8.1.2 Romulus Wasn't Built In A Day (Bonus Post)

In which two academics expand the boundaries of maths by expanding the boundaries of the Romulan Star Empire.

Krause’s Kranium

Let's start with an introduction. This post was co-written with Dr Andrew Krause, who works in the Department of Mathematical Sciences at the University of Durham. Andrew is also co-creator of the VisualPDE website we use here to visualise the mathematical models discussed in the post (as well as a bunch of other science and mathematics you can find on that website). Andrew has been entirely cool about me going to his office to talk excitedly about my half-baked theories about an entirely imagined universe, and turning them into mathematical models, despite not even really being a Trekker.

The result of these conversations was this post, which takes a model for the spread of an interstellar civilisation proposed by William Newman and Carl Sagan (yes, that Carl Sagan) in 1981, and uses it to consider the expansion of humanity and Romulans in the Beta Quadrant, and what happens when they meet. To our knowledge, this is the first time this has been done - indeed the application of the Newman/Sagan model to a visual PDE programme would seem to be novel more generally.

We will use the historical context of this model as our starting point. The key question with which Newman and Sagan were concerned was the speed at which a hypothetical interstellar society would expand. This in turn was part of a larger academic conversation about whether the fact humanity had not (and still has not) become aware of such a culture can be considered evidence of absence. Simply put, the faster we might expect an interstellar civilisation to expand, the more the fact that no such civilisation’s expansion has reached us suggests those cultures do not actually exist.

Newman and Sagan argue in their paper that other models (for instance, that of astronomer Eric M Jones) dramatically overestimate the speed of such an expansion, and hence, argue too strongly against the possibility of interstellar societies. In particular, they objected to models which suppose a central system launches repeated waves of colony ships to deal with otherwise uncontrollable population growth. Their argument against such a model is that the amount of effort required to achieve interstellar travel, particularly in a safe and sustainable form, is massively greater than that needed to lower the birth rate. As a result, the paper assumes that, far from interstellar travel being a plausible method of population control, population control is almost certainly a prerequisite to interstellar travel. The need to achieve zero population growth (ZPG) will almost inevitably arrive in a civilisation’s development earlier than whatever scientific breakthroughs are needed to make travel between the stars possible, and focus on the Malthusian issues of population expansion is more likely to delay the development of interstellar travel than it is to expedite it.

We recognise that this assumption, while surely extremely plausible in our own reality, is of course not one used in the universe of Star Trek itself. We shall return to this issue later in the post, but for now we simply note that the Federation, at least, is not expanding due to the demands of a rising population, and all suggestions are that the Earth of Trek is, if anything, even less populous than the one we live upon today.

Obviously, ZPG is only a concern for a world in which further population growth is not desirable. For newly colonised planets, it makes perfect sense that the population will keep increasing. This is only true up to a point, however, represented in the model as the carrying capacity of a planet. The assumption of the model is that a planet’s population will grow until this capacity is reached, at which point it will stop, as whatever policies and processes are used by this society to ensure ZPG come into play. Population growth within this civilisation therefore happens only on comparatively newly-colonised worlds, where the carrying capacity has not yet been reached. In order to keep the model simple, it is assumed that each newly colonised location has the same carrying capacity, though relaxing such an assumption is relatively straightforward if one has some idea of the distribution of planets in terms of what sized populations they could support.

These assumptions are neither to say that birth rates drop to zero on planets reached longer ago, nor that emigration from those worlds will not be part of the mechanism by which ZPG is maintained. The assumption is simply that, except for the most newly-reached star systems, the birth and immigration rates match the death and emigration rates balance out in each system, and that emigration is not the only method by which this balance is achieved. 

The model Newman and Sagan propose, then, does not assume either the need for nor the fact of widespread migration from a central point. Instead, interstellar travel is considered as being primarily (comparatively) a short-range affair. The paper considers such travel as being essentially random in nature, taking place according to the ultimately unknowable whims of individual people and the ships they pilot. Fortunately, our inability to predict the behaviour of each individual ship does not prevent us from being able to consider the likely effect in aggregate, just as how we need not understand the individual path each water droplet will take to know how water will run through a channel. This idea that unpredictable individual elements can be well-predicted in terms of their behaviour in the aggregate is crucial to the mathematical approach known as diffusion models, one example of which the Newman/Sagan paper presents and discusses.

In this specific example, the behaviour of spaceships is modelled as follows. Spaceships can choose to move in any direction at any time, completely irrespective of where it has moved before – this last part is known as the memoryless property. This might seem like a ridiculous assumption to make, but it’s been proved to be a useful way to model all kinds of movement, from the individual molecules inside a fluid to, rather more pertinently, the spread of muskrat populations across hospitable environments. In particular, this assumption results in the desirable property that the population overall spreads outwards – the high density of populations on planets compared to the vast expanse of uncolonised space essentially guarantees that, overall, we see people spreading outward until they reach new star systems, start to reproduce there, and start the whole process over again. We might think of this as putting a few drops of food colouring in a bowl of water – the concentrated droplets spread outwards, colouring more and more of the surrounding liquid. Yes, the tint gets fainter and fainter as the food colouring becomes ever more diffuse, but in our case we keep periodically adding additional droplets (each time a new colony is founded).

What this means is that, even if we consider each spaceship to be travelling in completely random directions, overall, ships will tend to move away from population centres rather than toward them, and they will do this, as with the muskrats this model can also be applied to, without any conscious desire to get away from it all.

As Newman and Sagan argue, though, “extra-terrestrials are not muskrats”, and neither are humans. We do have a sense of when things are getting a little too crowded in the abstract, as opposed to simply needing to travel further for resources when those resources are being used up by a large local population. The paper therefore introduces an additional assumption, which is that spaceflights become more common as the local population increases, and in a non-proportional way – doubling the local population results in more than double the number of spaceflights. This behaviour, combined with the diffusion principle, describes how our interstellar society grows. Spaceship trips occur across the civilisation, but generally only over comparatively short distances. We don’t see long trips from long-established colonies out into the frontier, rather that frontier is pushed outwards by increasingly frenetic short-range flights; drops of food colouring into water happening ever-more frequently as the surrounding water becomes ever-darker.

We can now divide our interstellar society, at any given time, into two areas. Any star system from which a ship cannot (given our assumptions about range) directly move to a system not at carrying capacity is referred to as being in the centre. Any system from which a ship can make such a direct trip is referred to as being on the border. In the former area, spaceships are equally likely to travel in any direction, and those trips overall make no difference to the population numbers – people leave (or die) at the same rate they arrive (or are born). At the border, though, ships are more likely to head towards less populous areas than more populous ones, which drives the outward expansion of the civilisation. Once a given system is populated, it becomes part of the border, and both population growth and emigration to other nearby systems will begin, with the latter focussed on systems even less populous. Eventually, that system will reach carrying capacity, as will all other nearby systems. Once that happens, the system will join the center, and ships leaving it will be equally likely to fly in any direction at all.

Note that the terms “centre” and “border” imply a single, unbroken area for our modelled civilisation, containing no lacunae, or areas of low population surrounded by systems at capacity. In fact, in order for this to be true we require two assumptions. The first is that each star system is habitable, which is an assumption already made by fixing each system as having an equal carrying capacity. The second is that population growth progresses identically in each system. That is, the time it takes for a system to reach carrying capacity is not affected by any specific local conditions. While this may in fact vary substantially in practice, if we zoom out far enough and consider broad averages across vast regions of space, then we do expect some level of same-ness (the technical term for this being ‘isotropy,’ which interestingly is also the fundamental claim of the cosmological principle stating that the Universe is on average isotropic). Assuming that star systems are, on a big enough scale, isotropic, then these assumptions above actually do become valid, and the model described an accurate representation of spreading of the population.

Pale Blue Dots

The result of these assumptions is a model predicting a narrow wave of expansion, radiating more or less symmetrically from a single region. Ahead of the wave is the emptiness of unclaimed space, and behind that wave is the unchanging population density of systems at their carrying capacity. Although it is perfectly possible to consider this model in the three dimensions within which the Milky Way actually resides, we will focus on how the model describes civilisational spread across the galactic plane, a two-dimensional representation of the galaxy, with a third dimension used to represent the population size at a given point.

At this point, we can introduce an analogy for how the model functions. Imagine that, in place of an approximate 2D map of the Milky Way on its bottom, with each star represented by a distinct dot (approximate because of the possibility of multiple stars sharing coordinates on the galactic plane, and differing only by height), we thought about trees in a forest. The trees, for whatever reason, are extremely sparse, and between them lies a vast expanse of grass or scrub. We can think of the trees as being analogous to star systems, with the lower-level vegetation surrounding them being the colossal tracts of interstellar space.

Now imagine someone walks to the tree representing our own solar system, and sets it on fire. As the tree burns, the fire will spread. Individual sparks/elements of flaming matter will travel more or less at random, according to the vagaries of air movement around the tree, but the spread of the fire itself will generally be away from what is already on fire, in favour of what is still to catch alight. Accordingly, the fire will spread to the surrounding vegetation (we will discount the possibility of fire-resistant vegetation, the possibility of rain, etc.). The vegetation on fire will then set more vegetation on fire, in an expanding wave that is roughly symmetrical in all directions. Eventually, the fire will reach another tree, which will catch alight in the same way as the original one, repeating the process. Since trees have more material to burn, each tree set alight will cause the fire to spread much faster than the ground-level vegetation which is carrying the flames from tree to tree.

Eventually, in any given location, the fire will run out of material to burn, and the fire will go out. This is analogous to a population reaching carrying capacity, and population growth ending. It is, we accept, rather grim to think of full population as being equivalent to burned out vegetation, with all possibility of growth and life at least temporarily impossible (though those implacably opposed to colonialism might well argue the analogy is rather more appropriate than many others would like to think). Either way, the analogy is limited in that fully burned-out locations will no longer see movement of spark or flammable elements, whereas in the Newman/Sagan model, interstellar travel within the civilisation’s centre remains common, it simply makes no difference to the overall population density. That said, the ideas of expansion, temporary growth (towards in this case “fully burned out”), and a general movement away from areas of greater growth (“more burned”) which are all present. So too is the idea that the more burned an area is, the faster the fire spreads (because of the heat generated by the growing fire, rather than because of the amount of burned material itself), as is the idea that it’s the star systems (trees) which are the primary drivers of the fire’s spread, and that spread will be fastest in the hottest areas.

There is another ancillary benefit here, which is the speed at which a tree burns will first increase, as more and more of it catches fire, and then decrease, as more and more of the available material to burn is burned. This is analogous to how Newman and Sagan assume the rate at which a population is increasing itself increases in the early stages of a colony’s development, only to start slowing down as the carrying capacity gets ever-closer, and maintaining population growth becomes ever-more difficult due to the limitations of the environment.

All clear? Good. We’re close to understanding what we call the discrete model, which thinks of individual stars as being the points at which population growth occurs. Newman and Sagan don’t actually use the discrete model, though Instead, they use a continuous model, which assumes people are just as likely to settle in interstellar space as they are within star systems. To stick with our analogy, the Newman/Sagan model assumes a forest containing nothing but trees (or equivalently, plains/scrubland/whatever that are entirely tree-free).

This is, obviously, not actually how the colonisation of space would work. What Newman and Sagan demonstrate, though, is that working as though it is makes their model far simpler.  Further, so long as we’re working with portions of the galactic plane containing stars in the thousands or higher, his continuous model will generally behave similarly compared to a much more complicated model using distinct star systems with vast spaces in-between. There is an exception to this; we would expect very different results between the discrete and continuous models if the width of the expansion wave is much narrower than the average distance between stars. This is because the continuous model allows for an agonisingly slow creep of habitation between systems as civilisation works to reach one star from another, with those habitations experiencing their own population growth. The discrete model, in an equivalent case, will allow no population growth between star systems until the system has been reached. In a case where the expanding wavefront takes a long time to travel between stars, then, the continuous model will handle population growth in the border region very differently to the discrete model.

Happily, Newman and Sagan are able to modify their model to compensate for this problem, and, even more happily, the existence of warp travel within the Trek universe means concerns that the expansion wave will be measured in fractions of a light-year can be discounted. 

Visions Of The 22^nd Century

The assumption of access to technology allowing travel at many times the speed of light does throw up other issues regarding the Newman/Sagan model. We’ve already mentioned the fact that in Trek, humanity at least discovers how to travel at light-speed without having figured out how to sensibly and sustainably achieve ZPG (though the horrifying death toll of the Eugenics Wars presumably made curbing population growth very much an issue of low concern during the period Zephram Cochrane was tinkering with the Phoenix). It also renders moot their argument that interstellar expansion would be much slower than previously estimated – even a Warp 5-capable starships Starfleet were beginning to use a few years before war with the Romulans measured their interstellar journeys in weeks, if not days. [1]

Ultimately, though, this issue need not concern us. Unlike Newman and Sagan, our interest here is not in how long it would take humanity, or any other race, to expand their holdings to a given size. What we are interested in here, rather, is what we might expect to see when two starfaring civilisations with different rates of growth encounter each other. 

The three main drivers of the spread of a civilisation through the galaxy are the population growth rate in areas which haven’t yet reached carrying capacity, what that the carrying capacity is, and what is known as the diffusion coefficient, which we will not consider in detail (that would be to very much take the “pop” out of “pop science”), but which can be thought of as describing the rate at which the civilisation is spreading into the areas it can spread into (that is, areas which are adjacent to occupied areas, but which have not yet reached carrying capacity). The diffusion coefficient in turn can be thought of as an expression linking how likely people are to want to emigrate to a new star system, how far apart on average star systems are, and how frequently new colony ships/fleets are being launched – basically, how frequent, quick, and attractive the opportunity for extra-stellar migration becomes. The larger this coefficient, the quicker the spread of a civilisation through interstellar space.

In our case, it is the ratio between the diffusion coefficients of humanity and Romulans which concerns us. Our assumption, based on the previous post, is that the Romulans have a markedly lower diffusion coefficient speed_r than that of humanity, speed_h, at the time the two powers meet. We further assume that, at some point after contact, the Romulans increase their diffusion coefficient as a response to the outbreak of war, and the fear of being outflanked to the galactic south. We further assume the new Romulan coefficient is even greater than that of Starfleet, given their risk of encirclement, and because this helps explain why the technologically superior Romulans ultimately lost the war.

The parameters our model uses, then, are as follows:

speed_r, the original diffusion coefficient for the Romulans;

speed_h, the diffusion coefficient for humanity/Starfleet;

T_h, the time (which we do not give a unit to) at which humanity starts spreading out from the Sol system (with the Romulans having started doing the same at time 0);

T the time (again without units) at which the Romulans change their approach to colonisation, in response to encountering Starfleet;

speed_rh, the new diffusion coefficient for the Romulans.

We also use (x_h,y_h) and (x_r,y_r) as coordinates for Earth and Romulus, respectively.

One set of values for the above are shown in the version of the model we have embedded here (Starfleet is in blue, the Romulans are in white, and the Klingon Empire - assumed constant here to keep things simple - is in purple).

As can be seen in this model, the resulting shape of the Romulan Empire is in line with the star charts seen in Picard and Section 31, an ellipse with Romulus itself toward one end. What we don’t see is the long axis of the ellipse lying perpendicular to the line between Sol and Romulus. This is due to this model not taking into account the fact that expansion into the unclaimed space just beyond human holdings would have been prioritised over expansion in directions away from the widening front line. Later models may take this into account. We might also want to consider humanity’s own response to first contact with what was to become their oldest enemies.

That’s for a different post, though. For now, feel free to click through to the model itself, and use the f(x) button (top left) to change any of the parameters listed above to create your own map of the Beta Quadrant, “Year Of Hell” style.

Have fun!

[1] Fellow dabblers in Trek stellar cartography will of course know that attaching consistent speed estimates to warp values is the kind of quixotic effort that makes working to understand star dates seem positively sensible.