Envelopes of Spacecraft Trajectories with a Single Impulse

  • Andrea CarusoEmail author
  • Lorenzo Niccolai
  • Alessandro A. Quarta
  • Giovanni Mengali
Original article


This paper analyzes reachable domains of spacecraft using a single impulsive maneuver. In particular, compact expressions are obtained of the envelopes of spacecraft trajectories in closed form, in both cases of either radial or tangential impulse. Suitable bounds are enforced on the magnitude of the velocity variation to obtain an elliptic transfer trajectory after the maneuver with a pericenter radius greater than the primary body’s radius. Three different cases are investigated: (1) the impulse point is fixed and the velocity variation may be varied; (2) a fixed impulse magnitude and free maneuver point; (3) both free impulse point and magnitude. Finally, some mission scenarios are analyzed to show the effectiveness of the proposed analytical method.


Impulsive maneuver Reachable domain Trajectory envelopes Radial impulse Tangential impulse 


A, B

Auxiliary variables, see Eq. (12)




Angular momentum, [\(\mathrm{km}^2/\mathrm{s}\)]


Primary’s center of mass


Semilatus rectum, [\(\,\hbox {km}\)]


Primary-spacecraft distance, [\(\,\hbox {km}\)]


Spacecraft orbital velocity, [\(\,\hbox {km/s}\)]

\(\varDelta u\)

Radial velocity variation, [\(\,\hbox {km/s}\)]

\(\varDelta v\)

Tangential velocity variation, [\(\,\hbox {km/s}\)]


Auxiliary variable, see Eq. (13)


Polar angle, [\(\,\hbox {rad}\)]


Primary’s gravitational parameter, [\(\hbox { km}^3/\hbox {s}^2\)]

\(\nu _m\)

True anomaly at maneuver point, [\(\,\hbox {rad}\)]


Auxiliary variable, see Eq. (6)



Value just before the maneuver


Value just after the maneuver


Maximum allowable value


Minimum allowable value



This work is partly supported by the University of Pisa, Progetti di Ricerca di Ateneo (Grant no. PRA_2018_44).

Compliance with ethical standards

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© AIDAA Associazione Italiana di Aeronautica e Astronautica 2019

Authors and Affiliations

  • Andrea Caruso
    • 1
    Email author
  • Lorenzo Niccolai
    • 1
  • Alessandro A. Quarta
    • 1
  • Giovanni Mengali
    • 1
  1. 1.University of PisaPisaItaly

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