# Optimization of Low-Thrust Limited-Power Trajectories in a Noncentral Gravity Field—Transfers between Orbits with  Small Eccentricities

Authors: Sandro da Silva Fernandes
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Numerical and first-order analytical results are presented for optimal low-thrust limited-power
trajectories in a gravity field that includes the second zonal harmonicJ2 in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity
vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical
transformations are performed and a suitable set of orbital elements is introduced. Hori method—a
perturbation technique based on Lie series—is applied in solving the canonical system of differential
equations that governs the optimal trajectories. First-order analytical solutions are presented for
transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary
orbits by solving the two-point boundary value problem described by averaged maximum Hamiltonian,
expressed in nonsingular elements, through a shooting method. A comparison between analytical and
numerical results is presented for some maneuvers.


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## Body

## 1. Introduction

This paper presents numerical and analytical results for optimal low-thrust limited-power transfers between orbits with small eccentricities in a noncentral gravity field that includes the second zonal harmonicJ2 in the gravitational potential. This study has been motivated by the renewed interest in the use of low-thrust propulsion systems in space missions verified in the last two decades due to the development and the successes of two space mission powered by ionic propulsion: Deep Space One and SMART 1. Several researchers have obtained numerical and sometimes analytical solutions for a number of specific initial orbits and specific thrust profiles in central or noncentral gravity field [1–26].It is assumed that the thrust direction is free and the thrust magnitude is unbounded [5, 27], that is, there exist no constraints on control variables. It is supposed that the orbital changes caused by the thrust and the perturbations due to the oblateness of the central body have the same order of magnitude. The optimization problem is formulated as a Mayer problem of optimal control with position and velocity vectors—Cartesian elements—as state variables. After applying the Pontryagin Maximum Principle [28], successive Mathieu transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series [29]—is applied in solving the canonical system of differential equations that governs the optimal trajectories. For transfers between close quasicircular orbits, new set of nonsingular orbital elements is introduced and the new Hamiltonian and the generating function, determined through the algorithm of Hori method, are transformed to these new elements through Mathieu transformations. First-order analytical solutions are obtained for transfers between close quasicircular orbits, considering maneuvers between near-equatorial orbits or nonequatorial orbits. These solutions are given in terms of systems of algebraic equations involving the imposed variations of the orbital elements, the duration of the maneuvers, the effects of the oblateness of the central body, and the initial values of the adjoint variables. The study of these transfers is particularly interesting because the orbits found in practice often have a small eccentricity, and the problem of slight modifications (corrections) of these orbits is frequently met [5]. For transfers between arbitrary orbits, a slightly different set of nonsingular elements is introduced, and the two-point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, is solved through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.


## 2. Optimal Space Trajectories

A low-thrust limited-power propulsion system, or LP system, is characterized by low-thrust acceleration level and high specific impulse [5]. The ratio between the maximum thrust acceleration and the gravity acceleration on the ground, γmax/g0, is between 10-4 and 10-2. For such system, the fuel consumption is described by the variable J defined as
(2.1)J=12∫t0tγ2dt,
where γ is the magnitude of the thrust acceleration vector γ, used as control variable. The consumption variable J is a monotonic decreasing function of the mass m of space vehicle:
(2.2)J=Pmax(1m-1m0),
where Pmax is the maximum power, and m0 is the initial mass. The minimization of the final value of the fuel consumption Jf   is equivalent to the maximization of mf.The general optimization problem concerned with low-thrust limited-power transfers (no rendezvous) will be formulated as a Mayer problem of optimal control by using Cartesian elements as state variables. Consider the motion of a space vehicleM  powered by a limited-power engine in a general gravity field. At time t, the state of the vehicle is defined by the position vector r(t), the velocity vector v(t), and the consumption variable J. The geometry of the transfer problem is illustrated in Figure 1. The control γ is unconstrained, that is, the thrust direction is free, and the thrust magnitude is unbounded.Figure 1
Geometry of the transfer problem.The optimization problem is formulated as follows: it is proposed to transfer the space vehicleM  from the initial state (r0,v0,0) at the initial time t0=0 to the final state (rf,vf,Jf) at the specified final time tf, such that the final consumption variable Jf is a minimum. The state equations are
(2.3)drdt=v,dvdt=g(r)+γ,dJdt=12γ2,
where g(r) is the gravity acceleration. For transfers between orbits with small eccentricities, the initial and final conditions will be specified in terms of singular orbital elements introduced in next sections. It is also assumed that tf-t0 and the position of the vehicle in the initial orbit are specified.According to the Pontryagin Maximum Principle [28], the optimal thrust acceleration γ* must be selected from the admissible controls such that the Hamiltonian function H reaches its maximum. The Hamiltonian function is formed using (2.3),
(2.4)H=pr•v+pv•(g(r)+γ)+12pJγ2,
where pr,pv, and pJ are the adjoint variables, and dot denotes the dot product. Since the optimization problem is unconstrained, γ* is given by
(2.5)γ*=-pvpJ.
The optimal thrust acceleration γ* is modulated [5], and the optimal trajectories are governed by the maximum Hamiltonian function H*, obtained from (2.4) and (2.5):
(2.6)H*=pr•v+pv•g(r)-pv22pJ.The consumption variableJ is ignorable, and pJ is a first integral. From the transversality conditions, pJ(tf)=-1; thus,
(2.7)pJ(t)=-1.
Equation (2.6) reduces to
(2.8)H=pr•v+pv•g(r)+pv22.
The consumption variable J is determined by simple quadrature.For noncentral field, the gravity acceleration is given by [30]
(2.9)g(r)=-μr3r+(∂V2∂r)T,
with the disturbing function V2 defined by
(2.10)V2=-μJ2(ae2r3)P2(sinφ),
where μ is the gravitational parameter, ae is the mean equatorial radius of  the central body, J2 is the coefficient for the second zonal harmonic of the potential, φis the latitude, and P2 is the Legendre polynomial of order 2. (∂V2/∂r) is a row vector of partial derivatives, and superscript T denotes its transpose.Using (2.8) and (2.9), the maximum Hamiltonian function can be put in the following form:
(2.11)H*=H0+HJ2+Hγ*,
where
(2.12)H0=pr•v-pv•μr3r,HJ2=pv•(∂V2∂r)T,Hγ*=pv22.H0 is the undisturbed Hamiltonian function, HJ2 and Hγ* are disturbing functions concerning the oblateness of the central body and the optimal thrust acceleration, respectively. HJ2 and Hγ* are assumed to be of the same order of magnitude.


## 3. Transformation from Cartesian Elements to a Set of  Orbital Elements

Consider the canonical system of differential equations governed by the undisturbed HamiltonianH0:
(3.1)drdt=v,dvdt=-μr3r,dprdt=μr3(pv-3(pv•er)er),dpvdt=-pr,
where er is the unit vector pointing radially outward of the moving frame of reference (Figure 2). The general solution of the state equations is well known in Astrodynamics [31], and the general solution of the adjoint equations is obtained through properties of generalized canonical systems [32], as described in the appendix. Thus,
(3.2)r=a(1-e2)1+ecosfer,(3.3)v=μa(1-e2)[esinfer+(1+ecosf)es],(3.4)pr=ar2{2apa+((1-e2)cosE)pe+(ra)sinfe(pω-(1-e3cosE)1-e2pM)}er+{sinfape-(e+cosf)ae(1-e2)pω+1-e2cosfaepM}es+1a1-e2{(ar)sinE[pIcosω+(pΩsinI-pωcotI)sinω]+1-e2(ar)cosE[pIsinω-(pΩsinI-pωcotI)cosω]}ew,(3.5)pv=1na1-e2{{2aesinfpa+((1-e2)sinf)pe-(1-e2)cosfepω+(1-e2)3/2e(cosf-2e1+ecosf)pM}er+{(1-e2)sinfe2a(1-e2)(ar)pa+(1-e2)(cosf+cosE)pe+(1-e2)sinfe(1+11+ecosf)(pω-1-e2pM)}es+{(ra)cos(ω+f)pI+(ra)sin(ω+f)(pΩsinI-pωcotI)}ew},
where es and ew are unit vectors along circumferential and normal directions of the moving frame of reference, respectively; a is the semimajor axis, e is the eccentricity, I  is the inclination of orbital plane, Ω is the longitude of the ascending node, ωis the argument of pericenter, f  is the true anomaly, E is the eccentric anomaly, M is the mean anomaly, n=μ/a3 is the mean motion, and (r/a),(r/a)sinf, and so forth are functions of the elliptic motion which can be expressed explicitly in terms of the eccentricity and the mean anomaly through Lagrange series [31]. The true, eccentric and mean anomalies are related through the equations:
(3.6)tanf2=1+e1-etanE2,M=E-esinE.
The unit vectors er,es, and ew of the moving frame of reference are written in the fixed frame of reference as
(3.7)er=(cosΩcos(ω+f)-sinΩsin(ω+f)cosI)i+(sinΩcos(ω+f)+cosΩsin(ω+f)cosI)j+sin(ω+f)sinIk,es=-(cosΩsin(ω+f)+sinΩcos(ω+f)cosI)i+(-sinΩsin(ω+f)+cosΩcos(ω+f)cosI)j+cos(ω+f)sinIk,ew=sinΩsinIi-cosΩsinIj+cosIk.Figure 2
Frames of reference.Equations (3.2)–(3.5) define a Mathieu transformation between the Cartesian elements (r,v,pr,pv) and the orbital ones (a,e,I,Ω,ω,M,pa,pe,pI,pΩ,pω,pM). Since the Hamiltonian function is invariant with respect to this canonical transformation, it follows, from (2.11) through (3.5), that
(3.8)H*=H0+HJ2+Hγ*,
where
(3.9)H0=npM,HJ2=2na∂V2′∂Mpa+1-e2na2e[-∂V2′∂ω+1-e2∂V2′∂M]pe+1na21-e2sinI[-∂V2′∂Ω+cosI∂V2′∂ω]pI+1na21-e2sinI∂V2′∂IpΩ+1-e2na2e[∂V2′∂e-ecotI(1-e2)∂V2′∂I]pω+1na[-2∂V2′∂a-(1-e2)ae∂V2′∂e]pM},Hγ*=12n2a2(1-e2){12(1-cos2f)[2aepa+(1-e2)pe]2+2(1-e2)sin2f[-apapω-(1-e2)2epepω]+4(1-e2)3/2sinf(-2e1+ecosf+cosf)[apapM+(1-e2)2epepM]+(1-e2)22e2(1+cos2f)pω2-2(1-e2)5/2e2(-2e1+ecosf+cosf)cosfpωpM+(1-e2)3e2(-2e1+ecosf+cosf)2pM2+4a2(1-e2)2(ar)2pa2+4a(1-e2)2(ar)(cosE+cosf)pape+(1-e2)2(cosE+cosf)2pe2+4a(1-e2)2e(ar)sinf(1+11+ecosf)×[papω-(1-e2)1/2papM]+2(1-e2)2e(cosE+cosf)(1+11+ecosf)×sinf[pepω-1-e2pepM]+[(1-e2)e(1+11+ecosf)sinf[pω-1-e2pM]]2+12(ra)2[pI2+(pΩsinI-pωcotI)2]+12(ra)2cos2(ω+f)[pI2-(pΩsinI-pωcotI)2]+(ra)2sin2(ω+f)pI(pΩsinI-pωcotI)},
with the disturbing function V2′ given by
(3.10)V2′=μa(aea)2J2{(ar)3[12-34sin2I]+34(ar)3sin2Icos2(ω+f)}.The new Hamiltonian functionH*, defined by (3.8)–(3.10), describes the optimal low-thrust limited-power trajectories in a noncentral gravity field which includes the perturbative effects of the second zonal harmonic of the gravitational potential.


## 4. Averaged Maximum Hamiltonian for Optimal Transfers

In order to eliminate the short periodic terms from the maximum Hamiltonian functionH*, Hori method [29] is applied. It is assumed that H0 is of zero-order and Hγ* is of the first-order in a small parameter associated to the magnitude of the thrust acceleration, and HJ2 has the same order of Hγ*.Consider an infinitesimal canonical transformation:(4.1)(a,e,I,Ω,ω,M,pa,pe,pI,pΩ,pω,pM)→(a′,e′,I′,Ω′,ω′,M′,pa′,pe′,pI′,pΩ′,pω′,pM′).
The new variables are designated by the prime. According to the algorithm of Hori method [29], at order 0,
(4.2)F0=n′pM′.F0 denotes the new undisturbed Hamiltonian. Now, consider the canonical system described by F0:
(4.3)da′dt=0,de′dt=0,dI′dt=0,dΩ′dt=0,dω′dt=0,dM′dt=n′,dpa′dt=32n′a′pM′,dpe′dt=0,dpI′dt=0,dpΩ′dt=0,dpω′dt=0,dpM′dt=0,
general solution of which is given by
(4.4)a′=a0′,e′=e0′,I′=I0′,Ω′=Ω0′,ω′=ω0′,M′=M0′+n′(t-t0),pa′=pa0′+32n′(t-t0)a′pM′,pe′=pe0′,pI′=pI0′,pΩ′=pΩ0′,pω′=pω0′,pM′=pM0′.
The subscript 0 denotes the constants of integration.Introducing the general solution defined above into the equation of order 1 of the algorithm of Hori method, it reduces to(4.5)∂S1∂t=HJ2+Hγ*-F1.
According to [29], the mean value of HJ2+Hγ* must be calculated, and, S1 is obtained through integration of the remaining part. F1 and S1 are given by the following equations:
(4.6)F1=-32n′J2(aea′)2(1-e′2)-2{cosI′pΩ′+12(1-5cos2I′)pω′}+a′2μ{(pΩ′sinI′-cotI′pω′)24a′2pa′2+52(1-e′2)pe′2+(5-4e′2)2e′2pω′2+pI′22(1-e′2)[(1+32e′2)+52e′2cos2ω′]+5e′2sin2ω′2(1-e′2)pI′(pΩ′sinI′-cotI′pω′)+12(1-e′2)(pΩ′sinI′-cotI′pω′)2[(1+32e′2)-52e′2cos2ω′]},S1=J2(aea′)2{{(1-32sin2I′)[(a′r′)3-(1-e′2)-3/2]+32sin2I′(a′r′)3cos2(ω′+f′)}a′pa′+{32sin2I′e′(1-e′2)[-12cos2(ω′+f′)-e′2(cos(2ω′+f′)+13cos(2ω′+3f′))]+(1-e′2)e′[(12-34sin2I′)[(a′r′)3-(1-e′2)-3/2]+34sin2I′(a′r′)3cos2(ω′+f′)}pe′+{34sin2I′(1-e′2)2[12cos2(ω′+f′)+e′2(cos(2ω′+f′)+13cos(2ω′+3f′))]}pI′+{32cosI′(1-e′2)2[-(f′-M′+e′sinf′)+12sin2(ω′+f′)+e′2(sin(2ω′+f′)+13sin(2ω′+3f′))]cosI′(1-e′2)2}pΩ′+{34(5cos2I′-1)(1-e′2)2(f′-M′+e′sinf′)+14(3cos2I′-1)e′(1-e′2)[(a′r′)2(1-e′2)+(a′r′)+1]sinf′+38sin2I′e′(1-e′2)[[-(a′r′)2(1-e′2)-(a′r′)+1]sin(2ω′+f′)+[(a′r′)2(1-e′2)+(a′r′)+13]sin(2ω′+3f′)]+38(3-5cos2I′)(1-e′2)2[(sin(2ω′+f′)+13sin(2ω′+3f′))sin2(ω′+f′)+e′(sin(2ω′+f′)+13sin(2ω′+3f′))]}pω′}+12a′5μ3{8e′sinE′a′2pa′2+8(1-e′2)sinE′a′pa′pe′-81-e′2e′cosE′pa′pω′+(1-e′2)[-54e′sinE′+34sin2E′-112e′sin3E′]pe′2+1-e′2e′[52e′cosE′-12(3-e′2)cos2E′+16e′cos3E′]pe′pω′+(1-e′2)-1[pI′2+(pΩ′sinI′-pω′cotI′)2]×[(-e′+38e′3)sinE′+38e′2sin2E′-124e′3sin3E′]+(1-e′2)-1×[pI′2cos2ω′+2pI′(pΩ′sinI′-pω′cotI′)sin2ω′-(pΩ′sinI′-pω′cotI′)2cos2ω′]×[(-54e′+58e′3)sinE′+(14+18e′2)sin2E′+(-112e′+124e′3)sin3E′]+(1-e′2)-1/2[-pI′2sin2ω′+2pI′(pΩ′sinI′-pω′cotI′)cos2ω′+(pΩ′sinI′-pω′cotI′)2sin2ω′]×[54e′cosE′-(14+14e′2)cos2E′+112e′cos3E′]+pω′2e′2[(54e′-e′3)sinE′+(-34+12e′2)sin2E′+112e′sin3E′]}.
Terms factored by pM′ have been omitted in equations above, since only transfers (no rendez-vous) are considered [2].It should be noted that the averaged maximum Hamiltonian and the generating function become singular for circular and/or equatorial orbits. In order to avoid these singularities, suitable sets of nonsingular elements will be introduced in the next sections.


## 5. Optimal Transfers between Close Quasicircular Orbits

In this section, approximate first-order analytical solutions will be obtained for transfers between close quasicircular orbits.


### 5.1. Transfers between Close Quasicircular Nonequatorial Orbits

Consider the Mathieu transformation [33] defined by
(5.1)a′′=a′,h′′=e′cosω′,k′′=e′sinω′,I′′=I′,Ω′′=Ω′,ℓ′′=M′+ω′,(5.2)pa′′=pa′,ph′′=pe′cosω′-(pω′-pM′e′)sinω′,pk′′=pe′sinω′+(pω′-pM′e′)cosω′,pI′′=pI′,pΩ′′=pΩ′,pℓ′′=pM′.
The new set of canonical variables, designated by the double prime, is nonsingular for circular nonequatorial orbits.Introducing (5.2) into (4.6) and using the expansions of the elliptic motion in terms of eccentricity and mean anomaly [31], one finds that the new averaged maximum Hamiltonian and the generating function are written up to the zeroth order in eccentricity as
(5.3)F1=-32nJ2(aea)2cosIpΩ+a2μ{4a2pa2+52(ph2+pk2)+12(pI2+pΩ2sin2I)},S1=32J2(aea)2{apasin2Icos2ℓ+ph[cosℓ+14sin2I(-5cosℓ+73cos3ℓ)]+pk[sinℓ+14sin2I(-7sinℓ+73sin3ℓ)]+14pIsin2Icos2ℓ+12pΩcosIsin2ℓ}+12a5μ3{8apa[phsinℓ-pkcosℓ]-12[3phpk+pΩsinIpI]cos2ℓ+14[3(ph2-pk2)+pI2-pΩ2sin2I]sin2ℓ}.
Double prime is omitted to simplify the notation.For transfers between close orbits,F1 and S1 can be linearized around a suitable reference orbit, and an approximate first-order analytical solution can be determined. This solution is given by
(5.4)Δx=Ap0+B,
where Δx=[ΔαΔhΔkΔIΔΩ′]T,p0 is the 5×1 vector of initial value of the adjoint variables, A is a 5×5 symmetric matrix concerning the optimal thrust acceleration and B is a 5×1 vector containing the perturbative effects of the oblateness of the central body. The variables α=a/a̅ and pα=a̅pa are introduced to make the adjoint and the state vectors dimensionless, and the variables Ω′=ΩsinI̅ and pΩ′=pΩ/sinI̅ are introduced to simplify the matrix A. The adjoint vector is constant. The matrix A and the vector B are given by
(5.5)A=[aααaαhaαk00ahαahhahk00akαakhakk00000aIIaIΩ′000aΩ′IaΩ′Ω′],(5.6)B=[bαbhbkbIbΩ′],
where
(5.7)aαα=4a̅5μ3ℓ̅|ℓ̅0ℓ̅f,aαh=ahα=4a̅5μ3sinℓ̅|ℓ̅0ℓ̅f,aαk=akα=-4a̅5μ3cosℓ̅|ℓ̅0ℓ̅f,ahh=a̅5μ3[52ℓ̅+34sin2ℓ̅]|ℓ̅0ℓ̅f,ahk=akh=-34a̅5μ3cos2ℓ̅|ℓ̅0ℓ̅f,akk=a̅5μ3[52ℓ̅-34sin2ℓ̅]|ℓ̅0ℓ̅f,aII=a̅5μ3[12ℓ̅+14sin2ℓ̅]|ℓ̅0ℓ̅f,aIΩ′=aΩ′I=-14a̅5μ3cos2ℓ̅|ℓ̅0ℓ̅f,aΩ′Ω′=a̅5μ3[12ℓ̅-14sin2ℓ̅]|ℓ̅0ℓ̅f,bα=εsin2I̅cos2ℓ̅|ℓ̅0ℓ̅f,bh=ε[cosℓ̅+14sin2I̅(-5cosℓ̅+73cos3ℓ̅)]|ℓ̅0ℓ̅f,bk=ε[sinℓ̅+14sin2I̅(-7sinℓ̅+73sin3ℓ̅)]|ℓ̅0ℓ̅f,bI=ε4sin2I̅cos2ℓ̅|ℓ̅0ℓ̅f,bΩ′=-ε2sin2I̅(ℓ̅-12sin2ℓ̅)|ℓ̅0ℓ̅f,
with ℓ̅f=ℓ̅0+n̅(tf-t0),t0 is the initial time, tf is the final time, and ε=(3/2)J2(ae/a̅)2. The overbar denotes the orbital elements of the reference orbit.The analytical solution defined by (5.4)–(5.7) is in agreement with the ones obtained through different approaches in [6, 10]. In [6] the optimization problem is formulated with Gauss equations in nonsingular orbital elements as state equations, and in [10] Hori method is applied after the transformation of variables. Equations (5.4)–(5.7) represent a complete first-order analytical solution for optimal low-thrust limited-power transfers between close quasicircular nonequatorial orbits in a gravity field that includes the second zonal harmonic J2 in the gravitational potential. These equations contain arbitrary constants of integration that must be determined to satisfy the two-point boundary value problem of going from the initial orbit at the time t0=0 to the final orbit at the specified final time tf=T. Since they are linear in these constants, the boundary value problem can be solved by simple techniques.An approximate expression for the optimal thrust accelerationγ* can be obtained from (3.5) and (5.2), by using the expansions of the elliptic motion, and it is given, up to the zeroth order in eccentricity, by
(5.8)γ*=1na{(phsinℓ-pkcosℓ)er+2(pα+phcosℓ+pksinℓ)es+(pIcosℓ+pΩ′sinℓ)ew}.According to Section2, the optimal fuel consumption variable J is determined by simple quadrature of the last differential equation in (2.3), with γ* given by (5.8). An approximate expression for J is given by
(5.9)ΔJ=12{aααpα2+2aαhpαph+2aαkpαpk+ahhph2+2ahkphpk+akkpk2+aIIpI2+2ahΩ′pIpΩ′+aΩ′Ω′pΩ′2}.For transfers involving a large number of revolutions, (5.4)–(5.7) can be greatly simplified by neglecting the short periodic terms in comparison to the secular ones, and, the system of algebraic can be solved analytically. Thus,
(5.10)pα=14μ3a̅5(ΔαΔℓ̅),ph=25μ3a̅5(ΔhΔℓ̅),pk=25μ3a̅5(ΔkΔℓ̅),pI=2μ3a̅5(ΔIΔℓ̅),pΩ′=2μ3a̅5[(ΔΩsinI̅Δℓ̅)+ε2sin2I̅].
Introducing these equations into (5.8) and (5.9), γ* and J are obtained explicitly in terms of the imposed variations of the orbital elements, the perturbative effects due to the oblateness of the central body and the duration of the maneuver:
(5.11)γ*=μa̅2{25(ΔhΔℓ̅sinℓ̅-ΔkΔℓ̅cosℓ̅)er+(12ΔαΔℓ̅+45(ΔhΔℓ̅cosℓ̅+ΔkΔℓ̅sinℓ̅))es+2(ΔIΔℓ̅cosℓ̅+[ΔΩΔℓ̅+εcosI̅]sinI̅sinℓ̅)ew},(5.12)ΔJΔℓ̅=14μ3a̅5{12(ΔαΔℓ̅)2+45(Δh2+Δk2Δℓ̅2)+4((ΔIΔℓ̅)2+[ΔΩΔℓ̅+εcosI̅]2sin2I̅)}.


### 5.2. Transfers between Close Quasicircular Near-Equatorial Orbits

Consider the Mathieu transformation [33] defined by
(5.13)a′′=a′,ξ′′=e′cos(ω′+Ω′),η′′=e′sin(ω′+Ω′),P′′=sin(I′2)cosΩ′,Q′′=sin(I′2)sinΩ′,λ′′=M′+ω′+Ω′,(5.14)pa′′=pa,pξ′′=pe′cos(ω′+Ω′)-(pω′-pM′e′)sin(ω′+Ω′),pη′′=pe′sin(ω′+Ω′)+(pω′-pM′e′)cos(ω′+Ω′),pP′′=pI′2cosΩ′cos(I′/2)-(pΩ′-pω′)sinΩ′sin(I′/2),pQ′′=pI′2sinΩ′cos(I′/2)+(pΩ′-pω′)cosΩ′sin(I′/2),pλ′′=pM′.
The new set of canonical variables, designated by the double prime, is nonsingular for circular equatorial orbits.Introducing (5.14) into (4.6) and using the expansions of the elliptic motion in terms of eccentricity and mean anomaly [31], one finds that the new averaged maximum Hamiltonian and the generating function are written up to the zeroth order in eccentricity and first-order in inclination as
(5.15)F1=32nJ2(aea)2(QpP-PpQ)+a2μ{4a2pa2+52(pξ2+pη2)+18(pP2+pQ2)},(5.16)S1=32J2(aea)2{pξcosλ+pηsinλ+12(QpP+PpQ)sin2λ+12(PpP-QpQ)cos2λ}+12a5μ3{8apa(pξsinλ-pηcosλ)+[34(pξ2-pη2)+116(pP2-pQ2)]sin2λ-[32pξpη+18pPpQ]cos2λ}.
Double prime is omitted to simplify the notation.For transfers between close orbits,F1 and S1 can be linearized around a suitable reference orbit, and an approximate first-order analytical solution can be determined. This solution is given by
(5.17)Δx=Cp0+D,
where Δx=[ΔαΔξΔηΔPΔQ]T,p0 is the 5×1 vector of initial value of the adjoint variables, C is a 5×5 matrix concerning the optimal thrust acceleration, and D is a 5×1 vector containing the perturbative effects of the oblateness of the central body. The matrix C and the vector D are given by
(5.18)C=[cααcαξcαη00cξαcξξcξη00cηαcηξcηη00000cPPcPQ000cQPcQQ],(5.19)D=[0dξdηdPdQ],
where
(5.20)cαα=4a̅5μ3λ̅|λ̅0λ̅f,cαξ=cξα=4a̅5μ3sinλ̅|λ̅0λ̅f,cαη=cηα=-4a̅5μ3cosλ̅|λ̅0λ̅f,cξξ=a̅5μ3[52λ̅+34sin2λ̅]|λ̅0λ̅f,cξη=cηξ=-34a̅5μ3cos2λ̅|λ̅0λ̅f,cηη=a̅5μ3[52λ̅-34sin2λ̅]|λ̅0λ̅f,cPP=18a̅5μ3[Δλ̅cosεΔλ̅+12sin((2+ε)λ̅f-ελ̅0)-12sin((2+ε)λ̅0-ελ̅f)],cPQ=18a̅5μ3[Δλ̅sinεΔλ̅-12cos((2+ε)λ̅f-ελ̅0)+12cos((2+ε)λ̅0-ελ̅f)],cQP=18a̅5μ3[-Δλ̅sinεΔλ̅-12cos((2+ε)λ̅f-ελ̅0)+12cos((2+ε)λ̅0-ελ̅f)],cQQ=18a̅5μ3[Δλ̅cosεΔλ̅-12sin((2+ε)λ̅f-ελ̅0)+12sin((2+ε)λ̅0-ελ̅f)],dξ=32J2(aea̅)2cosλ̅|λ̅0λ̅f,dη=32J2(aea̅)2sinλ̅|λ̅0λ̅f,dP=P0{[cosεΔλ̅-12εcos(ελ̅f-(2+ε)λ̅0)+12εcos(ελ̅0-(2+ε)λ̅f)]-1}+Q0[sinεΔλ̅+12εsin(ελ̅f-(2+ε)λ̅0)-12εsin(ελ̅0-(2+ε)λ̅f)],dQ=Q0{[cosεΔλ̅+12εcos(ελ̅f-(2+ε)λ̅0)-12εcos(ελ̅0-(2+ε)λ̅f)]-1}+P0[-sinεΔλ̅+12εsin(ελ̅f-(2+ε)λ̅0)-12εsin(ελ̅0-(2+ε)λ̅f)].
The adjoint variables are
(5.21)pa=pa0,pξ=pξ0,pη=pη0,pP=pP0cosεΔλ̅+pQ0sinεΔλ̅,pQ=-pP0sinεΔλ̅+pQ0cosεΔλ̅,
where Δλ̅=λ̅f-λ̅0=n̅(tf-t0). The overbar denotes the orbital elements of the reference orbit. Equations (5.20) do not contain higher order terms in ε  as appear implicitly in the solution presented in [7].The analytical solution defined by (5.17)–(5.21) is in agreement with the ones obtained through different approaches in [7, 10]. In [7] the optimization problem is formulated with Gauss equations in nonsingular orbital elements as state equations, and in [10] Hori method is applied after the transformation of variables. Equations (5.17)–(5.21) represent a complete first-order analytical solution for optimal low-thrust limited-power transfers between close quasicircular near-equatorial orbits in a gravity field that includes the second zonal harmonic J2 in the gravitational potential. These equations contain arbitrary constants of integration that must be determined to satisfy the two-point boundary value problem of going from the initial orbit at the time t0=0 to the final orbit at the specified final time tf=T. Since they are linear in these constants, the boundary value problem can be solved by simple techniques.As described in Section5.1, approximate expressions can be obtained for the optimal thrust acceleration γ* and the fuel consumption J. These expressions are given by
(5.22)γ*=1na{(pξsinλ-pηcosλ)er+2(pα+pξcosλ+pηsinλ)es+12(pPcosλ+pQsinλ)ew},ΔJ=12{cααpα2+2cαξpαpξ+2cαηpαpη+cξξpξ2+2cξηpξpη+cηηpη2+cPPpP2+(cPQ+cQP)pPpQ+cQQpQ2}.For transfers involving a large number of revolutions, (5.17)–(5.21) can be greatly simplified by neglecting the short periodic terms in comparison to the secular ones, and, the system of algebraic can be solved analytically. Thus,
(5.23)pα=14μ3a̅5(ΔαΔλ̅),pξ=25μ3a̅5(ΔξΔλ̅),pη=25μ3a̅5(ΔηΔλ̅),pP=μ3a̅58Δλ̅[(P0+ΔP)cos(εΔλ̅)-(Q0+ΔQ)sin(εΔλ̅)-P0],pQ=μ3a̅58Δλ̅[(Q0+ΔQ)cos(εΔλ̅)+(P0+ΔP)sin(εΔλ̅)-Q0].
The subscript denoting the constants is omitted. Introducing these equations into (5.22), γ* and J are obtained explicitly in terms of the imposed variations of the orbital elements, the perturbative effects due to the oblateness of the central body and the duration of the maneuver:
(5.24)γ*=μa̅2{25(ΔξΔλ̅sinλ̅-ΔηΔλ̅cosλ̅)er+(12ΔαΔλ̅+45(ΔξΔλ̅cosλ̅+ΔηΔλ̅sinλ̅))es+4Δλ̅[Pfcos(λ̅+θf)-P0cos(λ̅+θ0)+Qfsin(λ̅+θf)-Q0sin(λ̅+θ0)]ew},ΔJΔλ̅=14μ3a̅5{12(ΔαΔλ̅)2+45(Δξ2+Δη2Δλ̅2)+16cos(εΔλ̅){(ΔP2+ΔQ2Δλ̅2)+4Δλ̅2sin(εΔλ̅2)×[Pf(P0sin(εΔλ̅2)-Q0cos(εΔλ̅2))+Qf(P0cos(εΔλ̅2)+Q0sin(εΔλ̅2))]}},
with θ0=εn̅(t-t0) and θf=εn̅(t-tf).


### 5.3. General Remarks on Optimal Transfers between Close  Quasicircular Orbits

Considering the analytical solutions described above, general remarks about the effects of the oblateness of the central body on optimal transfers between close quasicircular orbits can be stated.(1)
MatricesA and C, defined by (5.5) and (5.18), respectively, can be decomposed into two square matrices 3×3 and 2×2. This fact shows that there exists an uncoupling between the in-plane modifications and the rotation of the orbital plane. Similar results were obtained by Edelbaum [1] and Marec [3] for transfers in a Newtonian central field.(2)
There is a normal component of the optimal thrust acceleration that counteracts the perturbative effects due to the oblateness of the central body. This normal component is proportional toε.(3)
In general, the maneuvers in a noncentral gravity field are more expensive than the maneuvers in Newtonian central field, taking into account that the perturbative effects due to the oblateness of the central body must be counteracted.(4)
Fuel can be saved for long-time maneuvers which involves modification of the longitude of the ascending node if the terminal orbits are direct (0°<I̅<90°) and ΔΩ<0, and if the terminal orbits are retrograde (90°<I̅<180°) and ΔΩ>0.(5)
The extra consumption needed to counteract the perturbative effects due to the oblateness of the central body reaches its maximum for maneuvers between orbits with45° or 135° of inclination and its minimum (null extra consumption) for maneuvers between equatorial or polar orbits (the gravity field is symmetric in these cases).(6)
As the transfer time increases, the extra fuel consumption increases.(7)
The extra fuel consumption is greater for transfers between low orbits.


## 5.1. Transfers between Close Quasicircular Nonequatorial Orbits

Consider the Mathieu transformation [33] defined by
(5.1)a′′=a′,h′′=e′cosω′,k′′=e′sinω′,I′′=I′,Ω′′=Ω′,ℓ′′=M′+ω′,(5.2)pa′′=pa′,ph′′=pe′cosω′-(pω′-pM′e′)sinω′,pk′′=pe′sinω′+(pω′-pM′e′)cosω′,pI′′=pI′,pΩ′′=pΩ′,pℓ′′=pM′.
The new set of canonical variables, designated by the double prime, is nonsingular for circular nonequatorial orbits.Introducing (5.2) into (4.6) and using the expansions of the elliptic motion in terms of eccentricity and mean anomaly [31], one finds that the new averaged maximum Hamiltonian and the generating function are written up to the zeroth order in eccentricity as
(5.3)F1=-32nJ2(aea)2cosIpΩ+a2μ{4a2pa2+52(ph2+pk2)+12(pI2+pΩ2sin2I)},S1=32J2(aea)2{apasin2Icos2ℓ+ph[cosℓ+14sin2I(-5cosℓ+73cos3ℓ)]+pk[sinℓ+14sin2I(-7sinℓ+73sin3ℓ)]+14pIsin2Icos2ℓ+12pΩcosIsin2ℓ}+12a5μ3{8apa[phsinℓ-pkcosℓ]-12[3phpk+pΩsinIpI]cos2ℓ+14[3(ph2-pk2)+pI2-pΩ2sin2I]sin2ℓ}.
Double prime is omitted to simplify the notation.For transfers between close orbits,F1 and S1 can be linearized around a suitable reference orbit, and an approximate first-order analytical solution can be determined. This solution is given by
(5.4)Δx=Ap0+B,
where Δx=[ΔαΔhΔkΔIΔΩ′]T,p0 is the 5×1 vector of initial value of the adjoint variables, A is a 5×5 symmetric matrix concerning the optimal thrust acceleration and B is a 5×1 vector containing the perturbative effects of the oblateness of the central body. The variables α=a/a̅ and pα=a̅pa are introduced to make the adjoint and the state vectors dimensionless, and the variables Ω′=ΩsinI̅ and pΩ′=pΩ/sinI̅ are introduced to simplify the matrix A. The adjoint vector is constant. The matrix A and the vector B are given by
(5.5)A=[aααaαhaαk00ahαahhahk00akαakhakk00000aIIaIΩ′000aΩ′IaΩ′Ω′],(5.6)B=[bαbhbkbIbΩ′],
where
(5.7)aαα=4a̅5μ3ℓ̅|ℓ̅0ℓ̅f,aαh=ahα=4a̅5μ3sinℓ̅|ℓ̅0ℓ̅f,aαk=akα=-4a̅5μ3cosℓ̅|ℓ̅0ℓ̅f,ahh=a̅5μ3[52ℓ̅+34sin2ℓ̅]|ℓ̅0ℓ̅f,ahk=akh=-34a̅5μ3cos2ℓ̅|ℓ̅0ℓ̅f,akk=a̅5μ3[52ℓ̅-34sin2ℓ̅]|ℓ̅0ℓ̅f,aII=a̅5μ3[12ℓ̅+14sin2ℓ̅]|ℓ̅0ℓ̅f,aIΩ′=aΩ′I=-14a̅5μ3cos2ℓ̅|ℓ̅0ℓ̅f,aΩ′Ω′=a̅5μ3[12ℓ̅-14sin2ℓ̅]|ℓ̅0ℓ̅f,bα=εsin2I̅cos2ℓ̅|ℓ̅0ℓ̅f,bh=ε[cosℓ̅+14sin2I̅(-5cosℓ̅+73cos3ℓ̅)]|ℓ̅0ℓ̅f,bk=ε[sinℓ̅+14sin2I̅(-7sinℓ̅+73sin3ℓ̅)]|ℓ̅0ℓ̅f,bI=ε4sin2I̅cos2ℓ̅|ℓ̅0ℓ̅f,bΩ′=-ε2sin2I̅(ℓ̅-12sin2ℓ̅)|ℓ̅0ℓ̅f,
with ℓ̅f=ℓ̅0+n̅(tf-t0),t0 is the initial time, tf is the final time, and ε=(3/2)J2(ae/a̅)2. The overbar denotes the orbital elements of the reference orbit.The analytical solution defined by (5.4)–(5.7) is in agreement with the ones obtained through different approaches in [6, 10]. In [6] the optimization problem is formulated with Gauss equations in nonsingular orbital elements as state equations, and in [10] Hori method is applied after the transformation of variables. Equations (5.4)–(5.7) represent a complete first-order analytical solution for optimal low-thrust limited-power transfers between close quasicircular nonequatorial orbits in a gravity field that includes the second zonal harmonic J2 in the gravitational potential. These equations contain arbitrary constants of integration that must be determined to satisfy the two-point boundary value problem of going from the initial orbit at the time t0=0 to the final orbit at the specified final time tf=T. Since they are linear in these constants, the boundary value problem can be solved by simple techniques.An approximate expression for the optimal thrust accelerationγ* can be obtained from (3.5) and (5.2), by using the expansions of the elliptic motion, and it is given, up to the zeroth order in eccentricity, by
(5.8)γ*=1na{(phsinℓ-pkcosℓ)er+2(pα+phcosℓ+pksinℓ)es+(pIcosℓ+pΩ′sinℓ)ew}.According to Section2, the optimal fuel consumption variable J is determined by simple quadrature of the last differential equation in (2.3), with γ* given by (5.8). An approximate expression for J is given by
(5.9)ΔJ=12{aααpα2+2aαhpαph+2aαkpαpk+ahhph2+2ahkphpk+akkpk2+aIIpI2+2ahΩ′pIpΩ′+aΩ′Ω′pΩ′2}.For transfers involving a large number of revolutions, (5.4)–(5.7) can be greatly simplified by neglecting the short periodic terms in comparison to the secular ones, and, the system of algebraic can be solved analytically. Thus,
(5.10)pα=14μ3a̅5(ΔαΔℓ̅),ph=25μ3a̅5(ΔhΔℓ̅),pk=25μ3a̅5(ΔkΔℓ̅),pI=2μ3a̅5(ΔIΔℓ̅),pΩ′=2μ3a̅5[(ΔΩsinI̅Δℓ̅)+ε2sin2I̅].
Introducing these equations into (5.8) and (5.9), γ* and J are obtained explicitly in terms of the imposed variations of the orbital elements, the perturbative effects due to the oblateness of the central body and the duration of the maneuver:
(5.11)γ*=μa̅2{25(ΔhΔℓ̅sinℓ̅-ΔkΔℓ̅cosℓ̅)er+(12ΔαΔℓ̅+45(ΔhΔℓ̅cosℓ̅+ΔkΔℓ̅sinℓ̅))es+2(ΔIΔℓ̅cosℓ̅+[ΔΩΔℓ̅+εcosI̅]sinI̅sinℓ̅)ew},(5.12)ΔJΔℓ̅=14μ3a̅5{12(ΔαΔℓ̅)2+45(Δh2+Δk2Δℓ̅2)+4((ΔIΔℓ̅)2+[ΔΩΔℓ̅+εcosI̅]2sin2I̅)}.


## 5.2. Transfers between Close Quasicircular Near-Equatorial Orbits

Consider the Mathieu transformation [33] defined by
(5.13)a′′=a′,ξ′′=e′cos(ω′+Ω′),η′′=e′sin(ω′+Ω′),P′′=sin(I′2)cosΩ′,Q′′=sin(I′2)sinΩ′,λ′′=M′+ω′+Ω′,(5.14)pa′′=pa,pξ′′=pe′cos(ω′+Ω′)-(pω′-pM′e′)sin(ω′+Ω′),pη′′=pe′sin(ω′+Ω′)+(pω′-pM′e′)cos(ω′+Ω′),pP′′=pI′2cosΩ′cos(I′/2)-(pΩ′-pω′)sinΩ′sin(I′/2),pQ′′=pI′2sinΩ′cos(I′/2)+(pΩ′-pω′)cosΩ′sin(I′/2),pλ′′=pM′.
The new set of canonical variables, designated by the double prime, is nonsingular for circular equatorial orbits.Introducing (5.14) into (4.6) and using the expansions of the elliptic motion in terms of eccentricity and mean anomaly [31], one finds that the new averaged maximum Hamiltonian and the generating function are written up to the zeroth order in eccentricity and first-order in inclination as
(5.15)F1=32nJ2(aea)2(QpP-PpQ)+a2μ{4a2pa2+52(pξ2+pη2)+18(pP2+pQ2)},(5.16)S1=32J2(aea)2{pξcosλ+pηsinλ+12(QpP+PpQ)sin2λ+12(PpP-QpQ)cos2λ}+12a5μ3{8apa(pξsinλ-pηcosλ)+[34(pξ2-pη2)+116(pP2-pQ2)]sin2λ-[32pξpη+18pPpQ]cos2λ}.
Double prime is omitted to simplify the notation.For transfers between close orbits,F1 and S1 can be linearized around a suitable reference orbit, and an approximate first-order analytical solution can be determined. This solution is given by
(5.17)Δx=Cp0+D,
where Δx=[ΔαΔξΔηΔPΔQ]T,p0 is the 5×1 vector of initial value of the adjoint variables, C is a 5×5 matrix concerning the optimal thrust acceleration, and D is a 5×1 vector containing the perturbative effects of the oblateness of the central body. The matrix C and the vector D are given by
(5.18)C=[cααcαξcαη00cξαcξξcξη00cηαcηξcηη00000cPPcPQ000cQPcQQ],(5.19)D=[0dξdηdPdQ],
where
(5.20)cαα=4a̅5μ3λ̅|λ̅0λ̅f,cαξ=cξα=4a̅5μ3sinλ̅|λ̅0λ̅f,cαη=cηα=-4a̅5μ3cosλ̅|λ̅0λ̅f,cξξ=a̅5μ3[52λ̅+34sin2λ̅]|λ̅0λ̅f,cξη=cηξ=-34a̅5μ3cos2λ̅|λ̅0λ̅f,cηη=a̅5μ3[52λ̅-34sin2λ̅]|λ̅0λ̅f,cPP=18a̅5μ3[Δλ̅cosεΔλ̅+12sin((2+ε)λ̅f-ελ̅0)-12sin((2+ε)λ̅0-ελ̅f)],cPQ=18a̅5μ3[Δλ̅sinεΔλ̅-12cos((2+ε)λ̅f-ελ̅0)+12cos((2+ε)λ̅0-ελ̅f)],cQP=18a̅5μ3[-Δλ̅sinεΔλ̅-12cos((2+ε)λ̅f-ελ̅0)+12cos((2+ε)λ̅0-ελ̅f)],cQQ=18a̅5μ3[Δλ̅cosεΔλ̅-12sin((2+ε)λ̅f-ελ̅0)+12sin((2+ε)λ̅0-ελ̅f)],dξ=32J2(aea̅)2cosλ̅|λ̅0λ̅f,dη=32J2(aea̅)2sinλ̅|λ̅0λ̅f,dP=P0{[cosεΔλ̅-12εcos(ελ̅f-(2+ε)λ̅0)+12εcos(ελ̅0-(2+ε)λ̅f)]-1}+Q0[sinεΔλ̅+12εsin(ελ̅f-(2+ε)λ̅0)-12εsin(ελ̅0-(2+ε)λ̅f)],dQ=Q0{[cosεΔλ̅+12εcos(ελ̅f-(2+ε)λ̅0)-12εcos(ελ̅0-(2+ε)λ̅f)]-1}+P0[-sinεΔλ̅+12εsin(ελ̅f-(2+ε)λ̅0)-12εsin(ελ̅0-(2+ε)λ̅f)].
The adjoint variables are
(5.21)pa=pa0,pξ=pξ0,pη=pη0,pP=pP0cosεΔλ̅+pQ0sinεΔλ̅,pQ=-pP0sinεΔλ̅+pQ0cosεΔλ̅,
where Δλ̅=λ̅f-λ̅0=n̅(tf-t0). The overbar denotes the orbital elements of the reference orbit. Equations (5.20) do not contain higher order terms in ε  as appear implicitly in the solution presented in [7].The analytical solution defined by (5.17)–(5.21) is in agreement with the ones obtained through different approaches in [7, 10]. In [7] the optimization problem is formulated with Gauss equations in nonsingular orbital elements as state equations, and in [10] Hori method is applied after the transformation of variables. Equations (5.17)–(5.21) represent a complete first-order analytical solution for optimal low-thrust limited-power transfers between close quasicircular near-equatorial orbits in a gravity field that includes the second zonal harmonic J2 in the gravitational potential. These equations contain arbitrary constants of integration that must be determined to satisfy the two-point boundary value problem of going from the initial orbit at the time t0=0 to the final orbit at the specified final time tf=T. Since they are linear in these constants, the boundary value problem can be solved by simple techniques.As described in Section5.1, approximate expressions can be obtained for the optimal thrust acceleration γ* and the fuel consumption J. These expressions are given by
(5.22)γ*=1na{(pξsinλ-pηcosλ)er+2(pα+pξcosλ+pηsinλ)es+12(pPcosλ+pQsinλ)ew},ΔJ=12{cααpα2+2cαξpαpξ+2cαηpαpη+cξξpξ2+2cξηpξpη+cηηpη2+cPPpP2+(cPQ+cQP)pPpQ+cQQpQ2}.For transfers involving a large number of revolutions, (5.17)–(5.21) can be greatly simplified by neglecting the short periodic terms in comparison to the secular ones, and, the system of algebraic can be solved analytically. Thus,
(5.23)pα=14μ3a̅5(ΔαΔλ̅),pξ=25μ3a̅5(ΔξΔλ̅),pη=25μ3a̅5(ΔηΔλ̅),pP=μ3a̅58Δλ̅[(P0+ΔP)cos(εΔλ̅)-(Q0+ΔQ)sin(εΔλ̅)-P0],pQ=μ3a̅58Δλ̅[(Q0+ΔQ)cos(εΔλ̅)+(P0+ΔP)sin(εΔλ̅)-Q0].
The subscript denoting the constants is omitted. Introducing these equations into (5.22), γ* and J are obtained explicitly in terms of the imposed variations of the orbital elements, the perturbative effects due to the oblateness of the central body and the duration of the maneuver:
(5.24)γ*=μa̅2{25(ΔξΔλ̅sinλ̅-ΔηΔλ̅cosλ̅)er+(12ΔαΔλ̅+45(ΔξΔλ̅cosλ̅+ΔηΔλ̅sinλ̅))es+4Δλ̅[Pfcos(λ̅+θf)-P0cos(λ̅+θ0)+Qfsin(λ̅+θf)-Q0sin(λ̅+θ0)]ew},ΔJΔλ̅=14μ3a̅5{12(ΔαΔλ̅)2+45(Δξ2+Δη2Δλ̅2)+16cos(εΔλ̅){(ΔP2+ΔQ2Δλ̅2)+4Δλ̅2sin(εΔλ̅2)×[Pf(P0sin(εΔλ̅2)-Q0cos(εΔλ̅2))+Qf(P0cos(εΔλ̅2)+Q0sin(εΔλ̅2))]}},
with θ0=εn̅(t-t0) and θf=εn̅(t-tf).


## 5.3. General Remarks on Optimal Transfers between Close  Quasicircular Orbits

Considering the analytical solutions described above, general remarks about the effects of the oblateness of the central body on optimal transfers between close quasicircular orbits can be stated.(1)
MatricesA and C, defined by (5.5) and (5.18), respectively, can be decomposed into two square matrices 3×3 and 2×2. This fact shows that there exists an uncoupling between the in-plane modifications and the rotation of the orbital plane. Similar results were obtained by Edelbaum [1] and Marec [3] for transfers in a Newtonian central field.(2)
There is a normal component of the optimal thrust acceleration that counteracts the perturbative effects due to the oblateness of the central body. This normal component is proportional toε.(3)
In general, the maneuvers in a noncentral gravity field are more expensive than the maneuvers in Newtonian central field, taking into account that the perturbative effects due to the oblateness of the central body must be counteracted.(4)
Fuel can be saved for long-time maneuvers which involves modification of the longitude of the ascending node if the terminal orbits are direct (0°<I̅<90°) and ΔΩ<0, and if the terminal orbits are retrograde (90°<I̅<180°) and ΔΩ>0.(5)
The extra consumption needed to counteract the perturbative effects due to the oblateness of the central body reaches its maximum for maneuvers between orbits with45° or 135° of inclination and its minimum (null extra consumption) for maneuvers between equatorial or polar orbits (the gravity field is symmetric in these cases).(6)
As the transfer time increases, the extra fuel consumption increases.(7)
The extra fuel consumption is greater for transfers between low orbits.


## 6. Optimal Long-Time Transfers between Arbitrary Orbits

In this section the two-point boundary-value problem for long-time transfers between arbitrary orbits is formulated. In Section7, this boundary-problem problem will be solved numerically through a shooting method, and the solution will be compared to the analytical ones in the case of orbits with small eccentricities.Since the averaged maximum Hamiltonian, given by (4.6), becomes singular for circular and/or equatorial orbits, a set of nonsingular elements is introduced.Consider the Mathieu transformation [33] defined by
(6.1)a′′=a′,ξ′′=e′cos(ω′+Ω′),η′′=e′sin(ω′+Ω′),P′′=sin(I′2)cosΩ′,Q′′=sin(I′2)sinΩ′,(6.2)pa′′=pa′,pξ′′=pe′cos(ω′+Ω′)-(pω′-pM′e′)sin(ω′+Ω′),pη′′=pe′sin(ω′+Ω′)+(pω′-pM′e′)cos(ω′+Ω′),pP′′=pI′2cosΩ′cos(I′/2)-(pΩ′-pω′)sinΩ′sin(I′/2),pQ′′=pI′2sinΩ′cos(I′/2)+(pΩ′-pω′)cosΩ′sin(I′/2).
This transformation of variables is the same one given by (5.12), excepting the fast-phase, which is unnecessary (short-periodic terms have been eliminated). Double prime designates the new variables.Introducing (6.2) into (4.6), F1 is written as
(6.3)F1=-32nJ2(aea)2(1-ξ2-η2)-2{(1-2(P2+Q2))[(ξpη-ηpξ)+(PpQ-QpP)]+12(1-5(1-2(P2+Q2))2)(ξpη-ηpξ)}+a2μ{4a2pa2+pξ2(52-52ξ2-2η2)+pη2(52-52η2-2ξ2)-ξηpξpη+12(1-ξ2-η2)-1{(1+32ξ2+32η2)×[14(pP2+pQ2)-14(PpP+QpQ)2+14(1-P2-Q2)-1×[(QpP-PpQ)2+4(ξpη-ηpξ)(PpQ-QpP)+4(P2+Q2)(ξpη-ηpξ)2]]-58(1-P2-Q2)-1[(ξ2-η2)(P2-Q2)+4ξηPQ]×[[2(ξpη-ηpξ)+(PpQ-QpP)]2+(1-P2-Q2)(pP2+pQ2)]-52ξηpPpQ+58(ξ2-η2)(pP2-pQ2)+52(ξpη-ηpξ)[-(ξ2-η2)(PpQ+QpP)+2ξη(PpP-QpQ)]}.
Double prime is omitted to simplify the notation. Note that (5.15) is a simplification of (6.3) for quasicircular near-equatorial orbits.The two-point boundary-value problem for long-time transfers between arbitrary orbits is formulated by the following system of canonical differential equations:(6.4)dxdt=(∂F1∂p)T,dpdt=-(∂F1∂x)T,
where x=[aξηPQ]T and p=[papξpηpPpQ]T, and the boundary-conditions
(6.5)a(t0)=a0,ξ(t0)=ξ0,η(t0)=η0,P(t0)=P0,Q(t0)=Q0,a(tf)=af,ξ(tf)=ξf,η(tf)=ηf,P(tf)=Pf,Q(tf)=Qf,
with initial time t0=0 and final time tf=T. The explicit form of state and adjoint equations is derived by using MAPLE Software 9, and they are not presented in text because they are extensive.


## 7. Results

In this section, numerical and analytical results are compared for maneuvers described in Tables1 and 2. Earth is the central body, such that J2=1.0826×10-3. In Table 1, orbital elements of initial and final orbits are defined for a maneuver of modification of all orbital elements considering different values of transfer durations T=tf-t0 and ratios ae/a0. In Table 2, orbital elements of initial orbit, values of transfer duration and ratio ρ=yf/y0, where y denotes the orbital element that is changed in the maneuver, are defined for small (orbit corrections) and moderate amplitude maneuvers of modification of semimajor axis and inclination of the orbital plane. Results are presented in Figures 3 through 8 using canonical units. In these units, a0=1.0 and μ=1.0. For example, in metric units, taking ae=6378.2km,ρ=0.950 corresponds to a0=6713.89km, and T=100 (time units) corresponds to 24.20 hours (1.008 day), and ρ=0.150 corresponds to a0=42521.34km, and T=250 (time units) corresponds to 964.45 hours (40.185 days). Thus, maneuvers described in the tables below have duration of 1 to 74 days, approximately.  On the other hand, considering that the value of |γ*| is approximately given by 2ΔJ/T (short periodic terms are neglected), |γ*| has values between 1.0×10-4 and 4.0×10-3 (canonical units) for all results presented in Figures 3–5. These values of magnitude of the thrust acceleration are typical for LP systems [5, 27].Table 1
Maneuver of modification of all orbital elements.
T=tf-t0ae/a0Maneuver of modification of all orbital elementsInitial orbitFinal orbitOrbital elements1000.9500.9000.8500.8001500.7500.7001.0001.150a0.6500.6000.00250.0050e2000.5500.50030.00°35.0°I0.4500.40030.00°37.5°Ω2500.3500.30060.00°62.50°ω0.2500.200—0.1500.100Table 2
Small and moderate amplitude maneuvers.
ae/a0T=tf-t0Orbital elements of initial orbitρ=yf/y00.975100a=1.01.051.10150e=0.00251.151.20200I=30.00°1.251.30250Ω=30.00°1.351.40—ω=60.00°1.451.50Figure 3
Consumption for maneuver of modification of all orbital elements.Consumption for maneuver of modification of semimajor axis.
(a)(b)(c)(d)Consumption for maneuver of modification of inclination.
(a)(b)(c)(d)Figure3 shows the consumption J as function of ratio ae/a0 and values of T for the maneuver of modification of all orbital elements described in Table 1. Figures 4 and 5 show the consumption J as function of ratio ρ=yf/y0 and T for the small and moderate amplitude maneuvers described in Table 2, considering that the maneuvers are performed in central or noncentral gravity field. In these figures, solid line represents analytical solution, and dashed line represents numerical solution. Analytical solutions, including short periodic terms, are obtained by solving the linear system of algebraic equations given by (5.4)–(5.7) (transfer between nonequatorial orbits) with the semimajor axis and the inclination of the reference orbit defined by a̅=(a0+af)/2 and I̅=(I0+If)/2, respectively. In order to include the short periodic terms, it is assumed that the initial position of the vehicle is the pericenter of the initial orbit. Variables I and Ω are transformed into nonsingular variables P and Q to compare the analytical results with the numerical results.  Numerical solutions are obtained by solving the two-point boundary value problem described by (6.4) and (6.5) through a shooting method [34].Considering the maneuvers described in Tables1 and 2, Figures 3–5 show that analytical and numerical solutions yield quite similar values of the consumption J in the following cases: (i) maneuvers of modification of all orbital elements with ratio ae/a0<0.550; (ii) maneuvers of modification of semimajor axis with ratios ρ<1.25 and ae/a0<0.550; (iii) maneuvers of modification of inclination with ratios ρ<1.50 and ae/a0<0.550, for all values of T. For maneuvers with very high ratio ae/a0, analytical and numerical solutions only yield similar results for the same maneuvers described above for T=100 (time units).Figures6–8 represent the time history of state variables (nonsingular orbital elements) for maneuvers of modification of all orbital elements (Table 1) considering three values of ratios ae/a0—0.150, 0.550, and 0.950—and two values of transfer duration T—100 and 250 (time units). According to these figures, the analytical solution yields a good representation of the numerical solution for all state variables, excepting the semimajor axis for the maneuver with ae/a0=0.950 and T=250. The numerical solution shows that the semimajor axis is strongly perturbed by the oblateness of earth, and its time evolution cannot be described by the linear approximation given by the analytical solution. On the other hand, the analytical solutions shows that the amplitude of the short periodic terms decreases as the transfer duration increases such that these terms can be omitted for very long-time transfers.Time history of state variables for maneuver of modification of all orbital elements withae/a0=0.950—T=100 and 250.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Time history of state variables for maneuver of modification of all orbital elements withae/a0=0.550—T=100 and 250.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Time history of state variables for maneuver of modification of all orbital elements withae/a0=0.150—T=100 and 250.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 9
Vectorse,h, and N.


## 8. Conclusion

In this paper an approach based on canonical transformations is presented for the problem of optimal low-thrust limited-power maneuvers between close orbits with small eccentricities in noncentral gravity which includes the perturbative effects of the second zonal harmonic of the gravitational potential. Analytical first-order solutions are derived through this approach for transfers between nonequatorial orbits and for transfers between near-equatorial orbits. These analytical solutions have been compared to a numerical solution for long-time transfers between arbitrary orbits described by averaged maximum Hamiltonian, expressed in nonsingular orbital elements. The results show that the analytical solutions yield good estimates of the fuel consumption for preliminary mission analysis considering maneuvers with moderate amplitude, ratioae/a0 and transfers duration. Numerical results also show that the semimajor axis is strongly perturbed by the oblateness of earth, and its time evolution cannot be described by the linear approximation given by the analytical solution.



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*Source: 503168-2009-07-19.xml*