Skip to main content

Optimal circle-to-rectilinear orbit transfer with circumferential thrust


This paper investigates the optimal transfer trajectories from a circular parking orbit towards the apocenter of a rectilinear ellipse, where the spacecraft reaches a quasi-stationary condition relative to an inertial reference frame. The spacecraft is equipped with a propulsion system that provides a circumferential continuous propulsive acceleration, that is, an acceleration whose direction is perpendicular to the primary body-spacecraft line. The performance index to minimize is the total flight time, and an indirect method is used to analyze the transfer trajectories. In this context, the optimal transfer performance is obtained as a function of the spacecraft propulsive acceleration magnitude through an interpolation procedure of numerical simulations. The results obtained with a continuous thrust propulsion system are also compared with those derived from a multi-impulse transfer. Finally, the paper investigates a heliocentric mission scenario in which the spacecraft minimizes the flight time required to reach a rectilinear ellipse with a given value of the aphelion radius.

This is a preview of subscription content, access via your institution.


  1. Tsien, H. S. Take-off from satellite orbit. Journal of the American Rocket Society, 1953, 23(4): 233–236.

    Article  Google Scholar 

  2. Battin, R. H. An introduction to the mathematics and methods of astrodynamics, revised edition. AIAA, 1999, 408–418.

    Book  Google Scholar 

  3. Petropoulos, A. E., Sims, J. A. A review of some exact solutions to the planar equations of motion of a thrusting spacecraft. In: Proceedings of the 2nd International Symposium on Low-Thrust Trajectory (LoTus-2), 2002.

    Google Scholar 

  4. Niccolai, L., Quarta, A. A., Mengali, G. Orbital motion approximation with constant circumferential acceleration. Journal of Guidance, Control, and Dynamics, 2018, 41(8): 1783–1789.

    Article  Google Scholar 

  5. Bombardelli, C., Baù, G., Peláez, J. Asymptotic solution for the two-body problem with constant tangential thrust acceleration. Celestial Mechanics and Dynamical Astronomy, 2011, 110(3): 239–256.

    MathSciNet  MATH  Article  Google Scholar 

  6. Quarta, A. A., Mengali, G. Analysis of spacecraft motion under constant circumferential propulsive acceleration. Acta Astronautica, 2014, 105(1): 278–284.

    Article  Google Scholar 

  7. Roy, A. E. Orbital motion. Advances in Design and Control, 2005, 87–89.

    Google Scholar 

  8. Dandouras, I., Pirard, B., Prado, J. Y. High performance solar sails for linear trajectories and heliostationary missions. Advances in Space Research, 2004, 34(1): 198–203.

    Article  Google Scholar 

  9. Mengali, G., Quarta, A. A. Optimal heliostationary missions of high-performance sailcraft. Acta Astronautica, 2007, 60(8–9): 676–683.

    Article  Google Scholar 

  10. Quarta, A. A., Mengali, G. Solar sail capabilities to reach elliptic rectilinear orbits. Journal of Guidance, Control, and Dynamics, 2011, 34(3): 923–927.

    Article  Google Scholar 

  11. Quarta, A. A., Mengali, G. Optimal solar sail transfer to linear trajectories. Acta Astronautica, 2013, 82(2): 189–196.

    Article  Google Scholar 

  12. Colombo, G., Lautman, D. A., Pettengill, G. An alternative option to the dual-probe out-of-ecliptic mission via Jupiter swingby. In: Proceedings of Symposium on the Study of the Sun and Interplanetary Medium in Three Dimensions, 1976, 37–47.

    Google Scholar 

  13. Mengali, G., Quarta, A. A., Romagnoli, D., Circi, C. H2-reversal trajectory: a new mission application for high-performance solar sails. Advances in Space Research, 2011, 48(11): 1763–1777.

    Article  Google Scholar 

  14. Zeng, X. Y., Baoyin, H. X., Li, J. F., Gong, S. P. New applications of the H-reversal trajectory using solar sails. Research in Astronomy and Astrophysics, 2011, 11(7): 863–878.

    Article  Google Scholar 

  15. Bryson, A. E., Ho, Y. C. Applied optimal control. Hemisphere, 1975, 71–89.

    Google Scholar 

  16. Stengel, R. F. Optimal control and estimation. Dover Publications, 1994, 222–254.

    MATH  Google Scholar 

  17. Mengali, G., Quarta, A. A. Optimal three-dimensional interplanetary rendezvous using nonideal solar sail. Journal of Guidance, Control, and Dynamics, 2005, 28(1): 173–177.

    Article  Google Scholar 

  18. Shampine, L. F., Gordon, M. K. Computer solution of ordinary differential equations: the initial value problem. W. H. Freeman & Company, 1975, 72(460): 155.

    MATH  Google Scholar 

  19. Shampine, L. F., Reichelt, M. W. The MATLAB ODE suite. SIAM Journal on Scientific Computing, 1997, 18(1): 1–22.

    MathSciNet  MATH  Article  Google Scholar 

  20. Sims, J. A., Flanagan, S. N. Preliminary design of low-thrust interplanetary missions. In: Proceedings of the AAS/AIAA Astrodynamics Specialist Conference and Exhibit, 1999, 99–338.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Alessandro A. Quarta.

Additional information

Andrea Caruso received his B.S. and M.S. degrees in aerospace engineering from University of Pisa, in 2013 and 2016, respectively. He is currently a Ph.D. student at the University of Pisa. His research interests are in spaceflight mechanics and spacecraft trajectory optimization.

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

Giovanni Mengali received his doctor engineer degree in aeronautical engineering, in 1989, from the University of Pisa. 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 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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Quarta, A.A., Mengali, G. & Caruso, A. Optimal circle-to-rectilinear orbit transfer with circumferential thrust. Astrodyn 3, 31–43 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • circumferential acceleration
  • rectilinear ellipse
  • optimal transfer
  • preliminary mission analysis
  • quasi-stationary condition