The fighters launch from their carrier! They jink and turn -- shooting bogeys down as they go. Finally, they approach the enemy capitol ships. They fly close, fire torpedoes -- and escape, leaving behind billowing fire. Wait. Is this World War II or a space opera confection?

In this article, we'll consider how the size of ships affects how they can fight -- and come to some surprising conclusions.

Throughout science fiction, we are treated to familiar tropes of war: the large ships with heavy guns, carriers, fighters, andd bombers. Their modes of attack, sizes and even names, are borrowed from recent history. Perhaps we have George Lucas to thank for this, more than anyone. Certainly, he has the artistic license to make space look like the Pacific. (Other periods could give intriguing inspirations: space like a medieval siege, perhaps?)

But the images we have raise many questions. Technology made warships and fighter planes possible, but basic physical laws establish what differs between a large ship and a small ship; and what advantages lie in either design. The benefits of being large seem obvious: you can carry more guns, more marines, and more fuel. But is one big ship better than two smaller ones, each half its size? Size changes many things, often subtly; we'll explore a few of them below.

### Insides to Outsides

We face one problem immediately. What does it mean for a ship to be big? Is it bigger in length? In volume? In mass? These may seem to be equivalent, but they are not.

To simplify things, consider the size of a square. On the left is one with sides of length 1 and an area of 1 (in feet or whatever units you like, it doesn't matter). We want to build a bigger square so, on the the right, we double its length to 2. What happens to its area? We can count 4 units. This is not a doubling, but a quadrupling! How surprising! When things get bigger, they do not get bigger in all ways at the same rate. This is the heart of scaling.

How can such a thing matter? Consider the square as a storage box. Manufacturing a box takes money; but it is not so much the space inside that matters, as the edges. The square on the left might cost $4 to produce because the length of its sides (its perimeter) is 4. We could use four such small boxes to pack four units of goods at a cost of $16. The box on the right could do the same task, but much more cheaply: its perimeter is 8, and thus it costs $8. Larger boxes contain relatively more interior space than do small ones, relative to their outside surface.

#### Another dimension

This trend becomes even more extreme when we consider 3-dimensional cubes instead of squares. The figure below shows a small cube with length 1 sides and a large one with length 2 sides. How do volume and the surface area scale up with length?

As shown in the table, both surface and volume increase with length, but volume increases faster; thus, the ratio of volume to surface increases.

Length | Volume | Surface | V:S Ratio |
---|---|---|---|

1 | 1 | 6 | 0.17 |

2 | 8 | 24 | 0.33 |

3 | 27 | 54 | 0.5 |

#### Notation

These rules of scaling can be characterized algebraically; and even if you don't plan on doing any calculations, these short-hands are useful. Volume can be calculated by multiplying all three dimensions of an object; for cubes, these are the all the same length, denoted by *l*. Thus volume:

V = l^{3}

Surface, meanwhile, is the area of each side (for squares, length times length) multiplied by the number of sides (6). We aren't really interested in calculating exact quantified though, only thinking about how surface is related to length; so the fact that it is six times larger because of being a cube can be ignored. Therefore the equation below is not one of equality, but a statement of proportionality, which is good enough for us:

S ∝ l^{2}

We can also show mathematically how much faster volume increases than does surface area (recalling the rules for exponents): it increases faster by one unit of length.

V ÷ S ∝ l^{3} ÷ l^{2} = l^{1} = l

#### The benefits of a big ship

This could mean various things for spaceships, depending on our assumptions about technologies. But consider that volume grants a ship much of its utility--be it in hauling cargo or carrying weaponry--while surface area offers mostly liability: it is the shootable part of a ship, and may be especially costly. Unless heat dissipation is very important, or exotic shielding technology depends on surface area, there are benefits to being large.

### Armor at the Edge

Starker results emerge when we consider armor. If we scale a ship by increasing its length, the lengths of all its components increase in lock-step, and so do their volumes. All ships thus devote the same portion of their volume to armor. But armor effectiveness depends on thickness, not volume, and this scales exactly with length (thickness ∝ l^{1}). So large ships have better armor than small ones:

#### Fighting Back

But armor is ever in a race with weaponry. Just how useful will doubled armor thickness be? It depends on the scaling of weapons, and particularly on their penetrative ability; which hinges on the technologies we imagine being used--lasers, missile, or something stranger.

Let's assume a minimum scaling effect of l^{1}; perhaps because the length of a ship's rail guns determines their power. In this case, weapon penetration scales to match armor thickness. There might still be an incentive to engineer ever-larger ships, because they would be immune to smaller ones, but all ships would function similarly.

Alternately, maybe if energy weapons are used, penetrative ability might scale with volume (l^{3}). That would mean that armor (a) does not keep pace with weapon power (p):

p ÷ a ∝ l ÷ l^{3} = l^{-2}

In such a case, large ships would be relatively more vulnerable to damage (from one another) than small ships, leading to highly destructive battles where aggressive offense and initiative predominate. Unless, of course, the ship engineers try to compensate for the effect, and add extra armor to large ships. If armor thickness were forced to scale with volume, though, a large ship would use more and more of its internal space for protection, leaving it with very little else:

This would place a cap on ship size, and diminish the other advantages of size. The trade-offs in engineering are outside the purview of this article, but some of the problems are obvious.

(I've ignored the fact that large ships can suffer more damage and still function. This is because they presumably pack more guns as well. Both these relationships should scale roughly with volume, canceling each other out.)

### Big and Slow?

One visual image that sticks with us is that small ships are fast and big ones are slow. On Earth, vessels in the water inevitably experience this problem because of drag (clearing the water out of the way, having it brush against the hull, and vortices that form around the back). But in space, there is no drag: all that matters is your engine power and how massive you are.

This is described by the familiar Newtonian formula, F = MA, where A is acceleration and F is the force generated by the engines. Mass, M, scales with volume, or l^{3}. Presumably, engine power does too, as the engines scale proportionally with ship size:

M and F scale identically with length, so acceleration does not depend on size. There is no intrinsic reason why large ships should be any slower than small ones.

#### Maneuvers

Similar logic applies to maneuverability, although the equations aren't as simple--skip the details below if you like. If a ship want to jink or change direction, it has to rotate. Rotation is described by an equation like that for straight movement:

α = τ ÷ I

Where α is angular acceleration, I is the ship's moment of inertia (describing how hard it is to rotate) and τ is the torque applied by thrusters. Torque is generated by thruster power multiplied by how far the thrusters are from the center of the ship, which depends on its length. Just as with engines, we can assume thrusters are made larger, and more powerful, with the rest of the ship:

τ ∝ F × l = l^{3} × l = l^{4}

The moment of inertia for a cube is proportional to its mass times its length squared, or l^{5}. Substituting I and Τ into the original equation:

α ∝ l^{4} ÷ l^{5} = l^{-1}

What does this mean? Thrusters can't keep pace with the increasing mass of large ships, making them less maneuverable. How much less? If a ship is doubled in size, it will be half as quick to turn.

### Ships at War

A picture of ships in space begins to emerge. While radical technologies might undermine or reverse any of these trends, they can be taken as the baseline. Ships in space will not behave as ships in the sea, and the disparities between small and large ships will be different too. What are these disparities? Consider what we gain and lose by building a larger ship than an enemy:

- Speed is unaffected. The large ship is no faster or slower than the small.
- Maneuverability is hurt, though not drastically (inversely proportional to length).
- Its armor gets a little thicker, and it presents less of a target, relative to its capabilities.
- The penetrative abilities of its weapons keep pace with, or exceed, the gains in armor. Large ships are safe from smaller ones, but must prepare for equally damaging, or more damaging, battles with peers.

As I said at the beginning, it's not my place to excoriate any imagined sci-fi world for being unrealistic because it doesn't adhere to these principles. Technology, economics and military tactics can reasonably be invoked to support any scheme. But neither is there reason to always stick with the tropes we've inherited. Capitol ships might be faster than small ships, and many interesting tactics could flow from that fact. Science fiction has never been a slave to physics, but it should take inspiration from it.