Space 21: On Payloads and Magic Mirrors

[wrog]

(…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:

1. 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.
   
   
2. 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.

And I've been loathe to get into detail on this, seeing as mirror design will ultimately rely on matters of structural engineering and materials science that I know little about, or rather, future 1000-years-from-now^* structural engineering and materials science that almost nobody in the present is going to know much about.

Best we can do for now is calculate the amount of abuse our mirrors will need to take, and then appeal to the existence of carbon nanotubes/buckyballs/etc and imagine that it will someday be possible to manufacture reflective sheets that are single big-ass molecules that span however many km², or discover some other insanely lightweight unobtanium that solves all of our problems. (And then we go back to speculating on the less boring stuff.)

It turns out, however, there is one more unexpected (to me, at least) technology-independent observation that sheds light (pardon the pun) on what we're going to be doing, i.e., assuming we will eventually get good enough at building mirrors/solar-sails to be using them for interstellar travel.

This entails crunching a few numbers (as always).

First a quick review of the math simplifications I use (just to get them all in one place, having described most of them before):

• (speed of light $c=1$)
  If you measure time in years, then you also have to measure distance in (light-)years; or if you measure distance in feet, then time should be in (light-)feet — what normal people would call nanoseconds; — and so on. Velocities then become dimensionless fractions of the speed of light, light rays in space-time go 45°, and other nice features fall out.
  
  This by itself does not, however, nail down particular distance or time units.
  
  
• (acceleration $g=1$)
  Normally people think of the standard gravity g as 9.80665 m/s², but if we pick the right distance(time) unit, we get g as 1 unit/unit². Or equivalently, $1/g$ will just be that distance(time).
  
  As it happens, g is ever so slightly more than 1 light-year/year² or 1 per year. Inverting that means the unit we need is slightly smaller, what I've been calling the Acceleration (light-)Year, roughly 354 (light-)days. So g is 1 per aY.
  
  (Alternatively, we could instead decide to accelerate our payloads at the slightly less harsh rate of $g_{\text{Gregorian}}=\text{9.4995 m/s²}$, for which the corresponding magic (distance)time unit is the usual Gregorian (light-)Year. This would reduce various numbers in what follows by about 5%, but won't change the overall picture qualitatively.)
  
  
• $g=1$ also means that 1 acceleration year is how long it takes, in proper time (i.e., time as experienced by the passengers/the payload itself), for the payload's velocity angle — as measured by some inertial observer, and interestingly it does not matter which observer — to increase by 1 (or decrease by 1 in the deceleration phase).
  
  Quick review of velocity angles: $$\tanh \left(\text{velocity angle}\right)=\text{velocity}$$ just like $$\tan \left(\text{angle}\right)=\text{slope}$$ in Euclidean plane geometry. We use velocity angles because they add nicely:  If you're going at velocity angle 4 (= 0.99933c) and you toss a rock in front of you at velocity angle 1 (= 0.7616c), then the rock will be going at velocity angle 5 (= 0.9999092c) w.r.t. whoever was measuring your v.a. as 4. As opposed to velocities themselves, which are like slopes in space-time and add stupidly (e.g., ½c + ½c = ⅘c).
  
  The speed of light corresponds to an infinite velocity angle, so velocity angles can go as high as you want, even though velocities themselves can't even reach 1 (speed of light), which explains how it is that we can keep accelerating forever at a rate of 1, i.e., keep adding 1 every year, without actually hitting lightspeed.
  
  Also, when things get really slow (non-relativistic), the velocity and velocity angle numbers are essentially the same, e.g., 70 miles/hour is around 0.0000001, and it doesn't really matter whether we call that the velocity or the velocity angle because the difference in this case is somewhere around 0.00000000000000005.
  
  
• Having $c^2=1$ means we use the same units for energy, mass, and momentum.
  
  Recall:  1 kg of energy is what we get if a terrorist takes ½kg of matter + ½kg of antimatter and lets them recombine, roughly 89 petajoules or 21 megatons of TNT, roughly the size of the biggest nuclear weapons ever tested — what you need if, say, you want the entire state of Rhode Island to not exist anymore.
  
  
• If we measure energy in kg, then we measure power in kg per acceleration year. 1 kg/aY actually comes out to a quite reasonable 2.94 gigawatts (cue Doc Brown), something that present day power plants can actually do (Doc Brown is off by a few orders of magnitude if Wikipedia is correct that lightning typically peaks at around 1 terawatt, then this is basically 1/300th the intensity of a lightning bolt).

Note that we don't necessarily have to use kg as our unit of energy. We might want something larger. For one thing, there's a whole range of possible payload sizes we might want to consider. Even if the Breakthrough Starshot folks will be happy enough getting a single kg out to Alpha Centauri, eventually we are going to want to do more:

payload vehicle	unit of …			possible payloads
	mass	energy	power
shotput	1 kg	89 PJ	2.94 GW	DNA samples, bacteria, enzymes, seeds
casket	100 kg	8.9 EJ	294 GW	eggs, insects, food, mona lisa, corpse
winnebago	104 kg = 10 t	890 EJ	29.4 TW	1-2 people
car ferry	1000 t	89 ZJ	2.94 PW	20-200 people**
cruise ship aircraft carrier	105 t	8.9 YJ	294 PW	2,000-20,000 people**

Quick review of prefixes: G=giga[10⁹], T=tera[10¹²], P=peta[10¹⁵], E=exa[10¹⁸], Z=zetta[10²¹], Y=yotta[10²⁴].

Also, again, that 21 megaton bomb in the energy column and the 1/300th lightning bolt in the power column is the first row. You need to multiply those by 100 million to get to the last row.

By the way, petawatt lasers are not actually completely ridiculous. Today's inertial confinement fusion people at Livermore have them. The catch is their lasers only operate for a billionth of a second, whereas for a laser ship in a transit tube having to fire at our payload over the course of an entire light-second or two while it's traveling near the speed of light, we'll need to raise that duration by a factor of about a billion (and also increase the aperture so that the laser can stay focused over distances of light-seconds).

Now, it turns out there's a really easy derivation of how much power the payload mirror has to receive in order to accelerate by $g=1$.

Pick a payload mass, 1kg (really doesn't matter; everything is linear in mass anyway), and look at things from the payload point of view. Payload is initially stationary (because you're always stationary in your own frame of reference) — meaning what follows is deep in the non-relativistic realm where we don't have to care about the distinction between velocity and velocity angle and thus acceleration works pretty much exactly the way Newton would have expected it to:

• we have an incoming infinitesimal light pulse, whose energy/momentum is dE kg,
• arriving during an infinitesimal time interval dt,
• hitting the mirror and setting the payload moving at infinitesimal velocity dv,
• except $\mathrm{dv}=\mathrm{dt}$ (since $\mathrm{dv}/\mathrm{dt}=g=1$)
• payload emerges with dt kg of momentum while
• the bounced light pulse is now dE kg headed back out in the opposite direction.

The conservation of momentum accountants will then tell us, $2\mathrm{dE}=\mathrm{dt}$, or

$$\frac{\mathrm{dE}}{\mathrm{dt}}=\frac{1}{2}\hspace{1em}\text{(kilograms per acceleration-year per kilogram of payload)}$$

this being constant over the whole time the ship is accelerating (or decelerating). Which means we need to be receiving a constant 1.47 GW per kg of payload, ideally evenly spread out over however big the mirror is. (Note this is very much not what the laser ships need to be sending; we'll get back to that in a future post.)

One imagines, at the very least, that being able to construct a payload ship of a particular size, implies being able to build a mirror of that size; this is then arguably the smallest mirror size we need consider. The largest payload on our chart, the cruise ship or aircraft carrier, at 10⁵ tonnes or 10⁸ kg will be something like 1/2 km long, so at the very least we should expect to be able to do a mirror that's at least 1km square or 1 million m². So that's 0.01 m²/kg as our lower bound on sizes of mirror to consider.

Note that a mirror this small will need to handle 147 GW/m² — simultaneous, continuous lightning strikes at every 10m² — and not melt. Not that I'm certain what 1000-years-from-now materials science will be able to provide, but you should probably have a bad feeling about this.

At the other end of the scale, the largest reflector we'd want to give to a 1kg payload would be something on the order of 100m², given that present day proposals for making solar sails out of mylar or kevlar have densities ranging from 3-12g/m², meaning this size sail by itself would have mass of around a kilogram, which would seriously kill our efficiency, though optimistic thoughts of carbon nanotube bullshit may get this down to 0.1g/m² and hence 10g for this size of sail, which I guess we could live with.

Or perhaps the 1000-years-from-now materials science people will come through for us and find a way to build the 1000m² per kg mirror, so, in the upcoming table we can add a row for that one, just to see what it looks like. Keep in mind this size mirror reduces the power flux to 1.47 MW/m² putting us within an order of magnitude of Ordinary Microwave Oven, so making the mirror this big is probably overkill.

At some point, I have to deliver on my promise to relate this to solar power satellites. So…

Let's talk about the Sun, whose total power output (w) is 382.8 yottawatts (yay, we get to use that prefix again; not quite big enough to get to RW, ronna- (R) being 10²⁷, a prefix which was only approved last year; oh well), or 1.302×10¹⁷ kg/aY. Notice how in order for this to be appearing in the power column on the previous chart, we'd need to be adding another row for a payload vehicle of mass 10 million tonnes, probably several kilometers across, with passenger capacity of up to maybe 2 million people, larger than any vehicle built in the entire history of the human race,…

…a row that I would probably be labeling "Portable City" or "Death Star",…

…and then there would be another three or four rows after that, each 100 times yet more massive, before we finally get to something large enough (e.g., Mars' moon Phobos) that getting it to accelerate at g would use up the entire power output of the Sun

… so that we're clear about just how ridiculous the Sun is.

Given that number it's fairly straightforward to come up with, for each of the various mirror sizes (a) per kg, the equivalent solar distance (r), i.e., the distance from the sun at which the mirror would be receiving that 1.47 GW/(that size) power flux, by solving

$$\frac{w}{4\pi r^2}a=\frac{1}{2}\text{\hspace{1em}\hspace{1em}(kg/aY/kg-payload)}$$

for r.

mirror size per kg payload	power flux	mirror diameter for		equivalent solar distance
		winnebago	cruise ship
1000.0m²	1.47MW/m²	3.57km	357km	4,550,000km
100.0m²	14.7MW/m²	1.13km	113km	1,440,000km
10.0m²	147.MW/m²	357m	35.7km	455,000km	"Um, guys? We're in the star." — Eli***
1.0m²	1.47GW/m²	113m	11.3km	144,000km
0.1m²	14.7GW/m²	35.7m	3.57km	45,500km
0.01m²	147GW/m²	11.3m	1.13km	14,400km

… the key column being the rightmost one. Just in case anyone was wondering, the solar photosphere — the thing you really don't want to be going inside if you don't have magical Ancient Technology protecting you — has a radius of around 700,000km. So,… yeah.

… the overall point of this exercise being that, however hard it might be to build the solar power satellite mirror that can live in the solar corona, building a payload ship mirror that will survive the trip to Tau Ceti will be more difficult, even in the largest/easiest case.

Or, conversely, if it turns out we really can build the 10.0m²/kg mirror capable of handling a 147MW/m² power flux, it's barely going to notice that it's in the solar corona (thus would qualify as badass civil engineering), at which point the solar power folks may well be just fine putting their satellites in a circular orbit at 1.5 million km.

My guess is there will be a sweet spot somewhere between 50 and 200 m²/kg, just large enough to be annoying.

Actually, this is a good result. It means that, of the tasks of building the interstellar transit tubes and making them work vs. building the chain of solar power satellites and making that work, the latter will be far easier, which is good because that's what we'll be needing first.

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^*There's an implicit assumption here that I already know isn't true:  This weird idea that all fields of endeavor will somehow progress at the same rate, the reality being that progress can be quite haphazard, happening when/where it needs to, sometimes not even then, and occasionally we go backwards.

Electric motor designs have largely not changed in the last 100 years; they got Good Enough in the 1920s. Propeller designs have been stable for even longer, and yet a few years ago somebody found a new one that beats the absolute living crap out out of all of the previous ones.

In other news, our concrete keeps getting better. Perhaps someday we'll finally rediscover the formula the Romans used 2000 years ago that still outlasts anything we can build today.

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^** I am being perhaps optimistic about passenger capacities, here, in using the capacities of present-day seagoing surface ships as a vague guide. The comparison with payload starships thousands of years from now has a certain apples to oranges quality to it. In theory, if we're really able to develop the sufficiently powerful lasers/etc we'll need, one imagines we'll have similar advances in recycling, life support, shielding tech, and the psychology of cramming lots of people into small spaces for a few years. But, see previous comment about various fields not progressing at the same rates.…

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^*** Much as I appreciate John Scalzi (science advisor for Stargate: Universe) trying to get as much right as he could, I do have to quibble about Rush's mention of the corona being "millions of degrees".

Temperature is essentially average kinetic energy per molecule: In a near-vaccuum (i.e., extremely few molecules per m³), temperature doesn't matter that much (the amount of energy those molecules will be imparting will be relatively low, at least as compared with all of the other shit going on that would be giving everybody cancer and trying to melt the ship). I.e., what Rush said was indeed technically correct but not something they needed to worry about.

Likewise for "gravitational stresses": It's true that the sun's gravity, on or near the surface —supposing for a moment that the sun had an actual surface you could stand on, which it just doesn't — is 28g, which indeed would be pretty nasty to try to stand up in. But the ship is in orbit, i.e., free-fall, where nobody should feel gravity at all, because falling.

What matters then is tidal forces (derivative of gravity) which diminish as 1/r³; the Sun being 100 times the size of the Earth would need to be a million times as massive to have the same magnitude tides on its surface, but the mass ratio w.r.t. Earth is only a few hundred thousand. Which then means that, for low solar orbit, tides are actually around 4 times weaker than low Earth orbit. And, since, we're seeing Destiny in low planetary orbits all the time, they shouldn't be worried about this, either. Never mind that these sorts of tides — which geophysicists actually do use to survey deep underground stuff — you can't even detect unless you either have very, very sensitive instruments, or something very, very, very large (e.g., an ocean).

You really need to be near a black hole or a neutron star before tides start ripping apart something as small as a spaceship, and oddly enough, it's the smaller black holes you have to watch out for (the bigger ones, you're already inside the event horizon before you even notice the tides).

I suppose, to be fair, one could argue that Rush was purposefully exaggerating the dangers to an audience that didn't know better, since that would be in character for him, however that wouldn't explain why the other knowledgeable folks (e.g., Voelker, who's supposed to be an actual astrophysicist, …) wouldn't be catching him at it and correcting him.

Comments

Mirrors don't have to handle all the heat

[pjz]

> Note that a mirror this small will need to handle 147 GW/m² — simultaneous, continuous lightning strikes at every 10m² — and not melt. Not that I'm certain what 1000-years-from-now materials science will be able to provide, but you should probably have a bad feeling about this.

A quick web search finds current mirrors can be four-nines efficient, so actual heat handling can be reduced by a factor of 10^4. Which helps, though admittedly 14.7MW/m² is still a large number.

Also! Progress on Roman concrete found

Re: Mirrors don't have to handle all the heat

[wrog]

I meant "handle" as in "be able to reflect". Seems to me there has to be a point where, if there's enough energy coming in, the metallic coating gets vaporized and then you don't have a mirror anymore. I'm not sure where that is, but my gut feeling is 147 GW/m² is well beyond.

oh, wait, we can use the Stefan–Boltzmann law.

See, the backside of the mirror, facing away from the sun, exposed to vacuum, — assuming the carbon-nanotube bullshit that's holding its structure isn't serving as an insulator — can radiate as a blackbody. In fact, that's really the only way we have for the mirror to dump excess heat that would otherwise build up and eventually cause it to vaporize.

The rule is that the power flux radiated will be proportional to the fourth power of the temperature. The mirror heats up until it reaches a temperature where it can radiate the heat away as fast as it's coming in.

So, e.g., in the 1 m²/kg-payload case, where the incoming flux is 1.47GW/m², assuming the mirror can reflect away 99.99% of that back towards the sun/lasership, we're left with 147kW/m² being retained as heat that needs to be radiated out the back. Dividing by the Stefan-Boltzmann constant (5.670374419×10⁻⁸ W/m²K⁴) and taking the 4th root gives a minimum temperature for the mirror such that said radiation will happen. Which gives us this chart:

mirror size per kg payload	10⁻⁴ power flux	operating temperature
1000.0m²	147 W/m²	226 K
100.0m²	1.47 kW/m²	401 K
10.0m²	14.7 kW/m²	714 K
1.0m²	147 kW/m²	1269 K
0.1m²	1.47 MW/m²	2256 K
0.01m²	14.7 MW/m²	4013 K

the last 3 rows being problematic, since the melting point of aluminum is around 900 K, and since we're in a vacuum where it can't exist as a liquid, that's probably game over right there.

Not that we necessarily have to use aluminum, but, on the other hand, I'm suitably amazed that the 10m²/kg case seems to actually vaguely work (assuming aluminum doesn't deform too badly at 714 K — then again, that's only 441℃, not something you want to touch with your bare hands, but still not all that hot in the grand scheme of things. Meanwhile, 401 K is basically a hot stove and 226 K is South Pole Station, so those work just fine).