Establishing cycler trajectories between Earth and Mars

Acta Astronautica 112 (2015) 114–125

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Establishing cycler trajectories between Earth and Mars$ Blake A. Rogers a,n, Kyle M. Hughes a, James M. Longuski a, Buzz Aldrin b a b

Purdue University, School of Aeronautics & Astronautics, 701 West Stadium Avenue, West Lafayette, IN 47907-2045, USA Buzz Aldrin Enterprises LLC, 11901 Santa Monica Blvd., Suite 496, Los Angeles, CA 90025, USA

a r t i c l e in f o

abstract

Article history: Received 27 January 2014 Received in revised form 15 December 2014 Accepted 3 March 2015 Available online 11 March 2015

Several cycler concepts have been proposed to provide safe and comfortable quarters for astronauts traveling between the Earth and Mars. However, no literature has appeared to show how these massive vehicles might be placed into their cycler trajectories. In this paper, trajectories are designed that use either V 1 leveraging or low thrust to establish cycler vehicles in their desired orbits. In the cycler trajectory cases considered, the use of V 1 leveraging or low thrust substantially reduces the total propellant needed to achieve the cycler orbit compared to direct orbit insertion. In the case of the classic Aldrin cycler, the propellant savings due to V 1 leveraging can be as large as 24 metric tons for a cycler vehicle with a dry mass of 75 metric tons, and an additional 111 metric tons by instead using low thrust. The two-synodic period cyclers considered benefit less from V 1 leveraging, but have a smaller total propellant mass due to their lower approach velocities at Earth and Mars. It turns out that, for low-thrust establishment, the propellant required is approximately the same for each of the cycler trajectories. The cycler concept may provide a crucial enabling technology that is safe, economical, and sustainable for the continuous habitation of Mars. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: Cycling trajectories Mars cyclers V 1 leveraging Low thrust

1. Introduction In the late 1960s and early 1970s, Hollister [1,2] and Hollister and Menning [3,4] discovered trajectories that use gravity-assists to repeatedly encounter the Earth and Venus. Rall [5] and Rall and Hollister [6] were the first to show that these types of trajectories also exist between Earth and Mars. Trajectories that repeatedly encounter the same planets on a regular schedule without stopping are now known as cycler trajectories, or cyclers. There are ballistic cyclers, which use

☆ Earlier versions of this paper were presented at the 2012 AIAA/AAS Astrodynamics Specialist Conference in Minneapolis, MN, USA and the 2013 AAS/AIAA Astrodynamics Specialist Conference in Hilton Head, SC, USA. n Corresponding author. Tel.: þ1 865 660 0507. E-mail addresses: [email protected], [email protected] (B.A. Rogers).

http://dx.doi.org/10.1016/j.actaastro.2015.03.002 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.

only a small amount of propellant, and powered cyclers, which require large propellant expenditures. There are also semicyclers, which involve occasional stops, that can be combined with cyclers to support continuous Mars habitation [7]. The advantage of a ballistic cycler is that the vehicle can be more massive than a powered cycler. Massive vehicles are preferable for long human missions because they allow for increased space, and therefore comfort, for the crew. Cycler vehicles can also be made safer by adding a significant amount of material between the source and the crew for radiation protection. Perhaps the simplest and most famous of the Mars cyclers is the Aldrin cycler, named for Buzz Aldrin [8–10]. Several other types of cycler trajectories have been offered as possible solutions by Niehoff [11,12]; Friedlander et al. [13]; Brynes et al. [14]; McConaghy et al. [15]; and Russell and Ocampo [16]. Each of these has their advantages and disadvantages, but the problem of getting cycler vehicles into their cycler orbits is significant, and establishing these

B.A. Rogers et al. / Acta Astronautica 112 (2015) 114–125

Nomenclature a e h K L M V1 ΔV

semi-major axis, AU eccentricity altitude, km number of Earth orbit revolutions number of spacecraft orbit revolutions spacecraft orbit revolution on which the maneuver performed hyperbolic excess velocity, km/s change in velocity, km/s

transfer trajectories has not been investigated. In order to effectively compare each of the cycling trajectories, the initial mass in low-Earth orbit (IMLEO) cost of establishing these cycler trajectories must be taken into account. 2. Methodology Assuming a circular co-planar model, the simplest way to get onto the desired trajectory would be to perform a single, large maneuver at Earth, but this method would be expensive, and so, a more efficacious method of constructing “establishment trajectories” is attempted. At this point, it is necessary to introduce new terminology for the trajectories involved. The combination of the establishment and cycler trajectory will henceforth be called the “establishment-cycler” trajectory. 2.1. V 1 leveraging V 1 leveraging is the well-known use of a small deepspace maneuver to modify the V 1 at a body. Hollenbeck [17] was the first to use this technique when constructing a ΔVEGA trajectory. The term V 1 leveraging was first used by Williams and Longuski [18] when describing ΔVVGA trajectories and then more generally by Sims and Longuski

Fig. 1. The V 1 leveraging maneuver consists of a (1) ΔV launch followed by a (2) ΔV aphelion to increases the V 1 at the subsequent Earth flyby, which may be chosen to occur before (3  ) or after (3 þ ) perihelion. It should be noted that the crew does not board the cycler vehicle until the Earth flyby.

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ΔV DSM deep space maneuver, km/s ΔV total total ΔV for V 1 leveraging technique, km/s ΔV direct ΔV required to launch directly from low Earth orbit into cycler trajectory, km/s superscripts þ, 

Earth encounter performed after ( þ), before ( ) the spacecraft passes the line of apsides

[19]. Recent advancements in the understanding of V 1 leveraging are presented by Capagnola and Russell [20]. Exterior V 1 leveraging is used for this analysis, and a detailed description is given by Sims et al. [21] as follows and is illustrated in Fig. 1. First, a nominal orbit is chosen that has a period that will intercept the Earth after an integer number of periods. To find such nominal orbits, one chooses the number of Earth orbit revolutions (K) and the number of spacecraft orbit revolutions (L). If the spacecraft is on this orbit and no other maneuver is performed, the spacecraft would intercept the Earth tangentially in K years. Next, an initial tangential ΔV (labeled as ΔV launch in Fig. 1) is performed so that the resulting V 1 is larger than the V 1 required to get onto the nominal orbit. On some revolution number (M) of the spacecraft's orbit, a retrograde maneuver is performed at aphelion. This maneuver causes the orbit of the spacecraft to intersect the orbit of the Earth non-tangentially twice (indicated by 3  and 3 þ in Fig. 1), meaning that a gravity-assist maneuver can be used to raise (or lower) the energy of the orbit. The notation used to specify a given V 1 leveraging technique is K: LðMÞ 7 . The maneuver is performed at aphelion because the shape of the orbit is most efficiently altered when the velocity is the lowest. Therefore, performing the maneuver at aphelion produces the largest encounter angle between the Earth and the spacecraft [22]. The trajectory after the flyby is determined by whether the encounter is selected to be before ( ) or after (þ) perihelion and by the altitude of the flyby. Sims found [21] that for aphelion radii at the orbital distance of Mars, flybys before perihelion require less total ΔV. The trajectories considered in this work will all have aphelia close to Mars. 2.1.1. Genetic algorithm Finding an appropriate resulting trajectory using V 1 leveraging can be time consuming, especially if an exhaustive search or trial and error approach is used. One must initially choose the K, L, M, the magnitude of the V 1 above the nominal orbit, and the altitude of the flyby (h) to determine the final characteristics and to see if they meet the requirements. A genetic algorithm is implemented (chosen because of the mixed integer and continuous design variables) to solve the minimization problem: X min J ¼ ΔV ¼ ΔV launch þ ΔV aphelion ð1aÞ

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subject to   adesired  agenetic  r aerror

ð1bÞ

  edesired  egenetic  reerror

ð1cÞ

to be between 1 and 15, a V 1 at launch above nominal is between 1 and 3 km/s (with a maximum error of 1 m/s), and a flyby altitude between 300 and 5000 km (with a maximum error of 1 km), there would be a total of approximately 4:76  1010 possible initial conditions.

with the variables x ¼ ðK; L; M; V 1 ; hÞ

ð1dÞ

K; L; M A Z

ð1eÞ

V 1; h A R

ð1f Þ

where adesired is the desired final semi-major axis of the cycler orbit, agenetic is the actual value of the semi-major axis of the cycler orbit found using the genetic algorithm, and aerror is the allowable difference between these two values. The eccentricity variables are defined similarly. MATLAB'ss genetic algorithm function, ga, from the Global Optimization Toolbox is used with a design point population of 40. There are three genetic algorithm operators that mimic the evolutionary process: selection, crossover, and mutation. The selection function (how the genetic algorithm chooses parents for the next iteration) is MATLAB's stochastic uniform function. The crossover function (how two parents are combined to form a child) is MATLAB's heuristic function. The mutation function (how small, random changes are made to children) is MATLAB's adaptive feasible function. To solve the minimization problem and enforce both the linear and the nonlinear constraints, MATLAB uses sequential minimization of a Lagrangian barrier function. (This method is discussed in detail in Ref. [23].) The range of values used for K, L, and M are varied throughout the process. At first, each of these variables is allowed to have a value between 1 and 15 to get a sense of the types of orbits that could be found. It becomes evident that diminishing returns set in at much smaller values than 15, i.e. the difference in minimum ΔV between two different large K values (e.g. 10 and 15) is negligible. Because K represents the number of years the establishment process will take, an upper limit of five is set to allow for a significant amount of savings to occur, but does not take an unreasonable amount of time in doing so. Each run through the genetic algorithm takes an average of 3 min with a single 2.53 GHz processor. Based upon the initial population, which is generated randomly, different trajectories are found. It is desirable to find trajectories with different K values, so eventually K is constrained to be just one number per run with no variation. The genetic algorithm is then run at least 20 times to find the other values that gave the lowest ΔV. More than 20 runs are required if there is significant deviations in the results. For a series of runs, the optimal ΔV values are usually within tens of meters per second of one another. Occasionally, there is more deviation among the top results and so more than 20 runs are performed until there is more of a consensus of what is the lowest ΔV achievable. An exhaustive search may appear appealing for this problem, however the brute force approach is not feasible because of the number of variables and the required level of orbit element accuracy on the post V 1 leveraging trajectory. For example, if K, L, and M values are allowed

2.1.2. Patched conic propagator analysis The establishment-cycler trajectory solutions found by the genetic algorithm in the circular, co-planar model can be verified by using the Satellite Tour Design Program (STOUR) developed by the Jet Propulsion Laboratory for the Galileo mission tour design [24]. STOUR uses an analytic ephemeris for the location of the planets and the well-known patched-conic method to calculate the three-dimensional gravity-assist trajectories. Incorporating the V 1 leveraging maneuver in STOUR was achieved with the ΔVEGA subroutine developed by A. J. Staugler in 1996 at Purdue University [25]. This subroutine allows the user to specify launch date, nominal resonance orbit, approximate deviation in launch V 1 from the nominal, location of the maneuver in terms of true anomaly (at aphelion for this study), and number of heliocentric spacecraft and Earth revolutions between the maneuver and the Earth flyby. With the parameters of the V 1 leveraging maneuver specified, STOUR then finds all possible patchedconic trajectories to the subsequent Mars encounter. To ensure that the desired cycler trajectory is found, launch dates are searched over a three-year period in order to encompass the 2 17 year Earth–Mars synodic period. The starting launch date is chosen to be no sooner than 2022. For some cycler trajectories, the aphelion is of a similar size to Mars' perihelion. Such cyclers require a search space of launch dates larger than three years in order to find a solution with the required relative orientation of Earth and Mars while also having Mars positioned near its perihelion when it is encountered by the cycler vehicle. Once STOUR calculates the trajectories for the specified span of launch dates, those trajectories that have orbit characteristics similar to the cycler trajectory of interest are identified. First, trajectories with the desired cycler shape are found by matching the aphelion and perihelion radii for the Earth to Mars arc to the theoretical aphelion and perihelion radii of the cycler (in the circular, co-planar model). The trajectory is then checked to ensure that the transfer occurs on the correct segment of the orbit. For example, on the Earth-to-Mars leg of the outbound Aldrin cycler, the spacecraft must encounter Earth just before perihelion on the postEarth flyby orbit and arrive at Mars before reaching aphelion. The final trajectories that are chosen from the STOUR output over the three-year search are those that most closely match the desired cycler orbit shape (i.e. aphelion and perihelion radii) while also providing the desired orbit segment on the trip from Earth to Mars for the first leg of the trajectory. 2.2. Low-thrust analysis Low-thrust propulsion is a well-known alternative to the impulsive methods described above for establishing a cycler vehicle on its cycling trajectory. Electric propulsion systems employ low thrust in order to increase the mass delivered

B.A. Rogers et al. / Acta Astronautica 112 (2015) 114–125

or reduce the time of flight compared to chemical propulsion systems that use impulsive maneuvers [26,27]. For this application, low thrust can be used to shape an orbit so that it acts similar to the V 1 leveraging trajectories. Instead of using two impulsive maneuvers, there are thrusting arcs spread out over large portions of the orbit. 2.2.1. Interplanetary trajectory The low-thrust trajectories are constructed using the Jet Propulsion Laboratory's Mission Analysis Low-Thrust Optimization (MALTO) tool [28], assuming that the V 1 of launch from Earth is 0 km/s. A series of low-thrust maneuvers are performed so that the cycler vehicle encounters the Earth at the proper time and with the correct V 1 to continue on the cycler trajectory to Mars. Unlike the case using the V 1 leveraging technique, multiple Earth encounters are used during the establishment phase, and these encounters are added until the launch V 1 is reduced to 0 km/s. MALTO approximates continuous thrust by dividing each planet–planet leg of the trajectory into segments with an impulsive ΔV at each of the midpoints. (The implementation of this algorithm is discussed by [28].) The propagation of the trajectory between these impulsive maneuvers assumes conic trajectories and also assumes that all flybys are instantaneous and modeled by the rotation of the V 1 vectors. This method results in a constrained nonlinear optimization problem, which MALTO solves by using the nonlinear programming software Sparse Nonlinear OPTimizer (SNOPT) [29,30]. MALTO solves for the cycler trajectory in which the final mass is maximized. The process of constructing these low-thrust cycler trajectories is labor intensive because the required orbits are very sensitive to deviations in the required encounter dates and V 1 values. Experience with MALTO has shown that after these low-thrust trajectories are constructed, it can be difficult to change spacecraft properties (i.e. input power and mass). For example, if a trajectory is constructed assuming a cycler vehicle of mass 75 mt, changing that mass to 50 mt would be difficult due to many of the MALTO variables being on their bounds. In order to change the mass, one has to slowly reduce the mass by one or two metric tons at a time, and many times MALTO will not be able to converge when changing these values. Instead, if a property must be changed, it is usually easier (or necessary) to restart the trajectory construction process. However, the resulting trajectories will be essentially the same for the different property values (i.e. similar encounter dates, propellant masses, and thrust directions). For example, if a solution is found for a 75 mt vehicle and one is sought for a 50 mt vehicle, the final trajectory will be essentially the same, but the labor involved is non-trivial. 2.2.2. Escape via low-thrust spiral To reach a V 1 of 0 km/s at Earth, the cycler vehicle uses low thrust to spiral out from its original low-Earth orbit. There are several types of thrust programs that can be used to escape; this work considers the circular and elliptical spiral trajectories. Circular spiral: To the first order, the time optimal scheme is a circular spiral where the thrust is aligned with the velocity vector. [31,32] The equations of motion

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for tangential thrust are r_ ¼ V sin γ

ð2aÞ

μ T V_ ¼  2 sin γ þ m r

ð2bÞ

γ_ ¼

  V μ  2 cos γ r r V

ð2cÞ

The thrust and mass of the spacecraft are functions of the exit velocity, power, specific impulse, and efficiency of the propulsion system as follows: V e ¼ gI sp γ _ ¼ m

ð3aÞ

2ηP

ð3bÞ

V 2e

_ e T ¼  mV

ð3cÞ

Elliptical spiral: The elliptical spiral scheme keeps the perigee of the orbit near-constant throughout, which allows for a non-zero V 1 at Earth escape, the ability to perform a chemical burn close to Earth where it is the most efficient, and, while not relevant in the present work, a way to place a crew on the spacecraft after the majority of the spiral is completed. The equations of motion for the elliptical spiral are r_ ¼ V r

θ_ ¼

ð4aÞ

Vθ r

ð4bÞ

μ V2 T V_ r ¼  2 þ θ þ sin α r m r

ð4cÞ

VrVθ T þ cos α V_ θ ¼ m r

ð4dÞ

where α is the control law described in Reference [32]. This control law is undefined at perigee and apogee, and it is also inefficient in the neighborhood of apogee. Consequently, thrusting is only performed when the true anomaly of the spacecraft is between 51 and 1201 and between 2401 and 3551.

Fig. 2. The Aldrin Cycler.

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3. Results 3.1. V 1 leveraging 3.1.1. Selected cyclers and their characteristics The trajectories considered for establishment via V 1 leveraging are the Aldrin Cycler [9] (Fig. 2), the VISIT cyclers [11–13](VISIT-2 shown in Fig. 3), and various two synodic period cyclers [14,15] (S1L1 shown in Fig. 4). The orbital elements of the first legs of the various cycling trajectories are presented in Table 1. The Aldrin cycler, and examples of a VISIT-2 and S1L1 cycler are illustrated in Figs. 2, 3, and 4, respectively. In Fig. 2,

Fig. 3. The VISIT-2 (7(0.95)10) cycler.

the Aldrin cycler vehicle launches from Earth (E1) and travels

Fig. 4. The S1L1(2.8277) cycler.

to Mars (M2). The spacecraft next encounters Earth (E3), one synodic period after launch. A gravity assist is performed, which rotates the line of apsides by 1/7 of a revolution. The cycler vehicle continues on to Mars (M4), and repeats the process ad infinitum. After 7 Earth–Mars synodic periods (15 years), the Earth is at the same inertial position as launch. In Fig. 3, the VISIT-2 cycler vehicle launches from Earth (E1) and travels to Mars (M2). After approximately 9 and a half revolutions (10 since launching from Earth) and 15 years, the vehicle encounters Earth (E3) at the same inertial position as E1. In Fig. 4, the S1L1 cycler vehicle launches from Earth (E1) and travels to Mars (M2). The cycler vehicle travels on this same trajectory for approximately one and a half more rotations about the Sun until it encounters Earth (E3). A gravity assist is performed that changes the orbit to the inner trajectory. The cycler vehicle travels approximately one and a half revolutions about the Sun on this orbit until it encounters the Earth (E4) again, two synodic periods after launch (E1). A gravity assist is performed at E4 which rotates the trajectory 2/7 of a full rotation about the Sun to start the next S1L1. The cycler vehicle continues on to Mars (M5), the dashed line, and the process continues ad infinitum. The VISIT-1 and VISIT-2 cycling trajectories are not specific trajectories, but a class. The nðRp Þr 7 notation [15] used specifies their particular characteristics. In this notation, the VISIT-1 and VISIT-2 cycling trajectories considered are 7 (0.94)127 and 7(0.95)107, respectively. Similarly, the S1L1 cycler [15] is also a class, and the one considered is the S1L1 (2.8277), where the time between E1 and E3 is 2.8277 years. Several of the trajectories have multiple legs that change orbit characteristics after flybys of the Earth. However, to establish the cycler vehicle, only the characteristics of the first leg are used (i.e. E1 to M2). In the case of the S1L1 cycler (Fig. 4), there are two Earth encounters (E1 and E3) before the cycler repeats. One might consider the possibility of using the E3 encounter to launch (instead of E1), but it turns out that both of these encounters have the same V 1 magnitude, and so there is no propellant advantage in starting at E3. Also, starting at E3 lengthens the time-of-flight for the astronauts aboard to reach Mars (M5). For this work, it is assumed that the Earth encounter (E1) before the Mars encounter (M2) is where the S1L1 cycler begins. The number of cycler vehicles required for a given cycler trajectory is the amount needed to take advantage of each

Table 1 Orbital elements and number of vehicles for selected cycler trajectories. Type

Number of vehicles

Semi-major axis (AU)

Eccentricity

Aphelion radius (AU)

Perihelion radius (AU)

Aldrin Cycler VISIT-1 VISIT-2 Case 1 Case 2 Case 3 S1L1 U0L1

2 14a 14a 4 4 4 4 4

1.60 1.17 1.31 1.22 1.21 1.30 1.30 2.05

0.393 0.193 0.275 0.238 0.202 0.268 0.257 0.563

2.23 1.40 1.67 1.51 1.45 1.65 1.64 3.20

0.97 0.94 0.95 0.93 0.96 0.95 0.97 0.90

a These cyclers repeat every 7 Earth–Mars synodic periods, which usually means that 14 vehicles are needed. However, the VISIT cyclers encounter Earth and Mars more often than once every 15 years, so fewer vehicles could be used [15].

B.A. Rogers et al. / Acta Astronautica 112 (2015) 114–125

Earth–Mars or Mars–Earth transfer. These transfer opportunities include both the outbound cycler (ascending or up escalator), where Earth to Mars is the shorter leg, and inbound cycler (descending cycler or down escalator), where Mars to Earth is the shorter leg. Each cycler trajectory passes through Earth orbit and Mars orbit at least twice each, so it is crucial to know at which passage the spacecraft would encounter each planet. The Aldrin and Case 1 cyclers launch from Earth before perihelion, the other after. All of the cyclers, with the exception of Case 1, encounter Mars before aphelion. To get an initial idea of how these different cyclers compare, an impulsive ΔV to get directly onto each is calculated. It is assumed that the Earth and Mars orbits are circular and co-planar and the initial orbit around Earth is circular at an altitude of 300 km. The values are shown in Table 2.

Table 2 Direct ΔV required to go from LEO to cycler orbit with one impulsive burn for selected cyclers. Type

Direct ΔV a km/s

Aldrin VISIT-1 VISIT-2 Case 1 Case 2 Case 3 S1L1 U0L1

5.026 4.118 4.564 4.524 3.971 4.495 4.208 8.052

119

3.1.2. Genetic algorithm results A subset of the establishment-cycler trajectories found from running the genetic algorithm are shown in Table 3. The solutions shown have a maximum K value of 5, corresponding to V 1 leveraging times of flight slightly less than 5 years. Longer V 1 leveraging trajectories (corresponding to larger K) were found, but diminishing returns of smaller overall ΔV made them less attractive because of their longer flight times. The tolerances on the semi-major axis and eccentricity were set at 0.09 AU and 0.009, respectively. As an example, in Table 3 the Aldrin 4:3(2)  launches from the Earth with a V 1 of 2.638 km/s on an orbit slightly more energetic than a 4:3 resonance orbit with the Earth. On the spacecraft's second orbit about the Sun, when only one orbit is fully completed, a 0.604 km/s retrograde maneuver is performed to lower the perihelion of the orbit. The spacecraft completes this orbit and nearly completes an additional orbit, where before periapse on this last orbit around the Sun, the cycler vehicle intercepts the Earth with a V 1 of 5.717 km/s. Then a 300-km flyby changes the spacecraft's heliocentric orbit to have a semi-major axis of 1.61 AU and an eccentricity of 0.384, which takes the cycler vehicle to Mars. The total ΔV to achieve this final orbit is 4.118 km/s, savings of 0.488 km/s over the 4.606 km/s required if the cycler were launched directly into the final orbit. 3.1.3. Patched conic results The trajectory data for the given establishment-cycler trajectories found using the patched-conic propagator with analytic ephemeris in STOUR are shown in Table 4. According to NASA's Mars Design Reference Architecture [33], the Earth-to-Mars time of flight should be 180 days or less for the health of the crew. The results from STOUR confirm that the Aldrin, Case 2, and U0L1 cyclers have

a Direct ΔV refers to launching directly from low Earth orbit (LEO) into the cycler trajectory without using any gravity assist maneuvers.

Table 3 Genetic algorithm results for the selected cyclers in the circular, co-planar model. Type

a K: LðMÞ 7 V 1;launch (km/s) ΔV DSM (km/s) V 1;flyby (km/s) hflyby (km) ΔV total (km/s) a (AU) e

4:3(2)  3:2(1)  VISIT-1 5:4(3)  4:3(2)  VISIT-2 5:4(3)  4:3(2)  3:2(2)  Case 1 5:4(3)  4:3(2)  3:2(2)  Case 2 5:4(3)  4:3(2)  Case 3 4:3(2)  3:2(2)  S1L1 5:4(3)  4:3(2)  3:2(2)  U0L1 4:3(3)  3:2(1)  2:1(1) 

Aldrin

a

2.638 3.506 2.009 2.503 2.052 2.550 3.341 2.033 2.523 3.340 2.013 2.507 2.541 3.341 2.042 2.539 3.340 3.188 3.752 5.148

0.604 0.520 0.099 0.047 0.310 0.233 0.181 0.215 0.126 0.160 0.116 0.064 0.196 0.180 0.258 0.190 0.160 1.586 1.417 1.320

5.717 6.578 2.664 2.848 3.754 3.993 4.611 3.300 3.378 4.479 2.767 2.968 3.794 4.607 3.514 3.760 4.479 9.801 10.362 12.634

300 644 1175 644 300 1493 1818 2179 338 4160 422 1056 306 2390 419 309 2212 722 300 300

4.118 4.268 3.482 3.530 3.701 3.726 3.880 3.602 3.614 3.859 3.500 3.548 3.688 3.880 3.647 3.660 3.859 5.241 5.244 5.672

1.61 1.57 1.22 1.22 1.36 1.35 1.33 1.29 1.29 1.26 1.24 1.23 1.35 1.32 1.33 1.33 1.31 2.11 2.07 1.96

0.384 0.384 0.184 0.184 0.266 0.266 0.266 0.229 0.229 0.233 0.193 0.193 0.259 0.259 0.248 0.248 0.253 0.554 0.554 0.565

ΔV direct b (km/s) ΔV savings (km/s) 4.606 5.028 3.520 3.565 3.827 3.907 4.133 3.688 3.711 4.083 3.545 3.596 3.840 4.132 3.751 3.829 4.083 6.952 7.333 8.977

0.488 0.759 0.038 0.035 0.126 0.181 0.253 0.085 0.097 0.224 0.045 0.048 0.152 0.252 0.104 0.169 0.224 1.711 2.089 3.305

Deep space maneuver that is performed during V 1 leveraging at the aphelion. Note that these Direct ΔV vary slightly from the values in Table 2. These values again assume that the spacecraft is launching from a 300-km circular orbit, but use the slightly different orbit characteristics found with the genetic algorithm. The tolerances on the semi-major axis and eccentricity are 0.09 AU and 0.009, respectively. b

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B.A. Rogers et al. / Acta Astronautica 112 (2015) 114–125

Table 4 STOUR results for the selected cyclers in the analytic ephemeris. Type Aldrin VISIT-1 VISIT-2 Case 1 Case 2 Case 3 S1L1 U0L1 a

K: LðMÞ 7 

4:3(2) 3:2(1)  4:3(2)  4:3(2)  4:3(2)  4:3(2)  4:3(2)  4:3(2)  3:2(1) 

LD (mm/dd/yyyy)

TOFa (days)

Periapse (AU)

Apoapse (AU)

V1;launch (km/s)

ΔV DSM (km/s)

V1;flyby (km/s)

hflyby (km)

01/03/2023 12/09/2023 06/12/2046 12/27/2022 12/14/2024 04/22/2029 12/20/2022 12/20/2022 11/25/2025

161 172 201 207 365 171 216 221 97

0.983 0.964 1.012 0.985 0.946 0.997 0.988 0.987 0.960

2.229 2.229 1.470 1.670 1.503 1.433 1.651 1.635 3.189

2.608 3.449 2.540 2.503 2.558 2.522 2.492 2.492 3.702

0.568 0.530 0.043 0.221 0.346 0.059 0.187 0.182 0.794

5.509 6.546 2.834 3.878 4.526 2.910 3.688 3.657 8.049

10483 20745 120625 7696 140286 56068 13291 12965 6633

Time of flight from Earth flyby to Mars.

times of flight that satisfy this requirement. The 4:3(2)  Case 3 and 4:3(2)  S1L1 establishment-cycler trajectories have the smallest Earth launch V 1 values, as well as some of the smallest maneuver ΔV values. The lowest maneuver ΔV values come from the 4:3(2)  VISIT-1 and 4:3(2)  Case 2 establishment-cycler trajectories, 0.043 km/s and 0.059 km/s, respectively. Unfortunately, the VISIT-1 cycler requires more than 180 days to reach Mars, and, additionally, requires an infeasibly large number of cycler vehicles (up to 14 for missions that return the crew back to Earth). The 4:3(2)  Case 2 establishment-cycler trajectory, on the other hand, has an Earth launch V 1 of only 2.522 km/s, and has an Earth–Mars flight time of 171 days. Another notable establishment-cycler trajectory is the 4:3(2)  Aldrin cycler, which has a time of flight of 161 days, an Earth launch V 1 of 2.608 km/s, and a maneuver ΔV of 0.568 km/s. It should also be noted that for all of the trajectories considered in Table 4, the closest approach flyby altitude is well beyond the safe altitude of 300 km. The savings from V 1 leveraging, in terms of both launch V 1 and overall ΔV, are given in Table 5. The 3:2 (1)  Aldrin establishment-cycler trajectory requires a launch V 1 of 6.546 km/s if a direct launch into the orbit is performed, but with V 1 leveraging, only 3.449 km/s is needed, a savings of 3.097 km/s (as indicated in the table). The launch V 1 of 3.449 km/s corresponds to a launch ΔV of 3.731 km/s when launching from a 300 km altitude at Earth. Adding this ΔV with a deep space maneuver ΔV of 0.530 km/s gives a total ΔV of 4.261 km/s, which is a ΔV savings of 0.750 km/s over the direct launch. When the establishment-cycler trajectories given in Table 4 were determined, the aphelion of the cycler trajectories was usually the first parameter examined to find a match to a particular cycler trajectory (out of the many trajectories found in STOUR). Once a match was found for the aphelion (usually to within a hundredth of an AU), the cycler trajectory with the closest to the desired perihelion was chosen as the candidate match. Because of this selection order (aphelion first, perihelion second), the aphelion values tend to match more closely to those determined by the circular, co-planar model than do the perihelion values, as observed in Tables 4 and 1. For the case of the 4:3(2)  Aldrin trajectory, two additional analyses are done to further verify that the STOUR result is a true match to the circular-coplanar

Table 5 Savings accrued by using V 1 leveraging. Type

a K: LðMÞ 7 V 1 savings (km/s)

Aldrin 4:3(2)  3:2(1)  VISIT- 4:3(2)  1 VISIT- 4:3(2)  2 Case 4:3(2)  1 Case 4:3(2)  2 Case 4:3(2)  3 S1L1 4:3(2)  U0L1 3:2(1)  a b

ΔV total (km/s)

ΔV direct (km/s)

ΔV savingsb (km/s)

2.901 3.097 0.295

4.075 4.261 3.535

4.510 5.011 3.562

0.435 0.750 0.027

1.375

3.704

3.868

0.164

1.968

3.851

4.124

0.274

0.388

3.546

3.581

0.035

1.196

3.668

3.806

0.138

1.165 4.347

3.668 4.604

3.796 5.845

0.128 1.241

The difference between V 1 flyby and V 1 launch. The difference between direct ΔV and V 1 leveraging total ΔV .

trajectory. First, the V 1 leveraging results from the STOUR trajectory are compared to those of Sims et al. [21]. They calculated the amount of V 1 increase (from Earth launch to Earth flyby) expected for a given amount of maneuver ΔV. In Table 4, the V 1 increase for the 4:3(2)  Aldrin cycler is about 2.90 km/s due to a maneuver size of 0.57 km/s, which agrees with the increase predicted by Sims et al. of about 3 km/s for the same sized 0.57 km/s maneuver. The second additional verification test on the 4:3(2)  Aldrin trajectory is to determine whether the second leg of the cycler trajectory can be found using STOUR. After departing Mars, this second leg must have the same orbit shape as the first leg, pass through the aphelion of the orbit, and then pass through the perihelion before arriving at Earth. Upon further simulations in STOUR, the second leg of the Aldrin cycler is found. Interestingly, this second leg has perihelion and aphelion values that more closely agreed with the theoretical values for the Aldrin cycler (as determined using a circular, co-planar model) than those of the first leg. The 4:3(2)  Aldrin trajectory schematic is presented in Fig. 5. The spacecraft launches (1) from the Earth onto the dashed orbit and completes one and a half orbits. At aphelion (2) of this orbit, a maneuver is performed to

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Table 6 Results of low thrust cycler establishment using 210 kW and 5000 s I sp . Encounter Parameter

Aldrin

S1L1/Case 3

VISIT-2

E-1

Date (m/d/y) V 1 (km/s) Mass (mt)

09/27/2024 0 95

01/24/2029 0 95

– – –

E0

Date V1 Mass

10/29/2025 2.57 91.2

01/30/2030 1.93 91.4

09/08/2031 0 95

E1a

Date V1 Mass

11/02/2026 5.64 87.3

03/12/2031 5.80 87.1

02/28/2033 4.47 89.3

Total Flight Time (d) Total Propellant (mt)

765

777

537

7.7

7.9

5.7

Fig. 5. The 4:3(2)  Aldrin establishment-cycler trajectory.

lower the perihelion and get onto the dotted orbit. The spacecraft completes one full orbit and returns to aphelion. The spacecraft continues inbound toward perihelion and flies (3) by the Earth, performing a gravity-assist maneuver. This maneuver raises the heliocentric energy of the orbit and puts the vehicle on the cycler trajectory (the solid line). Mars is encountered (4), but does not significantly affect the trajectory. The spacecraft completes its orbit and encounters (5) the Earth once more in order to continue on the cycler trajectory. The 3:2(1)  Aldrin trajectory is similarly verified.

3.2. Low thrust Unlike the V 1 leveraging case, the savings of lowthrust establishment cannot be measured in terms of ΔV savings so a spacecraft and its propulsion system need to be considered. The cycler vehicle used in the low-thrust analysis is partially based upon the vehicle design by Nock [34,35]. Nock proposed a vehicle that is 70 mt when fully fueled (50 mt dry) with an ion propulsion system that has an input power of 148 kW, specific impulse of 5000 s, efficiency of 73%, specific mass of the propulsion system of 3.8 kg/kW and 4 kg/kW for the solar arrays. The cycler vehicle considered in this study will be slightly larger than Nock's to provide additional shielding, more crew space, or artificial gravity. The vehicle has a mass of 95 mt at Earth escape, 75 mt of which is dry mass, leaving 20 mt of propellant for the interplanetary phase of the establishment. In order to compensate for this larger mass, the power systems input power is increased to 210 kW. The propulsion and solar array specific masses are assumed to be the same, this means the SEP system mass increases from 1154 kg to 1638 kg, which is still a small fraction of the overall mass. The specific impulse is unchanged, but the efficiency is lowered slightly to 70%. The propulsion system above is based on a NIAC study, and may be somewhat futuristic. The use of other (less futuristic) SEP systems is also considered. As an example of a recent solar electric propulsion system, the Dawn spacecraft uses 3 NSTAR ion engines that have a specific impulse of 3100 s and 10 kW of power (at Earth) for the 1.2 mt wet mass. Another similar example is the Boeing 702 communications satellite that uses

a

Cycler trajectory begins.

XIPS ion thrusters with a specific impulse of 3500 s and 4.4 kW of power for 6.2 mt of mass [36,37]. The proposed cycler vehicle is much more massive than either the Dawn or Boeing 702 spacecraft, but they are representative of the current state of the technology. Another option is to use a propulsion system similar to that proposed by Price et al. [38] for a SEP tug that would take cargo to Mars in support of human missions. The trajectories are found using this similar propulsion system, which assumes a specific impulse of 2500 s, 100 kW of power, and a 70% efficiency. These trajectories are in addition to the trajectories found assuming the aforementioned spacecraft based upon the NIAC study. 3.2.1. Interplanetary trajectory The trajectories considered for low-thrust establishment are the Aldrin cycler, the VISIT-2 cycler, the Case 3 cycler, and the S1L1 cycler. The start date of each of the cycling trajectories is needed. The start dates for the Aldrin, S1L1, and Case 3 cyclers are taken from References [39,15], and [14], respectively. The dates and velocities for the S1L1 and Case 3 cyclers are similar. Therefore, these two cyclers will have nearly the same establishment, so the analysis is performed for the S1L1 case and then assumed to apply for Case 3 as well. The VISIT-2 cycler dates do not appear in the literature for the desired time range, and are found using STOUR. The trajectory details found from MALTO for the vehicle based upon Nock's design (210 kW, 5,000 s) are shown in Table 6. To ensure that the cyclers are in fact on the correct orbits, the trajectories are propagated until the next Mars encounter (M2) and the details of that leg are shown in Table 7. The masses reported include only the cycler vehicles and their propellant, but the trajectories between E1 and M2 are calculated assuming that a 35 mt fully fueled taxi vehicle is attached to ferry the crew from the spacecraft to the surface of Mars. An example of a low-thrust establishment trajectory is shown in Fig. 6. The arrows indicate where thrusting occurs. The standard nomenclature for the planetary encounters

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Table 7 Mars arrival properties after low thrust cycler establishment using 210 kW and 5000 s Isp . Encounter

Parameter

Aldrin

S1L1 / Case 3

VISIT-2

M2

Date V1 Mass

04/18/2027 9.39 87.3

07/15/2031 7.76 87.1

08/05/2033 7.81 89.3

Fig. 7. Establishment of the S1L1 and Case 3 cyclers using low thrust.

Fig. 6. Establishment of the Aldrin cycler using low thrust.

starts with the planets first letter and the encounter number. For example, traditionally the spacecraft would be launched at E1, travel to M2, etc. However, in this case, the cycler vehicle is not launched directly into the cycler orbit. Therefore, the encounters in Fig. 6 prior to establishment in the cycler orbit are labeled E-1 (launch) and E0. Launch from Earth begins at E-1, a gravity assist occurs at E0, and placement into the cycler orbit occurs at E1. Between E-1 and E0, more than one full revolution of the Sun is completed. This is also true between E0 and E1. The E1-toM2 leg is the first leg of the Aldrin cycler. The next Earth-toMars leg (E3-to-M4) occurs one synodic period later. The low-thrust establishment for the other cyclers are shown in Figs. 7 and 8. These trajectories, found by using MALTO, are similar to those described by Meissinger [40] and Sauer [41]. The trajectory details found for the propulsion system based upon Price's less futuristic design (100 kW, 2,500 s) are shown in Table 8.

Fig. 8. Establishment of the VISIT-2 cycler using low thrust. Table 8 Results of low thrust cycler establishment using 100 kW and 2500 s Isp . Encounter Parameter

Aldrin

E-1

Date (m/d/y) V 1 (km/s) Mass (mt)

09/14/2024 01/15/2029 0 0 95 95

– – –

E0

Date V1 Mass

09/24/2025 02/09/2030 1.58 90.4 89.0

09/02/2031 0.00 95

E1a

Date V1 Mass

10/31/2026 03/10/2031 5.35 82.3 81.1

02/28/2033

Total Flight time (d) 777 Total Propellant (mt) 12.7

3.2.2. Low-thrust spiral The initial orbit of each of the spacecraft is assumed to be circular and has an altitude of 300 km around Earth. Because each of the cycler trajectories has the same mass of 95 mt at Earth escape, the low-thrust spiral of a given type is the same for each (i.e. all circular spirals will be the same, all elliptical spirals will be the same). The low-thrust propulsion engine for the spiral stage of the trajectory is assumed to be the same as for the interplanetary phase. For the NIAC inspired propulsion system, using the circular spiral to achieve the constraint of 95 mt at escape, the starting mass of the spacecraft must be 110.5 mt (i.e. 15.5 mt of propellant is required). The circular spiral takes

a

S1L1/Case 3 VISIT-2

784 13.9

84.0 545 11.0

Cycler trajectory begins.

4 years, 5 days to complete, which is in addition to the establishment times displayed in Table 6. The elliptical spiral takes much longer to escape the Earth, so a time limit of five years is (arbitrarily) imposed. After the five years, the low thrust is turned off and when the cycler spacecraft comes back to perigee, an impulsive maneuver (with a specific impulse of 300 s) is used to escape. In this scenario, the cycler vehicle starts off with a mass of 186.0 mt. After five years of thrusting, the mass is

B.A. Rogers et al. / Acta Astronautica 112 (2015) 114–125

175.4 mt (10.6 mt of propellant is used). The chemical propulsion then uses 80.4 mt of propellant to achieve a ΔV of 1.81 km/s to allow the spacecraft to escape Earth. For the less futuristic propulsion system, the circular spiral requires 33.5 mt of propellant (i.e. the vehicle has a mass of 128.5 mt in LEO). The elliptical spiral requires a mass of 203.1 mt in LEO (after 5 years, 20.6 mt of propellant has been used, followed by chemical propulsion using 87.6 mt of propellant to achieve a ΔV of 1.92 km/s to escape Earth.) If no low thrust is used and instead there is only a singular, impulsive maneuver performed by a chemical rocket, the ΔV required is 3.2 km/s. Using the rocket equation and assuming a specific impulse of 450 s, 101.2 mt of propellant is required (i.e. the initial mass is 196.2 mt). Note that this direct mass is less than the elliptical spiral mass for the propulsion system based on the design by Price because of the differences in I sp values for impulsive ΔV. 3.2.3. Non-targeted Earth encounters It appears from the North-Pole trajectory plots (e.g. Fig. 6) that the Earth and spacecraft might be near one another at times other than the targeted gravity assists, such as when the spacecraft crosses the Earth's orbit. If the spacecraft is within Earth's Hill sphere (of radius 1.5  106 km or 0.01 AU) at times other than the targeted gravity assists, then the gravitational forces from Earth could significantly perturb the trajectories found in this paper. In order to determine if such a significant perturbation exists, the distance between the Earth and spacecraft is found throughout each cycler trajectory. Fig. 9 shows the distance between the Earth and spacecraft for the Aldrin cycler. It can be seen that the spacecraft does not encounter the Earth at any time other than the intended encounters at E0 and E1; therefore, no “non-targeted” encounters of Earth occur. A similar conclusion holds for each of the other cyclers. 4. Discussion of results 4.1. V 1 leveraging The ΔV savings of the STOUR results can be converted to mass savings with a few assumptions and the rocket equation. Following the discussion above about the size of a cycler vehicle, the dry mass of a reasonable cycler vehicle is

Fig. 9. Radial distance between the Aldrin Cycler and the Earth.

123

75 mt. The low-thrust propulsion system is assumed to have ion engines with an Isp of 5000 s. However, the maneuvers in this study are done instantaneously, so a smaller Isp needs to be assumed. For the maneuvers performed at Earth, an Isp of 450 s is used, corresponding to a chemical rocket using liquid oxygen, liquid hydrogen. Due to boil off, this type of engine cannot be used for the deep space maneuvers, which can occur years after launch. For this maneuver, something comparable to the Galileo spacecraft is assumed, a monomethylhydrazine (MMH) and mixed oxygen and nitrogen (MON) thruster, which has a specific impulse of 300 s. The propellant masses required are found using the rocket equation and Table 9 shows the results. It should be noted that for the V 1 leveraging case, first the propellant required for the deep space maneuver is calculated, then that mass is added to the cycler mass to provide the initial mass for the launch maneuver. The propellant requirements for some of the cyclers are significantly reduced, while others are only slightly improved. For example, the propellant needed for the 3:2(1)  Aldrin cycler is reduced by 24.3 mt, or about 15%. On the other hand, the 3:2(1)  VISIT-1 cycler reduces its propellant needs by only 0.2 mt, or less than 1%. This small improvement does not seem to justify the extra approximate three years needed to establish the cycler. One may wonder why there is a 25 mt difference in the direct ΔV propellant required between the 4:3(2)  and 3:2(1)  Aldrin cyclers. The reason for this difference is that the final orbits for each case (after the flyby at E1) are slightly different, as shown in Tables 3 and 4. It is also the case that these Earth encounters (E1) occur when the Earth is close to its perihelion, which can cause spacecraft with small differences in encounter dates to have significantly different flyby V 1 values. 4.2. Low thrust The amounts of propellant required to establish each of the cyclers considered in this work are correlated to their V 1 values at E1, but, for a given propulsion system, each of the cyclers takes approximately the same amount of propellant to establish after Earth escape (between 5.7 and 7.9 mt for the more futuristic propulsion system and between 11.0 and 13.9 for the less futuristic propulsion system). As expected, the more futuristic propulsion system based on Nock's work outperformed the less futuristic system. For the spiral out from Earth, the circular spiral is preferred because it takes less time to complete than the elliptical spiral and, in large part due to the arbitrary fiveyear maximum spiral time, the circular spiral scheme is much more propellant efficient. (Whether it is desirable to significantly extend the spiral time is a matter not addressed here and is left up to future mission designers.) If a larger electrical propulsion system is used, the elliptical spiral may become more attractive. However, that would require a larger mass and would decrease the dry mass that could be used elsewhere. A trade study could be done to find the optimal propulsion system for the low-thrust establishment method that takes into consideration both the elliptical spiral and the interplanetary trajectory. To further improve the elliptical

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Table 9 Propellant mass savings of the various cyclers from V 1 leveraging. Type

K: LðMÞ 7 

Aldrin Cycler 4:3(2) 3:2(1)  VISIT-1 4:3(2)  VISIT-2 4:3(2)  Case 1 4:3(2)  Case 2 4:3(2)  Case 3 4:3(2)  S1L1 4:3(2)  U0L1 3:2(1) 

Launch V 1 savings (km/s) ΔV savings (km/s) Direct ΔV prop. (mt) V 1 leveraging prop. (mt) Prop. mass savings (mt) 2.901 3.097 0.295 1.375 2.030 0.388 1.196 1.165 4.347

0.435 0.750 0.027 0.164 0.274 0.035 0.138 0.128 1.241

spiral, lunar flybys could also be considered, as described by Landau et al. [42] and McElrath et al. [43]. Both of these analyses are outside the scope of the present work. The time required to establish the cyclers, including escaping from Earth, is between 5.5 and 6 years (The VISIT2 cycler establishment taking about half a year less than the others). This compares to the V 1 leveraging establishment technique which takes around three or four years to establish for each of these cyclers [44]. 4.3. Comparison of V 1 leveraging and low thrust For every cycler considered, the propellant masses required for the low-thrust establishment are significantly less than the V 1 leveraging establishment. For example, the Aldrin cycler requires 126.4 mt (Table 9) using V 1 leveraging, but only 23.2 mt using low thrust (Table 6). Including the circular spiral, it takes 6 years to establish the cycler vehicle on its trajectory using low thrust, but only 4 years using V 1 leveraging. However, in either case the vehicle will remain unoccupied until E1, and the time savings of the V 1 leveraging do not seem to make up for the increased propellant costs. Note that the V 1 at Earth (E1) differ between the V 1 leveraging case and the low thrust case because of the different methods used to determine the orbits. For example, in Table 4, the flyby V 1 for the Aldrin cycler established with 4:3(2)  V 1 leveraging is 5.509 km/s. On the other hand, Table 6 shows that the flyby V 1 for the Aldrin cycler established with low thrust is 5.64 km/s. A similar scenario results for the S1L1 and VISIT-2 cycler cases. Because low-thrust establishment was found to be superior to V 1 leveraging in every instance (and because the V 1 at E1 was found to be larger for the low-thrust cases than for the V 1 leveraging cases), an analysis that uses identical V 1 values at E1 (for both low-thrust and V 1 leveraging establishment) would show the low-thrust-establishment method to be even more advantageous.

133.4 158.5 93.1 105.2 116.0 93.8 102.7 102.3 207.0

126.4 134.2 92.9 103.0 111.8 93.6 100.9 100.9 157.9

7.0 24.3 0.2 2.2 4.1 0.2 1.8 1.4 49.1

cyclers. Though significant analyses have been devoted to ballistic and powered cyclers, no work has been presented on the problem of establishing a cycler orbit, i.e. transferring the cycler vehicle from LEO into the cycler trajectory. In order for a manned Mars mission to be successful, a comfortable and safe environment needs to be provided for the astronauts. These safe havens necessarily need to be massive, and the advantage of cycler trajectories is that after a large investment is placed into the establishment of the cycler trajectory, there would be significant returns on the investment as the cycler vehicle is used over and over again. The reduction of the initial investment is a major stepping stone to make this type of mission a reality. The results presented here suggest that a modest reduction in propellant mass is possible for establishing some of the cycler trajectories via V 1 leveraging. For example, the establishment of an Aldrin cycler using the 3:2 (1)  V 1 leveraging technique could potentially reduce the propellant mass (compared to an impulsive injection into the cycler trajectory by an impulsive burn from LEO) by 24.3 mt (i.e. to 134.2 mt from 159.5 mt). Using low-thrust propulsion (with a propulsion system based upon Nock's study), the propellant is reduced to a more significant degree as only 23.2 mt is required in total (a further reduction of 111 mt). Regardless of the cycler architecture used for a mission, a significant conclusion can be made: low-thrust establishment should be used (when large scale low-thrust propulsion systems become available) and the cost of establishment will be roughly equivalent for all cycler trajectories considered. While the V 1 leveraging technique reduced the propellant over the impulsive technique, the low-thrust approach is superior. Humans will one day live on Mars. The year 2019 will mark the 50th anniversary of the first human landing on the Moon. As Buzz Aldrin has suggested, that historic anniversary would present an ideal time for a future President to announce a commitment, similar to that of President Kennedy's, to establish a permanent human presence on the planet Mars within the following two decades.

5. Conclusion The concept of cycler vehicles, which would provide safe havens for astronauts traveling to and from Mars on repeating trajectories, has been discussed since Hollister introduced the idea in 1969. Several examples of Earth– Mars cycler trajectories exist, including the Aldrin cycler, the VISIT's 1 and 2, and various two-synodic period

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