Comparison between direct and indirect approach to solar sail circle-to-circle orbit raising optimization


This paper deals with the optimization of the transfer trajectory of a solar sail-based spacecraft between circular and coplanar heliocentric orbits. The problem is addressed using both a direct and an indirect approach, while an ideal and an optical force model are used to describe the propulsive acceleration of a flat solar sail. In the direct approach, the total flight time is partitioned into arcs of equal duration, within which the sail attitude is assumed to be constant with respect to an orbital reference frame, and a nonlinear programming solver is used to optimize the transfer trajectory. The aim of the paper is to compare the performance of the two (direct and indirect) approaches in term of optimal (minimum) flight time. In this context, the simulation results show that a direct transcription method using a small number of arcs is sufficient to obtain a good estimate of the global minimum flight time obtained through the classical calculus of variation.

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a :

propulsive acceleration vector (mm/s2)

a c :

characteristic acceleration (mm/s2)

a r :

radial component of propulsive acceleration (mm/s2)

a θ :

circumferential component of propulsive acceleration (mm/s2)

a :

gravitational acceleration at 1AU (mm/s2)

b 1, b 2, b 3 :

sail force coefficients

ê r :

radial unit vector

ê θ :

circumferential unit vector

\(\mathcal{H}\) :

Hamiltonian function

i f :

inclination of the target orbit (deg)

J :

performance index

N :

number of arcs

\(\hat{n}\) :

sail normal unit vector

O :

Sun’s center of mass

r :

Sun-spacecraft distance (AU)

\(\mathcal{T}\)(O; r, θ):

polar reference frame

]t :

time (days)

u :

radial component of spacecraft velocity (km/s)

v :

circumferential component of spacecraft velocity (km/s)

x :

state vector

z :

nonlinear programming parameter vector

α :

sail pitch angle (deg)

α n :

sail cone angle (deg)

β :

lightness number

δ :

nonlinear equality constraints

δn :

sail clock angle (deg)

ε :

dimensionless error

Δt :

flight time (days)

θ :

polar angle (deg)

λn, λθ, λu, λv :

adjoint variables

μ :

Sun’s gravitational parameter (km3/s2)






initial guess


numerical result




time derivative


evaluated using direct approach


evaluated using indirect approach


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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 a 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 Ph.D. 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 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.

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Caruso, A., Quarta, A.A. & Mengali, G. Comparison between direct and indirect approach to solar sail circle-to-circle orbit raising optimization. Astrodyn 3, 273–284 (2019).

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  • flat solar sail
  • circle-to-circle transfer
  • heliocentric mission analysis
  • trajectory optimization