The Breakthrough Starshot System Model

The inputs to the mission point design are summarized in table 1. A $1.06\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength is consistent with ytterbium-doped fiber amplifiers. An initial sailcraft displacement of $60\,000\text{\,}\mathrm{km}$ is consistent with a low-thrust non-Keplarian orbit [20, 21] that keeps the sailcraft (and associated spacecraft) stationary in the sky relative to the target star.

A $1\text{\,}\mathrm{g}$ payload is bookkept separately from sail mass and reserved for scientific instrumentation and associated support systems. Sail mass is calculated by the system model using the value of $D_{s}$ chosen by the optimizer combined with the input areal density. If the sail were an optimal-thickness layer of silicon dioxide, for example, it would have an areal density closer to $0.3\text{\,}\mathrm{g}\text{\,}{\mathrm{m}}^{-2}$, a room temperature absorptance of less than $10^{-10}$ [22], and a normal reflectance of 12%. This point design uses a lower areal density of $0.2\text{\,}\mathrm{g}\text{\,}{\mathrm{m}}^{-2}$, higher absorptance of $10^{-8}$, and a higher reflectance of 70%, consistent with a hot two-dimensional nanohole photonic crystal that is somewhat tuned for high reflectance [23]. Because the system model does not yet incorporate a photonic crystal model, the stratified layer optical model is turned off and absorptance and reflectance remain constant throughout trajectory integration.

A maximum temperature of $625\text{\,}\mathrm{K}$ is placed on the sail to prevent thermal runaway and to prevent damage to non-operating electronic or photonic components that are part of the sail or its payload. An assumed total hemispherical emittance of 0.01 accounts for the energy that is radiated by the sail in both the forward and backward directions at $625\text{\,}\mathrm{K}$. This emittance is much lower than that of typical materials because the sailcraft is so thin and because its absorptance is nearly zero at $1.06\text{\,}\mathrm{\SIUnitSymbolMicro m}$. In comparison, a blackbody at $625\text{\,}\mathrm{K}$ emits most strongly at $4.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This means that to approach the radiative performance of a blackbody, the sail must switch from virtually invisible to strongly absorbing/emitting within as small a wavelength range as possible.

Cost factors are chosen such that beamer capex is less than $10B. In particular, the laser cost is $0.01\text{\,}\mathrm{[}$$]\per, four orders of magnitude lower than the present cost, consistent with nearly automated production and high yields. Automated production lines for microwave oven magnetrons long ago reached this cost. The optics cost is $500\text{\,}\mathrm{[}$$]\per\squared, three orders of magnitude lower than the present cost for diffraction-limited optics, again consistent with nearly automated production and high yields. This value is comparable with the retail cost of computer monitors. $50\text{\,}\mathrm{[}$$]\per energy storage is greater than current materials costs for some, but not all, energy storage technologies, and is more optimistic than current experience curve projections of future electrical energy storage costs [24].

Upon running the system model using the inputs in table 1, the optimizers converge to the values given in table 2. The sail diameter that minimizes beamer capex is found to be $4.1\text{\,}\mathrm{m}$, corresponding to a sailcraft mass of $3.6\text{\,}\mathrm{g}$. At $0.2\text{\,}\mathrm{c}$, this mass has a relativistic kinetic energy of $1.9\text{\,}\mathrm{G}\mathrm{W}\mathrm{h}$, whereas the beamer expends $63\text{\,}\mathrm{G}\mathrm{W}\mathrm{h}$, corresponding to 2.9% system energy efficiency. This energy costs only $6M at a price of $0.1\text{\,}\mathrm{[}$$]\per, making the energy cost three orders of magnitude lower than the $8.0B beamer capex.

The $8.0B beamer capex is comprised of three unequal expenses with, surprisingly, laser cost being the smallest at $2.0B. This is for a $200\text{\,}\mathrm{GW}$ maximum transmitted power, which is far below the petawatt-class lasers now in existence but many orders of magnitude greater in pulse energy (and duration). The stored pulse energy of $63\text{\,}\mathrm{G}\mathrm{W}\mathrm{h}$ contributes the most to the beamer capex at $3.1B, but is only one fortieth of the $2.5\text{\,}\mathrm{T}\mathrm{W}\mathrm{h}$ annual market for electrical energy storage that is projected for 2040 [24], with electric vehicles being the dominant source of demand. Though it would be impractical to convene two million electric vehicle owners and drain their batteries for each starshot, it may be possible to use second-hand or donated battery packs that no longer perform well enough for electric vehicles.

The remainder of the beamer capex is incurred by the optics; a filled array of telescope elements that form a $2.7\text{\,}\mathrm{km}$ effective diameter. This effective diameter is the same as that of the Crescent Dunes solar energy concentrator, located in the California desert. Of course, the array modules for the beamer will be very different from solar concentrator mirrors. When operating at maximum power, the beamer’s radiant exitance is $37\text{\,}\mathrm{kW}\text{\,}{\mathrm{m}}^{-2}$ (a spatial average obtained by diving the power output by the effective area of the primary optic). This exitance is forty times that of the solar concentrator operating at its peak, but three orders of magnitude lower than that of military laser beam directors. In the plane of the sail, the beam converges to a much higher irradiance. By limiting the beamer’s power output, the system model reduces the irradiance at the sail to its temperature-limited value of $8.7\text{\,}\mathrm{GW}\text{\,}{\mathrm{m}}^{-2}$, which is three orders of magnitude lower than the flux of a $1\text{\,}\mathrm{kW}$ laser in a $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ fiber, and five orders of magnitude lower than the non-thermal ablation threshold [25].

The $8.7\text{\,}\mathrm{GW}\text{\,}{\mathrm{m}}^{-2}$ sail irradiance produces only $40\text{\,}\mathrm{Pa}$ photon pressure, equivalent to a moderate breeze. But the sail is very thin and the breeze moves at the speed of light, resulting in $14\,900\text{\,}\mathrm{g}\mathrm{{}^{\prime}}\mathrm{s}$ initial acceleration. Such acceleration is experienced by bullets and artillery shells, but for fractions of a second. Even as the sailcraft reaches $0.2\text{\,}\mathrm{c}$ after $9\text{\,}\mathrm{min}$, it is still accelerating at $2500\text{\,}\mathrm{g}\mathrm{{}^{\prime}}\mathrm{s}$. Such is the cost-optimum truncation point for the trajectory.

During the sailcraft’s acceleration, a satellite that transits the beam in low Earth orbit would see a flash of little more than $37\text{\,}\mathrm{kW}\text{\,}{\mathrm{m}}^{-2}$ for a fraction of second. This is because the high-flux part of the beam is the focus, and the focus is initially past geostationary orbit and only gets further away as the sail moves off. The beam and the ultra-high-acceleration sailcraft, behaving as if a speck of dust in optical tweezers, may be able to dodge satellites in medium Earth orbits or supersynchronous orbits. As is current practice, beam-satellite conjunction analyses would be performed to help schedule times at which lasing is allowable. Also, there would be interlocks to dim or douse the beam if needed to protect unexpected aircraft or flocks of birds. If the interruption were short enough, the trajectory could resume.

The 1-D sailcraft trajectory and associated quantities are plotted as functions of time in fig. 8. This trajectory corresponds to the cost-optimal point design summarized in table 2. Starting at the top left plot, the sailcraft begins accelerating from a distance that would be just past the radius of geostationary orbit. If the destination star were in the plane of the solar system, then the sailcraft would pass the Moon’s orbit within the first minute, and could be a third of the way to Mars⁵⁵5at its minimum distance from Earth by the end of acceleration around the $9\text{\,}\mathrm{min}$ mark. Alpha Centauri is at $-60\text{\,}\mathrm{\SIUnitSymbolDegree}$ declination, so its trajectory would be downwards out of the plane of the solar system.

The sailcraft speed increases almost linearly for the first half of the acceleration time. Thereafter, the speed rises more gradually, until cutoff occurs precisely as $0.2\text{\,}\mathrm{c}$ is reached.

Essentially none of the beam is spilled until the trajectory reaches the $3\text{\,}\mathrm{min}$ mark. Until this time, the beamer power is throttled back to prevent the sail from overheating. In the sail frame, this means that the incident and absorbed power, force, photon pressure and acceleration are all constant. As seen from the beamer frame, relativistic effects cause the apparent acceleration to pull slightly downward as speed increases.

After the $3\text{\,}\mathrm{min}$ mark, the beam begins to spill around the sail as it moves further away. The beamer power gradually ramps up to compensate for increasing losses. Evidently, it minimizes beamer capex to oversize the beamer power. At some point, the beamer can no longer compensate for the increasing beam spillage, and the sail cools. By the end of the trajectory, less than 20% of the transmitted power reaches the sail.

Note that $P_{1}$ as plotted in fig. 8 does not include finite photon travel time between the beamer and sailcraft, hence its time index is distorted. The undistorted plot would be shifted and compressed left, then drop to zero at $8\text{\,}\mathrm{min}$. To avoid the need to post-calculate the undistorted $P_{1}$, pulse energy $Q_{0}$ is obtained by integrating $\left(1-\beta\right)P_{1}$ along with the trajectory.

The sail’s angular extent as seen from the beamer is initially within the pointing stability of telescopes such as the Hubble Space Telescope, but exceeds this as the sail gets further away. Even at the speed of light, the round trip time between the beamer and sail varies from $0.4\text{\,}\mathrm{s}$ at the beginning of the trajectory to $134\text{\,}\mathrm{s}$ at the end. Given also the high sail accelerations, it is obvious that the beamer cannot actively point to follow the sail. Instead, the sailcraft must be beam riding, seeking the axis of the beam by active or preferably passive stabilization schemes.

The Doppler shift plot shows the $1.06\text{\,}\mathrm{\SIUnitSymbolMicro m}$ beam redshifting as seen from the sailcraft frame, eventually reaching $1.30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength. The reflected light again redshifts, eventually reaching $1.60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength. If the sailcraft were to transmit at the $0.85\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength commonly used by VCSEL laser diodes, it would be received on Earth at $1.04\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The sailcraft’s relativistic kinetic energy reaches $6.7\text{\,}\mathrm{T}\mathrm{J}$ ($1.9\text{\,}\mathrm{G}\mathrm{W}\mathrm{h}$). Per unit mass, this is $2\text{\,}\mathrm{PJ}\text{\,}{\mathrm{kg}}^{-1}$. In comparison, the heat produced by Pu-238 alpha decay is three orders of magnitude lower at $2\text{\,}\mathrm{TJ}\text{\,}{\mathrm{kg}}^{-1}$. One way to tap the sailcraft’s kinetic energy could be via the interstellar medium: The Local Interstellar Cloud is primarily composed of $0.3\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{3}}$ of partially ionized hydrogen [26]. From the sailcraft’s perspective traveling at cruise velocity, this manifests as a monochromatic hydrogen beam that is incident from the direction of travel, having a combined kinetic energy of $55\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$, or $0.7\text{\,}\mathrm{kW}$ over the sail’s area if it faces the direction of travel.

In the highly unlikely event that a sailcraft were to collide with a planetary atmosphere, then the energy released would be equivalent to nearly $2\text{\,}\mathrm{k}\mathrm{t}$ of TNT. On average, one asteroid per year enters the Earth’s atmosphere with this energy, though asteroids are orders of magnitude slower and heavier. The sailcraft would vaporize before it got nearly as low into the atmosphere as asteroids do.

Sail diameter is a variable that the system model varies to minimize beamer capex. As shown in fig. 7, it is varied by the GSS algorithm of the outermost iteration (iteration 1). By turning off this iteration, the design space as seen by the optimizer is plotted in fig. 9. Starting at the top left plot, the beamer (total) capex is minimum at the expected sail diameter of $4.1\text{\,}\mathrm{m}$. On every plot, this diameter is marked by a vertical line. The line shows that the cost optimum does not exactly correspond to any other extrema; the optimum is a true tradeoff between storage, lasers and optics, as depicted in fig. 6.

A striking feature in most of the plots in fig. 9 is that the mission parameters vary in a qualitatively different way as sailcraft exceed $8\text{\,}\mathrm{m}$ diameter. This is because smaller sailcraft operate in a temperature-limited regime, whereas larger sailcraft operate in a beamer flux-limited regime. On each trajectory integrator timestep, the system model calculates the sail temperature resulting from maximum beamer power. If this temperature exceeds the maximum, this maximum becomes the boundary condition, and the corresponding beamer power (less than its maximum) is calculated.

Intuitively, one might expect that, relative to the baseline point design in table 2, money is saved by using a larger sail that is slowly accelerated by low power coming from a larger beamer over a longer period of time. This limit corresponds to the right-hand side of each plot in fig. 9. For sails that exceed $8\text{\,}\mathrm{m}$ diameter, the optimum beamer diameter does indeed increase with sail diameter, as does beam duration and range. Consistent with expectations, acceleration decreases in this limit. However, the laser power that is needed increases instead of decreases in the large sail diameter limit, and so capex also increases due to the laser cost and the increased cost of energy storage. Thus, economics does not favor the slow acceleration school of thought.

The $0.2\text{\,}\mathrm{c}$ point design assumes particular cost performance figures of merit for the laser, optics and storage. These are referred to as $k_{l}$, $k_{a}$, and $k_{e}$ in the simple cost model of eq. 26. But what happens if the lasers’ cost performance, for example, is lower than expected? Also, what happens if the material absorbs more, or less than assumed, or the payload is heavier or lighter?

Table 3 summarizes point designs, each of which varies a technology figure of merit relative to the baseline. The table lists the change in beamer capex, beamer diameter, beamer maximum transmit power, and sail diameter, relative to the baseline of table 2. In the first entry, the laser cost per watt is increased by 10 times. The resulting capex increases only by 2.4 times because the cost-optimum power $P_{1}$ halves and the cost-optimum beamer diameter $D_{b}$ increases (increases in transmission efficiency) to compensate. Similarly, increasing the optics cost per unit area by 10 times causes the capex to increase by only 2.4 times. In this case, the cost-optimum solution halves the beamer diameter and dramatically increases the power to compensate. Increasing the storage $/kWh by 10 times causes the capex to nearly quadruple, the biggest increase so far because storage is the dominant cost, as seen in the top left plot of fig. 9. The cost-optimum point design has an increased system energy efficiency, achieved by increasing the product of beamer and sail diameters, as described by the Goubau beam propagation eqs. 1 and 2.

In the happy event that a technology is even cheaper than needed, the capex is reduced less than might be intuitively expected. Reducing laser cost to a tenth of its baseline has the least effect, reducing capex to 70% of its baseline. This is because, as seen in fig. 9, laser cost is subdominant and further reductions do not significantly affect the leading costs of storage and optics. For both optics and storage, reducing their costs to a tenth of the baseline has the effect of halving the capex.

If the sail material absorbs 10 times more energy than expected⁶⁶6Reducing absorptance is a good way to reduce trajectory duration given that the sailcraft is mostly temperature-limited, as seen in fig. 8. Shorter durations might be needed if the beam cannot be interrupted by passing satellites. Absorbing 0.1x increases the temperature-limited irradiance on the sailcraft by 10x, acceleration increases by 10x, and the duration falls from $9\text{\,}\mathrm{min}$ to less than $3\text{\,}\mathrm{min}$., then the capex is more than quadruple that of the baseline, the largest increase of all. In comparison, a tenfold increase in payload mass does not quite triple the capex. The three order of magnitude absorptance range considered here is smaller than the five order of magnitude absorptance range of the current sail substrate material candidates [23]. For this reason, table 3 lists results for higher absorptances at the end.

In all, a tenfold increase in any figure of merit does not increase the capex by nearly as much. Nor does a tenfold decrease in a technology cost decrease the capex by nearly as much. This is because there are three constituent costs that are of comparable magnitudes after cost minimization. The minimization process trades away large increases in one cost for smaller increases in others or all. Therefore, cost minimization dampens the solution’s sensitivity to unexpected changes in technology figures of merit.

Sail reflectance variations are also listed in table 3. They show that if the baseline mission had used a single-layer dielectric sail with a reflectance of 25%, as shown in fig. 5, then the capex would triple. This $18B difference is why the baseline mission uses a photonic crystal sail with a higher reflectance of 70%. It is well known that nanostructuring the sail can improve the reflectance [23], and with such a large financial return, it is inconceivable that this would not be done. Though almost perfect reflectors are possible in principle, table 3 shows diminishing financial returns much above 90% reflectance. Such performance is attainable at a single wavelength, but the trajectory-averaged value will likely be lower as the Doppler shift lengthens the beam wavelength by 20% over the course of the sail acceleration. Also, the sail material’s optical properties may be leveraged for communications, sensing, or processing, and these competing uses would likely be worth a modest decrease in reflectance and/or increase in absorptance during sail acceleration.

Cruise velocity $\beta$ is specified to be $0.2\text{\,}\mathrm{c}$ in the Breakthrough Starshot objectives. Nevertheless, it is interesting to vary this velocity to see how the cost-optimal mission changes. The resulting plots are shown in fig. 10. Each quantity is plotted from $0.001\text{\,}\mathrm{c}$ to $0.99\text{\,}\mathrm{c}$. The beamer (total) capex is shown in the top left plot. Also shown are constituent costs of storage, optics, and lasers. Choosing a mission that has half the cruise velocity, $0.1\text{\,}\mathrm{c}$, decreases the beamer capex to a quarter of its baseline, whereas doubling it to $0.4\text{\,}\mathrm{c}$ quintuples it. That is to say, it costs $29B extra to halve the trip time or saves $6B to double it. In the case of Alpha Centauri, this shortens the $4.37\text{\,}\mathrm{l}\mathrm{y}$ trip time from 22 years to 11 years, or lengthens it to 44 years.

Figure 10 also shows that cost-optimal missions are limited by sail temperature if the cruise velocity is faster than $0.03\text{\,}\mathrm{c}$. This value of cruise velocity corresponds to minimum beam duration. The minimum beamer diameter reaches $10\text{\,}\mathrm{km}$ at $0.7\text{\,}\mathrm{c}$ and $100\text{\,}\mathrm{km}$ at greater than $0.99\text{\,}\mathrm{c}$.

The inputs to the precursor mission point design are summarized in table 4. The $1.06\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wavelength is consistent with ytterbium-doped fiber amplifiers. An initial sailcraft displacement of $300\text{\,}\mathrm{km}$ is consistent with a low Earth orbit from which the sailcraft is entrained by a low-power beam.

A $1\text{\,}\mathrm{mg}$ payload (non-sail mass) is reserved for one or two sensors and associated support systems. Sail mass is calculated by the system model based on the value of $D_{s}$ chosen by the optimizer combined with the areal density given as an input. Similar to the $0.2\text{\,}\mathrm{c}$ point design, this design assumes a photonic crystal sail material with the same thermal, mass and optical properties, except for a less ambitious reflectance of 40%. Again, the stratified layer optical model is turned off, and absorptance and reflectance remain constant throughout trajectory integration.

Cost factors are chosen to lie between present values and those of the $0.2\text{\,}\mathrm{c}$ point design. The laser cost is $1\text{\,}\mathrm{[}$$]\per, two orders of magnitude lower than the present cost, consistent with requirements for laser-powered launch vehicles, yet two orders of magnitude greater than microwave oven magnetrons. The optics cost is $10k/m², two orders of magnitude lower than the present cost for diffraction-limited optics, consistent with significant production line automation. This value is comparable with the cost of radio telescope aperture. $100\text{\,}\mathrm{[}$$]\per energy storage is above current material cost floors for many energy storage technologies and in line with experience curve projections for electric vehicle battery packs circa 2035 [24].

Upon running the system model using the inputs in table 4, the optimizers converge to the values given in table 5. The cost-optimum sail diameter is found to be $19\text{\,}\mathrm{cm}$, corresponding to a mass of $6.6\text{\,}\mathrm{mg}$.

At $0.01\text{\,}\mathrm{c}$, the sailcraft has a relativistic kinetic energy of $8\text{\,}\mathrm{k}\mathrm{W}\mathrm{h}$, whereas the beamer uses $78\text{\,}\mathrm{M}\mathrm{W}\mathrm{h}$. This yields 0.01% system energy efficiency; two orders of magnitude lower than the $0.2\text{\,}\mathrm{c}$ mission. However, this energy costs only $8k at a price of $0.1\text{\,}\mathrm{[}$$]\per, making it five orders of magnitude cheaper than the beamer capex of $517M.

At $169\text{\,}\mathrm{m}$ effective diameter, the $0.01\text{\,}\mathrm{c}$ beamer is 16 times smaller than the $0.2\text{\,}\mathrm{c}$ beamer. Also, it has one third the maximum radiant exitance at $13\text{\,}\mathrm{kW}\text{\,}{\mathrm{m}}^{-2}$ vs. $37\text{\,}\mathrm{kW}\text{\,}{\mathrm{m}}^{-2}$. The temperature-limited irradiance in the sailcraft frame is the same as that of the $0.2\text{\,}\mathrm{c}$ mission because the sailcraft’s absorptance, emittance, and temperature limit are the same. The initial sailcraft acceleration is two thirds that of the $0.2\text{\,}\mathrm{c}$ mission because of the lower reflectance, but the cost-optimal trajectory ends at only $7\text{\,}\mathrm{g}\mathrm{{}^{\prime}}\mathrm{s}$, squeezing almost everything it can from the diminishing beam. Even then, the energy storage costs only $8M because storage is cheap relative to lasers and optics, and because there is a limit to the extent that laser and optics costs can be reduced by increasing the pulse length.

The sailcraft’s relativistic kinetic energy reaches $30\text{\,}\mathrm{M}\mathrm{J}$ ($8\text{\,}\mathrm{k}\mathrm{W}\mathrm{h}$). Per unit mass, this is $4\text{\,}\mathrm{TJ}\text{\,}{\mathrm{kg}}^{-1}$. In comparison, the heat produced by Pu-238 alpha decay is half as much at $2\text{\,}\mathrm{TJ}\text{\,}{\mathrm{kg}}^{-1}$. At $1\text{\,}\mathrm{A}\mathrm{U}$, the solar wind is primarily composed of $9\text{\,}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{c}\mathrm{m}\mathrm{{}^{3}}$ flowing radially outward from the Sun at $0.001\text{\,}\mathrm{c}$ [27]. From the perspective of a sailcraft cruising away from the Sun, this manifests as a $0.009\text{\,}\mathrm{c}$ proton beam that is incident from the direction of travel, having a combined kinetic energy of $0.15\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-2}$, or only $4.2\text{\,}\mathrm{mW}$ over the sail’s area if it faces the direction of travel.

The inputs to the vacuum tunnel system model are detailed in table 6. Relative to the inputs for the $0.01\text{\,}\mathrm{c}$ mission, the vacuum tunnel has a much shorter initial sail displacement of $1\text{\,}\mathrm{m}$, consistent with a sail positioned close to an optic at the start of the tunnel. Also, the payload is reduced to a token $10\text{\,}\mathrm{ng}$, the mass of a few cells, and the reflectance is reduced to a near-term value of 35% per the stratified layer reflectance predictions in fig. 5. Such a reflectance is consistent with commercially-available Si₃N₄ membrane x-ray windows, for example. Again, the stratified layer optical model is turned off, and the absorptance and reflectance remain constant throughout trajectory integration.

To equal the acceleration of the $0.2\text{\,}\mathrm{c}$ mission, the sail ideally has the same areal density, reflectance, absorptance, and temperature limit as the $0.2\text{\,}\mathrm{c}$ sail. For the vacuum tunnel, point designs assume higher areal density and lower reflectance to be consistent with the idea that for the time being, a sail is a $\sim$ $1\text{\,}\mathrm{mm}$ diameter high-purity film that is not a photonic crystal. Correspondingly, the absorptance is assumed to be an order of magnitude lower than that of the $0.2\text{\,}\mathrm{c}$ sail. If the absorptance were not lower, then the reflectance would need to be increased to obtain sufficient acceleration.

The cost factors are chosen to be consistent with current market values. In addition to laser, optics, and storage costs, there is an additional vacuum tunnel cost of $10k/m. This tunnel cost factor is consistent with the cost of the LIGO vacuum tunnels [28] and includes the beam tunnel and its enclosure, as well as vacuum equipment. To account for differences from the $1\text{\,}\mathrm{m}$ LIGO tunnel diameter, the tunnel cost is multiplied by the relative tunnel diameter. The tunnel cost does not include items that do not scale with length such as R&D, detectors, project management, laboratory construction, and operations.

Upon running the system model using the inputs in table 6, the model converges to the values plotted in fig. 11. This figure is comprised of a family of point designs spanning the range of 1-$500\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ sail speed at cutoff. Slower than $20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, laser cost is of primary importance and tunnel cost is secondary. Faster than $30\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, tunnel cost is of primary importance and laser cost is secondary. In all cases, optics cost is tertiary. Storage cost turns out to be trivial, not even reaching $1000 at $500\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$, so the cost plot is not scaled to show it.

If the sail had no payload, it would have a temperature-limited acceleration exceeding $80\,000\text{\,}\mathrm{g}\mathrm{{}^{\prime}}\mathrm{s}$ regardless of tunnel length; this acceleration is more than five times that of the $0.2\text{\,}\mathrm{c}$ mission. When payload mass is taken into account, the temperature-limited acceleration exceeds that of the $0.2\text{\,}\mathrm{c}$ mission only in tunnels longer than $50\text{\,}\mathrm{m}$.

For $20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ sail speed at cutoff, sufficient to escape the solar system starting from Earth, the model infers: A $0.4\text{\,}\mathrm{km}$ tunnel, $22\text{\,}\mathrm{kW}$ of lasers, and a $0.6\text{\,}\mathrm{m}$ diameter telescope, costing a total of $5M.

For $200\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ sail speed at cutoff, which equals or exceeds the fastest human-made craft, the model infers: A $28\text{\,}\mathrm{km}$ tunnel, $4.6\text{\,}\mathrm{MW}$ of lasers, and a $3.1\text{\,}\mathrm{m}$ diameter telescope, costing a total of $1.3B.

There is a small but noticeable bump toward the upper right corner of the sail diameter and other plots in fig. 11, near the $50\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ mark. This bump is accompanied by a subtle change in gradient of sail diameter vs. speed. Decreasing the integration step size and convergence tolerances does not remove the bump, nor does the model hit any kind of assumed limit on an internal variable. Also, plotting costs vs. sail diameter for the speed at which the bump occurs shows that the minimum (found by iteration 1 of the solution procedure shown in fig. 7) is a global minimum and that there are no competing local minima that could cause the solution to jump from one value to another. It turns out that the source of this bump is the Goubau beam model where the two efficiencies described by eq. 2 are spliced together. Though this splice preserves the continuity of the function, it introduces a discontinuity in its gradient. Smoothing this function in the vicinity of the splice has the effect of removing the bump in fig. 11.