Research Article  Open Access
Yongjun Song, YoungJoo Song, Seongwhan Lee, KapSung Kim, Ho Jin, "Potential Launch Opportunities for a SmallSat Mission around the Moon Injected during a Lunar Flyby En Route to Mars", Mathematical Problems in Engineering, vol. 2019, Article ID 1245213, 14 pages, 2019. https://doi.org/10.1155/2019/1245213
Potential Launch Opportunities for a SmallSat Mission around the Moon Injected during a Lunar Flyby En Route to Mars
Abstract
In this work, the concept of a multipurpose mission that can explore both the Moon and Mars with a single launch is proposed, and potential launch opportunities are analyzed to establish an earlyphase trajectory concept. The proposed mission applies the concept of a piggyback ride to a smallsized lunar probe, i.e., the daughtership, of the main Mars orbiter, i.e., the mothership. For the trajectory design, the EarthMoonMars gravity assist (EMMGA) trajectory is adopted for the mothership to reach Mars, and the daughtership is assumed to be released from the mothership during lunar flyby. To investigate the earlyphase feasibility of the proposed mission, the launch windows have been analyzed and the associated deltaVs have been directly compared with the solutions obtained for typical EarthMars direct (EMD) transfer options. The identified launch windows (in the years 2031 and 2045) could be the strongest candidates for the proposed conceptual mission. Under the current assumptions, up to approximately 15% (in 2031) and 9% (in 2045), more dry mass is expected to be delivered to Mars by appropriately selecting one of the currently available launch vehicles, regardless of whether the EMMGA transfer option is used. For missions around the Moon using a SmallSat in 2031, the feasibility of a lunar orbiter case and an impactor case is briefly analyzed based on the deltaVs required to divert the SmallSat from the mothership. Although the current work is performed under numerous assumptions for a simplified problem, the narrowed candidate launch window from the current work represents a good starting point for more detailed trajectory design optimization and analysis to realize the proposed conceptual mission.
1. Introduction
For many scientific reasons, interest in interplanetary exploration for scientific discovery purposes is currently rising. The National Aeronautics and Space Administration (NASA) announced its goal of a human exploration mission to Mars and is first focusing on the return of humans to the surface of the Moon. As a part of this plan, NASA will construct a lunar orbital platform called the Gateway, which is a modular space station located in near rectilinear halo orbit (NRHO) between the Earth and the Moon [1]. The Gateway will serve as the gateway for lunar landing and further deep space missions [2]. By utilizing the Gateway, NASA plans to land the first female astronaut on the Moon by 2024 and aims to establish a sustainable human presence on the Moon by 2028 [3]. As the paradigm for planetary exploration has shifted, many fascinating studies have been conducted on orbital stabilities and lowenergy transfers [4–12]. Further details on the relevant literature are left for the readers to discover, as there exists a significant amount of research.
Also of interest when designing interplanetary missions is the reduction of the overall mission cost while reliably achieving the original scientific and technological mission objectives. In general, the total mission costs for interplanetary missions can be reduced by using a number of approaches, such as by minimizing the required overall deltaVs to accomplish the mission, by designing the mission trajectory to target multiple planets within a single launch, or by designing missions that can be performed with a smallsized spacecraft.
Most of the launch windows for interplanetary missions are usually designed to minimize the required deltaVs, and the flyby technique is occasionally applied to reduce the overall mission cost. A flyby, which is a gravityassisted flight, utilizes the gravity of a celestial body to change the spacecraft’s direction or speed, thus allowing the spacecraft to gain or lose energy without any additional fuel consumption. The flyby technique can be adapted to design transfer trajectories between other planets or central body bounding trajectories directed toward another planet by using the natural moon [13]. This flyby technique has enabled interplanetary missions requiring high inclination changes relative to the ecliptic plane with limited performance of the launch vehicle [14]. Moreover, by applying the flyby technique, not only can deltaV budgets be minimized, but the opportunity to visit additional planets in a single launch can be gained. This concept is called a multipurpose space mission because it allows for the exploration of several targets in a single mission [15].
In addition to minimizing overall mission costs by finding the best launch windows, either for direct or flyby missions, the overall mission costs and riskiness will be greatly reduced if interplanetary missions are performed with a smallsized spacecraft. With current technological improvements, diverse scientific data can be collected at an extremely low cost if a smallsized spacecraft is used for solar system exploration missions [16]. Indeed, the NASA Innovative Advanced Concepts Program (NIAC) has planned small, lowcost missions beyond the low Earth orbit using CubeSats [17]. Additionally, the first Mars exploration CubeSat, MarCO (Mars Cube One), was launched with a Mars lander called InSight on May 2018, and it sent an image from Mars [18]. Moreover, conceptual CubeSat missions for other solar system objects, such as Io [19] and Deep Space Gateway (DSG) [20], have been proposed recently.
One of the most important concerns during the early design stage of a Mars mission is finding a solution that can reduce the overall mission cost while maximizing the scientific and technological gains. One of the solutions could be building a mothership that can carry a small satellite as an onboard payload to achieve a lunar flyby en route to Mars, which certainly will provide additional opportunities to explore the Moon with a single launch. During the mothership’s lunar flyby phase, a small satellite could be released to orbit the Moon and perform its own mission to achieve its designated scientific goals. After the release of a small satellite, the mothership continues its journey to Mars to accomplish its main mission objectives. This concept also benefits the system reliability of a small satellite’s bus compared with an interplanetary mission planned with only the small satellite itself because a small satellite’s bus can be effectively protected from planetary hazards, such as thermal and radiation hazards, during the journey to the Moon [21, 22].
Many authors have revealed the usefulness of utilizing a lunar flyby when designing an interplanetary mission. Farquhar and Dunham [23] reported that double lunar swingby maneuvers increase the C3 value from –0.5 to 4.5 km^{2}/s^{2} and suggested that double and triple lunar flybys could potentially offer trips to the planets. Hanson et al. [24] analyzed the potential of an EarthMars trajectory by using single and multiple lunar flybys to improve the available payload gain. Kawaguchi et al. [25] designed a real EarthMars trajectory for the Japanese Mars exploration spacecraft, NOZOMI, using a similar double lunar flyby method. Hernandez and Barbee [26] considered using a single lunar flyby for the NearEarth Asteroids (NEA) mission, and their main objective when adapting lunar flybys to explore NEA was to reduce the total required deltaVs during the overall mission sequences. They concluded that a lunar flyby is more effective than a direct transfer for exploring NEA. All these studies concluded that lunar flyby methods allow a spacecraft to depart Earth at a lower energy than that needed to escape and then proceed to planetary encounters. Although many studies apply the concept of lunar flyby to design Earth escape trajectories, most of these studies applied multiple lunar gravity assists to maximize deltaV savings. Similar work was conducted by Hanson et al. [24] in the 1990s. In Ref. [24], many lunar flyby options were considered, but multiple lunar gravity assists were focused on rather than a single flyby to reduce the deltaV budget. The application of multiple lunar gravity will lead to very complex mission operations, which have already been discussed as a disadvantage of the multiple gravity assist [24]. Furthermore, feasibility studies on Mars transfer trajectory design with a single lunar flyby have rarely been performed. Most importantly, no recent work has proposed and analyzed possible Mars mission launch opportunities using single lunar flyby for the upcoming decades. Providing such future launch opportunities is very important in further realizing the proposed conceptual mission. Therefore, the current work investigates suitable launch opportunities of a Mars mission by using only a single lunar flyby to support the emerging SmallSat piggyback ride planetary mission concept. Utilization of single lunar flyby will ultimately ease the mission operation and minimize overall mission riskiness, which will be a driving factor of mission success.
The main objective of the current study is to perform an earlyphase feasibility study on a proposed conceptual multipurpose mission. The realization of such a mission first requires trajectory design studies that estimate the overall mission deltaV budgets. The current study investigates launch opportunities for an EarthMars mission utilizing a flyby of the Moon, and it includes an estimation of the required overall deltaV budgets. For the first step, to roughly estimate overall mission deltaV budgets, the current work employed a simplified methodology, the patched conic method with simplified equations of motion, to determine potential launch years. The obtained deltaV budgets are compared with typical deltaV values obtained using EarthMars direct transfer options. The mothership’s mass budget characteristics are also roughly investigated using currently available data on heavy launch vehicle performance. Additionally, the mass budgets for a SmallSat mission around the Moon deployed from a mothership were estimated. Although the current work utilized wellknown methodologies to derive results, it is expected that the current work can make the following contributions: First, both mission designers and supervisors who are interested in SmallSat application to interplanetary missions can gain insight while establishing similar mission concepts. Second, the current work can serve as a reference for further work related to launch opportunities in the near future. Finally, based on the results of the current work, detailed trajectory design solutions can be advanced by implementing an optimization algorithm with Nbody equations of motion and with more realistic mission constraint parameters. The remainder of this manuscript is organized as follows: Detailed simulation procedures, including equations of motion based on the patched conic method, are described in Section 2. The numerical implications and assumptions for setting up the simulation are provided in Section 3. Detailed simulation results and an associated discussion are presented in Section 4. The conclusions are discussed in Section 5.
2. Models and Methods
This section describes the models and methods used to design an EarthMoonMars gravity assist (EMMGA) trajectory and determine the orbit of a lunar SmallSat. In this paper, we use a single lunar flyby method similar to that described by Hernandez and Barbee [26] to calculate the EMMGA trajectory. The EMMGA trajectory is designed using the patched conic method. This method uses a series of conic sections patched together to form a trajectory bounded by the central body, which primarily affects the motion of the spacecraft. For a detailed discussion of this method, readers can refer to the papers by Barbee et al. [27] and Hernandez and Barbee [26]. In this method, the EMMGA trajectory is divided into three phases: Earthcentered, Suncentered and Marscentered. In the Earthcentered phase, the mothership releases the SmallSat during the lunar flyby phase, and the released SmallSat is then captured by the Moon with an onboard thruster. After the flyby, the mothership is ejected to an Earthescape hyperbola that leads to Mars after a Suncentered interplanetary cruise phase. During the Marscentered phase, the mothership arrives at the Mars sphere of influence and enters the Mars parking orbit. To design the Earth’s sphere of influence (SOI) escape trajectory, it is assumed that the mothership uses only a single, unpowered lunar gravity assist. However, multiple lunar gravity assist techniques can be applied for this phase rather than a single flyby to reduce the deltaV budget. Indeed, reducing overall deltaV cost by utilizing multiple gravity assist is another important factor that should be considered during the early mission design phase. However, as discussed, the current work is focused on investigating suitable launch opportunities under a single lunar flyby. Lastly, the deltaVs in the current work are assumed to be impulsive burns during each phase.
2.1. Lunar Flyby Trajectory Generation En Route to Mars
2.1.1. EarthCentered Phase (Stage 1)
The first step in designing the trajectory consists of targeting a lunar flyby in the Earthcentered phase. In this stage, the trajectory of the mothership is designed using fourbody equations of motion. Fourbody equations of motion can be replaced with twobody equations of motion in this Earthcentered phase for preliminary trajectory design purposes. However, the current work adopted fourbody equations of motion to increase the maturity of the trajectory solutions obtained through this work, and these equations can be used as a basis of further work regarding the detailed optimization of highfidelity dynamic models. The equations of motion of the mothership with Earth as the primary central body and the Moon and Sun as perturbing bodies in the Earthcentered Earth mean equator and equinox of epoch J2000 (EEME2000) coordinate system are expressed in [28, 29]:where , , and denote the position, velocity, and acceleration vectors of the mothership with respect to the Earth, respectively, is the gravitational constant of the Earth, and and are the perturbing gravitational forces of the Moon and Sun with Battin’s q function, respectively, which can be expressed as follows [28, 30]:where and are the gravitational constants of the Moon and Sun, respectively, and and are the position vectors from the Moon and Sun to the mothership, respectively. The q functions for the Moon, , and the Sun, , are expressed as follows:
The initial conditions of equations (1a) and (1b) can be defined with the initial position () and velocity vectors () of the mothership as follows:where and are the position and velocity vectors of the mothership on the initial Earth parking orbit, respectively, and is defined as the deltaV vector required to transfer the mothership to the Moon under fourbody dynamics.
To derive , , and , the current work used Ramanan’s pseudostate method [31] and slight corrections were made to the obtained results for application in fourbody dynamics. Ramanan’s method is a very powerful and effective method based on twobody dynamics, and it solves Lambert problem iteratively to generate initial conditions of transfer trajectories for moon missions. Of course, the best way to obtain , , and is to apply an optimization algorithm under a highfidelity dynamic model. However, the current work used Ramanan’s pseudostate method to focus on the preliminary analysis. To use Ramanan’s pseudostate method, the user must first define the following parameters: the translunar injection (TLI) maneuver time (), the EarthMoon transfer time (), and initial Earth parking orbit elements (i.e., semimajor axis (), eccentricity (), inclination (), and argument of the perigee ()). In addition, the user should select a preferred TLI maneuver location. Usually, there are two possible maneuver execution locations on the initial parking orbit for a given : one during the ascending motion of the parking orbit and the other during the descending motion. Once these parameters are given, the Ramanan’s pseudostate method can be used to calculate the remaining parking orbit elements (i.e., the right ascension of the ascending node , the true anomaly (), and the minimum TLI maneuver vector (). For additional details on Ramanan’s pseudostate method, refer to Ref. [31].
After applying Ramanan’s pseudostate method, the state vectors of the mothership in the Earth parking orbit ( and ) and are obtained, where is the minimum TLI maneuver vector derived under twobody dynamics. As the current work used fourbody equations of motion during the Earthcentered phase, , as shown in equations (4a) and (4b), cannot be directly replaced with . Therefore, is obtained in relation to , as follows:
As shown in equations (5a) and (5b), a small velocity magnitude correction, , is made to the magnitude of while maintaining the same direction as that of . The direction of will not actually be exactly the same as that of , but the current work accepted this assumption, as the deltaV magnitude change rate plays a more significant role in the resultant trajectory solutions than the direction change rates unless the estimated deltaV directions are completely insufficient. The current work obtained proper ranges of based on trial and error by investigating the behaviors of the resultant trajectory. Within the given ranges of , that minimizes while satisfying the lunar flyby condition is selected as a candidate .
After the initial state vectors required to perform a lunar flyby ( and ) are determined, the trajectory of the EarthMoon system is propagated with stopping conditions for the mothership when it reaches the Earth’s SOI after a single lunar flyby over a time span of . The radial distance of the Earth’s SOI, , is given as 924,500 km. When the trajectory reaches the Earth’s SOI after a lunar flyby, the mothership’s state vectors ( and ) are located on the boundary of the Earth’s SOI at , which can be simply calculated as . During propagation, two different constraints are used to ensure that the generated trajectory is a lunar flyby. The first constraint is given as , where is the altitude of the mothership from the surface of the Moon. If is less than 0 km during the simulation, then the mothership will impact the lunar surface. The second condition is a constraint, . The maximum of is given as 20 days during the simulation because the typical mothership’s flight time in the Earth’s SOI is approximately 10 days when a single lunar flyby is applied. If the mothership cannot reach the Earth’s SOI until a of 20 days, then we consider that the mothership is unable to escape from the Earth’s SOI after flyby. If the trajectory does not satisfy one or both of these constraints during propagation, then we assume that the EMMGA trajectory is impossible to be generated. Herein, additional constraints should be considered to ensure a practical lunar flyby for realflight operation. For instance, “faceon” geometry during lunar flyby should be guaranteed for the purpose of Earth communication, and lower limits on flyby altitude and velocity should be set, as these conditions are strongly related not only to the mothership but also to SmallSat’s bus design, especially the fuel budget. Nevertheless, the current work omits details of these additional constraints, as they can be derived only after further establishment of detailed mission concepts and goals.
2.1.2. SunCentered Phase (Stage 2)
When the mothership escapes from the Earth’s SOI, the gravity of the Sun becomes the dominant force. In this phase, the motion of the mothership is primarily affected by the Sun, and the perturbations of other planetary bodies are neglected. To describe the motion of the mothership in the Suncentered phase, the state vectors of the mothership are transformed into a heliocentric ecliptic coordinate system, thus producing the following equations:where and are the state vectors of the mothership with respect to the Earth, as previously discussed, and are the position and velocity vectors of the mothership with respect to the Suncentered frame, respectively, and and are the Earth’s state vectors with respect to the Sun, which are calculated from the precise ephemerides. The matrix is the rotation matrix that converts the state vector from equatorial coordinates to ecliptic coordinates at [32–34].
After converting the states, the trajectory from the Earth’s SOI to Mars’ SOI can be simply calculated via the Lambert problem in the heliocentric coordinate system. To solve the Lambert problem, the previously calculated and as well as the Mars arrival time () and the Suncentered position of Mars () at are required. By solving the Lambert problem, the required velocity at the Earth’s SOI () and at Mars’ arrival () can be easily calculated. Once the Lambert solution is found, the required deltaV for leaving the Earth’s SOI, , can be simply calculated as follows:
The method adapted to compute in the current study is mainly used to adapt the patched conic method to focus on the preliminary analysis. Notably, computing deltaV at the SOI may not be efficient from an energetic point of view, requiring further improvement through optimization using Nbody dynamics. For example, instead of imparting the deltaV at the moment of SOI crossing, the following options can be considered in upcoming work. First, a powered lunar flyby that imparts the deltaV at the moment just after periapsis passage of the Moon can be considered. Imparting the deltaV just after periapsis seems to be a reasonable option under the current mission concepts with regard to deployment of the SmallSat during the lunar flyby of the mothership. Performing a maneuver before the lunar flyby may be another option, but this approach could result in a higher deltaV requirement to release the SmallSat due to the increase in the periapsis velocity and certainly requires further study. Finally, the utilization of a powered Earth flyby can be considered. For this option, the mothership can fly back to Earth after completion of a single lunar flyby and impart the deltaV to depart to Mars at the moment of periapsis passage of Earth, which still satisfies the utilization of a single lunar flyby to avoid mission complexities.
2.1.3. MarsCentered Phase (Stage 3)
When the mothership arrives at Mars’ SOI, the Mars arrival hyperbolic excess velocity with respect to Mars () is calculated as follows:where is Mars’ velocity with respect to the Sun at , which is calculated directly from the ephemerides, and was previously calculated in Section 2.1.2.
If the mothership targets a circular Mars mission orbit, then the required deltaV from the hyperbolic entry orbit into a target circular orbit () can be simply calculated as follows:where is the semimajor axis of the Mars mission orbit, and is the gravitational constant of Mars. Using equations (5a), (7), and (9), the total deltaV required for the EMMGA trajectory () is calculated as in equation (10). The overall conceptual diagram of the mission segment through stage 1–3 and the deltaV maneuver points is schematically shown in Figure 1.
(a)
(b)
2.2. Released SmallSat Trajectory Generation
The equations of motion of the SmallSat released from the mothership during lunar flyby are described with twobody motion as follows:
In equation (11), and represent the acceleration and position vectors of the SmallSat in the MoonCentered Moon mean equator and International Astronomical Union (IAU) vector of the epoch J2000 (MMME2000) coordinate system, respectively. Note that the SmallSat is assumed to be released from the mothership at the moment of the mothership’s perilune passage () during flyby. The value of can easily be computed during propagation of the mothership’s trajectory under the condition of a flight path angle of 0 degrees.
To evaluate equation (11), the initial state of the SmallSat (, ) at can be expressed as follows:where and are the position and velocity vectors of the mothership at the perilune, respectively, and is the deltaV vector required to divert the SmallSat from the mothership. To determine the required total deltaV to perform a SmallSat mission around the Moon (), the Hohmann transfer method is used to simplify the given problem; therefore, the direction of the applied deltaV to divert the SmallSat into the mission orbit around the Moon is always opposite the direction of the mothership’s velocity. If the SmallSat is an orbiter, a burn () other than is necessary to circularize the SmallSat’s orbit to a target altitude. For this case, can be , where is the magnitude of the second burn directly obtained by solving the Hohmann transfer problem. However, if the final mission of a SmallSat is an impactor mission, then will have the same magnitude as , which can be calculated with the target altitude condition of 0 km while solving the Hohmann transfer problem. Setting a 0km target altitude while solving the Hohmann transfer problem may result in a very shallow impact angle for an impactor, and the resultant deltaVs may differ from the deltaVs derived from the current study if a larger impact angle is necessary. However, the current study accepted this assumption to roughly estimate relevant deltaVs. If the goal of the impactor mission is refined, then the required impact angle condition will certainly be another constraint to consider for more indepth trajectory design and analysis.
2.3. Mass Budget Estimation
From the obtained deltaVs, the propellant and dry mass of the mothership and the SmallSat can be estimated. In the EMMGA trajectory, two burns ( and ) will be executed by the mothership based on our design concept. Therefore, the required propellant and dry mass of the mothership ( and ) can be easily calculated with the initial mass of the mothership (), and the specific impulse of the mothership () is as follows:where is the gravitational acceleration of Earth.
For the SmallSat, the onboard propulsion system of the SmallSat itself should be used to generate to orbit or impact the Moon. The required propellant mass of the SmallSat () during lunar capture can be calculated as in equation (13a) with the values of , , and , where is the overall initial mass of the SmallSat and is the specific impulse of the SmallSat’s onboard thruster.
3. Simulation Setup
During the simulation, the launch period of the mothership is assumed to be between the years 2026 and 2045. To calculate the precise position, velocity vectors, and physical parameters of the celestial bodies, Jet Propulsion Laboratory (JPL) DE405 ephemerides [32] is used to obtain the gravitational constant, equatorial radius, etc. The hours, minutes, and seconds of and for the given dates are assumed as 00 : 00 : 00 in the Universal Coordinate Time (UTC) timescale and converted to the Barycentric Dynamical Time (TDB) timescale to adapt DE405. To perform a simulation, the initial earth departure date, namely, , is given from Oct. 2026 to Dec. 2045. This time span covers the same analysis duration as that in the work of Burke et al. [35], which analyzed launch opportunities of the EMD transfer option. After the initial of each year is selected, the corresponding is calculated as . Additionally, to search for the best departure and arrival dates, the search bounds for are given by initial and by ±180 days for the initial , which leads to a total time of flight (TOF) from Earth to Mars ranging from 125 to 605 days. The and perilune altitude for the flyby were calculated during the process of patching the problem and were directly used to simulate the current problem.
For numerical integration, the DOPRI8 integrator is used [36, 37], which is based on the 7^{th} to 8^{th} order Runge–Kutta–Fehlberg method with an adaptation of the error control technique from Press et al. [38] with a truncation error tolerance of . For the initial parking orbit conditions during the Earth departure phase, namely, to perform the TLI maneuver, the circular orbit is assumed to have a 300km altitude with an inclination of 80.0 degrees. The remaining orbital elements and are properly selected during the simulation, as mentioned in Section 2.1.1. is given in the range from –2.0 to 2.0 m/s and increased with 1 m/s steps which were selected based on trial and error during the simulation to satisfy given flyby conditions of the current work. If the search range of is smaller than –2.0 m/s, then most of the discovered trajectories are similar to the trajectories obtained with a of –2.0 m/s. When is larger than 2.0 m/s, the mothership will crash into the Moon or simply return to the vicinity of Earth. The transfer flight time from Earth to the Moon, , is varied within the range from 4.0 to 6.0 days with 0.1 day searching steps. When the search range of is extended, no remarkable changes in the simulation results are discovered. For example, when is set to less than 4 days, meaningful deltaV changes throughout the mission are not discovered, and if is set to less than 3 days, then the constraints discussed in Section 2 (h > 0, TOF in the Earth SOI < 20 days), which are quite critical for finding a feasible EMMGA transfer option, are not guaranteed. All the parameters selected based on trial and error, as in the current study, must be given as constraints or free parameters. Additionally, with proper control variables nested within the optimal problem with complex dynamics, to improve the efficiency of imparting the deltaV before Earth SOI escape, the overall mission deltaV can be further minimized and more detailed trajectory design tradeoff studies can be performed. Finally, the mission orbit of the mothership after arriving on Mars is assumed to be circular with an altitude of 100 km.
4. Results and Analysis
4.1. EMMGA Trajectory Characteristics
4.1.1. Launch Opportunities
To investigate the feasibility of the proposed conceptual mission, the launch opportunities of a mothership that utilizes the EMMGA transfer option should first be analyzed. Indeed, the total deltaV budget between the EMMGA and EMD transfer options should be compared because typical Mars missions utilize the Hohmannshaped EMD transfer option due to its simplicity and optimality. In Table 1, the major trajectory design parameters for the period from 2026 to 2045 are compared between the EMMGA and EMD transfer options, including the Earth departure date, the Mars arrival date, the deltaVs at each phase (, , and ), the C3 energy, and the transfer type of the EMD trajectory. Note that every value for the EMD transfer options has been directly obtained and regenerated from the work of Burke et al. [35]. Burke et al. [35] showed the best launch windows of the EMD transfer option with four types of EMD trajectory solutions: Type IA, Type IB, Type IIA, and Type IIB. Nominally, Type I and Type II trajectories have heliocentric travel angles that are less than and greater than 180 degrees, respectively. In addition, because the orbits of Earth and Mars are neither exactly circular nor coplanar, one launch opportunity may require less departure energy (categorized as Type IA and Type IIA) or have a lower (categorized as Type IB and Type IIB) than another opportunity. While adapting launch opportunities from the work of Burke et al. [35] for comparison, launch windows exceeding a C3 of Earth departure of 12 km^{2}/s^{2} are neglected to consider the limits of launcher performance. For this reason, the transfer data for 2037 and 2039 are not presented in Table 1.

As expected, the total deltaV magnitude required to accomplish the proposed mission, namely, utilization of the EMMGA transfer options, generally must be greater than that required for the EMD option, regardless of EMD transfer types. Specifically, a deltaV of approximately 7.148–8.006 km/s is required if the EMMGA transfer options is used, and a deltaV of approximately 5.678–7.493 km/s is necessary if the EMD option is adopted. Notably, Table 1 shows that the yearly total deltaV difference between the EMMGA and EMD transfer options varies greatly, even if the Earth departure and Mars arrival timeframes are similar, except for some years within 1 or 2month differences. For example, if the Mars mission is scheduled for 2026, then the EMMGA transfer option would require approximately 1.44 km/s more deltaV than the EMD transfer. However, if a similar mission is scheduled for 2031, then approximately 0.5 km/s more deltaV can provide another opportunity to visit the Moon and perform another meaningful science mission.
These phenomena appear to be caused by differences in the launch and arrival geometries, including the dates, between the EMMGA and EMD transfer options. The launch and arrival geometry differences will certainly change the orbital energy of the derived mission orbit and the total deltaV. Typically, , which is the deltaV required for a spacecraft to reach target planets (to Mars for the EMD option and to the Moon for the EMMGA option), is within the range from 3.571–3.689 km/s for the EMD transfer options and from 3.101–3.116 km/s for the EMMGA transfer options, showing that the EMD cases require greater departure deltaVs than the EMMGA cases. This result again confirms that a stronger upper stage of the launch vehicle should be supported for a spacecraft headed to Mars than for those headed to the Moon. However, if the EMMGA transfer option is selected, the Earth departure C3 is approximately –2 km^{2}/s^{2} and increases to approximately 1.1–1.3 km^{2}/s^{2} after lunar flyby, which is far lower than the typical C3 magnitude, which is approximately 89 km^{2}/s^{2} for the EMD option. Regarding onboard propulsion deltaV generation, approximately 0.972–2.323 km/s more deltaV should be generated regardless of the mission timeframe difference. This change is primarily due to the additional deltaV requirement, , for the EMMGA cases, which is approximately 1.846–2.324 km/s to insert the mothership into the transMars trajectory.
However, because of different overall TOFs between the EMMGA and EMD transfer cases as well as the mothership’s final approach conditions for Mars, the value of for the EMMGA transfer option was almost equal or slightly less than that for the EMD transfer option. As shown in Table 1, the directly calculated using the of the EMMGA case is slightly less than that of the EMD case, except for the years 2028 and 2043. For the mission period of 2031, the difference in between the EMMGA and EMD cases was approximately 1.03 km/s, which ultimately reduced the total deltaV requirement difference between the EMMGA and EMD options to less than approximately 0.5 km/s. As shown in Table 1, the Earth departure C3 value of the year 2045 is considerably larger (more than 10 km^{2}/s^{2}) than that of other years. Recalling that there are launch opportunities with a lower departure energy (Type A) or a lower (Type B), Table 2 compares the deltaV characteristics of the EMD and EMMGA transfer options with inclusion of both Types A and B for the year 2045. In terms of , Type IIB is clearly a better option than Type IIA for EMD transfers, as the of Type IIA substantially increases due to the increase in , namely, . At this point, a Mars mission can be expected to be carried out most efficiently with an Earth departure C3 of Type IIA and a of Type IIB for the year 2045. In this case, one of the candidate launch options that can be considered is EMMGA transfer. Although the EMMGA option requires significantly more deltaVs than the Type IIB EMD option, selecting the EMMGA option for a 2045 mission may be another very attractive option in terms of launch capabilities and the maximum return of scientific data from a single launch. In fact, the difference in total deltaVs required between the EMMGA option and Type IIA of the EMD option for this year is only approximately 0.6 km/s, which is slightly greater than that for the year 2031. This phenomenon can again be explained by differences in launch and arrival geometries, including the dates, between the EMMGA and EMD transfer options. If the EMMGA transfer option is selected for the year 2045, as shown in Table 2, then the Earth departure date and arrival date are much closer, with 8 days for departure and 4 days for arrival, to those of Type IIB than to those of Type IIA, which can ultimately reduce the amount of .

Importantly, the current results are derived from a simplified model with several assumptions that must be resolved in more detail. In particular, the total deltaV difference between the EMMGA and EMD options is expected to be further minimized compared to the current findings by applying a more efficient energetic strategy to impart deltaVs before escaping the Earth’s SOI using an implementing optimization algorithm. Consequently, the proposed mission can ultimately reach Mars with an additional opportunity to explore the Moon. The discovery of such a launch window that can perform the proposed mission, even with a similar Earth departure and Mars arrival timeframe, can benefit a wide range of routine work required to realize this mission. Because of the simplicity of the proposed mission, the adoption of a single lunar flyby will certainly reduce the trajectory designers’ time and effort required to establish the trajectory concept during the early stage and perform the relevant feasibility analysis. Simultaneously, realtime operation efforts can be minimized to reduce the overall mission risks, which are mostly caused by human factors. Most significantly, the possibility of performing the proposed mission based on the technical readiness of the currently available launch vehicle’s performance has been confirmed, and the details of the mass budget estimations will be discussed in the following subsection.
4.1.2. Spacecraft Mass Budget Estimation
This subsection analyzes and compares the expected spacecraft’s mass budget characteristics between the EMD and EMMGA transfer options based on the deltaVs obtained in the previous section. To estimate the mass budget of the spacecraft, the performances of three different currently available heavy launch vehicles are directly adapted as example cases. In Table 3, the representative payload delivery capabilities to the Moon and Mars are shown for Ariane 5, Proton M, and Atlas V. To compare the spacecraft mass budgets between the EMD and EMMGA transfer options, the payload capabilities for targeting Mars (EMD case) and the Moon (EMMGA case) shown in Table 3 are used. Notably, however, the mass budgets derived in the current section are all rough estimates and may differ for different launch constraints and conditions, including the finalized values of the missiondependent overall deltaVs. The payload delivery capabilities shown in Table 3 are derived under the assumption that launchers have the best performance and certain aspects of injection C3 and escape declination [39]. According to the work of Biesbroek [39], the payload delivery capabilities of the launchers shown in Table 3 are all derived under the conditions of 10 km^{2}/s^{2} of C3 for Mars and –2 km^{2}/s^{2} of C3 for the lunar mission. However, wide ranges of launch constraints should be additionally considered for more realistic mass budget estimation. is assumed to be approximately 330 s, which is the same as the performance of the recent ExoMars onboard main engine [40]. In the following discussion, the “spacecraft mass” for the EMD and EMMGA cases directly refers to the mass of the Mars probe itself and the mass of the mothership, including the SmallSat’s mass, respectively.

In Figure 2, the ratios of a spacecraft’s estimated dry mass that can be delivered to Mars using the EMD and EMMGA transfer options are compared for each launcher. The spacecraft’s dry mass ratio is calculated with the simple equation, , where and are the spacecraft’s dry mass that can be delivered to Mars using the EMD and EMMGA transfer options, respectively. The results indicate that a greater spacecraft dry mass can be delivered to Mars using the EMMGA transfer option if the ratio is greater than 100%, and the EMD option allows more mass to be carried to Mars if the ratio is less than 100%. Figure 2 reconfirms that the EMD transfer option can deliver a greater mass to Mars than the EMMGA case for most launch years, regardless of the launch vehicle used. However, for the year 2031, the given dry mass ratio is greater than 90% with the Proton M (95.24%) and Atlas V (92.09%) launch vehicles. This result indicates that the final mass delivery capability between the EMD and EMMGA transfer options can be minimized to less than 10% by selecting an appropriate launch date. Interestingly, the dry mass ratio for the year 2031 is greater than 100% with the Ariane 5 launcher (115.20%), which indicates that more mass (up to approximately 15% more) can be delivered to Mars when the EMMGA transfer option is used instead of the EMD transfer option. For example, if a Mars mission is planned for launch in 2031 for scientific purposes, then the proposed EMMGA transfer approach can be considered as an option. A dry mass ratio of 15% corresponds to approximately 200 kg, which is sufficient mass for a SmallSat to perform an additional mission around the Moon. Although not depicted in Figure 2, the dry mass ratio for the 2045 mission is also compared for the EMMGA and Type IIA of EMD options. As expected, the dry mass ratios are approximately 109.81% (Ariane 5), 90.78% (Proton M), and 87.78% (Atlas V) higher than those of the Type IIB case. For Ariane 5, a dry mass ratio of 9.81% corresponds to approximately 145 kg of additional mass. The suggested masses in the current subsection are all rough estimates and may be different if detailed launch constraints, such as mass penalties for injection inclinations and dependencies of C3 values on launch date changes, are considered. For example, an additional mass of 200 kg for a 2031 mission may lead to different results if mass penalties for injection inclinations are considered, although the assumed C3 value (10 km^{2}/s^{2}) to derive the payload delivery capabilities is somewhat greater than the C3 value (8.237 km^{2}/s^{2}, shown in Table 1) estimated for the 2031 mission. Details of the roughly estimated spacecraft dry mass budgets for each launch vehicle are compared in Table 4, and the values include the propellant masses.

4.2. SmallSat Mission Capabilities
Based on the launch opportunities of the previously analyzed EMMGA transfer option, this subsection further analyzes the capabilities of SmallSat missions. Indeed, the deltaV magnitude to deploy a SmallSat to be captured around the Moon, , represents a major design factor when establishing the feasibilities of current conceptual missions. Therefore, the current analysis is performed based on the magnitude of SmallSat separation deltaVs. To deploy a SmallSat into the orbit around the Moon, the deltaVs generated from the SmallSat’s separation mechanism and onboard propulsion system are required. However, this study ignores the magnitude of the deltaV generated from a SmallSat separation mechanism because the magnitude is small compared with the deltaV generated by the onboard propulsion system. As previously discussed, two different lunar capture scenarios, i.e., orbiting and impaction cases, are considered for the analysis. For the orbiting case, the SmallSat is ultimately assumed to be inserted into a circular orbit around the Moon with a 100km altitude, and it has the same orbital inclination established during the hyperbolic approach of the mothership in the lunar SOI. In fact, a different range of inclinations is achievable during the lunar flyby, which can be accomplished by setting the inclination as the target constraint while formulating a trajectory optimization problem for more detailed analysis. However, the current analysis accepts this assumption to focus on the preliminary analysis. For an impacting case, the SmallSat is assumed to directly impact the lunar surface after being deployed from a certain altitude of the mothership’s hyperbolic periapsis passage.
Among the several previously analyzed EMMGA launch opportunities, 2031 is selected as the year with the best mass budget performance. In Table 5, the associated deltaV characteristics for this year, including the periapsis altitude and velocity of the SmallSat at the moment of release, are shown. In the previous section, we showed that a total SmallSat mass of approximately 200 kg can be deployed around the Moon with an Ariane 5 launch vehicle using the EMMGA transfer option, which is comparable to that of the EMD transfer option. Assuming 277 s of [41], which is one of the highest I_{sp} values for currently available miniaturized thrusters, approximately 143.8 kg of dry mass is allowed when performing a mission around the Moon (143.876 kg for an orbiter and 143.823 kg for an impactor). Although the current analysis is performed using several assumptions that simplified the current problem, sending an approximately 200kgclass SmallSat with a dry mass of approximately 140 kg with a single launch would certainly fulfill the requirements of the worldwide scientific community.

4.3. Example of a Transfer Trajectory
This subsection presents an example transfer trajectory of a 2031 mission that utilizes the EMMGA transfer option. Figure 3 depicts the entire transfer trajectory from Earth departure to Mars arrival. The left side of Figure 3 represents the trajectory of the 1^{st} stage of the proposed conceptual mission in which the mothership departs the Earth on Jan. 8, 2031, performs a flyby of the Moon on Jan. 11, 2031, and escapes the Earth’s SOI on Jan. 17, 2031. The total flight time of the mothership escaping the Earth’s SOI is approximately 9.88 days. After the mothership escapes the Earth’s SOI, it continues its journey to Mars (right side of Figure 3) and arrives on Nov. 9, 2031, which is approximately 296 days from its departure from Earth. The total TOF for the mothership is approximately 305 days, and the required is approximately 8.006 km/s. In Table 6, detailed trajectory information for the 2031 mission is summarized. As shown in Table 6, the values of t_{tof_E–M} and ΔV_{Add.} are located at the lower limits of the search range under the current simulation conditions. Due to the simulation conditions discussed previously, the results of the 2031 case may not significantly differ for lower search range limits. However, the limits of these values can be further tuned for more detailed analysis by adapting an optimization scheme. During the mothership’s lunar flyby, the SmallSat is released to perform its own mission. The SmallSat is deployed at an altitude of approximately 198.1 km at the periapsis passage of the mothership. The corresponding time is Jan. 11, 2031, 20 : 57 : 26.770 (TDB), and the SmallSat’s mission around the Moon is then performed. Figure 4 shows an example of the SmallSat’s mission trajectory around the Moon for the orbiter (top) and impactor cases (bottom). For the orbiter mission case, the SmallSat is inserted into a final 100kmaltitude circular orbit around the Moon after 61.2 min of transfer, and for the impactor case, approximately 58.8 min will be required to impact the lunar surface.
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(b)

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5. Conclusions
The current work proposes the concept of a multipurpose mission that can explore both the Moon and Mars with a single launch and analyzes potential launch opportunities for earlyphase design work. For the mothership to reach Mars, the EMMGA trajectory is adapted for the trajectory design basis, and the daughtership is assumed to be released from the mothership during the lunar flyby to perform its own mission. To investigate potential launch opportunities, the associated deltaVs have been derived and compared with the solutions provided by typical EMD transfer options with a simplified dynamic model to focus on the preliminary design studies. Additionally, two different lunar capture scenarios (orbiting and impacting cases) have been considered to investigate the SmallSat mission capabilities. The launch opportunities for the years 2026 through 2045 have been investigated. The analysis confirms that, regardless of transfer type and mission period, the EMMGA transfer options generally require a greater deltaV than the EMD options. However, two candidate launch years have been identified; the year 2031, and possibly 2045, may be the strongest candidate for a mission with the EMMGA option. In both years, the proposed conceptual mission is expected to be performed with only slightly greater deltaVs, approximately 0.5 km/s and 0.6 km/s, respectively, than those of the Type IIA EMD transfer option of the 2031 and 2045 missions, respectively. To further improve the deltaV efficiency by adapting the strategies discussed, the overall mission deltaV difference between the EMD and EMMGA options are expected to be minimized compared to the current estimated values. The differences in deltaVs are mainly caused by differences in the launch and arrival geometries, especially the geometry at the Mars arrival date, for the EMMGA and EMD transfer options. Three different currently available heavy launch vehicles (Ariane 5, Proton M, and Atlas V) were selected as candidate launchers, and the associated mass budgets were roughly analyzed. Interestingly, up to approximately 15% more dry mass can be delivered to Mars by using the EMMGA transfer option instead of the Type IIA EMD transfer option with the Ariane 5 launcher in the year 2031. For the 2045 mission, approximately 9% more dry mass can be delivered when the EMMGA and Type IIA of EMD options are compared for the Ariane 5 launcher. Although the estimation of the current launcher’s payload delivery capabilities did not consider detailed launch constraints, rough dry mass ratio estimates of 15% and 9% correspond to approximately 200 kg and 145 kg, respectively, which appear to be sufficient for a SmallSat class mission around the Moon. Indeed, as the current work was performed under several assumptions to simplify the problem and focus on the preliminary analysis, many challenges still require resolution for this conceptual mission to be realized, and substantial research may be required. However, the narrowed candidate launch window provided by the current work is a good starting point for more detailed optimal trajectory design and analysis to realize the proposed conceptual mission. Moreover, enabling a mission capable of reducing the overall mission costs while maximizing scientific merit will certainly be attractive to space scientists and supervisors who are willing to plan and propose such a mission.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by NRF2014M1A3A3A02034761, NRF2016M1A3A9913306, and the BK21 plus program through the National Research Foundation (NRF) funded by the Ministry of Education of Korea.
References
 J. Crusan, J. Bleacher, J. Caram et al., “NASA’s gateway: an update on progress and plans for extending human presence to cislunar space,” in Proceedings of the IEEE Aerospace Conference, Big Sky, MT, USA, March, 2019. View at: Google Scholar
 T. D. Haws, J. S. Zimmerman, and M. E. Fuller, “SLS, the gateway, and a lunar outpost in the early 2030s,” in Proceedings of the IEEE Aerospace Conference, Big Sky, MT, USA, March 2019. View at: Google Scholar
 National Aeronautics and Space Administration (NASA), NASA Taps 11 American Companies to Advance Human Lunar Landers, National Aeronautics and Space Administration (NASA), 2019, Washington, DC, USAhttps://www.nasa.gov/pressrelease/nasataps11americancompaniestoadvancehumanlunarlanders.
 M. Xu and S. Xu, “Exploration of distant retrograde orbits around the moon,” Acta Astronautica, vol. 65, no. 56, pp. 853–860, 2009. View at: Publisher Site  Google Scholar
 M. Xu and S. J. Xu, “Stability analysis and transiting trajectory design for retrograde orbits around moon,” Journal of Astronautics, vol. 30, no. 5, pp. 1785–1791, 2009. View at: Publisher Site  Google Scholar
 M. Xu, Y. Wei, and S. Xu, “On the construction of lowenergy cislunar and translunar transfers based on the libration points,” Astrophysics and Space Science, vol. 348, no. 1, pp. 65–88, 2013. View at: Publisher Site  Google Scholar
 M. Xu, Y. Liang, and K. Ren, “Survey on advances in orbital dynamics and control for libration point orbits,” Progress in Aerospace Sciences, vol. 82, pp. 24–35, 2016. View at: Publisher Site  Google Scholar
 Y. Liang, M. Xu, and S. Xu, “Lowenergy weak stability boundary transfers to pluto’s moons: preliminary trajectory design via triangular libration point,” Acta Astronautica, vol. 156, pp. 219–233, 2019. View at: Publisher Site  Google Scholar
 G. Mengali, A. A. Quarta, C. Circi, and B. Dachwald, “Refined solar sail force model with mission application,” Journal of Guidance, Control, and Dynamics, vol. 30, no. 2, pp. 512–520, 2007. View at: Publisher Site  Google Scholar
 A. A. Quarta and G. Mengali, “Semianalytical method for the analysis of solar sail heliocentric orbit raising,” Journal of Guidance, Control, and Dynamics, vol. 35, no. 1, pp. 330–335, 2012. View at: Publisher Site  Google Scholar
 L. Niccolai, A. A. Quarta, and G. Mengali, “Analytical solution of the optimal steering law for nonideal solar sail,” Aerospace Science and Technology, vol. 62, pp. 11–18, 2017. View at: Publisher Site  Google Scholar
 A. Caruso, A. A. Quarta, and G. Mengali, “Comparison between direct and indirect approach to solar sail circletocircle orbit raising optimization,” Astrodynamics, vol. 3, pp. 1–12, 2019. View at: Publisher Site  Google Scholar
 S. Kemble, Interplanetary Mission Analysis and Design, Praxis Publishing, Chichester, UK, 2006.
 C. R. Cassell and P. A. Penzo, “LAMDA: a software tool for lunarassist escape missions,” in Proceedings of the AAS/AIAA Astrodynamics Specialist Conference AAS 95405, Halifax, Nova Scotia, Canada, August 1995. View at: Google Scholar
 A. V. Labunsky, O. V. Papkov, and K. G. Sukhanov, Multiple Gravity Assist Interplanetary Trajectories, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1998.
 A. Poghosyan and A. Golkar, “CubeSat evolution: analyzing CubeSat capabilities for conducting science missions,” Progress in Aerospace Sciences, vol. 88, pp. 59–83, 2017. View at: Publisher Site  Google Scholar
 R. L. Staehle, D. Blaney, H. Hemmati et al., “Interplanetary CubeSats: opening the solar system to a broad community at lower cost,” Journal of Small Satellites, vol. 2, pp. 161–186, 2013. View at: Google Scholar
 National Aeronautics and Space Administration (NASA), Mars Cube One, National Aeronautics and Space Administration (NASA), 2019, Washington, DC, USAhttps://www.jpl.nasa.gov/cubesat/missions/marco.php.
 D. A. Williams, R. M. C. Lopes, J. CastilloRogez et al., “CubeSats to support future io exploration,” in Proceedings of the 49th Lunar and Planetary Science Conference 2018, The Woodlands, TX, USA, March 2018. View at: Google Scholar
 A. Shaw, R. Remblas, and P. Fulford, “Advantages of science cubesat and microsat deployment using DSG deep space exploration robotics,” in Proceedings of the Deep Space Gateway Science Workshop 2018, Denver, CO, USA, February 2018. View at: Google Scholar
 I. GarrickBethell, R. P. Lin, H. Sanchez et al., “Lunar magnetic field measurements with a CubeSat,” in Proceedings of the SPIE 8739, Sensors and Systems for Space Applications VI, Baltimore, MD, USA, May 2013. View at: Publisher Site  Google Scholar
 Y.J. Song, D. Lee, H. Jin, and B.Y. Kim, “Potential trajectory design for a lunar CubeSat impactor deployed from a HEPO using only a small separation deltaV,” Advances in Space Research, vol. 59, no. 2, pp. 619–630, 2017. View at: Publisher Site  Google Scholar
 R. W. Farquhar and D. W. Dunham, “Librationpoint staging concepts for earthmars transportation,” in Proceedings of the Manned Mars Mission Working Group Papers, a Workshop at Marshall Space Flight Center, NASA/TM89320, Huntsville, AL, USA, June 1985, 1, 6677. View at: Google Scholar
 M. Hanson John and A. W. Deaton, “Use of multiple lunar swingby for departure to mars,” in Proceedings of the AAS/AIAA Astrodynamics Conference AAS 91499, Durango, CO, USA, August 1991. View at: Google Scholar
 J. Kawaguchi, H. Yamakawa, T. Uesugi, and H. Matsuo, “On making use of lunar and solar gravity assists in LUNARA, PLANETB missions,” Acta Astronautica, vol. 35, no. 9–11, pp. 633–642, 1995. View at: Publisher Site  Google Scholar
 S. Hernandez and B. W. Barbee, “Design of roundtrip trajectories to nearearth Asteroids utilizing a lunar flyby,” in Proceedings of the 21st AAS/AIAA Space Flight Mechanics Meeting AAS 11181, New Orleans, LA, USA, February 2011. View at: Google Scholar
 B. W. Barbee, T. Esposito, E. Piñon III et al., “A comprehensive ongoing survey of the nearearth asteroid population for human mission accessibility,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference AIAA 20108368, Toronto, Ontario, Canada, August 2010. View at: Publisher Site  Google Scholar
 J. T. Betts, “Optimal lunar swingby trajectories,” The Journal of the Astronautical Sciences, vol. 55, no. 3, pp. 349–371, 2007. View at: Publisher Site  Google Scholar
 D. A. Vallado, Fundamentals of Astrodynamics and Applications, Microcosm Press, Hawthorne, CA, USA, 4th edition, 2013.
 R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, AIAA, New York, NY, USA, 1987.
 R. V. Ramanan, “Integrated algorithm for lunar transfer trajectories using a pseudostate technique,” Journal of Guidance, Control, and Dynamics, vol. 25, no. 5, pp. 946–952, 2002. View at: Publisher Site  Google Scholar
 E. M. Standish, “JPL planetary and lunar ephemerides, DE405/LE405,” in Jet Propulsion Laboratory Interoffice Memorandum, Jet Propulsion Laboratory, La Cañada Flintridge, CA, USA, 1998, IOM 312.F98048. View at: Google Scholar
 D. D. McCarthy and G. Petit, “IERS conventions (2003),” Tech. Rep., 2004, IERS Technical Note No. 32, https://www.iers.org/IERS/EN/Publications/TechnicalNotes/tn32.html?nn=94912. View at: Google Scholar
 P. K. Seidelmann, B. A. Archinal, M. F. A’hearn et al., “Report of the IAU/IAG working group on cartographic coordinates and rotational elements: 2006,” Celestial Mechanics and Dynamical Astronomy, vol. 98, no. 3, pp. 155–180, 2007. View at: Publisher Site  Google Scholar
 L.M. Burke, R. D. Falck, and M. L. McGuire, Interplanetary Mission Design Handbook: EarthToMars Mission Opportunities 2026 to 2045, National Aeronautics and Space Administration, Washington, DC, USA, 2010, NASA/TM2010216764.
 P. J. Prince and J. R. Dormand, “High order embedded RungeKutta formulae,” Journal of Computational and Applied Mathematics, vol. 7, no. 1, pp. 67–75, 1981. View at: Publisher Site  Google Scholar
 O. Montenbruck and E. Gill, Satellite Orbits: Models, Methods and Applications, Springer, Berlin, Germany, 2000.
 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN 77, Cambridge University Press, Cambridge, UK, 2nd edition, 1992.
 R. Biesbroek, Lunar and Interplanetary Trajectories, Springer, Berlin, Germany, 2016.
 R. Lescouzères, M. Wolf, B. Wollenhaupt et al., “Propulsion system development and verification activities for the 2016 exomars trace gas orbiter,” in Proceedings of the Space Propulsion Conference 2014 SP2014_2968951, Cologne, Germany, 2014. View at: Google Scholar
 K. L. Zondervan, J. Fuller, D. Rowen et al., “CubeSat solid rocket motor propulsion systems providing deltavs greater than 500 m/s,” in Proceedings of the AIAA/USU Conference of Small Satellites SSC14X1, 2014. View at: Google Scholar
Copyright
Copyright © 2019 Yongjun Song et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.