@Article{FernandesCarv:2021:EfMaZo,
               author = "Fernandes, Sandro da Silva and Carvalho, Francisco das Chagas",
          affiliation = "{Instituto Tecnol{\'o}gico de Aeron{\'a}utica (ITA)} and 
                         {Instituto Nacional de Pesquisas Espaciais (INPE)}",
                title = "Effects of the main zonal harmonics on optimal low-thrust 
                         limited-power transfers",
              journal = "Journal of the Brazilian Society of Mechanical Sciences and 
                         Engineering",
                 year = "2021",
               volume = "43",
               number = "12",
                pages = "e523",
                month = "Dec.",
             keywords = "Low-thrust trajectories, Zonal harmonics of gravitational field, 
                         Transfers between arbitrary orbits.",
             abstract = "This work considers the development of a numerical-analytical 
                         procedure for computing optimal time-fixed low-thrust 
                         limited-power transfers between arbitrary orbits. It is assumed 
                         that Earth's gravitational field is described by the main three 
                         zonal harmonics J(2), J(3) and J(4). The optimization problem is 
                         formulated as a Mayer problem of optimal control with the state 
                         variables defined by the Cartesian elements-components of the 
                         position vector and the velocity vector-and a consumption variable 
                         that describes the fuel spent during the maneuver. Pontryagin 
                         Maximum Principle is applied to determine the optimal thrust 
                         acceleration. A set of classical orbital elements is introduced as 
                         a new set of state variables by means of an intrinsic canonical 
                         transformation defined by the general solution of the canonical 
                         system described by the undisturbed part of the maximum 
                         Hamiltonian. The proposed procedure involves the development of a 
                         two-stage algorithm to solve the two-point boundary value problem 
                         that defines the transfer problem. In the first stage of the 
                         algorithm, a neighboring extremals method is applied to solve the 
                         {"}mean{"} two-point boundary value problem of going from an 
                         initial orbit to a final orbit at a prescribed final time. This 
                         boundary value problem is described by the mean canonical system 
                         that governs the secular behavior of the optimal trajectories. The 
                         maximum Hamiltonian function that governs the mean canonical 
                         system is computed by applying the classic concept of {"}mean 
                         Hamiltonian{"}. In the second stage, the well-known Newton-Raphson 
                         method is applied to adjust the initial values of adjoint 
                         variables when periodic terms of the first order are included. 
                         These periodic terms are recovered by computing the Poisson 
                         brackets in the transformation equations, which are defined 
                         between the original set of canonical variables and the new set of 
                         average canonical variables, as described in Hori method. 
                         Numerical results show the main effects on the optimal 
                         trajectories due to the zonal harmonics considered in this 
                         study.",
                  doi = "10.1007/s40430-021-03229-5",
                  url = "http://dx.doi.org/10.1007/s40430-021-03229-5",
                 issn = "1678-5878",
             language = "en",
           targetfile = "fernandes_effects.pdf",
        urlaccessdate = "04 jun. 2024"
}