The application of directed energy to spacecraft mission design is explored using rapid transit to Mars as the design objective. An Earth-based laser array of unprecedented size (10 m diameter) and power (100 MW) is assumed to be enabled by ongoing developments in photonic laser technology. A phased-array laser of this size and incorporating atmospheric compensation would be able to deliver laser power to spacecraft in cislunar space, where the incident laser is focused into a hydrogen heating chamber via an inﬂatable reﬂector. The hydrogen propellant is then exhausted through a nozzle to realize speciﬁc impulses of 3000 s. The architecture is shown to be immediately reusable via a burn-back maneuver to return the propulsion unit while still within range of the Earth-based laser. The ability to tolerate much greater laser ﬂuxes enables realizing the combination of high thrust and high speciﬁc impulse, making this approach favorable in comparison to laser-electric propulsion and occupying a parameter space similar to gas-core nuclear thermal rockets (without the requisite reactor). The heating chamber and its associated regenerative cooling and propellant handling systems are crucial elements of the design that receive special attention in this study. The astrodynamics and the extreme aerocapture maneuver required at Mars arrival after a 45-day transit are also analyzed in detail. The application of laser-thermal propulsion as an enabling technology for other rapid transit missions in the solar system and beyond is discussed. Keywords: Laser-thermal propulsion, directed energy, mission design, rapid transit, Mars missions 1. Introduction Recent developments in photonics—in particular, the emergence of inexpensive ﬁber-optic laser am- pliﬁers—have revitalized interest in directed-energy propulsion. The ability to phase-lock large arrays of ﬁber-optic laser ampliﬁers together in a modular fash- ion, enabling them to operate as a single optical ele- ment of arbitrarily large size and power, has now been demonstrated at laboratory scales [1, 2]. The appli- cation of atmospheric compensation techniques origi- nally developed for astronomy (i.e., adaptive optics that can e ectively remove the beam distortions caused by Earth’s atmosphere) would allow the laser array to be built on earth as opposed to in space [3, 4]. Past work has been done on applying these developments to inter- stellar ﬂight. Indeed, dense laser arrays on the scale of kilometers with ﬂuxes on the order of 1 kW /m2leav- ing the array would enable true interstellar missions, Corresponding author Email addresses: emmanuel.duplay@mail.mcgill.ca (Emmanuel Duplay), andrew.higgins@mcgill.ca (Andrew Higgins)wherein the photon pressure of the laser would quickly propel 1-m-scale lightsails to 20–30% of lightspeed. If directed toward nearby solar systems, such a lightsail could return images from neighboring exoplanets within a 25-year mission [5, 6]. More near-term applications of directed energy for interplanetary ﬂight are better suited to using a reaction mass to couple the delivered laser energy to change the momentum of the spacecraft. Options using laser-based directed energy are laser-electric propulsion [7, 8] and laser-thermal propulsion (LTP1). Laser-thermal propul- sion is further classiﬁed into (1) laser ablation propul- sion using an initially condensed-phase reaction mass [9] and (2) laser-thermal propulsion with a gaseous propellant (typically hydrogen) that is heated and ex- panded through a nozzle. The second approach is well- matched to the continuous-wave nature of phased-array 1AU=Astronomical Unit, CME =Coronal Mass Ejection, GCNR =Gas-Core Nuclear Rocket, GCR =Galactic Cosmic Rays, LE =Laser Electric, LEP =Laser-Electric Propulsion, LSC =Laser- Supported Combustion, LTP =Laser-Thermal Propulsion, LTPS = Laser-Thermal-Propulsion System, NTR =Nuclear Thermal Rocket, PPS=Power and Propulsion System, PV =photovoltaic Preprint submitted to Acta Astronautica January 1, 2022 lasers employing atmospheric correction. The applica- tion of this mode of laser-thermal propulsion using a large phased-array laser to deep-space mission design is the subject of the present paper. Laser-thermal propulsion was extensively studied starting in the 1970s when the ﬁrst continuously burning hot plasma sustained by a laser was observed under lab- oratory conditions as reported in [10]. This discovery was soon followed by a speech by Arthur Kantrowitz [11] suggesting the use of lasers to directly heat propel- lant within a rocket, springboarding the development of LTP for the following three decades. Despite promis- ing preliminary results, according to a historical record [12] of laser propulsion research at the Marshall Space Flight Center, the lack of funds to maintain the com- plex laser systems and the lack of political interest con- tributed to the termination of experimental research be- ing conducted in this ﬁeld in the mid-1980s. Studies from this period usually considered gasdy- namic CO 2lasers operating at a 10.6- µm-wavelength, the most powerful lasers at the time. This longer wave- length and the meter-scale monolithic optics then avail- able limited consideration of laser-thermal propulsion to orbit transfer in near-Earth space applications [13]. The transition to the 1- µm operating wavelength of present- day ﬁber-optic lasers and the ability to combine them into a massively parallel, phased array of large e ec- tive optical diameter means that the focal length over which the laser can deliver energy (i.e.,2dfDeDr=) can be extended by two orders of magnitude or more, making the application of laser-thermal propulsion for deep-space missions of interest. Thus, a revisit to mis- sion design applications of laser-thermal propulsion is warranted. A recent NASA solicitation seeking revolutionary propulsion for rapid, deep-space transit identiﬁed a number of candidate missions of interest: traversing the distance between Earth orbit and Mars orbit in no more than 45 days, traversing a distance of 5 AU in no more than one year, traversing a distance of 40 AU in no more than ﬁve years, and traversing a distance of 125 AU in no more than ten years [14]. The Mars-in-45-day requirement is presumably motivated by concern over astronaut exposure to galactic cosmic rays (GCRs) and the potential threat of coronal mass ejections (CMEs) in transit. Recent in-situ measurements by the NASA Cu- riosity Rover have shown that the radiation environment on the surface of Mars is a factor of two lower than that experienced in transit to Mars once outside the Earth’s 2See Nomenclature at end of paperprotective magnetosphere [15]. This ﬁnding suggests an emphasis should be placed on propulsion technologies coupled to mission architectures that minimize the tran- sit time to Mars for crewed missions. In a larger sense, rapid missions to Mars have become a convenient metric in comparing di erent propulsion technologies [16], including nuclear thermal [17], nu- clear electric [18, 19], solar electric [20], and other high speciﬁc impulse and high speciﬁc power technologies such as fusion [21]. For this reason, we have selected a rapid-transit-to-Mars mission as the baseline design for this study. In this paper, the implications of using a 10-m-scale phased-array laser based on Earth and applied to rapid transit missions within the solar system and rapid tran- sits to Mars in particular are explored. The use of large inﬂatable reﬂectors with high reﬂectivity and the ability to tolerate intense laser ﬂux to focus the laser power delivered to the spacecraft into a hydrogen heat- ing chamber is able to generate speciﬁc impulses and thrust-levels (upon expansion through a nozzle) com- parable to advanced gas-core nuclear thermal rockets (Isp2500–3000 s). The design of the heating chamber is identiﬁed as the crucial element of the architecture and is explored in detail in the present study. The pro- pellant storage, regenerative cooling, and delivery sys- tem are also considered. Optimization of the transfer orbit utilizing the large vavailable in near-Earth space (within the range of the 10-m-scale laser) is conducted. Since a laser will not be available for deceleration upon arrival at Mars, considerable attention is also focused on advanced aerocapture techniques necessitated by the large approach velocities. The capability of having the Mars-injection propulsion stage to return to Earth by ef- fectuating a burn-back maneuver while still within the focal range of the Earth-based laser is studied in com- parison to a one-o -use scenario. Trade-o s between lower vand greater payload fraction missions (e.g., for cargo delivery) will also be explored. The speciﬁc mis- sion requirements for the design study reported here are derived from the NASA solicitation discussed above, however, this architecture could be used for a number of missions, including missions to the ice giant plan- ets (<5 year transit time), missions into the interstellar medium at 125 AU ( <10 year transit time), and mis- sions to the solar gravitational focus starting at 550 AU (<50 year transit time) [22]. 2 Figure 1: Concept of Operation diagram for a reusable Laser-Thermal Propulsion System 2. Rapid Mars Transit Mission Design 2.1. Architecture An LTP system for interplanetary transfers would re- quire a 10-m-diameter laser array operating at up to 100 MW, allowing the array to focus on a target up to 50 000 km away. LTP maneuver durations for missions considered in this study would range from several min- utes to an hour (depending on the mission), eliminating the need to build several arrays around the planet or in orbit to ensure the continuous supply of laser power, as would be the case for laser electric propulsion. This feature of laser-thermal propulsion makes this proposed architecture an attractive application to early prototype laser arrays, serving as a stepping stone toward the more ambitious infrastructure (kilometer-scale arrays) required for directed-energy interstellar travel. 2.2. Mission Proﬁle The mission concept of operations is presented in Figure 1. The mission begins with the launch of the laser-thermal-propulsion system (LTPS), hydrogen pro- pellant, and payload, either separately or as a single launch of a Falcon 9 or Atlas-class launch vehicle. The fueled LTPS and payload, depicted in Figure 2, are placed in an elliptical orbit with apogee above the van Figure 2: Conceptual render of a Laser-Thermal-Propulsion System, carrying a 1 ton payload for a 45-day Mars transfer Allen belts (Trans-MEO, 13000 500 km). This or- bit permits the required dwell time over the ground- based laser such that the laser-powered propulsive maneuver—with a duration on the order of one hour— can occur during a single pass over the laser ground site. At the end of this propulsive maneuver, the pay- load is released into the high-energy transfer orbit to Mars. Following payload release, the LTPS—still in 3 view and within the focal range of the ground-based laser—performs a second laser-powered maneuver to return it to the original elliptical orbit; this enables the entire LTPS hardware to be quickly reused following on-orbit propellant transfer. The payload requires only minor propulsive corrections to the trajectory during the short-duration ballistic transfer to Mars, followed by aerocapture upon arrival to Mars. Given the large ap- proach velocity upon arrival at Mars, direct entry is not deemed feasible, however, aerocapture can be followed either by entry and landing or by insertion into a parking orbit around Mars. Figure 3: Porkchop curves of hyperbolic excess velocity for Earth departure (left) and arrival at Mars (right). Targeted region for a fast Mars transfer is identiﬁed with a dashed rectangle. 2.3. Astrodynamics To ensure that the spacecraft remains within focal range and line-of-sight of the laser array throughout the propulsive maneuver, its trajectory was simulated assuming the start of the maneuver 30 minutes before reaching apogee. Given the propellant mass calculated in Table 4 (line 1, propellant mass mpr=700 kg) for a single-use mission and a constant mass ﬂow rate of 0.2 kg /s, the maneuver is expected to last 58 minutes. Accounting for the rotation of the Earth, the angle swept by the spacecraft in the sky relative to a ground observer is estimated to be 60°, with the maneuver ending before reaching geostationary altitude, well within the range of the 10-m-diameter laser array. A custom MATLAB implementation of the algo- rithms presented in [23] to solve Lambert’s problem was used to determine the precise vrequirements of this Figure 4: Mars transfer trajectory comparison. Orbits are to scale. 45-day transfer during a 2020 launch window, in or- der to compare the mission time to recently launched Mars missions. Astrodynamic solutions for this launch window are plotted in Figure 3. An optimal 45-day transfer is depicted in Figure 4 for comparison to the Perseverance rover mission launching a month and a half earlier yet arriving three and a half months later than an LTP-launched payload. This particular trajec- tory, which would have launched on 20 Sept. 2020, re- quired 13.95 km /s ofvfrom the LTPS parking orbit, and serves as a performance target for our vehicle de- sign. 2.4. Arrival Although the performance targets presented in NASA’s solicitation for “Revolutionary Propulsion for Rapid Deep Space Transit” could have been satisﬁed with a ﬂyby of Mars within 45 days, the value of such a mission is limited. This study therefore considered the feasibility of capturing the payload in Martian orbit, despite the lack of a laser array on Mars to provide the necessary v. Capture at Mars is a considerable chal- lenge: As can be seen in Table 1, the vrequired to in- sert the vehicle in a 150 km orbit about Mars is compa- rable to that of departure. Performing such a maneuver Table 1: 45-day Mars transfer parameters Parameter Value Unit Departure orbit apogee 13 000 km Final Martian orbit altitude 150 km Time of ﬂight 45 days Earth v1 16.00 km /s Mars v1 15.41 km /s Earth departure v 13.95 km /s Mars capture v 12.65 km /s 4 with chemical propellant (assumed Isp=451 s) is not feasible, as this would reduce the useful payload mass to less than 6% of the original 1000 kg. Without a laser- array at our destination, the only other way to decelerate is to perform an aerocapture maneuver. 2.4.1. Aerocapture Modeling A simple two-dimensional (2D) numerical aerocap- ture model was implemented to search for viable tra- jectories that would leave the spacecraft in an elliptical orbit once it exits the atmosphere, without imparting ex- cessive thermal or acceleration loads. As shown in Fig- ure 5, the entry vehicle is placed at a given aiming radius RMars+Rimpact and allowed to dive into the atmosphere to dissipate energy. The approximation of the 40NO Martian atmospheric model by Kozynchenko [24] was deemed su cient given its ease of implementation. The e ects of gravity, drag, and lift were validated against classical orbital dynamics and 2D simulations of atmo- spheric re-entry presented in [25]. Thermal loads were modeled by computing both the stagnation point con- vective [26] and radiative [27] heating rates, assuming a nose radius of 3.5 m. When possible, known constants for Mars were used in calculating the radiative heat ﬂux. As no data for the radiative heating velocity function (tabulated in [27]) was available for high velocity en- tries in the Martian atmosphere, values for Earth were used instead as radiative heating rates on either planet become comparable at velocities greater than 10 km /s [28]. Given the sensitivity of suitable trajectories to several factors (ballistic coe cient, periapsis, lift to drag ratio), an analytical approach was devised to guide trajectory optimization. Lift pointed towards the Martian surface (i.e., negative lift) can be used to force the vehicle’s trajectory to remain within the atmosphere longer than would be possible with gravity and drag alone. This ef- fect would be desirable to dissipate the energy of the spacecraft over a longer period, reducing the average thermal load and acceleration felt by the craft. Push- Figure 5: 2D numerical aerocapture model setuping this e ect to its limit leads to solving the following, simpler problem, illustrated in Figure 6: For a given spacecraft ﬂying at a particular orbital speed and alti- tude within the atmosphere, what lift force is necessary to provide enough centripetal acceleration to ensure the craft follows a circular trajectory that matches the curva- ture of the planet? This model of course is not a faithful representation of atmospheric entry but is useful to nar- row the parameter space. Solutions to this problem can be veriﬁed through more intensive simulations, rather than testing for all possible conﬁgurations. Considering motion in the centripetal direction, Equation 1 can be derived for the achievable circular velocity vcircas a function of the spacecraft’s ballistic coecient CBand its lift-to-drag ratio L=D. In addition, vglimit, the maximum vcircfor the allowable accelera- tion felt within the spacecraft, can also be derived by considering the forces transmitted by the spacecraft to its contents (astronaut, rover, etc.). vcirc=s g 1 r 2CBL D(1) vglimit=vut2amaxCB q 1+L D2(2) Equating both expressions allows us to solve for the at- mospheric density required for this maneuver when im- posing an acceleration limit (Equation 3). The required density and associated altitude can be determined by it- erating between Equation 3 and the 40NO atmospheric model, and can then be used to solve for the vehicle ve- locity vcirc. Again, these pseudo-steady-state equations are not meant to be complete solutions to aerocapture— a time-dependent problem—they are used to determine Figure 6: Idealized aerocapture model 5 the limits of this maneuver and provide a starting point for simulations. Figure 7: Idealized aerocapture solutions for an acceleration limit of 8 g. The orange circle identiﬁes the parameters used for trajectory simulation, whose results are plotted in Figure 8. glimit=2amaxCB r gq 1+L D2+amaxL D! (3) Figure 7 shows a set of velocity and altitude solutions for an acceleration limit of 8 g, suggesting that while there appear to be viable trajectories below 16 km /s, much greater entry speeds are not feasible with these constraints. This plot was used to estimate and select a subset of parameters in our 2D simulation. With a hyperbolic excess velocity of 15.4 km /s, the incoming spacecraft would enter the Martian atmosphere at ap- proximately 16 km /s, pointing to a required lift-to-drag ratio of two. In practice, a vehicle attempting to perform this would have to continuously adjust its lift-to-drag ratio as it slows down to maintain altitude, then even- tually allow itself to rise out of the atmosphere once it has decelerated below the escape velocity. Accordingly, simple L=Dmodulation laws were implemented in the simulation. Finally, for a 1000-kg conical payload, a ballistic coe cient as low as 40 kg /m2is thought to be achievable with deployable heat shield [29] or ballute technology [30]. With these selected properties, tra- jectories for a range of impact parameters Rimpact were propagated to ﬁnd a precise aerocapture solution.2.4.2. Simulation Results Figure 8 presents plotted data for one of several valid trajectories, showing some agreement with the ideal- ized model (Figure 7). The experienced acceleration does not drastically exceed the 8-g limit, and the re- quired lift-to-drag ratio remains below two throughout the maneuver. Although the maximum heat ﬂux, esti- mated at 2200 W /cm2, is far greater than typical atmo- spheric entries [31] and exceeds the capabilities of tra- ditional thermal protection system (TPS) materials [32], high-performance TPS are under active development: For example, the Heatshield for Extreme Entry Envi- ronment Technology (HEEET), tested at ﬂuxes of up to 3600 W /cm2[33], appears capable of withstanding the thermal loads of this maneuver. Figure 8: Aerocapture simulation results. Contact with the atmo- sphere occurs at approximately 624 minutes of simulation time. Sharp changes in lift-to-drag ratio are due to the use of a simpliﬁed control law. Although these results are encouraging, the practical- ity of such a maneuver is still uncertain. Solutions to decrease the ballistic coe cient, sustain intense thermal loads, and modulate a vehicle’s lift-to-drag ratio exist independently but may be challenging to integrate into a single vehicle without sacriﬁcing payload capacity. In addition, while the 8-g limit is respected, this load is sustained for several minutes, approaching the limits of human g-force tolerance [34], potentially restricting the use of this maneuver to unpiloted systems. 3. Spacecraft Architecture The proposed design for the LTPS is based on a 45- day transfer to Mars with a 1-ton payload, although 6 Figure 9: Spacecraft architecture overview several aspects of the design (e.g., propulsion, reﬂec- tor) are applicable to any given mission. Key subsys- tem masses are estimated to provide a lower bound for the LTPS dry mass and its speciﬁc mass parameter (kg/kW). This parameter is a useful metric to compare power-limited propulsion systems such as laser-electric or nuclear-electric propulsion, and it must be properly matched to speciﬁc impulse to maximize payload ca- pacity. Ideally, for the missions considered in this study, a 3000-s- Isppropulsion system should not exceed an of 0.005 kg /kW. This requirement is described in detail in Appendix A. Figure 9 identiﬁes key subsystems designed for this study and illustrates the basic concept of laser-thermal propulsion: The incoming laser emitted by the array on the ground is collected by a 10-m-wide reﬂector and fo- cused into the thrust chamber, where the hydrogen pro- pellant is heated then expelled through the nozzle. 3.1. Propulsion 3.1.1. Working Principle The laser thermal system considered for this space- craft uses a laser to heat a core of hydrogen plasma. As shown in Figure 10, this core is sustained as a Laser- Supported Combustion (LSC) wave and can reach tem- peratures of 30 000 to 40 000 K in the region of laser deposition. Heat radiating from the core heats up the surrounding ﬂow of gaseous hydrogen, which is then expelled through a high area-ratio nozzle optimized for vacuum operation. This concept is similar in many aspects to nuclear thermal rocket (NTR)—speciﬁcally Figure 10: Conceptual LSC wave thrust chamber schematic gas-core nuclear rocket (GCNR)—propulsion: Al- though the power source is di erent, both systems rely on a plasma core to heat up propellant by radiation, and both would occupy the same performance niche in terms of speciﬁc impulse (1000–3000 s). We thus beneﬁted from the extensive literature on NTR and GCNR [35– 38], as LTP faces similar engineering challenges regard- ing radiation absorption and wall cooling in particular. For a recent review of NTR and GCNR literature, see [39]. For the purposes of this study, heat transfer and propulsion analyses were performed assuming the stag- nation temperature of the ﬂow as equal to its bulk tem- perature after heat addition. Considering the e ects of ionization and dissociation of hydrogen at high temper- atures, and assuming complete expansion, it was deter- mined that up to 3000 s of speciﬁc impulse could be achievable with a bulk chamber temperature of 10 000 K at 1 atm, requiring a net power input of 90.4 MW for a 0.2 kg /s mass ﬂow rate (see Appendix B for more de- tails). A previous study by Nored [40] on laser propul- sion has found that a 100:1 expansion ratio, for the same chamber temperature at 10 atm, could achieve up to 2500 s of speciﬁc impulse. Research on GCNR propulsion suggests that 2500 s is readably achievable at 8300 K [41]. The exact value of speciﬁc impulse depends upon factors such as di erent assumptions re- garding nozzle area-ratio or varying chamber pressure. Chamber pressure in particular has a signiﬁcant e ect on speciﬁc impulse [42]: Lower pressures allow the hy- drogen to dissociate and ionize at lower temperatures, increasing its resulting exhaust velocity. This mission and spacecraft design assumes a 3000-s- Ispachieved at 10 000 K, but the e ects of lower speciﬁc impulses (1000–3000 s) are discussed in Appendix A. In sum- mary, a reduced Ispfrom 3000 to 2000 s could lead to an 8 to 19% decrease in payload mass ratio depending on the propulsion system speciﬁc mass, should all other as- pects of the mission and spacecraft remain unchanged. 7 3.1.2. Wall Cooling A key aspect in the design of a laser-thermal thrust chamber is ensuring that most of the radiation emanat- ing from the plasma core is absorbed by the hydrogen, letting as little heat reach the chamber walls as possi- ble. Minimizing this loss is beneﬁcial to maximizing speciﬁc impulse and to ensure the integrity of the cham- ber. GCNR systems tackle an identical issue by seeding the hydrogen propellant with absorbing particles such as carbon or tungsten. This method appears suitable to contain GCNR’s 55 000 K uranium plasma cores and provide speciﬁc impulses of up to 7000 s [41, 43]. Shoji and Larson [44] have applied this concept to laser propulsion and shown that seeding the (otherwise trans- parent) hydrogen ﬂow with carbon particles can reduce radiation losses to the walls down to 5% of the incom- ing laser power (i.e., 4.7 MW). This loss is used as a starting point for the design of cooling solutions for the thrust chamber. While this study assumes the use of propellant-seeding to determine cooling requirements, a speciﬁc carbon or tungsten seeding implementation was not considered at this stage of design. In order to match the conditions of our LSC model, a nominal chamber geometry of a 1-m-long, 1-m- diameter cylinder made of Inconel X-750 is assumed, with a maximum allowable temperature set to 1000 K to avoid incurring a signiﬁcant decrease in yield strength [45]. Cooling can be ensured in two ways: by run- ning cold propellant in channels in the chamber walls, like many conventional, regeneratively cooled rocket engines, or by forcing the propellant through a porous wall into the chamber, as proposed by some GCNR en- gine designs [37]. There is a concern that injecting cold propellant downstream of the thrust chamber would ef- fectively reduce the stagnation temperature of the ﬂow, decreasing the theoretically achievable speciﬁc impulse. In either case, ideally, the heat radiated to the walls is no longer lost and is instead used to pre-heat the propellant. Cooling requirements at the nozzle throat must also be considered, as it is typically subjected to signiﬁcant thermal loads. The heat transfer coe cient in the noz- zle was estimated using the closed-form correlation pre- sented by Bartz [46], assuming an approximate coni- cal nozzle proﬁle. The analysis suggests a minimum heat transfer coe cient of around 1400 W /m2-K at the throat to prevent exceeding the thermal limits of Inconel X-750, which would be achievable using conventional regenerative cooling with cryogenic hydrogen. How- ever, due to the low mass ﬂow rate of this design, it may be impractical to use this approach for both the thrust chamber and the nozzle: Should the propellant absorbmost of the heat transferred to the walls, its temperature could approach or exceed chamber wall limits, negating its ability to cool the system. We thus propose the combined use of transpiration and regenerative cooling in an expander cycle, as de- picted in Figure 11. Pressurized cryogenic hydrogen is fed into cooling channels along the nozzle walls and is then expanded through a turbine to power the pump. The exiting gas is injected into the thrust chamber by transpiration through a porous wall. This hybrid ap- proach has two beneﬁts: — Using regeneration at the nozzle throat instead of transpiration prevents signiﬁcant reductions in spe- ciﬁc impulse since no cold propellant is injected downstream of the LSC core. — As alluded to at the start of this section, cold hydro- gen requires seeding with carbon particles to ab- sorb thermal radiation. The porous wall can serve a similar purpose, absorbing radiation to then trans- fer it to the cold propellant by convection. Thermodynamic and heat-transfer calculations were used to determine the performance of this design. By integrating the heat ﬂux computed from Bartz’s corre- lation [46] (Equation 50), assuming complete coolant- side heat absorption, the maximum outlet temperature of the cooling jacket was calculated to be 522 K, low enough to suggest that it could plausibly maintain the throat temperature below the 1000 K limit. Isentropic expansion to 1 atm through the turbine lowers the propellant temperature to 382 K, after which it is transpired into the thrust chamber through a porous wall. Previous studies [37, 38] have investigated the use of transpiration cooling for GCNR engines operating at greater radiation ﬂuxes, and have found the method suitable, providing conﬁdence in its application to laser- thermal propulsion. For the purpose of mass estimation, a simple porous wall heat transfer model was devised, where the complex pore geometry was approximated as numerous thin tubes stacked next to each other, as shown in Figure 12a. This one dimensional model could then be solved as a pipe heat transfer problem (Figure 12b), with one pipe boundary condition set to a ﬁxed heat ﬂux of radiated heat from the LSC core. The other boundary’s temperature was set to the propellant inlet temperature. One ﬁnal simpliﬁcation was the assump- tion of a constant ﬂuid temperature ¯Tprat the average between the inlet temperature Tpr, inand the maximum possible temperature Tpr, max attainable should the pro- pellant absorb 100% of the radiated heat. While several parameters a ect the resulting inner wall temperature, 8 Figure 11: LTPS feed system diagram (a) Simpliﬁed model of porous wall structure (b) 1D porous wall cooling heat transfer model Figure 12: Porous-wall cooling model such as pore geometry and wall thickness, it was found that some conﬁgurations prevented the maximum tem- perature from drastically exceeding the stated 1000 K limit. For instance, a 4-mm-thick, 50% porous shell was found to reach a maximum temperature of 1044 K. With the addition of an external pressure shell, such a thrust chamber design would weigh 26.8 kg. 3.2. Feed System The LTPS could operate at low (atmospheric) cham- ber pressures with a small turbopump, powered by anexpander cycle, as described earlier. For the 45-day transfer mission with a vof 13.95 km /s, assuming an LTPS dry mass of 165 kg (from previous design itera- tions) and a 1-ton payload, 706 kg of propellant is re- quired, occupying 9.94 m3. Cryogenic hydrogen would be stored in a Kevlar-Epoxy composite tank to minimize mass. Assuming a 2-m-long tank with a 1.3 factor of safety, calculations suggest a dry tank mass of 7.75 kg, or about 1% of the propellant mass. This lower bound is based on a hoop stress calculation and approximates Kevlar-Epoxy strength assuming a 90° angle between ply ﬁbers, using the expression presented by Bourchak and Harasani [47]. This hoop stress analysis does not account for a loss of sti ness as the tank empties or vi- bration loads expected at launch. Although cryogenic propellant poses long term stor- age challenges, this is not relevant in the single-use sce- nario. A laser-thermal spacecraft could be launched, inserted in its parking orbit, and powered on within a matter of hours, leaving little time for signiﬁcant pro- pellant leakage. In the re-usable scenario, as shown in Figure 1, a refueling procedure would be necessary for each stage-o , performed either by launching the pay- load with its propellant or with a secondary launch ded- icated to refueling. Some form of insulation would nev- ertheless be necessary, which can be achieved with con- ventional multi-layer insulation: Assuming 10 layers of aluminized Mylar [48], the insulation increases the tank mass by 8.88 kg. Low system pressures and mass ﬂow rate suggest that the turbopump mass will be low compared to conven- tional thrusters. Given that turbomachinery design is out of scope at this stage, an upper bound for its mass was 9 Figure 13: Reﬂector design and nominal dimensions [m] found from existing and proposed turbopump designs operating at a comparable regime. Notably, the estimate generated in [49] indicates that a turbopump with oper- ating conditions on the same order of magnitude would weigh approximately 60 kg, serving as an upper bound for our own mass estimate. Assuming complete absorp- tion by the regeneration system of the heat transferred to the chamber walls, only a small fraction of the heated propellant would need to be expanded through the tur- bine to power the propellant feed system. 3.3. Reﬂector The largest element of the LTPS, its reﬂector, must be capable of redirecting and focusing the 10 m wide laser spot into the thrust chamber. This task can be performed with an o -axis parabolic mirror, whose op- timal focal-length was estimated with the assumption that longer lengths would lead to a linear cost in ad- ditional truss elements, while shorter focal lengths in- creased mirror material costs quadratically. This model yielded a 1011.6 m paraboloid mirror with a 6-m focal length. The reﬂector would also feature an active bea- con, providing the laser array with a co-operative target to facilitate tracking. Of course, a rigid mirror of this size would be too heavy and cumbersome to deploy or ﬁt inside a payload fairing. Our design considered the use of a lightweight, inﬂatable reﬂector instead. Inﬂatable space structures are under active development due to their low mass, compact launch package and simple deployment. Sev- eral inﬂatable reﬂectors have already been constructed [50], ﬂown, and tested in orbit [51], for telecommunica- tions and solar thermal propulsion applications [52].Table 2: Reﬂector materials and their equilibrium temperature Material Function  Teq[°C] PI-6FDA Canopy 0.0017 0.600 195 TiO 2/SiO 2Mirror 0.0050 0.685 230 The reﬂector follows a similar design to o -axis con- centrators studied for solar thermal spacecraft [53, 54], with two identical paraboloid membranes forming the main lenticular body. The mirror-side membrane would be coated in several dielectric layers (TiO 2/SiO 2or Ta2O5/SiO 2) to achieve 99.5% to 99.8% reﬂectivity, respectively [55]. The other membrane, the canopy, only exists to form an enclosed pressurized space, and must be transparent to 1.06- µm radiation both to re- duce losses and remain within its operational tempera- ture range. Table 2 lists the absorptivity , the emissiv- ity, and the approximate equilibrium temperature Teq of potential canopy and mirror materials3under 90 MW of laser power, for a thickness of 0.0254 mm. Note that the canopy is exposed to both the incoming laser ﬂux and its reﬂection from the mirror, doubling the e ec- tive incident power to 180 MW. Fluorinated polyimide ﬁlms (PI-6FDA) were selected as a transparent canopy material for their low absorptivity, allowing them to tol- erate the intense laser ﬂux while remaining well below their high glass-transition temperature of 321°C [56]. This heat-resistance also makes this polyimide suitable as a substrate for the dielectric mirror coatings. The lenticular mirror would be supported by inﬂatable struts and tensioning torus made of the same material. Mod- est internal pressures from 2 to 150 Pa are su cient to maintain the shape of the mirror in the vacuum of space, and are preferable to minimize the ﬂow rate of potential leaks. A concern with the use of an inﬂatable reﬂector is the impact of wrinkles and other deformations on op- tical performance. Optical performance tests of proto- type inﬂatable solar concentrators reported slope errors of 3 milliradians rms [53], which would translate to a focal spot radius of approximately 3 cm for the reﬂector envisioned in this study, which is satisfactory. Another concern, particularly for propulsion applications, is the ability of the inﬂatable structure to retain its shape and position while accelerating. For the 45-day Mars trans- fer, spacecraft acceleration is not expected to exceed 1 g. 3PI-6FDA absorptivity and emissivity taken from [56], where absorptivity was scaled for the reﬂector’s membrane thickness. TiO 2/SiO 2absorptivity was computed from [55] assuming no trans- mission and its emissivity value was taken from [57]. 10 The inﬂatable reﬂector must be able to maintain its fo- cus close to the center of the thrust chamber under this acceleration. Inﬂatable beam deﬂection equations [58] were used to provide an order of magnitude estimate of the deﬂection of the reﬂector at its tip under 10 m /s2 of acceleration. This deﬂection was found to be 4 mm, negligible given the overall size of the reﬂector, and re- sults in a proportionally small change in focal point. A more complete study of reﬂector deformation would be needed to understand its e ects on the stability of its focal point, and whether actuators could provide su - cient control to compensate. Active control mechanisms for the reﬂector would play a critical role in a fully op- erational LTPS. In addition to stabilizing the reﬂector, orientation mechanisms and secondary optics would be necessary to allow the spacecraft to decouple its thrust direction from the incoming beam direction. The design presented in Figures 2 and 9 clearly presents the work- ing principles of laser-thermal propulsion, but is limited to thrusting along the incoming laser’s direction. Figure 13 indicates nominal reﬂector dimensions, al- lowing for a mass estimate based on its computed sur- face area and volume. Required membrane thicknesses were selected based on the 2 and 150 Pa pressures in the lenticular body and torus, respectively, and commer- cially available ﬁlm thicknesses. Inﬂated with helium, the reﬂector would add 39.6 kg to the LTPS mass. 3.4. Mass and Speciﬁc Mass Table 3 provides a mass summary of our design, to- taling at 143 kg, which is of course an absolute lower bound for the overall propulsion system mass. Several components are missing from this calculation: such as regeneration jackets, truss elements, piping, and more. Nevertheless, this allows for an initial calculation of the LTPS speciﬁc mass parameter , keeping in mind that a Table 3: Dry LTPS mass summary. This omits several spacecraft sub- systems necessary for a complete design, but is used to establish a lower bound for the speciﬁc mass of the LTPS. Component Mass [kg] Thrust chamber 26.80 Propellant tank 7.75 Tank insulation 8.88 Turbopump assembly 60.00 Reﬂector membrane 39.59 Reﬂector pressurization gas 0.03 Total 143.05value of 0.005 kg /kW should not be exceeded to maxi- mize payload capacity as shown in Appendix A. For a gross power input of 90 MW and a 143-kg dry system mass, we ﬁnd an of 0.0016 kg /kW, far lower than needed. An ample mass margin is thus left for neglected subsystems mentioned earlier. Indeed, the LTPS mass could increase by an order of magnitude and still remain competitive. In fact, any below 1 kg /kW could enable rapid-transit missions within the solar sys- tem [16]. 4. Alternative Missions Although preliminary calculations show promise in the design and performance of an LTP transfer stage, simulations suggest that using this architecture to reach Mars within 45 days requires—at a minimum—a very delicate aerocapture maneuver if no laser array is avail- able to e ect the deceleration maneuver. Should this fail, while the trajectory discussed in Section 2 could be tuned to provide a free return (orbital period of 3.96 years), the duration of such a return makes it impracti- cal. Nevertheless, the high speciﬁc impulse of LTP still makes this system attractive to increase the payload ca- pacity of a mission using more conventional Hohmann transfers. Table 4 features propellant ( mpr) and payload ( mpl) mass data for three missions, powered by chemical or LT propulsion. Mars 1 represents the 45-day mission described in detail in this paper. Mars 2a considers a similar mission for a piloted spacecraft, including ad- equate life support systems for the outgoing trip, esti- mated to weigh at least 40 tons (based on Orion cap- sule and European Service Module wet mass). Finally, Mars 2b attempts to carry as much payload as pos- sible with the propellant available in a single-engine Centaur, utilizing a typical Hohmann transfer. Mass data was calculated for three propulsion systems: the LTPS as described in this study, a heavier LTPS with =0:005 kg /kW, and a single-engine Centaur upper stage [59] for comparison. The use of LTP could lead to a 10-fold increase in payload capacity compared to chemical thrusters for the same propellant mass. These capabilities make this architecture especially attractive in the context of long-term Martian settlements, where large quantities of specialized equipment, habitats, and consumables sent from Earth will be needed to support a colony. Should such a settlement construct its own directed- energy system, laser-thermal transfer stages could be- come the workhorse of an interplanetary economy, pro- 11 Table 4: Alternative Mars transfers with single use or re-usable stages. mpl: payload mass, v: mission delta-v from medium Earth orbit, : speciﬁc mass, mps: propulsion system mass, Isp: speciﬁc impulse, mpr: propellant mass. MissionTime [days]mpl[kg]v [km/s]Propulsion SystemPower [MW] [kg/kW]mps[kg] Isp[s]mpr[kg] (Single)mpr[kg] (Reuse) Mars 1 45 1 000 13.95LTPS 100 0.0016 160 3 000 700 860 LTPS 100 0.0050 500 3 000 910 1 400 Centaur N /A N /A 2 247 451 73 000 1 300 000 Mars 2a 45 40 000 13.95LTPS 4 000 0.0016 6 400 3 000 28 000 34 000 LTPS 4 000 0.0050 20 000 3 000 36 000 56 000 Centaur N /A N /A 2 247 451 950 000 2 100 000 Mars 2b 180130 000 4.18LTPS 3 000 0.0016 4 800 3 000 20 830 22 000 120 000 LTPS 3 000 0.0050 15 000 3 000 20 830 23 000 11 000 Centaur N /A N /A 2 247 451 20 830 30 000 viding means for both fast transit and large cargo ship- ments between Earth and Mars. In addition, accessibility to more distant targets be- yond the asteroid belt and to the edge of the solar sys- tem could be greatly improved with the use of LTP. Ex- ploration missions to gas giants could be performed via a direct Hohmann transfer, reducing mission time and increasing launch window frequency. Furthermore, the LTPS as described here could be used as a solar-thermal spacecraft with little modiﬁcation. A solar Oberth ma- neuver could be performed with the same spacecraft, potentially enabling the ﬂyby of interstellar objects such as 1I/’Oumuamua [60, 61]. Finally, this architecture is also suitable for interstellar precursor missions, such as placing a spacecraft at the solar gravitational focus. Beyond this point, 550 AU away, light from distant star systems is focused by the Sun’s gravity, potentially enabling megapixel resolution imaging of exoplanets [22]. Such interstellar precursor missions require v’s of 30–50 km /s, which is within the capability of laser- thermal propulsion featuring Isp3000 s. These mis- sions—which are typically ﬂybys in nature and do not require rendezvous with a target—are better suited to the laser-thermal architecture explored in this study than payload delivery to Mars, since ﬂybys only require sig- niﬁcant vat departure from Earth. 5. Discussion and Further Work Enabled by shorter laser wavelength and the abil- ity to operate as a phased array of unprecedented op- tical dimensions, laser-thermal propulsion can now be extended two orders of magnitude deeper into cislu- nar space than previously considered in the 1970s and1980s. A second advantage that this proposed architec- ture capitalizes upon is the laser ﬂuxes that are permis- sible upon the inﬂatable reﬂector, which exceed by two orders of magnitude the ﬂux limitations on laser-electric propulsion with no active cooling [7, 8]. These high ﬂuxes allow laser-thermal propulsion to “burn hard” early in the mission, while the spacecraft is still within the focal length of the laser, enabling high vmissions with 10-m-scale lasers. A further beneﬁt of the abil- ity to perform high thrust burns in near-Earth space is the propulsion stage (which includes the hydrogen tank, heating chamber, nozzle, and reﬂector) can be imme- diately brought back to Earth, where it can be rapidly refueled and reused during the same launch window. While incurring a modest loss in payload capacity, the re-usability of the LTPS o ers signiﬁcant advantages over allowing the hardware to proceed to Mars, where it would have little utility. By borrowing and building upon concepts developed for solar-thermal and gas-core nuclear propulsion, our proposed design for an LTPS appears plausible and promises an unprecedented mass-to-power ratio. In fact, the values of found in this study are so low that they no longer inﬂuence the mission design; even if values must increase by an order of magnitude as the design of the propulsion system is further reﬁned (see Table 4), the implications for the ability to meet the mission objectives with a signiﬁcant payload fraction are neg- ligible. In fact, a speciﬁc impulse of 3000 s is opti- mal for a propulsion system three to four times heavier ( 0:005 kg /kW), as detailed in Appendix A. The stability and radiative heating properties of the laser-supported plasma should be further stud- ied. Plasma instability in particular is a complex phe- nomenon and is critical to this propulsion system, as the 12 inﬂatable reﬂector and tracking inaccuracies could be sources of instability. Numerical simulation and experi- mental work is underway in our research group to study the radiative properties of hydrogen plasma, the imple- mentation of particle-seeding, and its e ects on heat- transfer. Small-scale experiments on laser-supported plasma have been performed for propulsion applications [62] and could be expanded to study plasma stability and its response to disturbances stemming from imper- fect optics. Such experiments, coupled with extensive simulations, would pave the way to small-scale thruster prototypes operated by individual lasers within a labora- tory setting. Given the existing work on laser-supported plasma, we believe the realization of such prototypes may be feasible within the next decade. Should continued research e orts lead to the devel- opment of functional prototype LTP thrusters, smaller- scale test missions could be envisioned within low to medium Earth orbit, evaluating the performance of laser-tracking and the inﬂatable reﬂector. Orbital Trans- fer Vehicle (OTV) applications, already considered for solar-thermal propulsion [50], could also be served by small-scale laser-thermal thrusters. Such applications provide a useful de-scoping option that can still com- pete against chemical propulsion in terms of propellant eciency. The scalability of LTPS spacecraft should be further studied, as the focusing limits (and therefore the maximum thrust duration) of the laser array will de- pend on the reﬂector size. Several other inter-dependent mission parameters such as laser power, parking orbit, or propellant ﬂow rates can be adjusted to compensate for focusing limits, motivating the need for an in-depth analysis. The trade space between laser-electric and laser- thermal propulsion (Appendix A) should be further explored: Laser electric has the signiﬁcant advantages of a greater speciﬁc impulse and a modular architec- ture—permitting all components to be tested and val- idated on a benchtop—but at the expense of a limited laser ﬂux that the photovoltaics that are used to con- vert laser power to electricity can tolerate due to ther- mal constraints. The study of Sheerin et al. used a value of 10 kW /m2(i.e., about seven suns), which was deemed feasible without the use of active cooling of the photovoltaics. As a consequence, the propulsive ma- neuver for laser-electric propulsion missions typically requires days to weeks, which for continuous power de- livery would necessitate multiple laser sites on Earth or construction of a laser array in space. The laser-electric missions considered in [8] also necessitated larger ar- rays (750 to 1000 m in e ective diameter) in compari- son to the present study’s 10-m array for the same classof payload. Our preliminary conclusion is that, as larger laser arrays become available, laser-electric o ers the greatest beneﬁts, but for early application of phased- array lasers with 10-m-scale laser arrays, laser-thermal may o er a greater potential to realize missions with signiﬁcant payloads. This study has also shown that LTP greatly beneﬁts from advances made in gas-core nuclear propulsion and breakthroughs in directed-energy concepts. These links to alternate propulsion systems and the application- agnostic nature of laser-arrays should motivate the study of possible synergies between LTP, LEP, and perhaps even NTR propulsion systems. Hybrid directed-energy systems could perhaps allow a spacecraft to beneﬁt from the best of both worlds: using a high-thrust, short dura- tion laser-thermal maneuver to quickly escape Earth’s sphere of inﬂuence, then discarding the LTP hardware to rely on lower-power but more e cient LEP for the rest of the mission. Common plasma stability and heat- transfer challenges are faced by both LTP and GCNR propulsion, and the development of solutions to these issues would beneﬁt both architectures. The renewed interest in nuclear-thermal propulsion [63, 64] could be an opportunity to resolve these shared developmental issues in parallel. Cooling technologies designed for GCNR propulsion can be re-purposed for LTP, and work on laser-supported plasma stability can beneﬁt GCNR development without requiring ﬁssile material for ex- periments. There are also some commonalities for both laser-thermal and laser-electric architectures considered in the present study: Since a laser of su cient power for directed-energy propulsion would not yet be available on Mars, any rapid LTP or LEP Mars mission performed with a sub-km laser array would have to resort to aero- capture in order to land or orbit the planet. The present study suggests that aerocapture with an approach hyper- bolic excess velocity of 15 km /s is feasible with current thermal protection materials and vehicle designs within the state of the art. Several of the LEP ﬂyby missions presented in [8] could thus be turned into orbit trans- fer missions. Aerocapture maneuvers could be consid- ered for any target with a su ciently dense atmosphere. More detailed aerocapture analyses, such as ones done for Venus [65], Mars [31], Titan [66], and Neptune [67], could be revisited within the context of directed-energy missions to greatly expand the range of missions achiev- able with single, sub-km directed-energy arrays. 6. Conclusion The implications of the emergence of phased-array lasers of 10-m-scale and 100-MW power for the de- 13 sign of a high vmission have been examined, and the results of this study suggest the potential for a disrup- tion in comparison to conventional chemical and solar- electric propulsion. The high speciﬁc impulse achieved with directed energy allows laser-thermal propulsion to perform interplanetary missions with less propellant than chemical systems and in shorter thrust durations than solar-electric propulsion. The preliminary design of critical subsystems nec- essary for such a spacecraft has not found fundamen- tal technological roadblocks to realize this propulsion system. Furthermore, the mass-to-power ratios ( ) val- ues that may be achieved via laser-thermal propulsion (0.001–0.010 kg /kW) are unparalleled, far below even those cited for advanced nuclear propulsion technolo- gies, due to the fact that the power source remains on Earth and the delivered ﬂux can be processed by a low- mass inﬂatable reﬂector. We reiterate that the Mars-in-45-days goal is only used as a convenient metric for propulsion architectures. This requirement is motivated by the ability to deliver astronauts to the lower radiation environment on the sur- face of Mars while absolutely minimizing exposure to GCRs and potential CME events in route. Cargo deliv- ery missions were explored as well as part of this study, showing a potential ten-fold increase in payload capac- ity compared to the Common Centaur cryogenic upper stage. Other missions of interest previously considered unfeasible with conventional chemical or solar-electric propulsion may be realizable via this architecture, such as: rapid missions to the outer ice giant planets, into the interstellar medium, intercepting interstellar objects passing through the solar system, and to the solar grav- itational focus. 7. Acknowledgements The authors would like to acknowledge the valuable assistance of Mathias Larrouturou, Lynn Cherif, Rahul Atmanathan, Samuel Smocot, and Alp Tanriover in de- veloping this design study. We would also like to thank Philip Lubin, Carl Knowlen, Adam Bruckner, M ´elanie T´etreault-Friend, and all participants to this study’s Pre- liminary Design Review held in August 2020 for their input on this project. Our thermodynamic analyses were made possible by the CoolProp Open-Source Thermo- physical Property Library [68] and NASA Chemical Equilibrium with Applications (CEA) data [69]. Pork- chop plots (Figure 3) were generated using resources provided in [70]. This work was supported by the Natural Sciences and Engineering Research Council ofCanada (NSERC) Discovery Grant “Dynamic Materi- als Testing for Ultrahigh-Speed Spaceﬂight” and the McGill Summer Undergraduate Research in Engineer- ing program. Appendix A. Speciﬁc Impulse Optimization Much like electric propulsion, a laser-thermal thruster is power-limited by its energy source, such that its payload mass fraction mpl=m0can be maximized as shown by Stuhlinger [71], based on the propulsive ma- neuver duration tmand mission v. The relevant equa- tion is reproduced here as Equation A.1: mpl m0=ev cc2 1ev c (A.1) Where normalized delta-v vand normalized exhaust velocity care expressed based on the characteristic ve- locity vch, deﬁned by Equation A.2 in terms of maneuver duration tmand the power and propulsion system (PPS) speciﬁc mass [kg/W]. It is assumed that the PPS mass is proportional to the resulting jet power. vch=r 2tm (A.2) v=v vch(A.3) c=c vch=Ispg0 vch(A.4) These equations quantify the trade-o between speciﬁc impulse and thrust, and imply that a greater Ispis not always desirable. For example, di erent propulsion systems, character- ized by their speciﬁc mass , can be compared for a 45-day Mars transit in terms of payload mass fraction and thrust duration. For a mission vof 15 km /s and an eciencyof 90%, the payload mass fraction of both laser-thermal and laser-electric propulsion systems are plotted in Figure A.1. Fixed speciﬁc impulses were as- sumed for this comparison, assuming 10 000 s for laser- electric systems as per [8]. The e ect of the speciﬁc mass on payload capac- ity is made clear in Figure A.1. Where laser-thermal- propulsion systems may be able to launch signiﬁcant payloads with one to two hours of thrust, heavier sys- tems such as laser-electric propulsion would require 10 to 100 hours to have a positive payload mass ratio. However, when such maneuver durations are accept- able and feasible, laser-electric systems quickly over- take LTP owing to their typically greater speciﬁc im- pulse. 14 Figure A.1: Comparison of payload capacity vs. thrust duration for di erent propulsion systems for v=15 km /s. Expected for laser- electric systems taken from [8] . Equations A.1 and A.2 can also be used to optimize speciﬁc impulse for a given PPS speciﬁc mass. For time constrained missions, the payload mass ratio reaches a maximum for an exhaust velocity close to the character- istic velocity. Beyond that, increased exhaust velocity prevents the propulsion system from providing enough thrust to accelerate the spacecraft within the time con- straint, reducing the payload mass fraction of the sys- tem. In the context of developing a laser infrastructure to eventually power interstellar missions, ﬁnding applica- tions for directed-energy propulsion feasible with a sin- gle, small array is desirable, motivating a 1-hour con- straint on the propulsive maneuver. The intended mis- sions for laser-thermal propulsion, such as rapid transit to Mars or direct transit to the edge of the solar system, require vranging from 10 to 15 km /s. These param- eters allow us to generate Figure A.2, where the pay- load mass ratio is plotted as a function of normalized exhaust velocity and speciﬁc mass. A range of speciﬁc impulse values is overlaid to reveal the sensitivity of the payload mass ratio to Isp. The design presented in this study, with an Ispof 3000 s and a speciﬁc mass close to 0.0015 kg /kW would achieve a payload capacity of about 50%. However, an even greater exhaust velocity would be desirable at this speciﬁc mass to maximize the performance of this system, and it is unlikely that a fully realized LTPS would be as light as estimated here. Fortunately, it appears as though a 3000-s speciﬁc im- pulse is close to the optimal exhaust velocity for a spe- Figure A.2: Payload mass ratio dependency on normalized speciﬁc impulse for v=15 km /s. Constant Isplines (dashed) reveal the sensitivity of the payload mass ratio to the speciﬁc impulse for a given ciﬁc mass of 0.005 kg /kW. Thus, an LTP system opti- mized for rapid transit missions in the solar system with a 1-hour thrust duration should have a speciﬁc mass of 0.005 kg /kW, hence its inclusion in Table 4 as an opti- mal LTPS. Should a practical LTPS have an even greater , such as 0.01 kg /kW, the speciﬁc impulse of the sys- tem could be easily reduced by increasing propellant mass ﬂow rate for the same input power, in order to maximize the payload mass ratio. An additional beneﬁt of a greater speciﬁc mass is a lower sensitivity to changes in speciﬁc impulse. For in- stance, a reduction in Ispfrom 3000 to 2000 s results in a 19% decrease in payload mass ratio for a 0.0015- kg/kW- , compared to an 8% decrease for a 0.0050- kg/kW- . While these performance losses are not neg- ligible, the payload capacity enabled by LTP remains considerable compared to chemical systems. Appendix B. Speciﬁc Impulse of Laser-Heated Hy- drogen The speciﬁc impulse of laser-heated hydrogen pro- pellant was determined by converting the enthalpy of the propellant in the heating chamber into kinetic en- ergy, i.e., velocity, of the exhaust. The performance was bounded by considering two cases: An ideal case with chemical equilibrium maintained throughout the nozzle expansion and a worst-case scenario where the chemi- cal composition throughout the nozzle is kept “frozen”, i.e., no recombination occurs within the nozzle. 15 The speciﬁc impulse can thus be calculated from con- servation of energy Ispg0=c=p 2(hchamberhexit) (B.1) where the enthalpy hincludes the enthalpy of forma- tion. The NASA polynomial ﬁts to the NIST–JANAF thermochemical data were used for these values [69]. Both dissociation and ionization of the hydrogen propellant are considered, with the composition repre- sented by H2)(1 )H2+2(1 ) H+2 H++2 e (B.2) where and quantify the degree of ionization and dis- sociation of the hydrogen. The value of these parame- ters can be found by simultaneously solving the equilib- rium equations of dissociation and ionization as follows Kpdis(T)=h2 (1 ) 1+ +2 i2 h1 1+ +2 i P Pref! (B.3) and Kpion(T)=h2 1+ +2 i2 h2 (1 ) 1+ +2 i P Pref! (B.4) where KpdisandKpionare the equilibrium constants for the dissociation and ionization reactions, respectively. The numerical values of the equilibrium constants can be found using the Gibbs function data provided in the NASA database. These coupled non-linear algebraic equations must be solved numerically via an iterative algorithm. Once the equilibrium composition is solved for, the enthalpy of the propellant can be found by sum- ming the product of the mass fraction of each compo- nent with its temperature-dependent enthalpy. In the case of equilibrium ﬂow, the composition con- tinuously varies to maintain equilibrium until the hydro- gen is fully expanded to molecular hydrogen, which is assumed to occur at 298.15 K, since the NASA polyno- mials do not extend below this temperature. In the case of frozen ﬂow, the mass fractions of the composition are held constant throughout the expansion, but the en- thalpy of each component at the exit is computed at the ﬁnal expansion temperature (again, taken at 298.15 K). Finally, the Ispcan be obtained by normalizing the ob- tained exit velocity by the gravitational acceleration g0. The results for speciﬁc impulse are plotted in ﬁgure B.3a as a function of chamber temperature (at various chamber pressures) and chamber pressure (at various chamber temperatures). Note that for chamber tem- perature values greater than 10,000 K, the equilibrium 5000 7500 10,000 12,500 15,000 17,500 20,000 Temperature [K]1000200030004000500060007000Specific Impulse [s]Equilibrium Frozen(a) E ect of temperature 0.1 1.0 10.0 Pressure [Bar]1000200030004000500060007000Specific Impulse [s] (b) E ect of chamber pressure Figure B.3: Speciﬁc Impulse of laser-heated hydrogen speciﬁc impulse is signiﬁcantly greater as the chamber pressure is decreased from 10 bar to 0.1 bar, in contrast to the behavior of conventional chemical rockets. The reason for this behavior is that the degree of ionization is greater at lower pressures for the same chamber tem- perature, and the enthalpy of recombination that occurs as the propellant is expanded in equilibrium results in e ectively greater energy being released into the ﬂow, and consequently a greater exhaust velocity is obtained. Appendix C. Nomenclature Symbols: Asurface area aacceleration CBballistic coe cient [kg /m2] 16 cexhaust velocity ddistance or length Ddiameter ggravitational acceleration g0standard gravity henthalpy [J /kg] Ispspeciﬁc impulse [s] Kpchemical equilibrium constant mmass Ppressure [bar] q00heat ﬂux [W /m2] Rradius rradial coordinate Ttemperature ttime Greek Symbols: speciﬁc mass [kg /kW] ionized hydrogen mole fraction ( only in Appendix B) dissociated hydrogen mole fraction emissivity eciency wavelength Subscripts: 0 initial (wet mass) ch characteristic e laser emitter or array f focal (length) g glass-transition in inlet of transpiration pore m maneuver out outlet of transpiration porepl payload pr propellant ps propulsion system r laser receiver (e.g. reﬂector or PV array) rad radiation