The Trouble With Ginormous

Image by Udo Reitter from Pixabay
This article contains material generated as background reference in Mike’s Doctor Who: A Vortex Of War campaign, but it holds relevance to most campaigns including those of the Fantasy genre.
Introduction
Space is big – really, really, big.
I’m sure most readers will have come across that phrase, or something very like it, on numerous occasions, and have taken its lesson to heart.
But I would be equally certain that comic book writers and sci-fi authors and scriptwriters would also have done so – especially given the practice of vetting for scientific accuracy inherent in the last category.
And yet, I have been let down repeatedly in this respect by those very groups, so sometimes you have to wonder…
Part of the problem is undoubtedly because the scales concerned are epic beyond our ability to comprehend them directly. Of necessity, they have to be abstracted and we have to learn to think in those abstract scales.
But doing so leaves us vulnerable when we have to step up to another scale again; we understand the first scale and think that gives us a handle on the second. And that confidence is frequently misplaced.
Today’s article is intended to bridge that gap.
And my chosen starting place is one of my favorite comics as a kid: Green Lantern, specifically, the Green Lantern Corps.
3600 Sectors Of Trouble
Part of the canon of the Green Lantern Corps is that there are 3600 Green Lanterns, each of whom patrols a different sector of the galaxy, and who are usually drawn from one of the inhabited worlds within that sector.
And, if you don’t appreciate how big the galaxy actually is, that sounds perfectly reasonable. But one look below the surface reveals trouble brewing.
The size of the galaxy
In the course of a previous article on astrophysics (both for within games, and in general), A Game Of Drakes and Detectives: Where’s ET?, I reported on the size of the milky way, and gave various other parameters that will be useful in this discussion.
Let’s start with the cross -section of the milky way.
To quote from the accompanying article:
The milky way is roughly 150,000-200,000 light years in diameter, giving it a radius of 75-100,000 light years. But most of that is outlying material; in terms of the parts we’re interested in, it’s about 100,000 light-years across and about 1,000 light-years thick. But that thickness is the average for the whole thing, and the core noticeably bulges; about three times the thickness of the arms. We also need to exclude that core from our calculation of the plan area of the disk if we hope to get a volume. Looking at the galactic cross-section, the core is about 1/5th of the total diameter across, so about 20,000 light-years.
When I do that, I get an average thickness of the disk section of 926 light years, and a torroidal area of 2,400 million pi square light years, so the arms contain roughly 7 million million cubic light years.
The size of a sector
For the moment, let’s ignore the central bulge. That means that out 3600 sectors contain 7 million million cubic light years. If each sector holds equal volume, then will have a volume of 1944444444 cubic light years.
While it would be inaccurate to do so, let’s ignore that inaccuracy, and project this volume as a square section of the milky way that’s 926 light years thick. That means that our square has to have an area of roughly 2099832 square light years, which is a square of sides 1449 light years to a side.
That means that a hypothetical ‘sector’ would look like the diagram to the right:
Appreciating the size of a sector
To get a real handle on how big this is, let’s assume that our hypothetical green lantern is based at the extreme bottom front left, labeled α, and that some emergency occurs just barely within his area of responsibility at the extreme top right back. In other words, he has to cross the sector from one corner to the other, the distance between the points α and γ.
We know that β is a right angled corner, so what we have is a simple triangle in which we know only one length – βγ, defined as 926 light years.
But, we can also see that αβ is also a triangle with a right-angle, and we know the length of both sides (1449 light years), so good old pythagoris tells us what we need to know:
αβ^2 = 1449^2 + 1449^2 = 2 &Times; 2,099,601 = 4,199,202; therefore,
αβ = √4199202 = 2049.195.
Now we have two sides of the triangle αβγ – 2049.195 and 926. So:
αγ^2 = 2049.195^2 + 926^2 = 4199202 + 857476 = 5056678;
αγ = √ 5056678 = 2248.706 light years.
Now, the green lantern corps can travel at FTL speeds, but the actual speed is rarely if ever stated out loud; it’s “as fast as their will permits”. So, let’s throw some reasonable FTL speeds around and see how long this theoretical corner-to-corner trip will take.
At 10c, 2248.706 / 10 = 224.87 years.
At 100c, 22.48706 years.
At 1000c, 2.248706 years.
Hmm, that’s not working too well. So, let’s press on to more radical speeds:
At 10,000c, 0.2248706 years = 82.13398665 days (defining a year as 365.25 days).
At 100,000c, 0.02248706 years = 8.213398665 days.
At 1,000,000c, 0.8213398665 days = 19hrs 42m 44s.
Based on those numbers, to make a sector patrolable in any practical sense, speeds of between 50,000 and 1,000,000 times the speed of light are required.
But, by scaling the problem to numbers that we can all comprehend, we start to get a real impression of just how big a region of space we’re talking about.
The Size of a sector, part 2
But wait a moment – how are the people of gamma supposed to tell alpha that there’s a problem? A radio message will take over 2,200 years to get there – and be very hard to even detect, as the earlier article points out. So it’s not just the green lanterns that have to travel at ridiculous speeds, it’s everyone else.
The alternative, since not all the people being protected even have space flight, is for the Green Lantern to visit regularly, showing the flag and looking for trouble. And that points us back toward those ridiculous travel speeds.
The size of the galaxy, part 2
Let’s imaging a green lantern on the outer rim of the galaxy. Every now and then he has to report back to Oa. Most of the time, he creates a space warp that conveniently gets him there, but every now and then, there can be reasons for doing the trip the long way around.
Before we can assess that, however, we need to know where Oa is located.
Well, there are three logical possibilities – it’s either at the outer edge of the galaxy, it’s in close to the galactic core, or it’s somewhere in the middle of the disc.
This diagram illustrates the worst-case that results. The three proposed locations for Oa are labeled β, γ, and ε, respectively, while α remains our point of origin. Even without the black hole at the center (4), there would be enough radiation sources that travel straight through the core would be inadvisable. So, to safely get to β, we need to go to point 1 first. Similarly, to get to γ we need to go to point 2 first; and to get to ε, we need to go to 2, then to 3. Five and Six denote the ‘edges’ (top and middle, respectively) of the bulge.
A little thought will show that α to 1 is the hypotenuse of a triangle, with 4 it’s other corner, and that β-to-1 will be exactly the same length, and so will α to 2. It’s only once past the dangerous central galaxy that the course is altered by the different locations of Oa.
According to the cross-section diagram shown earlier, distance 1-4 is going to be 10,000 light years, and alpha to 4 will be 10,000 + 40,000 = 50,000 light years. That means that the first-leg distance is:
α-to-1 = 1-to-β = &alpha-to-2 = √ (10,000^2 + 50,000^2) = 50990 light years.
Therefore, α to β is twice this, or 101,980 light years.
2-4-γ forms a triangle with the same 4-2 measurement as 4-to-1, 10,000 light years, but the long axis is 20,000 light years less than the 50,000. So the distance from 2 to γ is about 31623 light years. So the total trip from α to γ will be 50,990 + 31,623 = 82,613.
α-to-two-to-three-to-ε is a more complicated problem, but we can easily calculate the distance direct from 2 to epsilon; while the additional deliverance to 3 will add to that, it would be a relatively small error. So, the length to ε from 4 is going to be 40,000 less than the 50,000, or 10,000; and therefore the direct distance from 2 to ε will be about 14142. Round it up to 14400, and that should be more than enough to compensate for the more complex course; and the total trip from α to ε is going to come to roughly 50,990 + 14,400 = 65,390 light years.
Now let’s apply those earlier speed estimates (50,000 and 1,000,000 times the speed of light, respectively, and calculate some travel times:
α-to-β @ 50,000c = 2.0396 years.
α-to-γ @ 50,000c = 1.65226 years, or about 20 months..
α-to_ε @50,000c = 1.3078 years, or about 15½ months.
α-to-β @ 1,000,000c = 0.10198 years = 37.248195 days.
α-to-γ @ 1,000,000c = 0.082613 years, or 30.2 days – call it a month.
α-to_ε @50,000c = 0.06539 years, or about 24 days.
The more ridiculously fast we make the travel, the less of a problem this becomes.
The Forest
There’s another saying – that you sometimes can’t see the forest for the trees. How many stars are likely to be present in a single sector?
In that earlier article, I calculated as a very rubbery best-guess that there were 220,000 million stars in the disc-region of the milky way. If there are 3600 sectors, that means that on average, each will contain 61,111,111 stars. From the earlier calculation of the volume of a sector (1,944,444,444 cubic light years), that means that each would occupy roughly 31.82 cubic light years, or a sphere 1.966 light years radius, on average. So the average gap between stars will be twice that, or one star every 3.933 light years.
Corner-to-corner in a sector? 2248.706 light years? That means running into (on average) 572 stars – but one is our departure point, and one our destination, so that’s 570 in the way, en route.
At 10c, that would be one every 224.87/570 = 0.3945 years.= 1 every 144 days. That’s doable.
At 100c, that becomes one every 14.4 days.
At 1000c, 1.44 days.
At 10,000c, 0.144 days = 3.456 hours.
At 100,000c, 0.3456 hours = 20.736 minutes.
At 1,000,000c, 2.0736 minutes. Constantly. For 20 hours or more.
I submit that with size, and radiation output, and potentially hostile residents, that anything faster than about 7,000 times the speed of light involves impossible speed of navigation – that would be a course correction every 5 hours or so, giving at least half-a night’s sleep. Drillers and fishermen have been operating on a four-hours-on, four-hours off schedule for years, and it’s not exactly unfamiliar territory for the military, either.
But if that’s our top speed, then the corner-to-corner sector trip will take about 117 days. And that’s far too long for a green lantern to be able to respond to an emergency.
But what’s the alternative?
Challenging assumptions
Okay, so let’s start by chucking the idea of 3600 sectors, and allow there to be more – many more. In fact, let’s look at stellar populations, make a few sci-fi-valid assumptions, and derive an estimate for just how big a sector should be – and use that to determine how many sectors there should be.
Let’s start by thinking about systems of significance – because some of them won’t be.
For a start, one of the inherent assumptions is that if life is possible, it will find a way; inhabited systems will be common. Next, let’s assume that for every inhabited system, there will be 1½ systems containing significant resources, but no life, giving those inhabited systems something to fight over, and something to kick-start interstellar expansion. And, because a system can have no significance other than being innately interesting for some reason, let’s say that such ‘scenic’ worlds are another ½
How many inhabited systems can one Green Lantern protect? Well, 1/3 aren’t advanced enough, technologically, to get themselves or anyone else into trouble; but that makes them an easy target for conquerors and would-be exploiters. 1/3 would be advanced enough to fend for themselves and enlightened enough not to try and exploit others (but they can still get into trouble occasionally). That leaves 1/3 as potential troublemakers.
Let’s assume that each of the troublemakers have to visited every year to keep an eye on them, and that such inspections take at least 3 days, not counting travel time. The more advanced and enlightened worlds might need to be visited once every 5 years for a day; and the primitive worlds once a year for a day.
So 1/3 of the stars need 3 days attention a year; 1/3 need 1 day’s attention; and 1/3 need 1/5 of a day. Add those up, and you get 4.2 days per interesting star. Throw in a couple of days of travel between them, and you get 8.2 days per star system of interest.
365 days in a year, divided by 8.2 days, gives 44.5 systems of interest. But there’s an assumed inefficiency here – sometimes you will be able to deal with one thing while en route to deal with another. So let’s increase that workload 300% and then allow for a little time off each year – giving 120 or so star systems.
With those numbers as a rough starting point, I get 61 inhabited systems, 93 worlds with significant resources, and 30 systems of other galactic significance, and a net stellar population of 1200 stars under one Green Lantern – on average.
Based on that premise, I divided the galaxy up so that green lanterns only had one galactic arm each within their sectors, and used stellar densities to divide the galaxy up into 305 regions, each of which would contain 400 sectors. I also found that I needed multiple strata or layers. In fact, when I counted them up, I got 350. Put those together, and you end up with 8,200,000 sectors, as the diagram below makes clear (the dots were my method of counting them, each color is 50 regions or strata).:
Click on the image for an even larger (more legible) version in a new tab.That really puts into perspective just how far wide of the mark that 3600 sectors was, doesn’t it?
Enhanced functionality
But this defines an average sector – as noted, some regions could have as many as 20 times these numbers, while others have less.
It can be presumed that 20 times the standard number of inhabited systems – 1220 of them – there would be twenty times the number of systems capable of provide a Green Lantern to the corps. Instead of one Green Lantern, they might have ten or twenty. Add in the fact that as stellar densities go up, travel time from one star to another goes down because the stars are closer together. Which means that fewer Green Lanterns are actually needed in such dense Sectors.
What about the sectors with fewer inhabited systems? Potentially, one Green Lantern could look after multiple adjacent sectors, but travel times form a significant restriction, so there are limits to this sort of thing. Fortunately, there’s an excess of Green Lanterns from the more densely-populated sectors, so a few of those can be “exiled” to the galactic periphery, perhaps as a temporary tour, eventually rotating back to their more-populated home sector.
The size of a sector, revisited
Instead of 3600 sectors, dividing the galaxy up into 8,200,000 makes them significantly smaller – so much so that it’s worth revisiting the physical size of a typical sector, and recalculating the corner-to-corner (worst case) travel times.
There are two possible approaches to the calculation: we could use the density of stars derived earlier, multiply by 1200, and get one answer for the volume; or we could take the estimated volume of the milky way and divide that by the number of sectors. In theory, both should give the same answer.
But I have the suspicion that the packing problem might be a source of significant error with the first approach.
Not familiar with the Packing problem? Consider a box of oranges. Your job is to arrange them to get as many as possible into the box, i.e. to minimize the wasted space.
If you simply stack them one on top of another (as shown above), there is a huge amount of empty space – each orange is taking up a cube of sides “2 orange-halves” long, a volume of 8o^3, but each orange only fills 4/3πr^3 = 4.19o^3. Almost half the space taken up by an orange is empty.
Instead, each row nests in the hollow created by the oranges of the layer below, effectively interleaving the layers of oranges. Calculating the difference isn’t particularly relevant, but ANY improvement is significant. And you can improve packing density even more by choosing slightly smaller oranges for the ‘indented’ layers.
I’m concerned that taking the spherical volume controlled by each star and simply multiplying by the number of stars might assume perfect stacking, or might assume linear stacking like the example shown, and any rounding error multiplied by 1200 is going to be significant.
So let’s do it in exactly the way we derived the size of a 3600th-sector.
7 million million cubic light years divided by 8,200,000 = 853658.5366 cubic light years each, =
a cube of sides 94.86 light-years across. Call it 95 light years for convenience.
Aside from the dimensions and proportions, the diagram representing a sector hasn’t changed.
α-to-β ^2 = 95^2 + 95^2 = 2 &Times; 9025 = 18050;
α-to-β = 134.35 light years.
α-to-γ ^2 = 95^2 + 134.35^2 = 9025 + 18050 = 27075;
α-to-γ = 164.545 light years.
At 10c, that’s 16.4545 years.
At 100c, that’s 1.645 45 years = 20 months..
At 1000c, that’s 0.164 545 years = 2 months.
At 10,000c, that’s 0.016 4545 years.= 6.010006125 days.
At 100,000c, that’s 0.001 645 45 years = 0.6 days = 14.424 hrs.
At 1,000,000c, that’s 16.4545 years = 0.06 days = 1.44240147 hours, = 86.544 minutes.
At 60,100c, that’s exactly 24 hours.
The Starfleet Problem
So, we have 8.2 million sectors that need Green Lanterns. Most need only one, but a significant number need between 1 and 20, and a significant number can’t supply even one, and so need to “borrow” one from one of the sectors with multiple GLs. Which means the average of those higher sectors isn’t going to be 10.5, it’s going to be more like 11.5 or 12.
If 20% of the sectors need to provide 12 GLs and 20% provide none, on average, that’s a total of 24.6 million GLs that need recruitment and training. Once trained, they need to maintain their proficiency, so that’s a further training burden.
How long does the average Green Lantern last? Maybe 20 years, maybe less? That means that 1.23 million need to be trained every year. And, if they have to renew their qualifications every 5 years, but that takes a fiftieth as long as the training, that’s another 0.0984 million ‘trainees’ a year. Total: 1.3284 million.
How many trainers are there to a trainee? How much allowance has to be made for trainees that wash out? How many administrators and other support staff are needed?
This brings us headlong into the Starfleet problem.
There is an episode of the Next Generation which follows Wesley Crusher to his being tested for entrance into the Starfleet Academy. Four gifted students have been preselected, but there’s only one space available. The other three are out of luck – for this year’s intake.
If you have an organization like Star Fleet, you are going to get millions upon millions of applicants per year – if not Billions. If there are 3,000 inhabited star systems in the Federation (a number plucked out of thin air) with an average of 1,000,000 inhabitants each (another number plucked from the ether with absolutely no justification), that’s 3000 million people. Earth alone, even after the calamities in the Star Trek history, is likely to have at least that number, and so are a number of other worlds. Kronos (the Klingon home world) and Vulcan come to mind, for example. All up, a minimum population of at least 12 billion people, and potentially considerably more.
If one percent a decade apply, that’s 120,000,000 applications, or 12 million a year. And if only 1% pass pre-application screening, that’s 120,000 applications. For how many openings? 30,000? 20,000? Ten?
It’s clear that the producers and writers of the episode in question had thought about this, and hence the 1-in-4 cut-off.
But here’s the rub: There is no certainty that the applicants from Moomba-III that are accepted are better than the applicants from Nonga-II that were rejected.
Starfleet is not an elitist organization, it’s not geared to recruit the best of the best – it’s geared to reject the excess while distributing it’s representation as broadly as possible.
And yet, in virtually every episode of TNG, and DS9, and Voyager, and more, Starfleet is portrayed as being the best of the best. So, while the portrayal of the recruitment process is logical, but flawed, it is also inconsistent with the portrayal of the organization outside of this episode.
The Starfleet problem is how do you recruit the best of the best when they are scattered throughout the Federation?
If instantaneous communications galaxy-wide are possible, as shown in Star Trek’s various incarnations, it becomes possible to do so – but that invalidates the entire premise of the drama within the episode in question. For this reason, I’ve never considered the episode as canonical; it falls through a logic hole.
The Green Lanterns – do they have such instantaneous communications? Some adventures suggest yes, others suggest no.
A bigger problem, though is the logistics required to actually train that many recruits. And house them. And feed them.
The Logistics Of Galactic Organizations
And therein lies the problem. These calculations, for the first time, create a practical appreciation of the size of the galaxy, and hence of the size of any galaxy-wide organization. And the results just don’t fit with the descriptions of those organizations in science fiction and other media.
What’s more, the questions scale – they apply just as reasonably to am organization like Star Fleet, even though that organization only operates in somewhat less than one quadrant of the galaxy.
They would scale to the local interstellar region, where small empires of 50-100 star systems might exist.
You can even scale them to be appropriate to an empire or kingdom in D&D terms – the questions are similar (small communities instead of stars), and the results are just as valid.
Once you can get a handle on the scale of your organization – be it a thief’s guild or a multinational church or the political organization of a nation – you can start to properly consider the logistics that are necessary for that organization to function.
There’s going to be an inherent logic that makes obvious sense to you. The consequences may well be surprising – who saw 8,200,000 sectors coming? – but they will be valid, and that will show.
Or, more accurately, the flawed extrapolations of incorrect assessments of scale will no longer be visible – romantic notions like 3600 sectors that look good on paper but make no sense in reality.
Questions Of Scale
But what, you may be wondering, if my assessments of the frequency of population of inhabited worlds is wrong? What if there aren’t 60-odd inhabited systems in a collection of 1200 stars, but only 30, or 20?
Obviously, the size of sectors would increase somewhat – but not be very much; distance between solar systems is unaffected, and that imposes a hard limit on what sounds plausible. Even 60,100 times the speed of light is pushing credibility to the limit.
Distance matters far more than most people appreciate. That’s why improvements in the technology of moving things around tend to have massive national and international repercussions; this is one of the most under-appreciated pillars of society.
If there’s one lesson from history that should be learned by all, it’s this: When people can do in days what would have taken weeks or months previously, society begins to change. When people can move freight around at the same pace, the transformation of society becomes inevitable.
- When humans had to carry everything on their own or their animal’s backs, mobility was limited, and so was the size of society.
- When the Romans introduced roads, it became far more efficient to move goods and people around. While carts had already existed, this was the change that enabled Empires to form.
- The age of Sail made international travel and commerce possible beyond one’s immediate neighbors.
- The age of Steam brought profound social impacts that altered every aspect of society, either directly or indirectly.
- The aircraft completely changed the rules of such trade. We’re still discovering and reacting to the ramifications of that – the most recent lesson being disrupted supply chains.
- But already, we can see the age of air freight coming to an end – not because of a lack of fuel, as was once thought to be the likely problem, but because of the climatic consequences. It seems likely that some reversion for cargoes of lesser importance will take place – unless we invent some sort of teleportation, of course.
Distances matter, and distances are a reflection of the proper appreciation of scale. This article has given everyone the basic tools that they need, and shown how to apply them; I consider that to be a very good day’s work.
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