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 1) 2) 1) By Akihito TOBA, Ikkoh FUNAKI, and Yoshiki YAMAGIWA 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 Earth’s gravitational sphere escape orbit Mars revolving 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 456 Trans. JSASS Aerospace Tech. Japan Vol. 17, No. 4 (2019) 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.