Extended analytical formulas for the perturbed Keplerian motion under a constant control acceleration

Federico Zuiani, Massimiliano Vasile

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)
169 Downloads (Pure)


This paper presents a set of analytical formulae for the perturbed Keplerian motion of a spacecraft under the effect of a constant control acceleration. The proposed set of formulae can treat control accelerations that are fixed in either a rotating or inertial reference frame. Moreover, the contribution of the (Formula presented.) zonal harmonic is included in the analytical formulae. It will be shown that the proposed analytical theory allows for the fast computation of long, multi-revolution spirals while maintaining good accuracy. The combined effect of different perturbations and of the shadow regions due to solar eclipse is also included. Furthermore, a simplified control parameterisation is introduced to optimise thrusting patterns with two thrust arcs and two cost arcs per revolution. This simple parameterisation is shown to ensure enough flexibility to describe complex low thrust spirals. The accuracy and speed of the proposed analytical formulae are compared against a full numerical integration with different integration schemes. An averaging technique is then proposed as an application of the analytical formulae. Finally, the paper presents an example of design of an optimal low-thrust spiral to transfer a spacecraft from an elliptical to a circular orbit around the Earth.

Original languageEnglish
Pages (from-to)275-300
Number of pages26
JournalCelestial Mechanics and Dynamical Astronomy
Issue number3
Early online date27 Dec 2014
Publication statusPublished - 31 Mar 2015


  • analytical solutions
  • first order expansions
  • low-thrust trajectories
  • orbit averaging
  • orbit circularisation

Fingerprint Dive into the research topics of 'Extended analytical formulas for the perturbed Keplerian motion under a constant control acceleration'. Together they form a unique fingerprint.

Cite this