© 2017, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.A study of Earth–Moon bi-impulsive trajectories is presented in this paper. The motion of the space vehicle is described by the classic planar circular restricted three-body problem. The velocity increments are computed through analytical expressions, which are derived from the development of the Jacobi Integral expression. To determine the trajectories, a new two-point boundary value problem (TPBVP) with prescribed value of Jacobi Integral is formulated. Internal and external trajectories are determined through the solution of this new TPBVP for several times of flight. A relation between the Jacobi Integral and the Kepler’s energy at arrival is derived and several kinds of study are performed. Critical values of the Jacobi Integral, for which the Kepler’s energy of the space vehicle on the arrival trajectory becomes negative, are calculated for several configurations of arrival at the low Moon orbit in both directions: clockwise and counterclockwise. Results show that the proposed method allows the estimation of the fuel consumption before solving the TPBVP, and it facilitates the determination of trajectories with large time of flight. However, increasing values of the time of flight are not necessarily related with the increase of the Jacobi Integral value, which means that the obtaining of new trajectories becomes more difficult as the Jacobi Integral increases. Moreover, the proposed method provides results to be used as initial guess for more complex models and for optimization algorithms in order to minimize the total fuel consumption. For this case, this paper presents an example where an internal trajectory with large time of flight is optimized considering the Sun’s attraction.

© 2018 IAAThe present work formulates an orbital transfer for an Earth-to-Earth mission between non coplanar orbits with different altitudes with a special feature: the occurrence of a lunar flyby during the transfer orbit. This lunar flyby is intended to help change the plane of motion of the spacecraft without fuel consumption. Only two-impulsive trajectories are considered with the velocity increments applied at the initial and final orbits. In order to solve this problem, a 3D patched-conic approximation associated with a two-point boundary value problem is proposed. The same transfer problem is formulated considering the spatial circular restricted three-body problem (SCR3BP). The results of the patched-conic approximation is compared with the results of the SCR3BP showing a good agreement between the models. This work also determines several trajectories in order to perform a study of the fuel consumption considering several inclinations and altitudes of both initial and final orbits around the Earth. The longitude of the ascending node of the initial orbit, and, the altitude of close approach with the Moon during the flyby are also analyzed. According to the total velocity increment analysis, the changing plane assisted by a lunar flyby can be very favorable. Despite the increase of the time of flight, the saving of fuel is considerable. Indeed, the total velocity increment of this kind of maneuver is in some cases better than the velocity increment provided by the bi-parabolic transfer.

© 2017, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.In this work, a study about minimum fuel trajectories in a round trip journey to the Moon is presented. It is assumed that the velocity changes are instantaneous, that is, the propulsion system is capable of delivering impulses such that the fuel consumption is represented by the total velocity increment applied to the space vehicle. It is also assumed that the velocity increments are applied tangentially to the terminal orbits, and, the outgoing trip and the return trip are analyzed separately such that the whole mission is performed with four impulses (two impulses in each trip). The mathematical models used to describe the motion of the space vehicle are three: the lunar patched-conic approximation; the classic planar circular restricted three-body problem, and, the planar bi-circular restricted four-body problem (PBR4BP). For computing the optimal trajectories, the Sequential Gradient-Restoration Algorithm with constraints is used. The influence of the Sun on round trip lunar missions is analyzed through the PBR4BP model. For all models, the trajectories studied are direct ascent maneuvers, and, both the outgoing and return trips are considered. The results obtained through the different models are compared with each other. The optimal results for the PBR4BP model show that a small reduction of the fuel consumption can be achieved if the initial phase angle of the Sun is chosen properly.

© 2018 Universidad Nacional Autonoma de Mexico. All Rights Reserved.A numerical-analytical procedure based on infinitesimal canonical transformations is developed for computing optimal time-fixed low-thrust limited power transfers (no rendezvous) between coplanar orbits with small eccentricities in an inverse-square force field. The optimization problem is formulated as a Mayer problem with a set of non-singular orbital elements as state variables. Second order terms in eccentricity are considered in the development of the maximum Hamiltonian describing the optimal trajectories. The two-point boundary value problem of going from an initial orbit to a final orbit is solved by means of a two-stage Newton-Raphson algorithm which uses an infinitesimal canonical transformation. Numerical results are presented for some transfers between circular orbits with moderate radius ratio, including a preliminary analysis of Earth-Mars and Earth-Venus missions.

© 2018, Journal of Aerospace Technology and Management. All rights reserved.In this paper, an analytical solution for time-fixed optimal low-thrust limited-power transfers (no rendezvous) between elliptic coaxial non-coplanar orbits in an inverse-square force field is presented. Two particular classes of maneuvers are related to such transfers: maneuvers with change in the inclination of the orbital plane and maneuvers with change in the longitude of the ascending node. The optimization problem is formulated as a Mayer problem of optimal control with the state defined by semi-major axis, eccentricity, inclination or longitude of the ascending node, according to the class of maneuver considered, and a variable measuring the fuel consumption. After applying Pontryagin’s maximum principle and determining the maximum Hamiltonian, short periodic terms are eliminated through an infinitesimal canonical transformation. The new maximum Hamiltonian resulting from this canonical transformation describes the extremal trajectories for long duration transfers. Closed-form analytical solution is then obtained through Hamilton-Jacobi theory. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition. Simplified solution is determined for transfers between close orbits. The analytical solution is compared to the numerical solution obtained by integration of the canonical system of differential equations describing the extremal trajectories for some sets of initial conditions. Results show a great agreement between these solutions for the class of maneuvers considered in the analysis. The solution of the two-point boundary value problem of going from an initial orbit to a final orbit, based on the analytical solution, is also discussed.

© 2017 IAAIn this work, a study about the influence of the Sun on optimal two-impulse Earth-to-Moon trajectories for interior transfers with moderate time of flight is presented considering the three-body and the four-body models. The optimization criterion is the total characteristic velocity which represents the fuel consumption of an infinite thrust propulsion system. The optimization problem has been formulated using the classic planar circular restricted three-body problem (PCR3BP) and the planar bi-circular restricted four-body problem (PBR4BP), and, it consists of transferring a spacecraft from a circular low Earth orbit (LEO) to a circular low Moon orbit (LMO) with minimum fuel consumption. The Sequential Gradient Restoration Algorithm (SGRA) is applied to determine the optimal solutions. Numerical results are presented for several final altitudes of a clockwise or a counterclockwise circular low Moon orbit considering a specified altitude of a counterclockwise circular low Earth orbit. Two types of analysis are performed: in the first one, the initial position of the Sun is taken as a parameter and the major parameters describing the optimal trajectories are obtained by solving an optimization problem of one degree of freedom. In the second analysis, an optimization problem with two degrees of freedom is considered and the initial position of the Sun is taken as an additional unknown.

© 2016, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.A numerical study of optimal time-fixed low-thrust limited power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is performed by means of two different approaches. The first approach uses a numerical method based on the second variation theory, usually known as neighboring extremals method, to solve the two-point boundary value problem obtained from the application of the Pontryagin Maximum Principle to the optimization problem formulated as a Mayer problem with the radial distance and the components of the velocity vector as state variables. The second approach is based on the solution of the two-point boundary value problem defined by a first-order analytical solution expressed in terms of non-singular orbital elements, which include short periodic terms, and derived through canonical transformations theory in a previous work. For transfers between close orbits, a simplified solution expressed by a linear system of algebraic equations is straightforwardly derived from this analytical first-order solution. In this case, the two-point boundary value problem can be solved by simple techniques. Numerical results are presented for transfers between circular orbits, considering several radius ratios and transfer durations. Some maneuvers involving orbits with arbitrary small eccentricities are also considered. The fuel consumption is taken as the performance criterion in comparison of the results.

© 2015, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.In this paper, a first-order analytical solution, which includes the short periodic terms, for the problem of optimal time-fixed low-thrust limited-power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is obtained through canonical transformation theory. Short periodic terms are eliminated from the maximum Hamiltonian, expressed in non-singular orbital elements, through an infinitesimal canonical transformation built through the Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton–Jacobi equation through the separation of variables technique. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition.

© 2015, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.A study of optimal bi-impulsive trajectories of round trip lunar missions is presented in this paper. The optimization criterion is the total velocity increment. The dynamical model utilized to describe the motion of the space vehicle is a full lunar patched-conic approximation, which embraces the lunar patched-conic of the outgoing trip and the lunar patched-conic of the return mission. Each one of these parts is considered separately to solve an optimization problem of two degrees of freedom. The parameters to be optimized are two: the phase angle of the point at which the space vehicle reaches the edge of the Moon’s sphere of influence and the initial velocity at departure. The Sequential Gradient Restoration Algorithm is employed to achieve the optimal solutions. Analytical and numerical derivatives of expressions describing the lunar patched-conic approximations are utilized to ensure the results. The results based on the patched-conic approximation show a good agreement with the ones provided by literature, and the solution trajectories proved to be consistent with the image trajectories theorem.

A semi-analytical approach is proposed to study resonances effects on the orbital motion of artificial satellites or space debris orbiting the Earth. Applying successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, we can choose the resonance to be considered as critical angle. Simulations are presented showing the variations of the orbital elements of bodies orbiting in the neighbourhood of the 2:1, 14:1 and 15:1 resonance condition. The half-width of the separatrix is calculated through a linearized model which describes the behavior of the dynamical system in a neighborhood of each critical angle. A semi-analytical approach is proposed to study resonances effects on the orbital motion of artificial satellites or space debris orbiting the Earth. Applying successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, we can choose the resonance to be considered as critical angle. Simulations are presented showing the variations of the orbital elements of bodies orbiting in the neighbourhood of the 2:1, 14:1 and 15:1 resonance condition. The half-width of the separatrix is calculated through a linearized model which describes the behavior of the dynamical system in a neighborhood of each critical angle. © 2013 2013 California Institute of Technology.

A numerical study of optimal low-thrust limited power trajectories for simple transfer (no rendezvous) between circular coplanar orbits in an inverse-square force field is performed by two different classes of algorithms in optimization of trajectories. This study is carried out by means of a direct method based on gradient techniques and by an indirect method based on the second variation theory. The direct approach of the trajectory optimization problem combines the main positive characteristics of two well-known direct methods in optimization of trajectories: the steepest-descent (first-order gradient) method and a direct second variation (second-order gradient) method. On the other hand, the indirect approach of the trajectory optimization problem involves two different algorithms of the well-known neighboring extremals method. Several radius ratios and transfer durations are considered, and the fuel consumption is taken as the performance criterion. For small-amplitude transfers, the results are compared to the ones provided by a linear analytical theory. © 2012 Sandro da Silva Fernandes et al.

Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. These simplified algorithms are applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation. Copyright © 2012 Sandro da Silva Fernandes.

In this work, the resonance problem in the artificial satellites motion is studied. The development of the geopotential includes the zonal harmonics J20 and J40 and the tesseral harmonics J22 and J42. Through successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, two critical angles are studied, 2201 and 4211. Numerical results show the time behavior of the semi-major axis and 2 angle.

The orbital dynamics of synchronous satellites is studied. The 2:1 resonance is considered; in other words, the satellite completes two revolutions while the Earth completes one. In the development of the geopotential, the zonal harmonics J 20 and J 40 and the tesseral harmonics J 22 and J 42 are considered. The order of the dynamical system is reduced through successive Mathieu transformations, and the final system is solved by numerical integration. The Lyapunov exponents are used as tool to analyze the chaotic orbits. Copyright © 2012 Jarbas Cordeiro Sampaio et al.

A study of optimal two-impulse trajectories with moderate flight time for Earth-Moon missions is presented. The optimization criterion is the total characteristic velocity. Three dynamical models are used to describe the motion of the space vehicle: the well-known patched-conic approximation and two versions of the planar circular restricted three-body problem (PCR3BP). In the patched-conic approximation model, the parameters to be optimized are two: initial phase angle of space vehicle and the first velocity impulse. In the PCR3BP models, the parameters to be optimized are four: initial phase angle of space vehicle, flight time, and the first and the second velocity impulses. In all cases, the optimization problem has one degree of freedom and can be solved by means of an algorithm based on gradient method in conjunction with Newton-Raphson method. © 2012 Sandro da Silva Fernandes and Cleverson Maranho Porto Marinho.

In the present work, the influence of the Sun on the fuel consumption of transfers from circular low Earth orbits (LEOs) to circular low Moon orbits (LMOs) is investigated. The class of two impulse trajectories is considered: a first accelerating velocity impulse tangential to the space vehicle velocity relative to Earth is applied at a circular low Earth orbit and a second braking velocity impulse tangential to the space vehicle velocity relative to Moon is applied at a circular low Moon orbit. The fuel consumption is equivalent to the total characteristic velocity which is defined by the arithmetic sum of velocity changes. Local optimal transfers are calculated through two different approaches: inner transfers and Belbruno-Miller transfers. In both cases, the optimization problem is solved by means of an algorithm based on gradient method in conjunction with Newton-Raphson method.

Numerical and first-order analytical results are presented for optimal low-thrust limited-power trajectories in a gravity field that includes the second zonal harmonic J 2 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 elementsposition and velocity vectorsas state variables. After applying the Pontryagin Maximum Principle, successive canonical transformations are performed and a suitable set of orbital elements is introduced. Hori methoda perturbation technique based on Lie seriesis 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. © 2009 Sandro da Silva Fernandes.

In this paper, some special features on the canonical version of Hori method, when it is applied to generalized canonical systems (systems of differential equations described by a Hamiltonian function linear in the momenta), are presented. Two different procedures, based on a new approach for the integration theory recently presented for the canonical version, are proposed for determining the new Hamiltonian and the generating function for systems whose differential equations for the coordinates describe a periodic system with one fast phase. These procedures are equivalent and they are directly related to the canonical transformations defined by the general solution of the integrable kernel of the Hamiltonian. They provide the same near-identity transformation for the coordinates obtained through the non-canonical version of Hori method. It is also shown that these procedures are connected to the classic averaging principle through a canonical transformation. As examples, asymptotic solutions of a non-linear oscillations problem and of the elliptic perturbed problem are discussed. © 2008 Elsevier Ltd. All rights reserved.

A complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elementsposition and velocity vectorsas state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.

A numerical and analytical study of optimal low-thrust limited-power trajectories for simple transfer (no rendezvous) between close circular coplanar orbits in an inverse-square force field is presented. The numerical study is carried out by means of an indirect approach of the optimization problem in which the two-point boundary value problem, obtained from the set of necessary conditions describing the optimal solutions, is solved through a neighboring extremal algorithm based on the solution of the linearized two-point boundary value problem through Riccati transformation. The analytical study is provided by a linear theory which is expressed in terms of nonsingular elements and is determined through the canonical transformation theory. The fuel consumption is taken as the performance criterion and the analysis is carried out considering various radius ratios and transfer durations. The results are compared to the ones provided by a numerical method based on gradient techniques.

An algorithm based on gradient techniques, proposed in a companion paper, is applied to numerical analysis of optimal low-thrust limited-power trajectories for simple transfer (no rendezvous) between coplanar circular orbits in a central Newtonian gravity field. The proposed algorithm combines the main positive characteristics of two well-known methods in optimization of trajectories: the steepest-descent method and the direct second variation method. The analysis is carried out for various radius ratios and transfer durations. The results are compared to the ones provided by a linear analytical theory. The performance of the proposed algorithm shows that it is a good tool in determining optimal low-thrust limited-power trajectories between close circular coplanar orbits in a Newtonian central gravity field.

In this paper the new approach for the integration theory of the canonical version of Hori method recently proposed is extended to the non-canonical one. It will be shown that the non-homogeneous ordinary differential equation with an auxiliary parameter t* associated with the mth order equation of the algorithm can also be replaced by a non-homogeneous partial differential equation in the time t. Using a generalized canonical approach, the general algorithm proposed by Sessin is then revised; as well as the Lagrange variational equations for the non-canonical version of Hori method. A simplified algorithm derived from Sessin's algorithm is presented for non-linear oscillations problem.

A first order analytical model for optimal small amplitude attitude maneuvers of spacecraft with cylindrical symmetry in an elliptical orbits is presented. The optimization problem is formulated as a Mayer problem with the control torques provided by a power limited propulsion system. The state is defined by Serret-Andoyer's variables and the control by the components of the propulsive torques. The Pontryagin Maximum Principle is applied to the problem and the optimal torques are given explicitly in Serret-Andoyer's variables and their adjoints. For small amplitude attitude maneuvers, the optimal Hamiltonian function is linearized around a reference attitude. A complete first order analytical solution is obtained by simple quadrature and is expressed through a linear algebraic system involving the initial values of the adjoint variables. A numerical solution is obtained by taking the Euler angles formulation of the problem, solving the two-point boundary problem through the shooting method, and, then, determining the Serret-Andoyer variables through Serret-Andoyer transformation. Numerical results show that the first order solution provides a good approximation to the optimal control law and also that is possible to establish an optimal control law for the artificial satellite's attitude. © 2003 COSPAR. Published by Elsevier Science Ltd. All rights reserved.

In this paper a slightly different approach is proposed for the process of determining the functions Sm and H*m of the algorithm of the canonical version of Hori method. This process will be referred to as integration theory of the mth order equation of the method. It will be shown that the ordinary differential equation with an auxiliary parameter t* as independent variable, introduced through Hori auxiliary system, can be replaced by a partial differential equation in the time t. In this way, the mth order equation of the algorithm assumes a form very similar to the one of other perturbation methods. In virtue of this new approach of the integration theory for Hori method, Lagrange's variational equations introduced by Sessin are revised. As an example, the Duffing equation is solved through this new approach.

The solution of the coast-arc problem is derived by using Sundman transformation and properties of generalized canonical systems. An universal closed-form solution, valid for all orbits, is obtained as function of a generalized anomaly. Simplifications for near parabolic orbits are also presented and the singularities for near equatorial orbits are eliminated by introducing a set of nonsingular orbits. © 2002 Elsevier Science Ltd. All rights reserved.

Some properties of generalized canonical systems - special dynamical systems described by a Hamiltonian function linear in the adjoint variables - are applied in determining the solution of the two-dimensional coast-arc problem in an inverse-square gravity field. A complete closed-form solution for Lagrangian multipliers - adjoint variables - is obtained by means of such properties for elliptic. circular, parabolic and hyperbolic motions. Classic orbital elements are taken as constants of integration of this solution in the case of elliptic, parabolic and hyperbolic motions. For circular motion, a set of nonsingular orbital elements is introduced as constants of integration in order to eliminate the singularity of the solution.

A closed-form solution of the coast-arc problem in a Newtonian central force field is derived applying properties of generalized canonical systems for elliptic, circular, parabolic, and hyperbolic motions. This generalized canonical approach involves a set of Mathieu transformations and requires the evaluation of only one integral related to the Kepler's classic equation.

In this note, properties of generalized canonical systems with first integrals are presented. These properties are applied in the analysis of a system of differential equations described by a Hamiltonian function whose integrable kernel is linear in coordinates and momenta.

In this note the solution of the coast-arc problem in Newtonian central field derived by means of properties of generalized canonical systems is revised. A different set of orbital elements is taken as arbitrary constants of integration that provides an unified approach for elliptical, hyperbolic and parabolic motions.

In this paper, we discuss a method of preliminary orbit determination for an artificial satellite based on the navigation message of the GPS constellation. Orbital elements are considered as state variables and a simple dynamic model, based on the classic two-body problem, is used. The observations are formed by range and range and range-rate with respect to four visible GPS. A discrete Kalman filter with simulated data is used as filtering technique. The data are obtained through numerical propagation (Cowell's method), which considers special perturbations for the GPS satellite constellation and a user satellite. © 1997 COSPAR. Published by Elsevier Science Ltd.

Some classic expansions of the elliptic motion - cos mE and sin mE - in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e-e*), where e* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence ρ(e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions - (a/r), (r/a), (r/a) sin v, ..., - derived straightforward from the expansions of E, cos E and sin E are also presented. © 1995 Kluwer Academic Publishers.

A complete first-order analytical solution is developed for the problem of optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements as state variables. After applying the Pontryagin maximum principle and determining the optimal thrust acceleration, an intrinsic canonical transformation is performed: the Cartesian elements are changed by suitable orbital elements. Hori's method is applied in determining a first-order analytical solution. Simple analytical solutions are obtained explicitly for long-time transfers. © 1995.

The classic Lagrange's expansion of the solution E(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solution E(e, M) is developed in powers of (e-e*), where e* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functions Jn(ne). The expansion is convergent for values of the eccentricity such that |e-e*|<ρ(e*), where the radius of convergence ρ(e*) is a positive real number, which is calculated numerically. © 1994 Kluwer Academic Publishers.

A complete first order analytical solution is developed for the problem of optimal low-thrust limited power transfers between neighbouring quasi-circular orbits in a noncentral gravity field generated by an ellipsoid. The optimization problem is formulated as a Mayer problem of optimal control with cartesian elements as state variables. After applying the Pontryagin Maximum Principle and determining the optimal thrust acceleration, successive canonical transformations are performed: the cartesian elements are changed by suitable orbital elements. Hori's method is applied in solving the new canonical systems which govern the optimal trajectories.

This paper deals with the study of the singular solutions of optimal impulsive transfer problem between close quasi-circular-equatorial orbits, considering the Earth oblateness. These solutions are similarly analysed as those performed by Marec for Keplerian orbits. © 1989.

This paper is an extension of the work by Marec, Edelbaum and Kouzmak in the sixties. It shows the influence of the oblateness of the Earth on the optimal impulsive transfer between close quasi-circular-equatorial orbits. © 1989.

A complete analytic study about the influence of Earth's oblateness on the optimal low-thrust limited power transfer of small amplitude (orbit correction) between quasi-circular orbits of small inclinations is carried out up to the first order in a small parameter defined by the nondimensional thrust acceleration. The coefficient for the second zonal harmonic J20 and the nondimensional thrust acceleration are supposed to be the same order of magnitude. Hori's method for generalized canonical systems is applied in order to obtain the analytical solution for adjoint and state differential equations. Simple analytic solutions are obtained explicitly for long-time transfer. © 1989.

The problem of optimal low-thrust, limited power transfer between quasi-circular orbits (e {reversed tilde equals} 0) around an oblate planet is analysed. It is assumed that the orbital changes due to thrust acceleration and Earth oblateness are of the same order. A first order solution to the problem is obtained by application of Pontryagin's Maximum Principle. Subsequently, by application of Hori's method for generalized canonical systems, a first order solution in a small parameter ε{lunate} is derived. Finally, three particular cases of long-time transfer and the orbit maintenance manoeuvre are considered. The results obtained are in agreement and represent an extension of the work done by Marec. © 1989.