Optimal transfers from an Earth orbit to a Mars orbit

PII:

Acta Astronautica Vol. 45, No. 3, pp. 119±133, 1999 # 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-5765/99 $ - see front matter S0094-5765(99)00109-5

OPTIMAL TRANSFERS FROM AN EARTH ORBIT TO A MARS ORBIT $ A. MIELE and T. WANG Rice University, Aero-Astronautics Group, Mail Stop 322, 6100 Main Street, Houston, TX 77005-1892, USA (Received 2 February 1999) AbstractÐThis paper deals with the optimal transfer of a spacecraft from a low Earth orbit (LEO) to a low Mars orbit (LMO). The transfer problem is formulated via a restricted four-body model in that the spacecraft is considered subject to the gravitational ®elds of Earth, Mars, and Sun along the entire trajectory. This is done to achieve increased accuracy with respect to the method of patched conics. The optimal transfer problem is solved via the sequential gradient-restoration algorithm employed in conjunction with a variable-stepsize integration technique to overcome numerical diculties due to large changes in the gravitational ®eld near Earth or near Mars. First, for given LEO and LMO radii, a basic optimization problem is considered: the minimization of the total characteristic velocity, the sum of the velocity impulses at LEO and LMO, assuming that the departure date is free, hence assuming that the planetary Mars/Earth phase angle di€erence at departure is free. At both departure and arrival, the optimal trajectory exhibits an asymptotic parallelism condition: at the end of near-Earth space, the spacecraft inertial velocity is parallel to the Earth inertial velocity; analogously, at the beginning of near-Mars space, the spacecraft inertial velocity is parallel to the Mars inertial velocity. The total characteristic velocity is DV = 5.652 km/s, corresponding to a ratio of payload mass to initial mass of 0.224 for typical speci®c impulse and structural factor. Then, for given LEO and LMO radii, a departure window is generated by changing the departure date, hence changing the planetary Mars/Earth phase angle di€erence at departure, and then reoptimizing the transfer. This results into an one-parameter family of suboptimal transfers retaining the asymptotic parallelism condition at arrival, but not at departure. For the suboptimal transfers, the phase angle travel and transfer time decrease with late departure and increase with early departure. Also, a change in departure date of +32 days [ÿ32 days] causes a characteristic velocity increase of 8% [5%], implying a payload mass decrease of 13% [8%]. # 1999 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION

The Mars mission is attractive and expensive [1±3]. An important component of a typical Mars mission is the transfer of a spacecraft from a low Earth orbit (LEO) to a low Mars orbit (LMO). The optimization of such transfer is of interest because optimal trajectories set a benchmark for minimum characteristic velocity, hence minimum propellant consumption; also, launch/arrival windows provide basic information for the development of robust guidance [4±8]. This paper addresses two problems: baseline optimal trajectory and departure window. (P1) Baseline Optimal Trajectory. For given LEO and LMO radii, a basic optimization problem is the minimization of the total characteristic velocity DV, the sum of the velocity impulses at LEO and LMO, equivalent under certain conditions to the minimization of the ratio of payload mass to initial mass. {Paper IAF-97-A.6.05 presented at the 48th International Astronautical Congress, 6±10 October, 1997, Turin, Italy.

Major parameters are four: (i) velocity impulse at departure; (ii) spacecraft vs Earth phase angle at departure; (iii) planetary Mars/Earth phase angle di€erence at departure; (iv) transfer time. These parameters must be determined so that DV is minimized, subject to the following requirements: tangential departure from circular velocity at LEO; tangential arrival to circular velocity at LMO. (P2) Departure Window. For given LEO and LMO radii, this study involves the computation of a one-parameter family of suboptimal transfers, obtained by changing the planetary Mars/Earth phase angle di€erence at departure, hence changing the departure date, vis-a-vis the optimal value determined under (P1). The objective is to determine how a change in parameter (iii), hence a change in departure date, a€ects the optimal values of the remaining parameters. Problems P1 and P2 are treated within the frame of the restricted four-body model: the spacecraft is considered subject to the gravitational ®elds of Earth, Mars, and Sun along the entire trajectory. This is done in order to achieve increased accuracy with respect to the method of patched conics. An

119

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important diculty in system integration and optimization is that the total gravitational force changes rapidly in near-Earth space or near-Mars space, but slowly in deep interplanetary space. Indeed, orbital periods are of order one hour to one day if Earth gravity or Mars gravity is dominant, but of order one year if Sun gravity is dominant. This diculty can be overcome by treating Problems P1 and P2 with the sequential gradient-restoration algorithm in conjunction with a variable-stepsize integration technique. 1.1. Remark The importance of LEO and LMO can be understood from the following considerations. For a spacecraft launched from the ground, LEO can be employed as a parking orbit before the spacecraft enters deep interplanetary space; this parking orbit is necessary to correctly position the spacecraft prior to the propulsive maneuver initiating the interplanetary transfer. For a spacecraft approaching Mars, LMO can be used as observation orbit or parking orbit. If the spacecraft has to land on Mars, the latter use is of interest to acquire data for precise landing and check preselected landing sites. 1.2. Content This paper is organized as follows. Section 2 deals with system description. Sections 3±4 contains the boundary conditions. Section 5 deals with the formulation of the optimal LEO-to-LMO transfer and outlines the computational method. Section 6 contains planetary and spacecraft data used in the numerical experiments. Sections 7±8 provide the numerical results for the baseline optimal trajectory and departure window. Finally, Section 9 contains the conclusions.

2. SYSTEM DESCRIPTION

We study the transfer of a spacecraft from a low Earth orbit (LEO) to a low Mars orbit (LMO) under the following scenario. Initially, the spacecraft moves in a circular orbit around Earth; an accelerating velocity impulse is applied tangentially to LEO, and its magnitude is such that the spacecraft escapes from near-Earth space into deep interplanetary space. Then, the spacecraft takes a long journey along an interplanetary orbit around Sun, enters near-Mars space, and reaches tangentially the low Mars orbit. Here, a braking velocity impulse is applied tangentially to LMO so as to achieve circularization of the motion around Mars. The following hypotheses are employed: (A1) the Sun is ®xed in space; (A2) Earth and Mars are subject only to Sun gravity; (A3) the eccentricity of the Earth and Mars orbits around Sun is neglected, implying circular planetary motions; (A4) the incli-

nation of the Mars orbital plane vis-a-vis the Earth orbital plane is neglected, implying planar spacecraft motion; (A5) the spacecraft is subject to the gravitational attractions of Earth, Mars, and Sun; (A6) the class of two-impulse trajectories, departing with a tangential velocity impulse from circular velocity at LEO and arriving with a tangential velocity impulse to circular velocity at LMO, is considered. Having adopted the restricted four-body model to achieve increased precision with respect to the patched conics model, we are simultaneously interested in ®ve motions: the inertial motions of Earth, Mars, and spacecraft with respect to Sun; the relative motions of the spacecraft with respect to Earth and Mars. To study these motions, we employ three coordinate systems. (SCS) Sun Coordinate System. This is a ®xed system with the origin at the Sun center. The xaxis is directed along the line SE0 connecting the Sun center S with the Earth initial position E0. The y-axis is orthogonal to the x-axis. The Sxy system is ®xed in space. (ECS) Earth Coordinate System. This is a moving system with the origin at the instantaneous Earth center. The x-axis is directed along the line SE connecting the Sun center S and the instantaneous Earth center E. The y-axis is orthogonal to the x-axis. (MCS) Mars Coordinate System. This is a moving system with the origin at the instantaneous Mars center. The x-axis is directed along the line SM connecting the Sun center S and the instantaneous Mars center M. The y-axis is orthogonal to the x-axis. To sum up, SCS is employed to describe the inertial motions of Earth, Mars, and spacecraft with respect to Sun. On the other hand, ECS and MCS are employed to describe the relative motions of the spacecraft with respect to Earth and Mars. Let E, M, S denote the centers of Earth, Mars, and Sun; let P denote the spacecraft. Let t denote the time, with 0 E t E t, where t is the ®nal time. With this understanding, we describe the motions of Earth, Mars, and spacecraft with respect to Sun; we employ the Sun coordinate system and polar coordinates. For details of coordinate system transformations, see [9]. 2.1. Earth Subject only to the Sun gravitational attraction, the motion of Earth (subscript E) around Sun is described by the following di€erential equations for the radial distance rE, phase angle fE, inertial velocity VE, and path inclination gE with respect to the local horizon: drE =dt ˆ VE sin gE ,

…1a†

Optimal Earth-to-Mars Transfers

dfE =dt ˆ …VE =rE † cos gE ,

…1b†

dVE =dt ˆ ÿ…mS =r2E † sin gE ,

…1c†

dgE =dt ˆ …VE =rE ÿ mS =VE r2E † cos gE ,

…1d †

dfP =dt ˆ …VP =rP † cos gP ,

fE ˆ t

p

mS =r3E …0†

VE ˆ

p



‡ …mE =r2PE † cos…CPE ÿ gP †

 mS =rE …0† ,

gE ˆ gE …0† ˆ 0:

…2d †

2.2. Mars Subject only to the Sun gravitational attraction, the motion of Mars (subscript M) around Sun is described by the following di€erential equations for the radial distance rM, phase angle fM, inertial velocity VM, and path inclination gM with respect to the local horizon: drM =dt ˆ VM sin gM ,

…3a†

dfM =dt ˆ …VM =rM † cos gM ,

…3b†

dVM =dt ˆ ÿ…mS =r2M † sin gM ,

…3c†

dgM =dt ˆ …VM =rM ÿ mS =VM r2M † cos gM ,

…3d †

where mS is the Sun gravitational constant. Neglecting the orbital eccentricity, we approximate the Mars trajectory with a circle; hence, rM ˆ rM …0†, fM ˆ t

p

 mS =r3M …0† ‡ fM …0†,

VM ˆ

p

dgP =dt ˆ …VP =rP ÿ mS =VP r2P † cos gP

 mS =rM …0† ,

gM ˆ gM …0† ˆ 0:

‡ …mE =VP r2PE † sin…CPE ÿ gP †

…2b† …2c†

Here mS, mE, mM are the gravitational constants of Sun, Earth, and Mars; rPE and rPM are the radial distances of the spacecraft from Earth and Mars; CPE and CPM are the inclinations of the vectors PE and PM with respect to the local horizon. These quantities are given by the following relations, obtainable via simple geometric considerations: r2PE ˆ r2P ‡ r2E ÿ 2rP rE cos…fP ÿ fE †,

…6a†

r2PM ˆ r2P ‡ r2M ÿ 2rP rM cos…fP ÿ fM †,

…6b†

and sin CPE ˆ ÿrP =rPE ‡ …rE =rPE † cos…fP ÿ fE †, cos CPE ˆ ÿ…rE =rPE † sin…fP ÿ fE †,

cos CPM ˆ ÿ…rM =rPM † sin…fP ÿ fM †,

…7b†

…7d †

with the implication that the trigonometric expressions in eqns (5c) and (5d) can be rewritten as sin…CPE ÿ gP † ˆ ÿ…rP =rPE † cos gP

…4b†

cos…CPE ÿ gP † ˆ ÿ…rP =rPE † sin gP ÿ …rE =rPE † sin…fP ÿ fE ÿ gP †, sin…CPM ÿ gP † ˆ ÿ…rP =rPM † cos gP ‡ …rM =rPM † cos…fP ÿ fM ÿ gP †, cos…CPM ÿ gP † ˆ ÿ…rP =rPM † sin gP

2.3. Spacecraft

…7a†

sin CPM ˆ ÿrP =rPM ‡ …rM =rPM † cos…fP ÿ fM †, …7c†

‡ …rE =rPE † cos…fP ÿ fE ÿ gP †,

…4d †

…5d †

‡ …mM =VP r2PM † sin…CPM ÿ gP †:

…4a†

…4c†

…5c†

‡ …mM =r2PM † cos…CPM ÿ gP †,

…2a† ‡ fE …0†,

…5b†

dVP =dt ˆ ÿ…mS =r2P † sin gP

where mS is the Sun gravitational constant. Neglecting the orbital eccentricity, we approximate the Earth trajectory with a circle; hence, rE ˆ rE …0†,

121

ÿ …rM =rPM † sin…fP ÿ fM ÿ gP †:

…8a†

…8b†

…8c†

…8d †

Subject to the gravitational attractions of Sun, Earth, and Mars, the motion of the spacecraft (subscript P) around Sun is described by the following di€erential equations for the radial distance rP, phase angle fP, inertial velocity VP, and path inclination gP with respect to the local horizon:

In light of eqns (8), the spacecraft equations of motion (5) become drP =dt ˆ VP sin gP ,

…9a†

drP =dt ˆ VP sin gP ,

dfP =dt ˆ …VP =rP † cos gP ,

…9b†

…5a†

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dVP =dt ˆ ÿ…mS =r2P ‡ mE rP =r3PE ‡ mM rP =r3PM † sin gP ÿ …mE rE =r3PE † sin…fP ÿ fE ÿ gP †

3.3. Spacecraft …9c†

In the Earth coordinate system (ECS), the spacecraft initial conditions (t = 0) are given by

ÿ …mM rM =r3PM † sin…fP ÿ fM ÿ gP †, dgP =dt ˆ …VP =rP ÿ mS =VP r2P ÿ mE rP =VP r3PE ÿ

mM rP =VP r3PM †

cos gP

‡ …mE rE =VP r3PE † cos…fP ÿ fE ÿ gP † ‡

…mM rM =VP r3PM †

…9d †

rPE …0† ˆ rLEO ,

…12a†

fPE …0† ˆ free,

…12b†

VPE …0† ˆ VLEO ‡ DVLEO ˆ

cos…fP ÿ fM ÿ gP †,

with rPE and rPM given by eqns (6). This form of the equations of motion allows one to avoid the computation of the angles CPE and CPM. 2.4. Remark In eqns (1)±(9), the f-angles are measured starting from the radial direction and are positive counterclockwise. The C-angles and g-angles are measured starting from the local horizon and are positive clockwise.

p …mE =rLEO † ‡ DVLEO , gPE …0† ˆ 0:

…12c† …12d †

Relative to Earth rPE, fPE, VPE, gPE are the radial distance, phase angle, velocity, and path inclination of the spacecraft; VLEO is the spacecraft velocity in the low Earth orbit prior to the application of the tangential velocity impulse; DVLEO is the accelerating velocity impulse; VPE is the spacecraft velocity after the application of the tangential velocity impulse. 3.4. Coordinate transformation

3. INITIAL CONDITIONS

Let t = 0 denote the initial time. With this understanding, we state the initial conditions for Earth, Mars, and spacecraft. 3.1. Earth In the Sun coordinate system (SCS), the Earth initial conditions (t = 0) are given by rE …0† ˆ given,

…10a†

fE …0† ˆ 0,

…10b†

VE …0† ˆ

p



mS =rE …0† ,

gE …0† ˆ 0:

At t = 0, the following coordinate transformation is employed to convert ECS variables present in the initial conditions (12) into SCS variables necessary to start the integration of the di€erential system (9): r2P ˆ r2E ‡ r2PE ‡ 2rE rPE cos fPE ,  fP ˆ arc sin …rE =rP † sin fE  ‡ …rPE =rP † sin…fPE ‡ fE † ,

if Ae0,

 fP ˆ p ÿ arc sin …rE =rP † sin fE  ‡ …rPE =rP † sin…fPE ‡ fE † ,

if A < 0,

…10d †

V 2P ˆ V 2E ‡ V 2PE ‡ 2VE VPE cos…fPE ÿ gPE †,

‡ fP ÿ fE ,

3.2. Mars In the Sun coordinate system (SCS), the Mars initial conditions (t = 0) are written as rM …0† ˆ given,

…11a†

fM …0† ˆ free,

…11b†

p

 mS =rM …0† ,

gM …0† ˆ 0:

…13b†

…13b†

…10c†

  gP ˆ ÿarc cos …V 2P ‡ V 2E ÿ V 2PE †=2VP VE

VM …0† ˆ

…13a†

…11c† …11d †

if Be0,

  gP ˆ arc cos …V 2P ‡ V 2E ÿ V 2PE †=2VP VE ‡ fP ÿ fE ,

if B < 0,

…13c†

…13d †

…13d †

where fE=gPE=0 and A ˆ rE cos fE ‡ rPE cos…fPE ‡ fE †, B ˆ fPE ÿ gPE :

…13e† …13f †

Optimal Earth-to-Mars Transfers

4.4. Coordinate transformation

4. FINAL CONDITIONS

Let t=t denote the ®nal time. With this understanding, we state the ®nal conditions for Earth, Mars, and spacecraft. 4.1. Earth In the Sun coordinate system (SCS), the Earth ®nal conditions (t=t ) are given by rE …t† ˆ rE …0†, fE …t† ˆ t VE …t† ˆ

p

p

…14a†

 mS =r3E …0† , 

mS =rE …0† ,

gE …t† ˆ 0:

123

…14b† …14c† …14d †

At t=t, the following coordinate transformation is employed to convert SCS variables resulting from the integration of the di€erential system (9) into MCS variables present in the ®nal conditions (16): r2PM ˆ r2P ‡ r2M ÿ 2rP rM cos…fP ÿ fM †,  fPM ˆ arc cos ÿ rM =rPM  ‡ …rP =rPM † cos…fP ÿ fM † ,

if Ge0,

 fPM ˆ ÿarc cos ÿ rM =rPM  ‡ …rP =rPM † cos…fP ÿ fM † ,

if G < 0,

In the Sun coordinate system (SCS), the Mars ®nal conditions (t=t ) are given by rM …t† ˆ rM …0†, p  fM …t† ˆ t mS =r3M …0† ‡ fM …0†, VM …t† ˆ

p

 mS =rM …0† ,

gM …t† ˆ 0:

…15a† …15b† …15c† …15d †

…17b†

…17b†

V 2PM ˆ V 2P ‡ V 2M ÿ 2VP VM cos…fP ÿ fM ÿ gP †,  gPM ˆ fPM ÿ arc sin …VP =VPM †

4.2. Mars

…17a†

  sin…fP ÿ fM ÿ gP † ,

if He0,

 gPM ˆ fPM ÿ p ‡ arc sin …VP =VPM †   sin…fP ÿ fM ÿ gP † ,

if H < 0,

…17c†

…17d †

…17d †

where gPM=0 and G ˆ fP ÿ fM ,

…17e†

H ˆ V 2P ÿ V 2M ÿ V 2PM :

…17f †

5. OPTIMIZATION PROBLEM

5.1. Performance index

4.3. Spacecraft In the Mars coordinate system (MCS), the spacecraft ®nal conditions (t=t ) are given by rPM …t† ˆ rLMO ,

…16a†

fPM …t† ˆ free,

…16b†

VPM …t† ˆ VLMO ‡ DVLMO ˆ

p …mM =rLMO † ‡ DVLMO , gPM …t† ˆ 0:

…16c† …16d †

Relative to Mars rPM, fPM, VPM, gPM are the radial distance, phase angle, velocity, and path inclination of the spacecraft; VLMO is the spacecraft velocity in the low Mars orbit after the application of the tangential velocity impulse; DVLMO is the braking velocity impulse; VPM is the spacecraft velocity before the application of the tangential velocity impulse.

The most obvious performance index is the characteristic velocity DV ˆ DVLEO ‡ DVLMO ,

…18†

which is the sum of the velocity impulses in low Earth orbit and low Mars orbit. The minimization of (18) is sought with respect to the functions rP(t ), fP(t ), VP(t ), gP(t ) satisfying the SCS di€erential system (9) on the time interval 0 E tE t, the ECS initial conditions (12), and the MCS ®nal conditions (16). Note that minimizing DV is the same as minimizing the mass of propellant consumed, since this mass is a monotonically increasing function of DV. In turn, for the case of a multistage con®guration with a uniform structural factor, minimizing the mass of propellant consumed is the same as maximizing the payload mass; see Section 5.7. 5.2. Baseline optimal trajectory For given LEO and LMO radii, Problem P1 consists in the minimization of the total characteristic

124

A. Miele and T. Wang

velocity DV. Major parameters are four: (i) velocity impulse at departure DVLEO, (ii) spacecraft vs Earth phase angle at departure fPE(0), (iii) planetary Mars/Earth phase angle di€erence at departure Df(0)=fM(0)ÿfE(0)=fM(0), and (iv) transfer time t. These parameters must be determined so that DV is minimized, subject to the following requirements: tangential departure from circular velocity at LEO; tangential arrival to circular velocity at LMO. 5.3. Departure window For given LEO and LMO radii, Problem P2 involves the computation of a one-parameter family of suboptimal transfers, obtained by changing the planetary Mars/Earth phase angle di€erence at departure Df(0), hence changing the departure date y vis-a-vis the optimal value determined under (P1). The objective is to determine how a change in Df(0), hence a change Dy in departure date, a€ects the optimal values of the remaining parameters. 5.4. Comments on the algorithm The sequential gradient-restoration algorithm (SGRA) is an iterative technique which involves a sequence of two-phase cycles, each cycle including a gradient phase and a restoration phase. In the gradient phase, the augmented performance index (performance index augmented by the constraints weighted via appropriate Lagrange multipliers) is decreased, while avoiding excessive constraint violation. In the restoration phase, the constraint error is decreased, while avoiding excessive change in the performance index. In a complete gradient-restoration cycle, the performance index is decreased, while the constraints are satis®ed to a preselected accuracy. Thus, a succession of feasible suboptimal solutions is generated, each new solution being an improvement over the previous one. SGRA is available in both mathematical programming format and optimal control format. For mathematical programming problems, SGRA was developed by Miele et al. in both ordinary-gradient version and conjugate-gradient version [10]. Variations of SGRA were also developed, but the basic form proved to be the more reliable, because of its robustness and stability properties. For optimal control problems, the development of SGRA by Miele et al. [11±15] has been parallel to that for mathematical programming problems. The resulting algorithm has proved to be a powerful tool for solving optimal trajectory problems of atmospheric and space ¯ight [16,17]. Applications and extensions of SGRA have been reported in the US, Japan, Germany, and other countries around the world; in particular, a version of this algorithm is currently used at NASA-JSC under the code name SEGRAM, developed by McDonnell Douglas Technical Service Company [18].

For the problem at hand (Earth-to-Mars optimal transfer), note that there are no control variables, only state variables and parameters. For this reason, it is preferable to use SGRA in mathematical programming format rather than optimal control format; either the ordinary gradient version or the conjugate gradient version can be used. Regardless of the version employed, the computation of gradients is much more complicated than in a standard mathematical programming problem. 5.5. Variable-stepsize integration scheme The optimization process requires multiple integrations of the system equations of the restricted four-body model. The integration process is computationally expensive and it is dicult to achieve the desired accuracy, owing to the fact that the total gravitational force changes rapidly in near-Earth or near-Mars space, but slowly in deep interplanetary space. Indeed, orbital periods are of order one hour to one day if Earth gravity or Mars gravity is dominant, but of order one year if Sun gravity is dominant. The above diculties can be overcome by properly designing a variable-stepsize integration scheme. Numerical experiments show that good results can be obtained by linking the integration stepsize to the total gravitational force, with the stepsize decreasing whenever the total gravitational force increases, and viceversa. 5.6. Computational environment The computations reported here were done on a Unix Sun Workstation using the C++ programming language. In particular, the integrations were executed via a ®fth-order Runge±Kutta±Fehlberg scheme. 5.7. Mass distribution As explained in Section 5.1, the characteristic velocity DV is related to the mass of propellant consumed, which in turn is related to the payload mass. For a two-stage con®guration, let m0 denote the initial mass; let p1, s1, m1 denote the propellant mass, structural mass, and payload mass of Stage 1; let p2, s2, m2 denote the propellant mass, structural mass, and payload mass of Stage 2. By de®nition, m0 ˆ m1 ‡ p1 ‡ s1 ,

…19a†

m1 ˆ m2 ‡ p2 ‡ s2 ,

…19b†

e ˆ s1 =… p1 ‡ s1 †,

…20a†

e ˆ s2 =… p2 ‡ s2 †,

…20b†

with

where e is structural factor, assumed to be uniform for Stages 1 and 2. The propellant masses are given

Optimal Earth-to-Mars Transfers

VE ˆ 29:78,

by p1 =m0 ˆ 1 ÿ Z1 ,

…21a†

p2 =m1 ˆ 1 ÿ Z2 ,

…21b†

Z1 ˆ exp…ÿDVLEO =gIsp †,

…22a†

Z2 ˆ exp…ÿDVLMO =gIsp †,

…22b†

with

g being the acceleration of gravity at sea level on Earth. Combining eqns (19)±(21), one obtains the sequential solutions p1 =m0 ˆ 1 ÿ Z1 ,

…23a†

s1 =m0 ˆ e…1 ÿ Z1 †=…1 ÿ e†,

…23b†

m1 =m0 ˆ …Z1 ÿ e†=…1 ÿ e†,

…23c†

p2 =m1 ˆ 1 ÿ Z2 ,

…24a†

s2 =m1 ˆ e…1 ÿ Z2 †=…1 ÿ e†,

…24b†

m2 =m1 ˆ …Z2 ÿ e†=…1 ÿ e†,

…24c†

with Z1 and Z2 given by eqns (22). These relations enable one to compute the mass distribution for Stages 1 and 2. In particular, the ratio of Stage 2 payload mass to initial mass is …25†

6. PLANETARY AND SPACECRAFT DATA

6.1. Planetary data The masses of Sun, Earth, and Mars are mS ˆ 1:989E30,

mE ˆ 5:976E24,

mM ˆ 6:418E23 kg,

…26a† …26b†

with corresponding gravitational constants mS ˆ 1:327E11,

mE ˆ 3:986E05,

mM ˆ 4:283E04 km3 =s2

rM ˆ 2:279E08 km:

…27b†

oM ˆ 0:524 deg =day:

…27c†

In particular, the angular velocity di€erence between Earth and Mars is Do ˆ oE ÿ oM ˆ 0:462 deg =day:

…28†

Therefore, Earth and Mars are in the same relative position every 779 days=2.13 years. 6.2. Spacecraft data The spacecraft is to be transferred from a low Earth orbit to a low Mars orbit with radii rLEO ˆ 6841,

rLMO ˆ 3597 km,

…29a†

corresponding to the altitudes hLEO ˆ 463,

hLMO ˆ 200 km,

RE ˆ 6378,

RM ˆ 3397 km:

…29b† …29c†

The circular velocities (subscript c) and escape velocities (subscript *) at LEO and LMO are given by …Vc †LEO ˆ 7:633, …Vc †LMO ˆ 3:451,

…V †LMO ˆ 10:795 km=s,

…30a†

…V †LMO ˆ 4:880 km=s:

…30b†

In the relative motion with respect to Earth, the ®rst of eqns (30a) is the spacecraft velocity prior to application of the velocity impulse; the second of eqns (30a) is a lower bound for the spacecraft velocity after the application of the velocity impulse. Analogously, in the relative motion with respect to Mars, the ®rst of eqns (30b) is the spacecraft velocity after the application of the velocity impulse; the second of eqns (30b) is a lower bound for the spacecraft velocity prior to the application of the velocity impulse. The spacecraft con®guration is assumed to have two stages with uniform structural factor e ˆ 0:1

…31a†

and engine speci®c impulse Isp ˆ 450 s:

…31b†

…26c† …26d †

.The Sun mass is nearly 330,000 times the Earth mass, and the Earth mass is nearly 10 times the Mars mass. Identical ratios hold for the gravitational constants. Earth and Mars travel around Sun along orbits with average radii rE ˆ 1:496E08,

VM ˆ 24:13 km=s,

since the Earth and Mars surface radii are given by

and

m2 =m0 ˆ …Z1 ÿ e†…Z2 ÿ e†=…1 ÿ e†2 :

oE ˆ 0:986,

125

…27a†

The associated translational velocities and angular velocities are given by

7. BASELINE OPTIMAL TRAJECTORY

The sequential gradient restoration algorithm for mathematical programming problems [10] was employed to solve Problem P1 so as to obtain the baseline optimal trajectory. Summary results are shown in Column 2 of Tables 1±3 and in Fig. 1. This ®gure includes three parts: the bottom part refers to deep interplanetary space and shows the spacecraft trajectory in Sun coordinates; the top left part refers to near-Earth space and shows the depar-

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Table 1. Departure window: Inertial and relative state variables at t = 0 Df(0)

29.00

43.86

59.00

74.00

94.00

deg

rP(0) fP(0) VP(0) gP(0) rPE(0) fPE(0) VPE(0) gPE(0)

1.496E08 0.00 30.09 22.19 6.841E03 ÿ99.58 11.53 0.00 C1

1.496E08 0.00 36.42 15.71 6.841E03 ÿ61.85 11.19 0.00 C2

1.496E08 0.00 41.03 3.29 6.841E03 ÿ11.88 11.43 0.00 C3

1.496E08 0.00 41.26 ÿ5.69 6.841E03 19.95 11.99 0.00 C4

1.496E08 0.00 39.92 ÿ13.65 6.841E03 46.27 13.04 0.00 C5

km deg km/s deg km deg km/s deg

Table 2. Departure window: Inertial and relative state variables at t=t Df(0)

29.00

43.86

59.00

74.00

94.00

deg

rP(t ) fP(t ) VP(t ) gP(t ) rPM(t ) fPM(t ) VPM(t ) gPM(t )

2.279E08 157.20 18.91 5.99 3.596E03 ÿ159.76 5.68 0.00 C1

2.279E08 179.02 20.13 10.02 3.597E03 ÿ140.97 5.55 0.00 C2

2.279E08 202.80 20.30 10.63 3.597E03 ÿ138.20 5.61 0.00 C3

2.279E08 222.66 20.08 10.64 3.597E03 ÿ139.97 5.75 0.00 C4

2.279E08 244.10 19.49 10.28 3.598E03 ÿ145.03 6.05 0.00 C5

km deg km/s deg km deg km/s deg

Table 3. Departure window: Planetary Mars/Earth phase angle di€erences, characteristic velocities, mass ratios, departure date, and transfer time Df(0) Df(t ) DVLEO DVLMO DV m0/mref m1/mref m2/mref Dy t

29.00 ÿ83.85 3.893 2.227 6.120 1.000 0.349 0.195 32.20 244.59 C1

43.86 ÿ75.13 3.552 2.100 5.652 1.000 0.386 0.224 0.00 257.88 C2

59.00 ÿ67.59 3.792 2.158 5.950 1.000 0.360 0.205 ÿ32.82 274.36 C3

ture portion of the trajectory in Earth coordinates; the top right part refers to near-Mars space and shows the arrival portion of the trajectory in Mars coordinates. 7.1. Results For the baseline optimal trajectory, the following comments can be made: 1. Relative to Earth, the spacecraft departs with an accelerating impulse DVLEO=3.552 km/s and a phase angle fPE(0)=ÿ61.85 deg. 2. Relative to Mars, the spacecraft arrives with a braking velocity impulse DVLMO=2.100 km/s and a phase angle fPM(t )=ÿ140.97 deg. 3. The total characteristic velocity is DV = 5.652 km/s. This corresponds to a ratio of payload mass to initial mass m2/m0=0.224 for a

74.00 ÿ56.86 4.355 2.301 6.656 1.000 0.303 0.166 ÿ65.33 283.62 C4

94.00 ÿ38.13 5.404 2.604 8.008 1.000 0.216 0.109 ÿ108.68 286.38 C5

deg deg km/s km/s km/s

day day

two-stage con®guration with uniform structural factor e=0.1 and engine speci®c impulse Isp=450 s; see Section 5.7. 4. The transfer time from LEO to LMO is t=257.88 days and the angular travel of the spacecraft with respect to Sun is fP(t )ÿfP(0)=179.02 deg. 5. The planetary Mars/Earth phase angle di€erence is Df(0)=43.86 deg at departure and Df(t )=ÿ75.13 deg at arrival. This means that Mars is ahead of Earth by 43.86 deg at departure and behind Earth by 75.13 deg at arrival.

7.2. Properties For the baseline optimal trajectory, the following properties can be stated:

Optimal Earth-to-Mars Transfers

127

Fig. 1. Baseline optimal trajectory, Df(0)=43.86 deg, Dy=0.00 days, t=257.88 days, DV = 5.652 km/s.

1. The spacecraft leaves the low Earth orbit tangentially with departure point chosen so that the phase angle is fPE(0)=ÿ61.85 deg. This means that, at departure, the angle between the spacecraft relative velocity and Earth inertial velocity is precisely 61.85 deg. 2. Due to Earth gravitational e€ects, the spacecraft inertial velocity changes direction rapidly in nearEarth space and becomes asymptotically parallel to the Earth inertial velocity at the end of nearEarth space. 3. The spacecraft reaches the low Mars orbit tangentially, with arrival point chosen so that the phase angle is fPM(t )=ÿ140.97 deg. This means that, at arrival, the angle between the spacecraft

relative velocity and Mars inertial velocity is precisely 140.97 deg. 4. Due to Mars gravitational e€ects, the spacecraft inertial velocity changes direction rapidly in nearMars space and is asymptotically parallel to the Mars inertial velocity at the beginning of nearMars space. 5. The angular travel of the spacecraft with respect to Sun is 179.02 deg, which is close to 180 deg, the value characterizing an idealized Hohmann transfer between the terminal radii (eqns (27a)) in the presence of only the Sun gravitational ®eld. Also, for the baseline optimal trajectory, the perihelion occurs nearly at departure from Earth and the aphelion occurs nearly at arrival on Mars,

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Fig. 2. Departure window suboptimal trajectory, Df(0)=94.00 deg, Dy=ÿ108.68 days, t=286.38 days, DV = 8.088 km/s.

consistently with the properties of an idealized Hohmann transfer trajectory. 6. In spite of the similarities noted in (e), there is an important qualitative di€erence between the baseline optimal trajectory of this paper and the idealized Hohmann transfer trajectory: in the baseline optimal trajectory of this paper, the velocity impulse relative to the planet is accelerating at departure but braking at arrival; in the idealized Hohmann transfer trajectory, the velocity impulses relative to the Sun are both accelerating.

employed to solve Problem P2 so as to generate the departure window. This is a one-parameter family of suboptimal trajectories, the parameter being the planetary Mars/Earth phase angle di€erence at departure Df(0), which is given for each trajectory. A change of Df(0) vis-a-vis the value associated with Problem P1 implies a change in departure date y. These changes are related by the simple expression   …32a† D Df…0† ‡ …oE ÿ oM †Dy ˆ 0, with D[Df(0)] is measured in degrees, Dy is measured in days, and

8. SUBOPTIMAL TRAJECTORIES AND DEPARTURE WINDOW

oE ÿ oM ˆ 0:462 deg =day:

The sequential gradient restoration algorithm for mathematical programming problems [10] was

Hence, an anticipated departure (Dy < 0) is characterized by increasing value of Df(0); conversely, a

…32b†

Optimal Earth-to-Mars Transfers

129

Fig. 3. Departure window suboptimal trajectory, Df(0)=29.00 deg, Dy=+32.20 days, t=244.54 days, DV = 6.120 km/s.

delayed departure (Dy > 0) is characterized by decreasing value of Df(0). For the departure window, summary results are shown in Tables 1±3. In these tables, Column 1 refers to Df(0)=29.00 deg, hence Dy=+32.20 days (delayed departure). Column 2 refers to Df(0)=43.86 deg, hence Dy=0.00 days; this is the baseline optimal trajectory of Section 7. Column 3 refers to Df(0)=59.00 deg, hence Dy=ÿ32.82 days (anticipated departure). Column 4 refers to Df(0)=74.00 deg, hence Dy=ÿ65.33 days (more anticipated departure). Column 5 refers to Df(0)=94.00 deg, hence Dy=ÿ108.68 days (extremely anticipated departure).

Particular trajectories are shown in Figs. 1±3. Figure 1 refers to Column 2 and shows the baseline optimal trajectory of Section 7. Figure 2 refers to Column 5 and shows the suboptimal trajectory associated with extremely anticipated departure (by more than 3.5 months). Figure 3 refers to Column 1 and shows the suboptimal trajectory associated with delayed departure (by about one month). In these ®gures, the bottom part shows the spacecraft trajectory in Sun coordinates and deep interplanetary space; the top left part shows the spacecraft trajectory in Earth coordinates and near-Earth space; the top right part shows the spacecraft trajectory in Mars coordinates and near-Mars space.

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Fig. 4. Departure conditions, Earth coordinates, gPE(0)=0.

Fig. 5. Departure conditions, Sun coordinates, fP(0)30.

Fig. 6. Arrival conditions, Sun coordinates.

Fig. 7. Arrival conditions, Mars coordinates, gPM(t )=0.

Optimal Earth-to-Mars Transfers

131

Fig. 8. Initial mass and characteristic velocity at LEO.

Fig. 9. Stage 1 payload mass and characteristic velocity at LMO.

Fig. 10. Stage 2 payload mass and total characteristic velocity.

For additional information on the departure window, see Figures 4±10.

vel, resulting in a 13 day decrease [16 day increase] in transfer time.

8.1. Results

8.2. Properties

For the one-parameter family of suboptimal trajectories belonging to the departure window, the following comments can be made:

For the one-parameter family of suboptimal trajectories belonging to the departure window, the following properties can be stated:

1. A change of ÿ15 deg [+15 deg] in planetary Mars/Earth phase angle di€erent Df(0) results in a departure date change Dy=+32 days [ÿ32 days]. 2. A change of ÿ15 deg [+15 deg] in Df(0) causes an 8% increase [5% increase] in total characteristic velocity, resulting in a 13% decrease [8% decrease] in payload mass. 3. A change of ÿ15 deg [+15 deg] in Df(0) causes a 12% decrease [13% increase] in phase angle tra-

1. In the relative motion, the spacecraft leaves the low Earth orbit tangentially, and the angle between the spacecraft relative velocity and the Earth inertial velocity is precisely fPE(0). The departure point position, hence the value of fPE(0), is not constant, but varies in a wide range. 2. Due to Earth gravitational e€ects, the spacecraft inertial velocity changes direction rapidly in nearEarth space, and is no longer asymptotically par-

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allel to the Earth inertial velocity at the end of near-Earth space. 3. In the relative motion, the spacecraft approaches the low Mars orbit tangentially, and the angle between the spacecraft relative velocity and the Mars inertial velocity is precisely fPM(t ). The arrival point position, hence the value of fPM(t ), is not constant but varies in a relatively narrow range. 4. Due to Mars gravitational e€ects, the spacecraft inertial velocity changes direction rapidly in nearMars space and is asymptotically parallel to the Mars inertial velocity at the beginning of nearMars space. 5. For the suboptimal transfers, the phase angle travel and transfer time decrease with late departure and increase with early departure.

9. CONCLUSIONS

From the analysis made within the frame of the restricted four-body model, the following conclusions emerge for a spacecraft being transferred from a low Earth orbit to a low Mars orbit: 1. Baseline Optimal Trajectory. In the relative motion, the spacecraft leaves tangentially the low Earth orbit and reaches tangentially the low Mars orbit. In the inertial motion, the spacecraft velocity is asymptotically parallel to the Earth velocity at the end of near-Earth space and also asymptotically parallel to the Mars velocity at the beginning of near-Mars space. 2. Baseline Optimal Trajectory. The spacecraft phase angle travel is 179 deg, which is close to the value associated with an idealized Hohmann transfer in the presence of only the Sun gravitational attraction (180 deg). The transfer time from LEO to LMO is 258 days. The planetary Mars/Earth phase di€erence is +44 deg at departure (Mars ahead of Earth) and ÿ75 deg at arrival (Mars behind Earth). The total characteristic velocity is DV = 5.652 km/s, corresponding to a ratio of payload mass to initial mass of 0.224 for a two-stage spacecraft with uniform structural factor e=0.1 and engine speci®c impulse Isp=450 s. 3. Suboptimal Trajectories and Departure Window. In the relative motion, the spacecraft leaves tangentially the low Earth orbit and reaches tangentially the low Mars orbit. In the inertial motion, the asymptotic parallelism between the spacecraft velocity and the planet velocity does not hold at the end of near-Earth space, but still holds at the beginning of near-Mars space. 4. Suboptimal Trajectory and Departure Window. The phase angle travel, transfer time, and planetary Mars/Earth phase angle di€erence at departure decrease with late departure and increase with early departure. Vis-a-vis the values associ-

ated with the baseline optimal trajectory, the characteristic velocity increases with both early and late departures; consistently, the ratio of payload mass to initial mass decreases with both early and late departures. 5. Suboptimal Trajectories and Departure Window. A late departure by +32 days [early departure by ÿ32 days], namely a change of ÿ15 deg [+15 deg] in planetary Mars/Earth phase angle di€erence at departure, causes an 8% increase [5% increase] in characteristic velocity and a 13% decrease [8% decrease] in payload mass.

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