the angle isn't too small.
For every positive distance there's a critical angle above which things meet (and at or below, they do not).
For every positive angle less than 90°, there's a critical distance below which things meet (and at or above, they do not).
Critical distance gets arbitrarily large as the angle gets closer to 90°.
Critical distance gets arbitrarily small as the angle gets closer to 0°.

There. You are caught up, now.

Implications for Crocodiles in Hyperbolic Africa

Let's start with a useful fun fact that follows from our observations so far:( Read more... )

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The Essence of Hyperbolic Geometry, Part 1

Wed, 02 Sep 2026 02:09:13 GMT

So this comes up in Relativity, it having been noted that, in a space-time diagram, velocities are essentially slopes and, as such, don't add/subtract the way you'd think, that the Right Thing with velocities in one dimension is to project them onto the 1-Year Later hyperbola and measure the distances along that — what we've been calling the "velocity angles" — i.e., adding/subtracting those quantities instead.

One is then inspired to ask what happens in two (and three) spatial dimensions where you can now have velocity vectors with some (ordinary) angle between them that's not 0° or 180°; how do you combine those? Normally, we'd be doing trigonometry at this point (two magnitudes, angle between them, Law of Cosines, turn the crank, done), but it's going to need to be a different kind of trigonometry, because the hyperboloid thing that the 2D velocities project onto isn't flat,…

… just like combining displacements along a sphere needs spherical trigonometry.

Except this isn't a sphere, but maybe there's some other kind of geometry out there.

Anyway, that would be a use-case. If you badly want to see a worked example up front, I did one a while back.

(Or you could just be curious about what's going on in all of those M. C. Escher drawings. Whatever works.)

But, in fact, we won't be depending on that. This will be more of an axiom wanking approach parallel to what we did here for Spherical Geometry,

… and, to be completely lazy about it, we'll just copy-paste in the initial bits of text we need from there. Like this:

The Geometry Axiom Everybody Hates

Start with a line ℓ and a point A not on it. How do you put a line through A that doesn't intersect ℓ?

(blah blah blah … feel free to go back and read everything here about the parallel postulate up to "Taking Red Pill #1" since it's probably been, like, 7 years since you read it.)

… fast forward through 2000 years of Greeks, Romans, Arabs, and basically everybody, beating their heads against the wall trying to prove [the proposition that there will always be exactly one line through A that doesn't intersect ℓ] … and failing … perhaps because (drumroll…) there exist ways to have this Not Be True.

Once you start thinking along these lines, it's not actually that hard to come up with alternatives — everything is, of course, easy once somebody has thought to do it and figured out how. So let's try out an alternative and see where it gets us. (… cue Twilight Zone music …)

Taking Red Pill #1

[There is no such line]. All pairs of lines intersect. No exceptions. [All straight lines are actually circles with radius 90°, etc… blah blah blah… Spherical Geometry. And there was much rejoicing.]

(… insert record-scratch noise.
     Uh oh. McCoy found the Guardian of Forever,
     went back in time,
     Edith Keeler is no longer road-kill,
     and now the Universe is completely different.
     Shit …)

Taking Red Pill #2

We'll give you multiple lines, i.e., we'll not only give you a second line, we'll let you tilt it through a whole range of angles, none of which will make it intersect the first line.

Oddly enough, this will also have consequences.

(…and now the pink elephants can really come screaming out of the walls.)

( Read more... )

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Relativity (6 of n): How to Fake Gravity

Sat, 14 Feb 2026 08:15:27 GMT

Special Relativity continued from part 5, in which we learned about translating coordinates, the unit-later hyperbola, and velocity angles

Or you can start from Part 1

Today's topic is constant acceleration, which, in Special Relativity, is much weirder than you might expect. ( Read more... )

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Relativity (5 of n): About coordinate translations and velocity angles

Wed, 11 Feb 2026 10:42:46 GMT

My Special Relativity series continued from part 4.

Today's topic is how to translate Moving People coordinates to Stationary People coordinates or to other Moving People and how it's really like a rotation except where it isn't, and how there's this thing that's like a rotation angle but different.

A Surveying Problem

So I needed a survey of my backyard to find out where everything is. The ISO (Stationary) People who I was originally going to hire were busy that day and also expensive, so I had the Rotated (Stationary) People do it, instead. ( Read more... )

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Relativity errata

Mon, 19 Jan 2026 02:50:26 GMT

I of course screwed up and left out the explanation of Lorentz Contraction from the obvious place in Part 3 where I should have talked about it.

So I've slotted it in (plus a new diagram to make super-clear what's happening).

Or if you want to get the flow/parallelism there's now a link to the earlier time dilation discussion (which I also cleaned up a bit), since that's really the same issue but turned on its side 90°.

so maybe the Stargate SG-1 example will be clearer now. or maybe not.

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Relativity (4 of n): The Storrow Drive Theorem

Fri, 16 Jan 2026 01:58:19 GMT

My explanation of Special Relativity, trying to keep it geometric, continued from part 3 wherein I describe the Interval between two events, a kind of "spacetime measurement" that everyone agrees on and that, depending on the trajectory between the events, can represent either a (proper) distance or a (proper) time lapse.

It is time to expand our universe a bit.

A New Dimension

Let's try adding a dimension. We move from the John Hancock Tower to the Flatiron Building (which, if you haven't seen it, is this mostly flat thing standing on end, hence the name). Each floor is a single south-to-north corridor of offices.

( Read more... )

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Relativity (3 of n): The Interval

Thu, 08 Jan 2026 08:27:37 GMT

My explanation of Special Relativity, trying to keep it almost completely geometric, continued from part 2 wherein I describe

what the Moving People spacetime grid looks like when mapped out by the Stationary People,
how all we have left to do is figure out their unit spacing, and
how there can exist Intermediate People who see the Moving and the Stationary People moving in opposite directions at the same speed.

We now take a moment to introduce, out of left field, a new definition that will turn out to be very useful:

Give a pair of events, separated by a distance $Δz$ and elapsed time $Δt$ , the interval between them is defined as the quantity ${Δz}^{2} - {Δt}^{2}$

which looks like a squared distance except for an annoying minus sign,
which seems to depend on whose coordinates we're using (Stationary People vs. Moving People vs. somebody else), but we will now show that:

For any given pair of distinct events, the interval between them is an invariant

( Read more... )

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Relativity (2 of n): Meet the Moving People

Sat, 03 Jan 2026 11:19:33 GMT

My revamped explanation of Special Relativity, trying to keep it completely geometric and not invoking Walls of Math, continued from here where I have now completely beaten to death the concept of "Stationary" and it's time to

Meet the Moving People

Actually, we're going to be particular about who we associate with. We want Moving People who will likewise be able to say that they don't think of themselves as moving, meaning they can't be accelerating or spinning, either. Which leaves having them coast at some constant velocity that is non-zero (because otherwise they'd be stationary and One of Us) and upwards (just to pick a direction).

We also want them moving slower than lightspeed (STL), for Reasons.

( Read more... )

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Relativity redux (1 of n): E pluribus immobiles

Sat, 03 Jan 2026 10:52:28 GMT

(yeah, ok, I don't actually know Latin)

So, here's another go at explaining Special Relativity. I remain annoyed at how few people really get it, even amongst avid SF consumers, entirely too accustomed to generations of SF writers papering over FTL issues with technobabble, 'cause we need that galactic empire, don'tcha know.

Also, it's long past time to dump poorly motivated 1930s pedagogy that real physicists abandoned long ago (e.g.,. "Your mass increases as you go faster", "What? why?", "Fuck you, it just does" [spoiler alert: just No; forget you ever heard that. And if, in 2026, anyone is still trying to teach it that way, someone needs to sit them down for A Talk]).

All we need is basic geometry you knew or could have learned about in 6th grade plus a bit of algebra (up to Pythagorean Theorem). I think I can get by without using a single square-root sign.

But there will be fewer handwaves this time. Because we have to be clear why things have to Not Be The Way You Expected and not leave wiggle room. Here goes:

The speed of light as a constant

Shine a flashlight off of a moving boat. How fast does the light go?

( Read more... )

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Testing a mastodon theory.

Wed, 02 Oct 2024 22:28:40 GMT

(...so, if I post this now link so that it is nearer the top of the page, will hachyderm.io light up the dreamwidth entry on my profile the way it's supposed to?)

Update: Survey sez, "Yes!" If my theory is correct, then the cutoff is somewhere between 30656 and 192123 characters, except the byte numbers are probably bigger since we're in UTF-8 Land. A 65536-byte limit (on how much Mastodon will read before giving up) would not surprise me.

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Deconstructing The Expanse (finally)

Tue, 02 Jan 2024 02:13:37 GMT

(this is Part 5; Part 1 through Part 4 contain possibly-necessary background, but maybe not)

It is time to start putting the pieces together.

I'll start with my first cardinal sin, which is that I've only seen the TV series, not read the books, so if I missed huge reams of infodump where the authors spelled out their actual numbers, so be it. Then again, authors who are not Andy Weir tend not to do this anyway, the cardinal rule of storytelling being that no matter how much work you put into world-building your iceberg, you need to leave most of it underwater, in order to not inspire your readers to beat you to death. At least not until after the series is hugely successful, to the point where publishers can be assured of there being enough rabid nutjobs out there that copies of a Chris-Tolkien-style dump of all research, drafts, cocktail napkins with half-begun stories on them, etc… accumulated during the life of the author will actually sell.

Perhaps what I'll be complaining about is where the TV writers indeed screwed things up. This seems unlikely to me, but if so, this wouldn't be the first time this sort of thing has happened (and, while you might feel sorry for the book authors, nobody was putting a gun to their heads and saying they had to take the TV deal).

Onward…

What is the Epstein Drive?

( Read more... )

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Kepler Orbit Cheat Sheet

Mon, 18 Dec 2023 11:27:18 GMT

(why I should not be allowed to write textbooks, part 342)

This is mainly because I wanted to have all of the formulas in one place. Also curious to see how much I can compress the derivation and still have it vaguely make sense. Also I wanted to learn more MathML.

The 2 body problem

We have two bodies with mass. Gravity attracts them to each other. How do they move?

( Read more... )

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Everything Sucks

Fri, 01 Dec 2023 23:54:52 GMT

a.k.a. Our Propulsion Methods Suck, Our Power Plants Suck, Fission Sucks, Fusion Sucks, and Antimatter May Also Suck

(This is Part 4; continued from Part 3, or you can start at Part 1

Recall how rockets work: Every $d t$ seconds we toss out some bit of mass $d m$ , with velocity $v_{exhaust}$ in the exact wrong direction, giving the rest of the ship, whose total mass is $m$ , a small kick $d v$ in the direction we want to go. The momentum accountants then tell us:

v_{exhaust} d m = - m d v

( Read more... )

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The Other End of the Scale

Fri, 01 Dec 2023 10:41:25 GMT

(this is Part 3, more background for critiquing The Expanse; here are Part 2 and Part 1)

Let's suppose we have Best Possible Rocket, as much free fuel/reaction-mass/whatever as we want, … and we just want to get there as fast as possible? Interestingly, there will still be limits.

( Read more... )

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The Truth About "Slingshot" Maneuvers

Thu, 30 Nov 2023 00:16:57 GMT

(This is "Deconstructing The Expanse, Part 2", continued from Part 1, here.)

As luck would have it, I decided to go back and watch part of an episode to check something, mainly to try to pinpoint where exactly the show goes off the rails, and aside from saving myself from a really embarrasing blunder, it also reminded me of a topic I meant to cover but forgot to.

It's Season 3, Episode 7, the one where Maneo, Speed Demon Belter Guy in his tiny racing pinnace decides he's going to be the first one through The Ring and arranges this whole elaborate sequence of "slingshots" to get there first (what happens when he gets there is a fun scene that I liked because I'm actually 12, but I won't spoil it).

And then I realized…

People have multiple misconceptions about gravitational maneuvers. The "slingshot" is badly named. I blame Star Trek.

( Read more... )

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Questions of Scale (Deconstructing The Expanse, part 1)

Sat, 25 Nov 2023 11:06:33 GMT

(other titles:)

How to do Solar-System-Travelog SF and Not Get It Completely Wrong

How Magical is The Expanse's "Epstein Drive" Anyway?

Why Nuclear Fusion Sucks and We Really Need Those Antimatter Cells

What Part of "Rockets Are Stupid" Did You Not Understand?

Just to get something out of the way up front:

I am now mostly convinced that, barring unexpectedly early, lucky results to our terraforming experiments, colonizing the solar system will for a long time remain an absurd enterprise that will only make sense if we manage to screw up the Earth really, really, really bad — at which point we've probably already gone extinct — and that SF premised on this should be regarded as a variety of steampunk (i.e. seeing what stories we can make out of improbable combinations of technology).

But never mind that. I do eventually expect an assortment of random human-crewed research/mining-supervision/etc stations scattered about needing to engage in commerce of a sort. E.g., just as today we maintain South Pole Station at fantastic expense and there's no way in hell anybody expects that to turn into a colony any time soon (or ever, or, at least, not until after the Antarctic ice-sheet has completely melted), but there will always be a few nutjobs wanting to live there. They may never comprise more than an infinitesimal percentage of Earth's population, but it will be sufficient to make the rest of this post/sub-series not entirely useless.

Having finished a re-watch of (the good parts of) The Expanse, I have to say, I really appreciate how they evidently did give thought to the solar system being way, way bigger than people give it credit for and are actually rather successful in getting this across in ways previous shows completely punted on — the Star Trek: TNG opening that zips by Mars, Jupiter, and Saturn in a matter of seconds being one of those "No, Just No" moments.

( Read more... )

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Space 21: On Payloads and Magic Mirrors

Sun, 12 Feb 2023 04:00:08 GMT

(…a curious observation about relativistic light-sails, … followed by a teentsy bit of obligatory, if undeserved, "Stargate: Universe" bashing.)

A couple of questions that come up in trying to imagine how the interstellar transit economy/infrastructure is all going to work:

Is there some ideal size for a solar power satellite?
This to some extent lies at the heart of my questions of what the solar power extraction regime is going to look like, what orbits we're going to prefer, how many satellites we'll be needing to build, how close to the sun they need to fly, etc.
How big a mirror/solar-sail are we going to need to propel our interstellar payloads?
This will be a central issue in transit tube design since it affects everything else, e.g., how far apart the laser ships need to be, how much antimatter they will need to be supplied with, etc.

Reviewing what I said earlier on the subject of mirrors/sails:

[We'll need them to be] extremely lightweight, totally reflective, micrometeoroid-tolerant, blah-blah-blah. There will be some engineering tradeoff to determine the size (i.e., the smaller we make it, the more accurate the rocket/laser thing has to be, and the more energy density it has to withstand, but the less vulnerable it'll be), which I'll leave the engineers to figure out.

My original thought was that our two applications for mirror-building don't really have much to do with each other beyond their reflecting (pardon the pun) the supply and demand sides of our energy economy, that once we got into the details, the differing goals (energy collection vs. propulsion) would assert themselves and ultimately there'd be very different requirements/designs.

For one thing, for solar satellites, why care about making them lightweight? We're just putting them in orbit, and, beyond occasional questions of servicing and refurbishment, they're not going anywhere. And, if anything, we want them sturdy enough to stand up to the stress of being in however close to the sun we need them to be.

Lightweight only matters in that the less material we use, the more we can make, and we will ultimately have to be making millions of them.

It's not that any of this is wrong; we're just missing an important detail is all.

( Read more... )

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Holy crap, MathML works here

Thu, 12 Jan 2023 01:40:24 GMT

It seems we can do shit like $a ⁢ x^{2} + b ⁢ x + c$ and maybe also $\frac{- b \pm \sqrt{b^{2} - 4 ⁢ a ⁢ c}}{2 ⁢ a}$ . Wheee…

(And if this isn't looking right, you may need to switch/update your browser: Google just released Chrome 109 which includes MathML support for the first time in a long while. Firefox has evidently had this since forever. Probably pointless waiting for Microsoft Edge to catch up (Update (2023-01-14): Edge caught up). And Opera is owned by the Chinese now, so you probably shouldn't be using it anymore. Bleah.)

Looks like I get to go update my math-heavy posts.

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the elephant thing

Tue, 22 Nov 2022 02:07:36 GMT

ok, so I'm on Mastodon (as @wrog@mastodon.murkworks.net, for now).

I will readily admit I never actually used Twitter (creating an account once upon a time was as far as I got) and still don't understand how it was ever supposed to work. I've got Local and Federated feeds spewing endless streams of spam; it's like sitting in the LambdaMOO living room but with 10,000 people there.

I'll be trying out the Lists feature (something similar worked well enough dealing with Livejournal at its height, but even that was far less traffic...).

I really miss Usenet and trn

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in which Yours Truly discovers the "Modules" Section and the "Custom Text" module.

Tue, 10 May 2022 02:12:12 GMT

I keep thinking I should put up a pinned post with pointers to my various "Space" posts and other things which people keep trying to find (or, rather, which I keep pointing people to when I get tired of explaining stuff in other forums)

And then I discover these Dreamwidth options I hadn't seen before (and maybe you hadn't either). And now, as the guy said in the Dr.Who "Blink" episde, "Look to your right -->" (*)

I am, of course, open to better ways to do this; this is still going to be cumbersome to maintain. (sure would be nice to be able to turn off auto-formatting for the Custom Text text; oh well...)

(*)Edit: well okay, not necessarily. You need to be viewing the full post on its own rather than seeing it as part of a feed (in which case it'll be whatever modules you have selected rather than me) — and I suppose if you're using your own format/stylesheet to view my posts, you may also lose... bleah.

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interlude on Lagrange points

Mon, 03 Jan 2022 05:41:38 GMT

promoting this out of my FailBook comments just because

So, there's this diagram which has been making the rounds now that the James Webb Telescope has been launched. It's good at showing where the Lagrange points are and, if you're used to reading topographic maps, what the gravity well generally looks like, that volcano with the 6000 mile deep crater that I talk about here, if you were having trouble picturing that (modulo the small matter that that was all about the Earth-Moon system, and this picture shows Sun-Earth L1-L5; but they both work the same way).

Granted, with L1, most people seem to be clear about the idea that, with Sun and Earth pulling in opposite directions, there's got to be some point in the middle where the forces are cancelling out, even if that's not quite what L1 is. But for L2-L5, people seem to be completely mystified, and I've been seeing far too many wacky attempts at explanations in the last few days.

Time to sort things out:

( Read more... )

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Space 19a: Solar Power Satellites, Take 3 Redux

Thu, 25 Mar 2021 08:53:44 GMT

goddammit.

… 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.

I really should know better.

If we're really going to depend on power satellite orbits following the Kepler rules, then we need to keep them far away from any actual planet. Mercury can't be there; end of story. If Mercury is there, then there's no way around having to solve a 3 body problem, and while it might be possible (like I've been assuming) to perturb the Kepler orbit into an actual orbit that works and stays in synch with Mercury, my gut feeling at the moment is that'll be a huge mess to get right.

Also Mercury's orbit is really eccentric which severely screws up the only 3-body solutions I know about.

And if Mercury is not there, then there's no way to hang a station at aphelion. By default, anything stationary at aphelion falls into the sun, unless we do weird space elevator shit and tether it to something (no idea); but, as with the Earth space elevator, that probably entails unobtanium cables with insane tensile strength and we're in Not Gonna Happen Land.

And I'm realizing now there's a much, much simpler solution available:

Put rockets on all of the antimatter cells.

That's it. Once a cell finishes charging, the satellite releases it and it flies off to wherever it needs to go.

(Yes, Rockets are Still Stupid, but if we put a mass driver on the satellite, then the satellite will nearly always have to do an orbit correction every time it launches a cell and we won't actually have saved anything. So rockets it is…)

Chances are, for the sake of administrative sanity if nothing else, we'll still want to have a collection point somewhere. And we'll probably want it to be a Lagrange point so as to have access to all of the best IPTS orbits. But this can be any of the ones available in the inner solar system.

Hell, we could even use Earth-Moon L2. That might even make sense if it's still fairly early in the agenda and lunar orbit is where we need to have antimatter arriving.

In case you were wondering, the energy cost of moving a kg from, say, Mercury's orbit to Earth's is about 2.5 billionths of a kg, which somebody has to be paying anyway. Unless we want to be really stingy and do some stupid funky Mercury/Venus flyby to bleed off momentum, which I suppose we could do, but bleah.

What to do with the empties is potentially more complicated. Maybe our software will be good enough that calculating ballistic trajectories to get them all back to some power satellite somewhere will be workable.

Or we can just make a point to never discharge a cell below the 2.5 billionth mark (or just keep an extra cell around with power in it to do infinitesimal recharges as needed), so that it'll always have enough energy to go find itself a power satellite. KISS.

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Space 20: Dyson Spheres

Tue, 23 Mar 2021 06:26:39 GMT

(you'll want to read about solar power satellites first, where we cover all of the sensible stuff)

On Dyson Spheres and the other ridiculous things truly advanced civilizations will be doing with all of their free time

I should note that the scenario that I fleshed out in the previous post, i.e., having a cloud of independently orbiting power satellites that would start blocking out the sun if we really were able to populate huge numbers of orbits, is a lot more what Freeman Dyson was talking about in his original paper. The fixed sphere that people tend to imagine when they hear about this stuff really makes no sense at all if you think about it (and Dyson knew this at the time and said so). I mean sure, let's waste a whole lot of effort and materiel fighting solar gravity directly; that should be oodles of fun.

Making only slightly less sense is the flotilla of stationary habitats supported by solar sails. We're doing this why? Because we have this pathological hatred of orbital mechanics and don't want to make use of it? So much better to be depending on a sail that can get a hole in it and then we go plummeting straight into the sun. That sounds so much like a place I would want to live; really. Sign me up, please.

At best, I could see it as a kind of Bite Me, Universe gesture by a civilization that really has Done Everything, is now completely bored, and just wants to build some kind of insane, completely pointless artifact, just for the hell of it, because they can. Sure. Why not?

But I can't see it as something we're going to do while we're still on our way to the stars.

And I still wonder:

Can we really build all that?

Forget the full Dyson sphere, let's consider just one of the solar power satellite orbits. The chart in my previous post is calling for anywhere between 600,000 and 10 million square kilometers of satellite area.

Now, at this point, I don't even know what we're going to be making them from or what the preferred tech is going to be for solar power generation. Photovoltaic? Big-ass mirror driving some kind of heat engine? Or maybe we'll be supplying light to a mini-farm that's going to grow megatons of gerbil food, and then, in the next module over, we have billions of gerbils running their little wheels at top speed (at which point our movie instantly loses its "No Animals Were Harmed..." designation)?

Yeah. No idea.

But we can always make some kind of half-assed estimate. Let's just build aluminum sheeting; whatever the solar collector is, we'll need someplace to mount it. How much sheeting can we make out of the asteroid belt? We can vaguely do this:

total mass of the asteroid belt (kg) is 2.3910×10²¹ kg.
people estimating relative abundances in the universe say aluminum is 58 ppm of everything. Fine, so asteroid belt might have 1.38678×10¹⁷ kg of aluminum.
let's be pessimistic and imagine that only 1% of it is mineable, or we lose 99% in the smelting process for whatever reason. Now we're at 1.38678×10¹⁵ kg.
the thinnest sheeting you can buy online is 1/32"=0.79375mm thick. Meaning we need 793.75 m³ of aluminum to make a 1 km² sheet.
Density of aluminum is 2.7g/cm³=2700kg/m³, so that's 2.143 kg to make a 1 m² sheet or 2.143 million kg to make a km² sheet.
So we get 650 million 1 km² aluminum sheets if we harvest the entire asteroid belt. Which is somewhere between 60 and 1000 times the number of km² we need to populate one of our orbits, i.e., if it were the case that collecting solar energy only needs the km² aluminum mirror and everything else on the satellite is really cheap and easily available.
So by this completely stupid measure, depending on which orbit we choose, anywhere between 60 and 1000 orbits are doable using the h= 36.635km⁻¹ separation and the Mercury L2 point.
Note that these orbits were designed to suck off 86400 kg/day which is 1/4.26×10⁹ of the sun's output. Meaning blotting out the sun will entail filling 4.26 billion orbits, assuming we get the geometry exactly right.

Which seems to suggest that we'll get our 86400 kg/day power production and maybe even be able to go up to 1000 times that, but as far as blocking out the sun goes, just forget about it.

But then we have this interesting fact that I only learned about recently:

The Asteroid Belt is way smaller than you think

On the off-chance that anybody's still trying to convince you that the asteroid belt is the remains of a planet that got destroyed, here's something that really makes that not work:

The total mass of the asteroid belt is about 3% of the mass of the moon.

To be sure, I always knew the asteroid scenes you see in The Empire Strikes Back were bogus (space is big), but I'm still surprised that there's not even remotely enough there for any kind of respectable planet. (Sorry, James Hogan and whoever else wrote SF stories that had a planet breaking up a million years ago)

Which calls into question some of the premises of asteroid mining.

I will grant that there's stuff that won't be available on the moon. But for what is available, which probably includes all sorts of building materials, why not just mine the moon?

We can obtain 3% of the moon by strip-mining the top 12 km of its surface, and we might even vaguely be able to do that with present-day tech. And it's right here; no needing to travel hundreds of millions of km to get to Ceres or wherever else. Probably get huge economies of scale, too.

Granted, this won't really help with the Dyson Sphere. Even consuming the entire moon is only giving us a factor of 30, which is a long way from the 4.26 billion we need. Even eating all of Jupiter only gets us a factor of 800,000.

I think I can safely say that there is not enough aluminum in the solar system to cover the sun. Switching to a more abundant metal like iron gains us another order of magnitude or two, but I suspect we're still hosed.

Though, again, I'm obliged to point out how dubious this particular estimate is. It's not an impossibility proof by any means, and if the materials science folks do manage to come up with some carbon-nanotube/ceramic bullshit unobtainium that's insanely lightweight and can be built out of anything — much like I'm expecting them to do for the laser ships and light-sails we need for the transit tube — then all bets are off.

But I still think I'm pretty safe in putting the Dyson Sphere on the Not Gonna Happen list. If not because it's impossible, then because I'm not convinced we're going to need it.

Granted it is a bit weird finding myself in the position of wanting to say that 86,400 kg of matter+antimatter per day really ought to be enough for anyone — sounds a little too like that apocryphal Bill Gates quote — and having to stop myself.

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Space 19: Solar Power Satellites, Take 3

Tue, 23 Mar 2021 06:11:48 GMT

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.

It also turns out that there will be nothing gained by creating a concentration of satellites at any point along the orbit, since that concentration will simply circulate around the orbit, just as any individual satellite will. In particular, there will be no way to, say, create a concentration close to the sun that stays there.

In fact, it's fairly easy to show (by taking the sum over a single orbital period in which every satellite visits every point of the orbit exactly once) that, assuming all satellites are the same size, for any (nonconstant) satellite density function you might try, there exists a constant density / "evenly spaced" rearrangement of the satellites that produces the same amount energy per unit time on average.

… we just need to nail down what we mean by "evenly spaced" in an eccentric orbit where, at any given time, individual bodies 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,

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
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)	ε	current (km²/day)
1,000,000	665,055	0.966176	20,724
2,173,763	1,000,000	0.927906	30,256
3,000,003	1,190,871	0.901848	35,303
13,764,313	3,000,000	0.617095	69,729
58,129,990	9,969,649	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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Space 18: A Holiday Card for the Kelvans

Sun, 03 Jan 2021 10:13:21 GMT

(continued from here)

So, now what?

Intergalactic Travel

This is where we get full-on, batshit crazy. (You knew this was coming, right?)

Quick reminder in case you forgot or didn't know: M31, i.e., item #31 on comet enthusiast Charles Messier's list of things that annoyed the shit out of him because they weren't actual comets, otherwise known as the Andromeda galaxy, is one of our nearer galactic neighbors, perhaps not the nearest, but it's the one everyone thinks of, so let's just go there.

It is 2.54 million light-years (2.62 million aLY) away. No problem, right? Let's have a look at the numbers:

accel / decel		total shiptime (aY)	mass Cost
shiptime (aY)	distance (aLY)	total shiptime (aY)	mass Cost
14.78	1,310,986.97	29.56	2,621,975.93
14.52	1,015,049.37	29.63	2,030,100.74
14.27	785,915.64	29.87	1,571,833.28
14.01	608,505.71	30.33	1,217,013.42
13.76	471,143.65	31.08	942,289.29
13.50	364,789.19	32.19	729,580.38
13.24	282,442.80	33.77	564,887.59
12.99	218,684.97	35.97	437,371.95
12.73	169,319.61	38.95	338,641.21
12.48	131,097.80	42.95	262,197.59

Yes, these mass costs do look a teentsy bit prohibitive.

And yes, you're reading that right: Even in the cheap case where we're opting to have the journey take 43 years — presumably, by the time we're contemplating this, we'll have figured out how to make an interstellar cruise ship tolerable for a few decades of travel rather than merely a few years (unless we're doing the Australia Thing, it's all prisoners, and we don't care) — we're still burning an entire Enterprise-class aircraft carrier (100,000 metric tons) for every five people we send.

On the other hand, it's a fair bet we won't even be considering this until we've developed, say, a substantial fraction of the Milky Way, say, 100 billion systems. Note that for this particular purpose "developed" means having gotten as far as building energy collection infrastructure and transit tubes; we don't need for them all to be inhabited. If they've all got the same 43,200 kg/day transit budget and we just declare a one day Transit Holiday for everybody, that's … rather a lot of aircraft carriers filled with antimatter, enough to send 200,000 people.

Do that a few days a year for a century and we're basically sending a Europe or 1/3 of an India. Which is not entirely bad for starting off a new galaxy.

But it's possible I'm being too conservative here. Because of something I haven't mentioned yet (it really only occurred to me today).

The Absolute Best Power Plant Ever

By the time our exploration sphere has expanded 25,000 light-years to include the center of our galaxy, we will then be encountering the Sagittarius black hole…

… which I originally thought we'd be wanting to stay the hell away from, but…

It spins, see, which means if you throw shit into it at just the right angle and arrange for it to break apart inside the ergosphere (weird-ass 2nd-event-horizon thing that spinning black holes have that you can actually get out of) at just the right time, then half of it will be spit out way, way, way, faster than anyone should reasonably expect, and then you collect all of that extra energy at some safe distance (hahahaha) and use it to charge up antimatter cells.

And then we're building the Really Big version of the solar energy conveyor thing to get all of that antimatter shipped out of there.

These guys say the rotational parameter for the SBH is around 0.44, which means that about 2.6% of its million-some-odd solar masses is rotational energy that can be completely extracted in this way, meaning there's about 5×10³⁴ kg theoretically available to be harvested.

That's half a billion billion billion aircraft carriers. Possibly more as long as there's crap continuing to fall in to speed up the hole. If we don't get greedy and, say, only take a trillion aircraft carriers per year, that's 3 trillion star systems worth of power generation right there.

Without having to explore and settle 3 trillion star systems.

… modulo certain engineering issues that I will not be attempting to address at this time. I suspect we will need to be using both sides of the paper for this one.

I'll grant that my original timeline has the explorers taking a few million years to reach the center of the galaxy and the SBH. It may be that once we have sufficient infrastructure in our immediate neighborhood, we may want to splurge and launch a few Really Fast explorers in the general direction of Sagittarius so that they're getting there a lot sooner, say 30,000 years.

This will be expensive: We're talking 5/6 lightspeed, which means mass cost ratio of around 11 and recall the explorer ships are big, ten aircraft carriers each, so that's 110 aircraft carriers to launch each one or 7000 times our annual single-system transit budget. Then again, recall that Earth is going to have thousands of years of downtime waiting for those first terraformed exoplanets to come on line. Or we could wait the 2500 years to have energy infrastructure set up in 100 systems so that launching the SBH-explorers won't be as big of a bite.

There's also be risk in launching explorers that far that fast in that we currently have no idea what the solar systems are like in closer to the center of the galaxy, whether the mining/reproduction stuff is even going to work out there; the advantage of the slower plan would be in knowing what's there long before we arrive.

But it may well be worth it if we can have this fucking huge energy pipeline up and running in a mere 60,000 years.

If we have the FHEP, we're definitely getting to Andromeda.

Or, if we want to be more ambitious some year and can extract just 0.00000000000002% of the SBH energy available, that's enough to send a billion people each to every single galaxy in the Local Group using the completely wasteful Andromeda-in-29.56-years schedule (first line of the chart above).

Doing the Long Range Exploring

The economics get a little weird, because there's no getting around spending a few million years on exploration and tube construction, and if we're willing to wait another million years, then the entire galaxy will be already terraformed and harvesting energy by the time we get there.

Also, Andromeda almost certainly has it's own big-ass black hole, and unlike with the Milky Way, our explorers can be heading straight there first thing to get that pipeline started. Which then gets us started on the next galaxy quite a bit sooner than you might think.

Never mind that we're now operating on a timescale where there'll be time to build some real Earths, not just the fake ones you can throw together in a few thousand years. Assuming we even care about that anymore (we might!).

We're clearly going to need to run the explorers differently, since proceeding at 1/100 lightspeed and waiting 250 million years for the explorers to get to Andromeda is really not optimal anymore, nor is the randomly expanding-in-all-directions thing going to work for us. We need to be a lot more goal-oriented now.

Also, intergalactic space is really, really thin. The most we're going to be able to expect is maybe 1 star every 1000 light-years and the mineralogical pickings may be slim once we're away from places where Type III supernovas have happened (though maybe building everything out of aluminum will work just fine). But there are indeed going to be stars. Assuming we can find 2500 systems vaguely lined up 1000 light years apart, we can then have the explorer taking, say, 500 years to build up some energy infrastructure in a system — again, all we need is a stupid little red dwarf — to accumulate enough energy to do the next 1000 light-year hop at 2/3 lightspeed and then begin construction of that portion of the transit tube backwards towards the Milky Way, so as to generally progress at 50% lightspeed and get there in 5 million years (starting up the Replication Thing and going back to the old explorer schedule once we get close to the end) with the transit tube mostly completed.

And we'll probably want to wait an additional 2.5 million years, because, among other things it might be nice to hear back and find out that our explorers actually made it there and were able to do stuff, and also confirm that the tube's been completed, before sending the first actual payload ships with Actual People, (though I suppose it'll be possible to have the transit tube start turning payloads around if the laser ships entirely lose contact with the far end, or receive an "oops", or attain some other reason to believe there isn't actually a far end).

Because once we get into the intergalactic realm, we're now operating on timescales where evolution can happen.

In other words, this is where we're more likely to be running into Opposition.

(next up: Aliens… oh wait, I'm lying; there are a few other things to cover first)

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ε = s − k/3s − 1	where s = ∛k(1 + √1 + k/27)
	and k = (2𝛑R/hAN)²