Closure
An open bubble gets steeply more expensive as it nears light speed, so practical open (Level 1) drives top out around 1% of c. To go faster, the bubble wall has to fold over into a horizon. A closed bubble is cut off from the outside universe: nothing inside can steer it, signal out of it, or see ahead of it. Each leg is a throw, not a drive.
Closing needs room. The horizon can't form with a radius under 15 m in its narrowest direction, which is why the smallest Level 2 ships are bus-sized, and why small craft hitchhike on carriers.
$$a \ge 15\ \text{m}$$Bubble geometry and stacking
Two inputs set the bubble. Bubble size \(s\) is the radius of a spherical bubble that would hold the ship, or the carrier and its hitchhikers, in units of the 25 m reference bubble. Bubble shape \(\lambda\) is its length-to-width ratio, from 1 (a sphere) up to about 3, the most a horizon will tolerate.
Lego ships are long and thin. Crammed into a sphere they waste most of the space; lined up nose to tail in a cigar-shaped bubble they pack tightly, so a stretched bubble needs less volume to carry the same ships. A lone ship has nothing to stack, so the benefit phases in as the bubble grows from one ship's worth (\(s = 1\)) to a carrier's (\(s = 2\)):
$$V = V_s \, \lambda^{-\gamma(s)}, \qquad V_s = \tfrac{4}{3}\pi\,(25\ \text{m} \times s)^3, \qquad \gamma(s) = 0.6 \times \operatorname{clamp}(s - 1,\ 0,\ 1)$$The bubble is a prolate spheroid of that volume, with short radius \(a\) and long radius \(c = \lambda a\):
$$a = \left(\frac{3V}{4\pi \lambda}\right)^{1/3}, \qquad c = \lambda a$$What costs power is the bubble wall. A stretched bubble has more wall than a sphere of the same volume, so the drive equations use \(R_A\), the radius of a sphere with the same surface area:
$$A = 2\pi a^2\left(1 + \frac{c}{a\,\varepsilon}\arcsin\varepsilon\right), \quad \varepsilon = \sqrt{1 - \frac{a^2}{c^2}}, \qquad R_A = \sqrt{\frac{A}{4\pi}}$$So stretching trades extra wall for tighter packing. For a lone ship it's a pure cost, and a sphere is best. For a carrier with hitchhikers, the packing wins and a 3:1 cigar cruises around 30% faster than a sphere. Every bubble is also limited by the 15 m closure rule, so small bubbles can't stretch far.
Power and the speed wall
A closed bubble has no top speed as such. Power rises in step with speed at first, then climbs without limit as the bubble nears a wall. Each bubble has a reference speed that the standard 5 GW reactor reaches, set by its wall area and its bubble factor \(\eta\):
$$v_\text{ref} = \eta \times 0.1\ \tfrac{\text{ly}}{\text{h}} \times \left(\frac{25\ \text{m}}{R_A}\right)^2$$With \(u = v / v_\text{ref}\), the power needed to hold speed \(v\) is
$$P(v) = P_a\,\frac{u}{1 - (u / u_w)^n}, \qquad P_a = P_2\left(1 - u_w^{-n}\right), \quad P_2 = 5\ \text{GW}, \quad u_w = 1.15, \quad n = 8$$The ship cruises where \(P(v)\) equals its reactor output. Because of the wall, doubling the reactor buys only about 9% more speed and burns fuel twice as fast. Ships fly at roughly the speed they do not because they can't go faster, but because going faster quickly stops being worth it.
The bubble factor
The bubble factor \(\eta\) measures how closely a drive's field matches the ideal wall profile. It scales the speed a given power buys. It is also very hard to raise: each step above 1 demands a far finer field geometry than the last, and the gains are fought for in thousandths.
| Bubble factor | What it means |
|---|---|
| below 0.95 | Worn, damaged or crude drives |
| 0.95 to 1.02 | Standard production drives |
| 1.02 to 1.06 | Precision drives: couriers, racers, the military |
| 1.06 to 1.12 | Exceptional: only a handful exist |
| above 1.12 | Theoretical: never built |
Fuel
In flight the reactor runs flat out, burning deuterium at a rate set by its output \(P\). A 10-minute, 5 GW burn uses about 10 g of deuterium, which gives \(\epsilon \approx 3 \times 10^{14}\) J/kg of usable energy:
$$\dot m = \frac{P}{\epsilon}, \qquad t_\text{endurance} = \frac{m_\text{fuel}}{\dot m}, \qquad \text{range} = v\,t_\text{endurance}$$When the tank runs low, a ship refuels by scooping from a star's atmosphere, which takes about 12 hours.
Heat
A closed bubble slowly heats its interior, in proportion to its wall area. Radiators still work inside the bubble, but poorly, since the heat has nowhere to go but the hot interior. They run at full power only once the ship surfaces:
$$H_\text{soak} = 50\ \text{MW} \times \left(\frac{R_A}{25\ \text{m}}\right)^2, \qquad \dot Q_\text{flight} = H_\text{soak} - \kappa D, \qquad \kappa = 0.3$$Here \(D\) is the radiators' rated output. When stored heat reaches the ship's capacity \(Q_\text{max} = 10\ \text{TJ}\), it must surface and stay down until it has cooled:
$$t_\text{cool} = \frac{Q}{D}$$Legs and star fixes
Navigation is dead reckoning, with an error of about 1 part in 100,000 of the distance travelled, so every ship surfaces at least once per light year to take a star fix. The fix takes about 1.5 hours, and the radiators dump heat while it's taken:
$$\Delta Q_\text{fix} = D \times 1.5\ \text{h}$$Bow load
A closed bubble sweeps interstellar gas into its front wall and carries it along. Every time the ship surfaces, that bow load is released as a flash of gamma rays and particles. It depends on the bubble's head-on cross-section, so a slim, stretched bubble arrives more quietly:
$$m_\text{bow} \approx 6\ \text{g} \times \left(\frac{a}{25\ \text{m}}\right)^2 \text{ per ly}, \qquad E = m_\text{bow}\,c^2 \approx 130\ \text{kt TNT per 6 g}$$