Trans. JSASS Aerospace Tech. Japan Vol. 17, No. 4, pp. 455-460, 2019
DOI: 10.2322/tastj.17.455
Analysis of the Orbital Transfer between the Earth–Mars Orbit using Electric Propulsion based on the Direct Collocation Method
By Akihito TOBA,1) Ikkoh FUNAKI,2) and Yoshiki YAMAGIWA1)
1)Shizuoka University, Hamamatsu, Japan 2) Institute of Space and Astronautical Science, JAXA, Sagamihara, Japan
(Received June 30th, 2017)
This paper discusses a transfer system between the Earth and Mars orbit using electric propulsion under the assumption of launching the H-IIA launch vehicle on a one-way trip. The spacecraft was assumed to be launched into the geocentric orbit, then transferred to the Earth’s heliocentric orbit using electric propulsion or a kick motor, and further transferred to Mars revolving orbit using electric propulsion. The dependency of the payload mass on the specific impulse of the propulsion and launch system were investigated by performing orbit optimization based on the direct collocation method. As a result, the most suitable transfer method to Mars orbit was the combination of a kick motor from the geostationary transfer orbit to 2 2 the Earth’s heliocentric orbit, and subsequently the use of electric propulsion to transverse to Mars orbit with C3 = 9 km /s . The maximum payload mass was 1,500 kg for a specific impulse of 3,000s.
Key Words: Electric Propulsion, System Design, Nonlinear Programming, Mars Exploration
Nomenclature f : final i : node number C : equation of constraints ip : initial at interplanetary transfer C3 : escape energy prop : propellant F : thrust pl : payload f : differential vector r : radial direction G : gravitational constant tot : total g : gravitational acceleration : circumferential direction ISP : specific impulse M : mass of the sun 1. Introduction m : spacecraft mass m : mass flow rate For a long time, Mars has been considered as an important P : electric power target for robotic missions and manned exploration. A roadmap r : radius for such missions is shown in Ref. 1). In most of the past studies t : time of these missions, only chemical propulsion was considered; u : control vector however, it has the disadvantage that the payload mass can only V : orbital velocity be very small. In contrast, spacecraft system using electric propulsion can achieve a larger payload mass ratio by applying x : state vector a high specific impulse than systems using only chemical : mass-to-power ratio propulsion. On the other hand, electric propulsion has the t : divided time drawback that its thrust is small. However, missions to mars V : change in velocity can become feasible by employing the high power and high : structure factor efficiency of electric propulsion.2) p : power processing unit efficiency In the past, numerous studies on the optimization of low- t : thrust efficiency thrust trajectories between Earth and Mars and mission analysis : angle were conducted.3-7) Those studies assumed the usage of heavy : inclination launch vehicles with the capability of launching a heavy Subscripts spacecraft into an interplanetary trajectory. The capability of 0 : initial Japanese launch vehicles is, however, limited in comparison 1 : first collocation point with that of other countries. 2 : second collocation point In recent years, the Japanese science community has been 3 : third collocation point discussing a Japanese Mars mission following the Martian e : earth Moons eXploration (MMX). The usage of electric propulsion will allow to transfer large payloads toward Mars even if the
Copyright© 2019 by the Japan Society for Aeronautical and Space Sciences and ISTS. All rights reserved. J-STAGE Advance published date: January 31st, 2019 455 Trans. JSASS Aerospace Tech. Japan Vol. 17, No. 4 (2019)
Initial launched orbit (=LEO or GTO) Transfer orbit
Earth
Sun Spacecraft
Earth’s heliocentric orbit Mars revolving orbit Earth’s gravitational sphere escape orbit Fig. 1. Orbit model of the inner part of the Earth’s gravitational sphere. Fig. 2. Orbit model of the interplanetary orbital transfer. capability of the Japanese launch vehicle is limited. transfer equation.8) In detail, this equation represents the change In this paper, a parametric study was conducted to clarify the in velocity for a transfer from a constant acceleration circle to superiority of electric propulsion over chemical propulsion for an inclined circle. The required propellant mass for the transfer transportation from Earth to Mars orbit using the Japanese from LEO to the Earth’s heliocentric orbit was calculated by launch vehicle for a one-way trip. Eq. (2) for each specific impulse using the change in velocity that could be estimated by Eq. (1). The mass of the spacecraft 2. Model and Method of the Numerical Simulations upon its arrival in the Earth’s heliocentric orbit was estimated by Eq. (3). The orbital transfer from Earth to Mars was assumed to 2 2 proceed the following way: V V0 Vf 2V0Vf cos (1) 1. The spacecraft is launched from the ground to the low Earth 2 orbit (LEO) or geostationary transfer orbit (GTO). V 2-a. The spacecraft traverses from LEO to the Earth’s gI sp heliocentric orbit using electric propulsions with a spiral m prop m0 1 e (2) transfer. 2-b. The spacecraft transfers to the Earth’s heliocentric orbit m m m (3) using a kick motor via LEO or GTO. ip 0 prop 3. The spacecraft is transferred across the Earth’s heliocentric Further, for the interplanetary orbital transfer, the motion of orbit to Mars revolving orbit using electric propulsions with a the sun and the spacecraft were considered as a two-body semi-spiral transfer, which is a spiral transition within one orbit. problem. Equations of motion for the radial direction and In this study, the transfer orbit was divided into two parts. circumferential direction were calculated by numerically The first part was the inner part of the Earth’s gravitational solving Eqs. (4) and (5), respectively. In this paper, the radial sphere, i.e., the orbital transfer from a geocentric orbit to the direction was defined as the direction that points away from the Earth’s heliocentric orbit. The second part was the outer part of sun, while the circumferential direction was defined as the the Earth’s gravitational sphere, i.e., the orbital transfer from counter-clockwise direction around the sun. The mass flow rate the Earth’s heliocentric orbit to the Mars revolving orbit. was estimated by Eq. (6), the total thrust was calculated by Eq. (7), and the payload mass when the spacecraft arrives in Mars 2.1. Analysis model revolving orbit was estimated by Eq. (8). The payload mass was A three-dimensional space with Earth as its origin was defined as the mass of the spacecraft at the time of arrival in assumed in the inner part of the Earth’s gravitational sphere and Mars revolving orbit which excludes the mass of the power a plane with the sun as origin was assumed for the supply system, propulsion system, propellant, and propellant interplanetary orbital transfer. The outlines of the orbit models tank from LEO. 2 are shown in Figs. 1 and 2. Figure 1 shows the inner part of the d 2r d GM m r F (4) Earth’s gravitational sphere. Figure 2 shows the interplanetary 2 2 r dt dt r orbital transfer model; both orbits were assumed to be circular. In this paper, the spacecraft was modeled as particle and 2 d dr d considered to generate electric power using a solar paddle. m r 2 F (5) dt2 dt dt Further, the spacecraft was considered to use power only to operate electric propulsions and that the power generation had 2t P no margins. m (6) g I 2 2.2. Basic equations sp In the inner part of the Earth’s gravitational sphere, the 2 P r 2 change in velocity required to transfer between orbits was F F 2 F 2 t e (7) r 2 calculated analytically by Eq. (1) because it can account for the g I sp r spiral orbit. Eq. (1) is called Edelbaum’s low-thrust orbit
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Table 1. Constraint conditions for the interplanetary transfer. Initial launched orbit (=LEO or GTO) Transfer orbit Initial Final Radius [AU] 1.000 1.524 Earth Angle [rad] 0 Free Radial velocity [km/s] 0 0 Sun Spacecraft Orbital speed [km/s] 29.78 24.13
Table 2. Spacecraft parameters. Earth’s heliocentric orbit Mass-to-Power ratio of solar paddle [kg/kW] 20 Power processing unit efficiency [-] 0.80 Earth’s gravitational sphere escape orbit Mars revolving orbit Conversion efficiency of solar paddle [-] 0.28 Fig. 1. Orbit model of the inner part of the Earth’s gravitational sphere. Fig. 2. Orbit model of the interplanetary orbital transfer. Structure factor of propellant tank 0.15 Fig. 3. Outline of system constraints of a fourth-degree Gauss–Lobatto Mass-to-Power ratio of the electric propulsion [kg/kW] 8 8) quadrature rule. capability of the Japanese launch vehicle is limited. transfer equation. In detail, this equation represents the change Thrust efficiency [-] 0.55
In this paper, a parametric study was conducted to clarify the in velocity for a transfer from a constant acceleration circle to superiority of electric propulsion over chemical propulsion for an inclined circle. The required propellant mass for the transfer P 1 m m tot mt (8) 1 transportation from Earth to Mars orbit using the Japanese from LEO to the Earth’s heliocentric orbit was calculated by pl 0 tot C 32 5 60 x 64 5x 32 5 60 x p 1 1 i 2 i1 launch vehicle for a one-way trip. Eq. (2) for each specific impulse using the change in velocity 120 (11) that could be estimated by Eq. (1). The mass of the spacecraft In this study, orbital optimization was conducted by above ti 5 3 5 fi 50 f1 5 3 5 fi1 0 2. Model and Method of the Numerical Simulations upon its arrival in the Earth’s heliocentric orbit was estimated equations. However, only the interplanetary orbital transfer by Eq. (3). part was optimized, while the inner part of the Earth’s 1 C3 32 5 60xi 64 5x2 32 5 60xi1 The orbital transfer from Earth to Mars was assumed to gravitational sphere was not optimized. For the specific 120 (12) 2 2 proceed the following way: V V0 Vf 2V0Vf cos (1) impulse, parametric calculations were conducted in the inner ti 5 3 5 fi 50 f3 5 3 5 fi1 0 1. The spacecraft is launched from the ground to the low Earth 2 part of the Earth’s gravitational sphere based on Eqs. (1), (2) In addition, control vectors, i.e., from the linear interpolation orbit (LEO) or geostationary transfer orbit (GTO). V and (3). The results from the calculations of the inner part of performed for the thrust direction, are defined in Eqs. (13) and 2-a. The spacecraft traverses from LEO to the Earth’s gI sp the Earth’s gravitational sphere were applied as the initial (14). heliocentric orbit using electric propulsions with a spiral m prop m0 1 e (2) conditions for the interplanetary orbital transfer part. Equations transfer. of motion are described by Eqs. (4), (5) and (6). The equation 1 1 1 u1 ui1 ui ui (13) 2-b. The spacecraft transfers to the Earth’s heliocentric orbit of thrust was described by Eq. (7). After orbital optimization, 2 2 5 mip m0 m prop (3) using a kick motor via LEO or GTO. the payload mass was calculated by Eq. (8). The thrust direction 3. The spacecraft is transferred across the Earth’s heliocentric Further, for the interplanetary orbital transfer, the motion of of the spacecraft was set as control parameter for orbital 1 1 1 u3 ui1 ui ui (14) orbit to Mars revolving orbit using electric propulsions with a the sun and the spacecraft were considered as a two-body optimization. The parameters were optimized by considering 2 2 5 semi-spiral transfer, which is a spiral transition within one orbit. problem. Equations of motion for the radial direction and the minimization of the transfer time between the Earth’s In this study, the optimization problems were solved by In this study, the transfer orbit was divided into two parts. circumferential direction were calculated by numerically heliocentric orbit to Mars revolving orbit. fmincon from the MATLAB Optimization Toolbox. The The first part was the inner part of the Earth’s gravitational solving Eqs. (4) and (5), respectively. In this paper, the radial 2.3. Direct collocation method fmincon function was used with the interior-point. direction was defined as the direction that points away from the sphere, i.e., the orbital transfer from a geocentric orbit to the Direct collocation with nonlinear programming is a method 2.4. Analysis conditions sun, while the circumferential direction was defined as the Earth’s heliocentric orbit. The second part was the outer part of for solving optimal control problems with any type of nonlinear The launching system from the ground was assumed to be counter-clockwise direction around the sun. The mass flow rate the Earth’s gravitational sphere, i.e., the orbital transfer from constraints and objective function. In this paper, the fourth- the H2A202 type of the H-IIA launch vehicle of the Japan was estimated by Eq. (6), the total thrust was calculated by Eq. the Earth’s heliocentric orbit to the Mars revolving orbit. degree Gauss–Lobatto quadrature rule and the fourth-degree Aerospace Exploration Agency (JAXA). It can launch 10,000 (7), and the payload mass when the spacecraft arrives in Mars Gauss–Lobatto system constraints were adopted for state kg to LEO with an angle of 30.4° at an altitude of 300 km and revolving orbit was estimated by Eq. (8). The payload mass was 2.1. Analysis model vectors and differential vectors.9,10) These methods use three 4,000 kg to GTO with an angle of 28.4° at an altitude of perigee defined as the mass of the spacecraft at the time of arrival in A three-dimensional space with Earth as its origin was collocation points t , t , and t within a given subinterval of 250 km and an altitude of apogee of 36,226 km. As a Mars revolving orbit which excludes the mass of the power 1 2 3 assumed in the inner part of the Earth’s gravitational sphere and [ t , t ]. x and x are discrete approximations for x(t) constraint condition for optimizing the spacecraft trajectory, supply system, propulsion system, propellant, and propellant i i 1 1 3 a plane with the sun as origin was assumed for the at t and t , respectively, as shown by Eqs. (9) and (10). The the initial state of the spacecraft was synchronized with the tank from LEO. 1 3 interplanetary orbital transfer. The outlines of the orbit models fourth-degree Gauss–Lobatto system constraints are Earth’s revolution, while the final state was synchronized with 2 are shown in Figs. 1 and 2. Figure 1 shows the inner part of the d 2r d GM formulated at the internal collocation points using x and Mars’s revolution. The constraint conditions are shown in m r F (4) 1 Earth’s gravitational sphere. Figure 2 shows the interplanetary 2 2 r x to evaluate the system equation resulting in the discrete Table 1. In this study, BPT-4000 Hall Thrusters were assumed dt dt r 3 orbital transfer model; both orbits were assumed to be circular. values f1 and f 3 , respectively, as shown in Fig. 3. The for electric propulsion. In this paper, the spacecraft was modeled as particle and 2 system constraints C and C , at x and x , For electric propulsion missions, the acceleration and d dr d 1 3 1 3 considered to generate electric power using a solar paddle. m r 2 F (5) respectively, are given by Eqs. (11) and (12). 2 dt dt electric power to mass ratio of the spacecraft system are dt 11) Further, the spacecraft was considered to use power only to 1 important parameters. The acceleration of the spacecraft x 7 5 9 x 32x 7 5 9 x operate electric propulsions and that the power generation had 2 P 1 i 2 i1 system at the Earth’s heliocentric orbit indicates the thrust to t 50 (9) no margins. m 2 (6) mass ratio of the spacecraft system. The electric power to mass g I 2.2. Basic equations sp ti 5 1fi 5 1fi1 ratio at the Earth’s heliocentric orbit is considered as a system 12) In the inner part of the Earth’s gravitational sphere, the 2 performance parameter. For an electric propulsion system, an 2 2 2t P re 1 change in velocity required to transfer between orbits was F F F (7) x 7 5 9x 32x 7 5 9x increase in specific impulse enhances the payload mass of the r 2 3 i 2 i 1 calculated analytically by Eq. (1) because it can account for the g I sp r 50 (10) spacecraft, which can be further boosted by increasing the spiral orbit. Eq. (1) is called Edelbaum’s low-thrust orbit ti 5 1fi 5 1fi1 amount of electric power. The acceleration of the spacecraft
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Table 3. Analysis conditions for the interplanetary transfer. Initial mass [kg] 10,000 Acceleration at the Earth’s heliocentric 310 5 g orbit [m/s2] Propellant mass from LEO to the Earth’s 5,450 4,085 3,255 2,702 2,309 heliocentric orbit [kg] Spacecraft mass when arrived at the Earth’s 4,550 5,915 6,745 7,298 7,691 heliocentric orbit [kg] Specific impulse [s] 1,000 1,500 2 ,000 2,500 3,000 Fig. 4. Initial thrust direction and optimized thrust direction for a Electric power [kW] 14.93 29.11 44.26 59.86 75.68 specific impulse of 3000 s. system at the Earth’s heliocentric orbit was fixed to examine specific impulse enhances both, the mass of the spacecraft these effects. It is to be noted that the mass of the power supply when it arrives in Mars revolving orbit and the payload mass. becomes small, while the mass of the propellant becomes large The enhancement of the mass of the spacecraft upon its arrival for a low specific impulse. On the other hand, the mass of the in Mars revolving orbit can be explained by the fact that the propellant becomes small and the mass of the power supply increase in specific impulse reduces the necessary mass of the system becomes large when a high specific impulse is applied. propellant and propellant tank. In turn, this reduction in Therefore, it is necessary to investigate the combination of propellant mass allows an increase of the payload mass. specific impulse and electric power to achieve the maximum Notably, the increase in specific impulse is accompanied by payload while keeping the acceleration of the spacecraft system only a relative small increase in the mass of the required power a fixed parameter. supply and propulsion system. As a result, the decrease in The acceleration is an important parameter for the motion of propellant mass is larger than the increase in system mass the spacecraft because when its value is large, the thrust for the related to the power supply and propulsion system. solar gravity grows also large. Therefore, it is thought that the These details on the mass of the spacecraft are shown in Fig. transfer time shortens, which in turn has a large influence on 6b. As seen in Fig. 6b, the increase in specific impulse the result of a mission to Mars. The acceleration of the decreases the mass of the propellant tank (orange bar) and the 5 spacecraft at the Earth’s heliocentric orbit was set to 310 g , propellant itself (yellow bar and blue bar), while it requires an which is 5% of the Sun’s gravity acceleration value at the increase in the mass of the power supply and propulsion system Earth’s heliocentric orbit. The spacecraft parameters are shown (gray bar). As can be clearly seen, the mass of the propellant in Table 2 and analysis conditions are listed in Table 3. for the orbital transfer from LEO to the Earth’s heliocentric orbit (yellow bar) has the greatest influence on the total 3. Analysis Results and Discussion spacecraft system mass, that is mass of the propellant, propellant tank, and power supply and propulsion system. 3.1. Payload mass and system mass As is shown in Fig. 6c, the yellow bar and green bar are Figure 4 shows the initial and optimized thrust directions. removed from Fig. 6b to consider relations of the mass of the The initial thrust direction was defined as being at an angle of propellant for the interplanetary transfer (blue bar), propellant 0° to the circumferential direction before the spacecraft was tank (orange bar), and power supply and propulsion system accelerated; the spacecraft was accelerated in the radial (gray bar). In Fig. 6c, the total mass is slightly increased by the direction for any positive angle between those directions. For increase in specific impulse. However, the payload mass the optimized thrust direction, at the beginning, the angle increased with the increase in specific impulse. Therefore, it is gradually increased to approximately 90°; afterward, thrust thought that the influence of increasing the total mass is smaller direction was changed to opposite direction and finally, the than the decrease in the propellant mass for the orbital transfer thrust direction became -20°. It is thought that the angle of the from LEO to the Earth’s heliocentric orbit (yellow bar) in Fig. thrust direction increased because of the increased acceleration 6b. For that reason, the payload mass increases with increasing in the radial direction. However, the speed of the spacecraft in specific impulse, although the system mass occupies at least the circumferential direction decreases by increasing thrust 70% of the total spacecraft mass. Figure 7 shows the transfer direction. In this configuration, it is thought that thrust in the time associated with each functional part of the spacecraft and negative direction was employed to reduce the speed in the the total transfer time. The transfer time from LEO to the radial direction and increase the speed in the circumferential Earth’s heliocentric orbit was calculated by dividing the mass direction of the spacecraft. Figure 5 shows the initial and of the propellant required for traversing from LEO to the optimized trajectories. For the initial trajectory, the final point Earth’s heliocentric orbit by the mass flow rate that is given by exceeds the radius of Mars revolving orbit. However, the Eq. (6). The transfer time was reduced because of the decreased optimized trajectory is synchronized with the Mars’s revolution mass of the propellant attributed to the increasing specific at its final point. Therefore, the spacecraft trajectory is impulse. The total transfer time amounted to about 650 days for considered as optimized. As is shown in Fig. 6a, the increase in each specific impulse. In all case, the BPT-4000 for the electric
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Payload mass Table 3. Analysis conditions for the interplanetary transfer. Power supply system and propulsion system mass Initial mass [kg] 10,000 Propellant tank mass Acceleration at the Propellant mass of interplanetary transfer Earth’s heliocentric 310 5 g Propellant mass from LEO to Earth's heliocentric orbit orbit [m/s2] 12000 Propellant mass from 10000 LEO to the Earth’s 5,450 4,085 3,255 2,702 2,309 8000 heliocentric orbit [kg] 6000 Spacecraft mass when arrived at the Earth’s 4,550 5,915 6,745 7,298 7,691 4000 heliocentric orbit [kg] 2000 Specific impulse [s] 1,000 1,500 2 ,000 2,500 3,000 Fig. 4. Initial thrust direction and optimized thrust direction for a 0 Electric power [kW] 14.93 29.11 44.26 59.86 75.68 specific impulse of 3000 s. 1000 1500 2000 2500 3000
Spacecraft Mass Details [kg] Specific Impulse [s] system at the Earth’s heliocentric orbit was fixed to examine specific impulse enhances both, the mass of the spacecraft Fig. 6b. Specific impulse dependency of the propellant mass from these effects. It is to be noted that the mass of the power supply when it arrives in Mars revolving orbit and the payload mass. LEO to the Earth’s heliocentric orbit, the propellant mass for the becomes small, while the mass of the propellant becomes large The enhancement of the mass of the spacecraft upon its arrival interplanetary transfer, mass of the propellant tank, mass of the power for a low specific impulse. On the other hand, the mass of the in Mars revolving orbit can be explained by the fact that the supply and propulsion system, and payload mass. propellant becomes small and the mass of the power supply increase in specific impulse reduces the necessary mass of the 6000 system becomes large when a high specific impulse is applied. propellant and propellant tank. In turn, this reduction in Therefore, it is necessary to investigate the combination of propellant mass allows an increase of the payload mass. Fig. 5. Initial trajectory and optimized trajectory for a specific impulse 5000 specific impulse and electric power to achieve the maximum Notably, the increase in specific impulse is accompanied by of 3,000s. 4000 3000 payload while keeping the acceleration of the spacecraft system only a relative small increase in the mass of the required power 2000 a fixed parameter. supply and propulsion system. As a result, the decrease in Details [kg] 1000 The acceleration is an important parameter for the motion of propellant mass is larger than the increase in system mass Spacecraft mass Spacecraft Mass 0 the spacecraft because when its value is large, the thrust for the related to the power supply and propulsion system. Payload mass 1000 1500 2000 2500 3000 solar gravity grows also large. Therefore, it is thought that the These details on the mass of the spacecraft are shown in Fig. 6000 Specific Impulse [s] transfer time shortens, which in turn has a large influence on 6b. As seen in Fig. 6b, the increase in specific impulse Fig. 6c. Specific impulse dependency of the mass of the propellant for the result of a mission to Mars. The acceleration of the decreases the mass of the propellant tank (orange bar) and the 5000 the interplanetary transfer, propellant tank, and power supply and 5 spacecraft at the Earth’s heliocentric orbit was set to 310 g , propellant itself (yellow bar and blue bar), while it requires an propulsion system. 4000 which is 5% of the Sun’s gravity acceleration value at the increase in the mass of the power supply and propulsion system Earth’s heliocentric orbit. The spacecraft parameters are shown Interplanetary transfer time (gray bar). As can be clearly seen, the mass of the propellant 3000 in Table 2 and analysis conditions are listed in Table 3. for the orbital transfer from LEO to the Earth’s heliocentric Transfer time from LEO to Earth's heliocentric orbit Total transfer time orbit (yellow bar) has the greatest influence on the total 2000 800 3. Analysis Results and Discussion spacecraft system mass, that is mass of the propellant, 600 propellant tank, and power supply and propulsion system. 1000 400 3.1. Payload mass and system mass As is shown in Fig. 6c, the yellow bar and green bar are Mass at Mars Revolving Revolving Mars Mass at Orbit [kg] 0 Figure 4 shows the initial and optimized thrust directions. removed from Fig. 6b to consider relations of the mass of the 200 The initial thrust direction was defined as being at an angle of 0 propellant for the interplanetary transfer (blue bar), propellant -1000 0° to the circumferential direction before the spacecraft was tank (orange bar), and power supply and propulsion system 1000 1500 2000 2500 3000 1000 1500 2000 2500 3000
Transfer Time [days] Time Transfer Specific Impulse [s] accelerated; the spacecraft was accelerated in the radial (gray bar). In Fig. 6c, the total mass is slightly increased by the Specific Impulse [s] Fig. 7. Specific impulse dependency of the transfer time from LEO to direction for any positive angle between those directions. For increase in specific impulse. However, the payload mass Fig. 6a. Specific impulse dependency of the arrival mass of the the Earth’s heliocentric orbit, interplanetary transfer time, and total the optimized thrust direction, at the beginning, the angle increased with the increase in specific impulse. Therefore, it is spacecraft at Mars revolving orbit and payload mass. transfer time. gradually increased to approximately 90°; afterward, thrust thought that the influence of increasing the total mass is smaller direction was changed to opposite direction and finally, the than the decrease in the propellant mass for the orbital transfer propulsions were considered to be operating for about 15,000 thrust direction became -20°. It is thought that the angle of the from LEO to the Earth’s heliocentric orbit (yellow bar) in Fig. section 3.1. The mass of the kick motor was set to 8,294 kg h. The BPT-4000 was operated for about 10,000 h in a previous thrust direction increased because of the increased acceleration 6b. For that reason, the payload mass increases with increasing (transfer via LEO) and to 1,537 kg (transfer via GTO) to study.13) However, there is no previous report on any operation 2 2 in the radial direction. However, the speed of the spacecraft in specific impulse, although the system mass occupies at least provide a C3 of 9 km /s . For the interplanetary transfer time, exceeding 15,000 h. Therefore, it is necessary to increase the the circumferential direction decreases by increasing thrust 70% of the total spacecraft mass. Figure 7 shows the transfer the same values as in section 3.1 were used. However, in this payload mass ratio and to shorten the total transfer time using direction. In this configuration, it is thought that thrust in the time associated with each functional part of the spacecraft and part, spacecraft parameters were optimized to maximize the a kick motor via the LEO or GTO. negative direction was employed to reduce the speed in the the total transfer time. The transfer time from LEO to the mass of the spacecraft with respect to the control parameters 3.2. Comparison of launch systems radial direction and increase the speed in the circumferential Earth’s heliocentric orbit was calculated by dividing the mass “thrust direction” and “throttling.” The analysis conditions for In this section, the spacecraft was assumed to receive a direction of the spacecraft. Figure 5 shows the initial and of the propellant required for traversing from LEO to the a launch with a kick motor via the LEO are shown in Table 4 certain C3 from a kick motor via LEO or GTO. The Kick motor optimized trajectories. For the initial trajectory, the final point Earth’s heliocentric orbit by the mass flow rate that is given by and those for a launch with a kick motor via the GTO are shown was assumed to be the KM-V2b of the Epsilon launch vehicle exceeds the radius of Mars revolving orbit. However, the Eq. (6). The transfer time was reduced because of the decreased in Table 5. of JAXA. Its specific impulse was assumed to be 301 s. The C3 optimized trajectory is synchronized with the Mars’s revolution mass of the propellant attributed to the increasing specific As is shown in Fig. 8, the increase in the specific impulse value was 9 km2/s2 to allow a comparison with solely using at its final point. Therefore, the spacecraft trajectory is impulse. The total transfer time amounted to about 650 days for generally enhances the payload mass for all launch systems. electric propulsion for traversing from the LEO discussed in considered as optimized. As is shown in Fig. 6a, the increase in each specific impulse. In all case, the BPT-4000 for the electric When using a kick motor, the payload mass is smaller than in
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Table 4. Analysis conditions for interplanetary transfer 2 2 use electric propulsion from LEO (given C3 =9 km /s via LEO). C = 9 km2/s2 C value [km2/s2] 9 3 with kick motor via LEO 3 C = 9 km2/s2 Initial mass [kg] 1,706 2500 3 with kick motor via GTO Acceleration at the Chemical Propulsion 2000 Earth’s heliocentric 3 10 5 g 2 orbit [m/s ] 1500 Specific impulse [s] 1,000 1,500 2,000 2,500 3,000 Electric power [kW] 4.478 6.717 8.956 11.19 13.43 1000
Table 5. Analysis conditions for interplanetary transfer 500 2 2 (given C3 =9 km /s via GTO). Payload Payload Mass [kg] 0 2 2 C3 value [km /s ] 9 Initial mass [kg] 2,463 -500 Acceleration at the 1000 1500 2000 2500 3000 5 Specific Impulse [s] Earth’s heliocentric 3 10 g orbit [m/s2] Fig. 8. Comparison of the payload mass between the case of using 2 2 Specific impulse [s] 1,000 1,500 2,000 2,500 3,000 electric propulsion from the LEO, C3 = 9 km /s with kick motor via LEO, 2 2 Electric power [kW] 6.465 9.697 12.93 16.16 19.39 C3 = 9 km /s with kick motor via GTO, and chemical propulsion. the case of electric propulsion from LEO for any specific to thank Editage (www.editage.jp) for English language editing. impulse value greater than 2,000s. Furthermore, in the case of a launch via GTO, a larger payload mass can be achieved than References through a launch via LEO for all specific impulse values. 2 2 Therefore, it is thought that a C3 of 9 km /s with a kick motor 1) ISECG: The Global Exploration Roadmap, NASA, 2013, p. 2. via the GTO is a more realistic system. In comparison with 2) Richard, R. H., Thomas, M. R, David, Y. O., and John, S. S.: Evaluation of a 4.5 kW Commercial Hall Thruster System for NASA chemical propulsion, using a kick motor allows an Science Missions, AIAA Paper 2006-4469, 2006. approximately 1.3–3.0 times larger payload mass than 3) David, Y. O., Richard, R. H., Ira, K., Jon, A. S., Noah, Z. W., chemical propulsion. Hence, with respect to the payload mass, Thomas, M. R., Ronald, T. R., and Robert, C. M.: Benefit of Using electric propulsion is superior to chemical propulsion. Hall Thruster for a Mars Sample Return Mission, Proceedings of 31st IEPC, IEPC-2009-217, 2009. 4) Robert, E. L., Zachary, J. B., Theresa, D. K., Erik, L. N., and 4. Conclusions Richard, L. M.: Mars Sample Return Orbiter Concepts Using Solar Electric Propulsion for the Post-Mars 2020 Decade, Proceedings of To implement a transportation system between the Earth – 2014 IEEE Aerospace Conference, 2014, pp. 133-142. Mars orbit using electric propulsion, the specific impulse, 5) Lianghui T., and Jian-ping Y.: Reentry Trajectory Optimization Using Direct Collocation Method and Nonlinear Programming, launch system, and acceleration dependency of payload mass IAC-06-C1.4.06, 2006. were investigated through performing orbit optimization based 6) Edward, J. N. and John, S. M.: Low Variable Thrust Interplanetary on the direct collocation method. As a result, the following Trajectory Data, NASA Technical Note, NASA-TN-D-4431,1968. conclusions were obtained. 7) Craig, A. K.: Efficient Computation of Optimal Interplanetary (1) For the orbital transfer from LEO using electric propulsion, Trajectories Using Solar Electric Propulsion, J. Guidance, Control, and Dynamics, 38 (2015), pp. 821-830. the transfer time from LEO to the Earth’s heliocentric orbit was 8) Vladimir, A. C., Hans, K. K., Chia-Chun, G. C., Jimmy, Y. M., too long when compared to the interplanetary transfer time. Thomas, J. L., and Jean, A. K.: AIAA Education Series Orbital (2) The comparison of launch systems indicated that a launch Mechanics, Third Edition, American Institute of Aeronautics and via GTO can achieve a larger payload mass than a launch via Astronautics, 2002, p. 349. 9) Kawabe, H.: Study on Application to Direct Elucidation and Glider LEO for each specific impulse. Flight of the Issue of Optimal Control, Ph.D. Thesis, Kyusyu (3) In comparison with chemical propulsion, of the use of a University, 1999 (in Japanese). kick motor allows an about 1.3–3.0 times larger payload mass 10) Albert, L. H. and Bruce, A. C.: Direct Optimization Using than chemical propulsion. Collocation Based on High-Order Gauss-Lobatto Quadrature Rules, J. Guidance, Control, and Dynamics, 19 (1996), pp. 592-599. As future work, the optimum acceleration values must be 11) John, W. D., Bob, V. and Andrew, V. I.: Fast Transits to Mars Using determined and moreover, it is necessary to examine a real Electric Propulsion, AIAA Paper 2010-6771, 2010. mission, such as a sample return mission. 12) David, Y. O. and Damon, L.: A Simple Semi-Analytic Model for Optimum Specific Impulse Interplanetary Low Thrust Trajectories, Acknowledgments Proceedings of 32nd IEPC, IEPC-2011-010, 2011. 13) Kristi, D. G., Alex, M., Ben, W., and Vadim, K.: Demonstration of
10,400 Hours of Operation on a 4.5 kW Qualification Model Hall The authors appreciate the fruitful discussion with Thruster, AIAA Paper 2010-6698, 2010. colleagues at Shizuoka University and JAXA. We would like
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