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Analytical solution to logarithmic spiral trajectories with circumferential thrust and mission applications


This study made use of a shape-based method to analyze the orbital dynamics of a spacecraft subject to a continuous propulsive acceleration acting along the circumferential direction. Under the assumption of a logarithmic spiral trajectory, an exact solution to the equations of motion exists, which allows the spacecraft state variables and flight time to be expressed as a function of the angular coordinate. There is also a case characterized by specific initial conditions in which the time evolution of the state variables may be analytically determined. In this context, the presented solution is used to analyze circle-to-circle trajectories, where the combination of two impulsive maneuvers and a logarithmic spiral path are used to accomplish the transfer. The determined results are then applied to the achievement of the Earth—Mars and the Earth—Venus transfers using actual data from a recent thruster developed by NASA.


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Correspondence to Marco Bassetto.

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Marco Bassetto received his Ph.D. degree in civil and industrial engineering at the Department of Civil and Industrial Engineering of the University of Pisa, Italy. From January 2020 to June 2021, he was the holder of a scholarship entitled “Dynamic analysis and control of an E-sail” at the same department, where he has been a research fellow since July 2021. His research activity focuses on the trajectory design and attitude control of spacecraft propelled with low-thrust propulsion systems such as solar sails and electric solar wind sails. E-mail:

Alessandro A. Quarta received his Ph.D. degree in aerospace engineering from the University of Pisa, Italy, in 2005 and, currently, he is a professor of flight mechanics at the Department of Civil and Industrial Engineering of the University of Pisa, Italy. His main research areas include spaceflight simulation, spacecraft mission analysis and design, low-thrust trajectory optimization, solar sail, and E-sail dynamics and control. E-mail:

Giovanni Mengali received his Doctor Engineer degree in aeronautical engineering in 1989 from the University of Pisa, Italy. Since 1990, he has been with the Department of Aerospace Engineering (now Department of Civil and Industrial Engineering) of the University of Pisa, first as a Ph.D. student, then as an assistant and an associate professor. Currently, he is a professor of space flight mechanics. His main research areas include spacecraft mission analysis, trajectory optimization, solar sails, electric sails, and aircraft flight dynamics and control. E-mail:

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Bassetto, M., Quarta, A.A. & Mengali, G. Analytical solution to logarithmic spiral trajectories with circumferential thrust and mission applications. Astrodyn (2022).

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  • circumferential thrust
  • logarithmic spiral trajectory
  • circle-to-circle transfer