Comparative Assessment of Human Missions to Mars

Comparative Assessment of Human Missions to Mars Damon Landau Ph.D Preliminary Exam How Shall We Go to Mars? The allure of people traveling to Mars has been the inspiration for numerous mission proposals.1–20 Despite decades of comparative analyses, 21–33 the debate continues on how to transport a crew to Mars and return them home safely Indeed, there are myriad design options when one considers the possible combinations of propulsion technologies, mission architectures, transfer vehicles, and timelines for a mission to Mars New trajectories (e.g Aldrin’s cycler) and technology developments (especially from NASA’s in-space propulsion program) provide tremendous leveraging effects for the human exploration of Mars However, a comparative analysis of the costs and benefits of these mission options is unavailable, and inconsistencies in crew number, vehicle masses, and allowable transfer times cloud previous comparative assessments To characterize the effects of these mission trades, I calculate the injected mass to lowEarth orbit (IMLEO) with the condition that the crew, payload, vehicles, propulsion systems, and transfer time of flight (TOF) must be consistent among different mission architectures I choose IMLEO as the metric because a change in mass is the most direct effect of varying a mission parameter and because IMLEO is strongly correlated to the dollar-cost of a given mission 34–37 In this way, the IMLEO benefit from developing a potential technology is known a priori to help direct the path of Mars exploration The goal of this research is to determine the best way of sending people to Mars by assembling and assessing the array of mission architectures and available technologies for interplanetary exploration Mars exploration architectures are differentiated by the placement of the interplanetary transfer vehicle at Earth or Mars (see Table 1) For example, NASA’s Design Reference Mission places the transfer vehicle into a parking orbit at Mars arrival (dubbed a semi-direct architecture).19, 20 Other ideas include parking orbits at both Earth and Mars (stop-over), 38, 39 a flyby at both Earth and Mars (cycler),40–45 a flyby at Earth and parking orbit at Mars (Mars-Earth semi-cycler),48, 49 and flyby at Earth and parking orbit at Mars (Earth-Mars semi-cycler) 50 A separate taxi vehicle then ferries the crew between the surface of a planet and the transfer vehicle to complete the crew transfer In a “direct” architecture the transfer vehicle lands on the surface of both planets, eliminating the need for a taxi vehicle Often, incorporating a parking orbit or flyby at planetary encounters lowers the IMLEO because the relatively massive transfer vehicle performs less maneuvers Table Placement of interplanetary transfer vehicle for different architectures Architecture Earth Encounter Mars Encounter Schemata Direct Surface Surface Semi-Direct Surface Parking Orbit Stop-Over Parking Orbit Parking Orbit M-E Semi-Cycler Flyby Parking Orbit E-M Semi-Cycler Parking Orbit Flyby Cycler Flyby Flyby The available propulsion technology plays a significant role in determining the IMLEO for a Mars mission For example, chemical propulsion [e.g liquid hydrogen with liquid oxygen (LH2/LOX)] has been the workhorse for space exploration, but the higher specific impulse of nuclear thermal rockets (NTR) or nuclear electric propulsion (NEP) can reduce propellant mass Another option is to make the propellant required for the return trip on Mars, or in-situ propellant production For example, a feedstock of terrestrial hydrogen may be combined with the carbon dioxide at Mars to produce methane and oxygen, eliminating the need to launch the return propellant from Earth Also, should it become practical to extract large quantities of water from Martian regolith, LH2/LOX propulsion systems may be used without the need for any propellant feedstock The atmosphere of Mars has been used to decelerate spacecraft for surface landing and to lower the energy of a parking orbit, but aerocapture, where the spacecraft is decelerated from the interplanetary transfer into a parking orbit, has yet to be attempted Mission architectures that rely on a parking orbit at Earth or Mars can benefit from aerocapture because the aerobrake mass fraction is usually less than the propellant mass fraction for orbit capture I plan to also assess the benefits of reusable propulsion systems and transporting propellant as cargo from one planet to the other Table provides a list of potential technologies along with an approximate technology readiness level (TRL).51 Table Current and near-tern technologies Approximate Technology Readiness Levela Chemical Propulsion Parking Orbit Rendezvous (Earth) Reusable Chemical Propulsion Parking Orbit Rendezvous (Mars) Refuel in Orbit (Earth) Cargo Nuclear Electric Propulsion Refuel in Orbit (Mars) Hyperbolic Rendezvous (Earth) Hyperbolic Rendezvous (Mars) Nuclear Thermal Rocket Aerocapture Transfer Vehicle NEP In-Situ Propellant Production Mars Launch Vehicle NTR Mars Water Excavation a For a definition of technology readiness levels see Ref 51 To further explore the design space I vary the mass of the transfer vehicle and taxi for a given crew size For example, a low-mass transfer vehicle is preferred from a IMLEO standpoint, but may be uncomfortable or even detrimental to the health of the crew By increasing the vehicle mass per person, additional radiation shielding or artificial gravity may be incorporated to reduce mission risk Another key factor to crew health is the time the crew must spend in space To examine the effect on IMLEO from reducing the flight-time, the allowed TOF between Earth and Mars may also be varied Finally, the change in IMLEO due to varying the mass of Mars cargo (e.g habitat, power plant, etc.) will also be examined Earth-Mars Trajectories Earth-Mars trajectories with low-energy requirements that also limit the (transfer) time a crew spends in interplanetary space are essential to the design of cost-effective, minimal-risk missions To examine the effects of limiting TOF, I compute optimal ∆V trajectories (to reduce mission cost) with constrained TOF (to reduce mission risk) Traditionally, Mars trajectories fall into two categories: 1) opposition class52–60 for short duration mission (600 days) with short Mars stay time (30 days), and 2) conjunction class56–66 for long Mars stay time (550 days) with long mission durations (900 days) Unfortunately, trajectories between Earth and Mars with short TOF, short mission duration, and low energy requirements (low ∆V) not exist; thus, I not examine opposition class trajectories further It is assumed that one mission (to Mars and back) occurs once every synodic period Since Earth-Mars trajectories approximately repeat every seven synodic periods (every 14.95 years), I compute short TOF trajectories over a seven synodic-period cycle (for 2009–2022 Earth departure years) The effects of powered versus aero-assisted planetary arrivals is also examined.67–70 Conjunction class trajectories (which are fairly well understood and documented) are suitable for direct, semi-direct, and stop-over missions because these architectures not require planetary flybys To compare semi-cyclers and cyclers with the other architectures on a consistent basis, however, requires further optimization than what is available in the literature For example, Mars-Earth semi-cycler trajectories have been computed previously, but not optimized across a range of TOF A variety of cycler trajectories was also available, but the ∆V or TOF was not ideal for Mars mission scenarios In the search for better cyclers, my research contributed to identifying two new families of cyclers for use in human mission to Mars Error: Reference source not found-45 Finally, I have designed four versions of Earth-Mars semi-cycler trajectories to complete the assessment of Mars mission architectures 50 An example trajectory for each trajectory type is presented in Fig 1–Fig Fig Outbound and inbound direct transfers a) E2-E3 near 2:1 Earth:spacecraft resonance b) E2-E3 near 2:3 resonance c) E2-E4 1.5 year transfer Fig Mars-Earth semi-cyclers a) M2-M3 near 3:4 resonance b) E1-E3 near 3:2 resonance, E3-E4 1:1 resonance, E4-E6 near 3:2 resonace Fig Earth-Mars semi-cyclers Fig Outbound cycler trajectory with E1-E3 near 3:2 resonance and E3-E4 near 1.5 year transfer Should an accident on the way to Mars preclude the crew from landing (e.g a propulsion system failure), a free-return trajectory71–78 would allow the crew to return to Earth without any major maneuvering (i.e zero deterministic ∆V) These trajectories are constructed such that if there is no capture maneuver at arrival, a gravity assist from Mars will send the crew and vehicle back to Earth I examine free-return trajectories for direct, semi-direct, and stop-over mission scenarios, though free-returns may be used (and are often incorporated) in semi-cycler and cycler scenarios An example free-return trajectory is presented in Fig While the free-return abort is available, the nominal mission only uses the Earth-Mars portion of the trajectory The crew would stay on Mars for about 550 days then take a short TOF inbound trajectory home (e.g the Mars-Earth transit in Fig 1) Fig Mars free-return trajectory with near 3:2 resonance from E1-E3 I model the heliocentric trajectories as point-to-point conics with instantaneous V∞ rotations at planetary encounters The minimum allowable flyby altitude at Earth and Mars is 300 km Deep space maneuvers are also modeled as instantaneous changes in the heliocentric velocity We not allow maneuvers within the sphere of influence of a flyby planet because of the operational difficulty in achieving an accurate ∆V during a gravity assist We assume that planetary departure and arrival maneuvers occur at 300 km above the planet’s surface, thus the ∆V for escape or capture is ∆V = V∞2 + µ / rp − 2µ / rp where µ is the gravitational parameter of the planet and rp is the periapsis radius of the escape or capture hyperbola (in this case 300 km above the surface radius) While Eq is explicitly the ∆V to achieve a V∞ magnitude from a parabola, it is sufficient to optimize interplanetary transfers that begin on the surface or in a parking orbit The difference between the true ∆V and Eq is found by subtracting the launch trajectory or parking orbit velocity at rp from the periapsis velocity of the parabola Because this difference is independent of the interplanetary transfer (both the parking orbit and the parabola are planetocentric trajectories), it does not affect the outcome of the optimal trajectory The sequence of maneuvers included in the optimal ∆V calculation is summarized in Table for each trajectory type If the crew taxi, transfer vehicle (TV), or both vehicles performs a maneuver (when the maneuvers are required is provided in Table 3), the weighting on the corresponding ∆V is unity, and if no maneuver is performed then the weighting is zero For this analysis, I assume the semi-cycler or cycler transfer vehicle is already in a parking orbit or on an interplanetary trajectory; thus, the initial transfer vehicle launch cost for these trajectories are ignored While these relative weightings (one or zero) not explicitly minimize IMLEO or cost, the resulting trajectories are representative of those that result from more detailed analyses (e.g one that includes the vehicle masses) A key benefit of this weighting system is that I only rely on natural parameters (planetary orbits and masses) for computations, yet retain trajectory features (i.e low V∞ and low ∆V) that are essential for effective integrated mission design Table Required maneuvers for each trajectory type Trajectory Earth Mars Mars Earth DSM departure ∆V arrival ∆V departure ∆V arrival ∆V Direct taxi & TV TVa or neitherb taxi & TV TVa or neitherb neither Free-return taxi & TV TVa or neitherb N/A N/A neither c a b M-E semi-cycler taxi TV or neither taxi & TV neither TV c a b E-M semi-cycler taxi & TV neither taxi TV or neither TV c Cycler taxi neither taxi neither TV a Powered capture Aero-assisted capture c The one-time transfer vehicle launch ∆V is ignored b I use a sequential quadratic programming algorithm 79,80 to compute minimum-∆V trajectories with bounded TOF (By bounded I mean the TOF may be less than or equal to the constrained value.) Similar methods have been used in the optimization of the Galileo trajectory to and at Jupiter and the Cassini trajectory to Saturn 81, 82 I have developed my own software that provides a user-defined objective function and constraints (with gradients) to a commercial (MATLAB) optimizer Earth-Mars trajectories are optimized so that the total ∆V over the entire 15-year cycle is minimized (as opposed to, say, minimizing the maximum ∆V during the cycle) Though the arrival V∞ for aerocapture is often limited (e.g below km/s at Earth68 and below km/s at Mars69), I did not constrain the V ∞; in this way the lowest possible ∆V trajectories are analyzed An initial guess for the timing and placement of deep-space maneuvers (DSMs) is obtained via Lawden’s primer vector analysis.83, 84 For example, in Fig a DSM would be placed where the primer magnitude is largest (in this case around 7.5 years time of flight) An augmented trajectory (with the new DSM) is then optimized for minimum ∆V Additional DSMs are added until a locally optimal trajectory is found Fig demonstrates an optimal trajectory, where the primer vector P satisfies the five conditions for optimality: d P are continuous dt P and P is aligned with ∆V at impulse times P = at impulse times P < on coasting arcs separating impulses d P = at impulses dt Fig Primer vector magnitude along sub-optimal cycler trajectory Fig Primer vector magnitude along optimal cycler trajectory Using the SQP algorithm in concert with primer vector analysis, I computed optimal-∆V direct, free-return, Mars-Earth semi-cycler, Earth-Mars semi-cycler, and cycler trajectories with the TOF constrained to below 120 to 270 days for both powered and aero-assisted planetary capture.85 A synopsis of the results is presented in Fig for the average ∆V over seven consecutive missions and in Fig for the maximum ∆V over the seven-mission timescale Fig Average ∆V over 15-year cycle Fig Maximum ∆V over 15-year cycle 10 & sp T = mgI The rocket equation is particularly useful in determining the ratio of initial mass (payload, hardware, tankage, and propellant) to payload mass m0 = mpay e 1− e ∆V gI sp ∆V gI sp α  ∆V gI  a0 gI sp − f t  e sp − 1÷ 2η   The spacecraft thrust is thus T = mpay a0 e 1− e ∆V gI sp ∆V gI sp α  ∆V gI  a0 gI sp − f t  e sp − 1÷ 2η   In order to minimize the initial mass [Eq ] or thrust [Eq ], an accurate means of determining the minimum ∆V for a given trajectory TOF and vehicle a0 and Isp is required Zola88 describes an approximate analytic method to calculate ∆V as a function of a0 and Isp, provided the ∆Voptimal burn time tb for a trajectory with the same TOF is known In contrast to the maximumacceleration trajectories provided in Fig and Fig 9, I present the ∆V for a direct transfer from Earth to Mars (and vice-versa) using the minimum possible acceleration during launch years 2009–2022 in Fig 10 In the specific case of Fig 10, I calculated trajectories that match the heliocentric orbit of Mars from the orbit of Earth and set the mass flow rate to zero (corresponding to infinite Isp) Because the acceleration is minimized, the burn time for these trajectories is equal to the TOF (i.e the thruster is always on) Using this data, the heuristic method of Ref 88 produces values for optimum payload mass fractions to within a few percent Alternatively, numerical optimization techniques provide higher fidelity results at the expense of longer computation time I also examined minimum-acceleration transfers for aerocapture missions (for arrival V∞ ≠ ) but found that the V∞ at arrival was impractical (i.e it is often at least double the impulsive transfer V∞) Instead, I optimized low-thrust trajectories with infinite Isp (i.e assuming constant mass) and set the acceleration to the levels found in Fig 10 for a given launch year and TOF combination The resulting ∆V are found in Fig 11 Because the same initial acceleration was used to calculate the powered and aero-assisted capture trajectories, the burn time is given by tb = (∆VA/∆VP)TOF where ∆VP is the powered arrival ∆V (from Fig 10) and ∆VA is the aeroarrival ∆V (from Fig 11) Generally, increasing the acceleration decreases the ∆V and burn time, and in the limit of infinite thrust and zero burn time, the ∆V approaches the sum of the departure and arrival V∞ for powered capture missions and becomes the departure V ∞ for aeroassisted capture 12 a) Earth to Mars b) Mars to Earth Fig 10 ∆V for minimum-thrust transfers with powered capture 13 a) Earth to Mars b) Mars to Earth Fig 11 ∆V for low-thrust transfers with aerocapture or direct entry Parking Orbit Reorientation 14 Because parking orbits are an essential ingredient to semi-direct, stop-over, and semi-cycler mission scenarios, an effective means of capturing into then departing from an orbit is required 100–106 In many cases the parking orbit is not conveniently oriented with respect to the interplanetary leg (e.g., returning to Earth from a parking orbit at Mars) One solution to this problem is to reorient the spacecraft’s orbit about the planet before departure This method includes a maneuver at apoapsis that rotates the parking orbit about the line of apsides to achieve the proper orientation at departure, thus coupling the effects of parking-orbit orientation with the interplanetary trajectories This method is thus termed the “apo-twist” maneuver because the parking orbit is “twisted” about the line of apsides via a ∆V at apoapsis.104, 106 I also account for the natural precession of the parking orbit (due to J2, lunar, and solar perturbations) during the stay time V∞ ,D V∞, A V∞, A V∞ ,D Case Case Fig 12 Orientation of required V∞ and parking orbit Fig 12 illustrates the problem of entering into and departing from a parking orbit by presenting an ideal situation (Case 1) and a non-ideal situation (Case 2) Case presents no difficulties because the planet provides sufficient bending to reach V∞ ,D with a tangential burn at periapsis Here, the approach hyperbola, parking orbit, and departure hyperbola are coplanar and have the same periapsis To calculate the ∆V (assuming conic trajectories) only the magnitudes of V∞ ,A and V∞ ,D are needed; the cost of entering the parking orbit is found explicitly via Eq , ∆V = V∞2, A + µ rp − µ ( rp − a ) where µ is the gravitational parameter of the planet, rp is the radius of periapsis, and a is the semi-major axis of the parking orbit The cost of departing is found by replacing V∞ , A with V∞ , D To achieve V∞ ,D in Case seems difficult, but is possible with the apo-twist maneuver In this case, V∞ , A , V∞ ,D , and the parking orbit not necessarily lie in the same plane Even in the case where V∞ , A and V∞ ,D are in the plane of the page, it will be necessary to target V∞ , A so that the arrival parking orbit is not in the plane of the page The selection of the arrival parking orbit plane, combined with a maneuver at apoapsis (to rotate the departure parking orbit along the line of apsides) are the key ingredients of the apo-twist method With this technique, the approach, parking orbit, and departure trajectories share a common periapsis, but the total ∆V will be greater than that provided by Eq The apo-twist is illustrated in Fig 13, where V∞ , A , 15 the arrival orbit, and V∞ ,D (without apo-twist) are in the plane of the page The desired V∞ ,D (which is out of the plane of the page) is achieved by rotating the parking orbit via the apo-twist maneuver A (hyperbolic ha V∞ , A φ φ V∞ , D V∞ , D (without apo-twist) (after apo-twist) Departure orbit (out of plane of page) Arrival orbit (in plane of page) Twist angle, φ Fig 13 Rotation of parking orbit about line of apsides by twist angle φ To assess the effectiveness of the apo-twist maneuver (which requires three burns), I compare it to the optimal two-burn scheme The first burn is the orbit insertion maneuver, which is performed tangentially to the velocity at periapsis Thus, the insertion maneuver may be replaced by atmospheric braking (aerocapture) if desired The second burn is the departure maneuver Generally, this is a three-dimensional maneuver (i.e not tangential at periapsis) because the parking orbit and V∞ ,D are not perfectly aligned (Case of Fig 12) The minimum ∆V and corresponding inclination, using both the apo-twist and two-burn methods, are provided in Table and Table The optimal time to perform the apo-twist maneuver for each arrival date is found under the t ∆V column in these tables We note that an apo-twist maneuver is generally available at any point in the stay time, but the lowest ∆V maneuvers often occur either at the beginning or end of the stay time The ∆VA and ∆VD columns give the cost of arriving and departing the orbit with a tangential periapsis burn using Eq We note that in the case of aerocapture the propulsive ∆VA ≈ The ∆Vadd columns under “Apo-twist” and “Two-burn” provide the additional cost of achieving the correct direction at departure for these methods Thus the total cost is ∆Vtotal = ∆VA + ∆VD + ∆Vadd Since the reorientation cost can be a significant fraction of the total ∆V, it cannot be neglected in mission design Moreover, an economical way of reducing this reorientation ∆V will lower the propellant cost of the mission I find that the apo-twist maneuver requires the least ∆V when compared to other methods 16 Table Stop-over cycler,39 Mars Ideala Apo-twist ∆VD t∆V Arrival Stay time ∆VA iA iD Date (days) (km/s) (km/s) (days) (deg) (deg) 7/13/2012 508 1.87 1.53 508 22.2 50.9 513 154.8 142.5 8/19/2014 513 1.74 1.01 10/5/2016 531 1.52 0.82 137.6 139.0 592 140.5 138.8 10/6/2018 592 2.03 1.38 521 146.1 144.5 2/20/2021 521 0.88 1.63 a For 300 km periapsis altitude and day period orbit ∆Vadd (km/s) 0.245 0.104 0.108 0.027 0.031 Two-burn ∆Vadd i (deg) (km/s) 30.1 0.564 150.8 0.181 134.5 0.315 141.7 0.061 145.7 0.035 Table Stop-over cycler, 39 Earth Ideala Apo-twist ∆VD t∆V Stay time ∆VA iA iD Arrival Date (days) (km/s) (km/s) (days) (deg) (deg) 7/1/2014 617 1.39 1.01 617 30.2 50.2 8/11/2016 637 1.22 0.88 637 140.7 140.4 10/16/2018 648 0.89 1.06 576 119.2 104.4 12/17/2020 640 0.96 1.32 356 117.8 68.5 2/21/2023 615 1.05 1.40 615 26.3 40.3 a For 300 km periapsis altitude and day period orbit ∆Vadd (km/s) 0.607 0.336 0.234 0.752 0.401 Two-burn ∆Vadd i (deg) (km/s) 36.3 1.273 136.9 0.795 115.9 0.141 94.2 1.698 33.1 0.979 Proposed Research My remaining research is to determine a method to safely dock the taxi and transfer vehicle during planetary flyby (hyperbolic rendezvous) and to calculate IMLEO across the Mars mission design space The IMLEO calculation integrates all my previous research by incorporating key results (i.e vehicle masses, ∆V, TOF, technology levels, etc.) from each branch Using an IMLEO metric I will rank the architectures and technologies as they apply to the exploration of Mars Hyperbolic Rendezvous When a parking orbit is used in a Mars mission, the rendezvous between the taxi and transfer vehicle occurs in the (captured) orbit During a flyby, however, the rendezvous occurs on a hyperbolic trajectory as the transfer vehicle passes one planet for the other (as in Fig 14) One rendezvous is not necessarily more difficult than the other, but hyperbolic rendezvous is more critical to mission success because the taxi has already committed towards the next planet During parking-orbit rendezvous the crew could abort to the surface because the taxi is still captured in a parking orbit Extra propellant could mitigate thrusting or navigational errors during hyperbolic rendezvous, but the safety margin requires additional research 17 from Mars   taxi transfer vehicle Fig 14 Hyperbolic rendezvous at Earth Because hyperbolic rendezvous is only required for semi-cycler or cycler missions (which are relatively new concepts) there has not been much research on how to guide the taxi to the transfer vehicle, especially when compared to circular or elliptical orbit rendezvous 107–109 There is at least one paper that addresses the issue of hyperbolic rendezvous, but only the deterministic ∆V is analyzed.42,108 In practice, the rendezvous problem requires analysis of both navigation and control algorithms so that the relative velocity between the taxi and transfer vehicle is near zero when they dock I plan to examine various guidance algorithms in a rotating frame (e.g the Hill-Clohessy-Wiltshire equations) in search of a scheme that can handle navigation and injection errors with modest ∆V Ranking of Mission Architectures and Technologies The optimized heliocentric trajectories, parking orbit reorientation, and hyperbolic rendezvous provide the ∆V information necessary to complete a Mars mission Yet several other parameters are required to estimate the mass required in LEO Table provides specific impulse and dry mass information for different propulsion systems For transfer vehicle NEP the acceleration and Isp vary as a function of TOF to provide the optimal ∆V for α /η = 10 kg/kW (which is the lowest value that still allows 120-day transfers) Cargo NEP will use the lowest initial acceleration that allows the cargo to reach Mars within a couple of synodic periods The cargo α /η is higher (30 kg/kW) for near-term technology and lower (10 kg/kW) if transfervehicle NEP is also assumed to exist Because the required thrust is usually on the order of tens of Newtons, magnetoplasmadynamic (MPD) thrusters are assumed 18 Table Propulsion system parameters minert α Propulsion System Isp (sec.) or mpropellant η Hydrogen/Oxygen (H2/O2) 450 0.15 Methane/Oxygen (CH4/O2) 380 0.12 Nuclear (NTR) 900 0.50 Cargo NEP 8,000 10–30 kg/kW Transfer Vehicle NEP 3,000–5,000 10 kg/kW mtank mpropellant 0.10 0.08 0.15 0.15 0.15 The initial-to-payload mass fraction is determined for a given maneuver for impulsive ∆V by m0 = mpay e 1− ∆V gI sp minert  ∆V gI sp  e − 1÷ mpropellant   and for low-thrust transfers via m0 = mpay e ∆V gI sp mtank  ∆V gI sp  α a0 gI sp − e − 1÷ 2η mpropellant   The sequence of stages may be combined and stacked to produce the IMLEO 1− e ∆V gI sp The payload mass depends on the assumed crew number, cargo mass, and vehicle masses I plan to normalize the IMLEO on a per person basis (i.e IMLEO is given in mt/person) Nominally the taxi and transfer vehicle masses are 1.25 mt/person and mt/person, respectively The cargo mass varies from 0–10 mt/person, and I also vary the taxi mass from 1–3 mt/person and the transfer vehicle from 3–10 mt/person Given these parameters (along with a few other assumptions) I can provide IMLEO tables of propulsion technology versus architecture for a given cargo mass, taxi, transfer vehicle, and TOF I can also produce plots of IMLEO versus TOF or transfer vehicle mass Finally, I can produce charts of the IMLEO-optimal architecture as a function of transfer vehicle mass and TOF Once the IMLEO data is collected, the technologies that provide the greatest IMLEO reduction for a given development 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Penzo, P A and Nock, K T., “Hyperbolic Rendezvous for Earth-Mars Cycler Missions,” Paper AAS 02-162, AAS/AIAA Space Flight Mechanics Meeting, San Antonio, TX, January 27–30, 2002, pp 763–772 93 25 Wang, P K C., “Non-linear guidance laws for automatic orbital rendezvous,” International Journal of Control, Vol 42, No 3, 1985, pp 651–670 109 26 ... minimal-risk missions To examine the effects of limiting TOF, I compute optimal ∆V trajectories (to reduce mission cost) with constrained TOF (to reduce mission risk) Traditionally, Mars trajectories... Earth -Mars portion of the trajectory The crew would stay on Mars for about 550 days then take a short TOF inbound trajectory home (e.g the Mars- Earth transit in Fig 1) Fig Mars free-return trajectory... have designed four versions of Earth -Mars semi-cycler trajectories to complete the assessment of Mars mission architectures 50 An example trajectory for each trajectory type is presented in Fig