Abstract
The solar radiation effects upon the orbital behaviour of an arbitrarily shaped spacecraft (or a solar sail in particular) in a general fixed orientation with respect to the local coordinate frame are investigated. Through introduction of a quasi-angle in the osculating plane, the motion of the orbital plane becomes uncoupled from the in-plane perturbations. Exact solutions in the form of conic sections and logarithmic spirals can readily be formulated for certain specific initial conditions. An effective out-of-plane spiral transfer trajectory is obtained by reversing the force component normal to the orbital plane at specified positions in the orbit. By choosing the appropriate control angles for the sail orientation, any point in space can be reached eventually. In the case of general initial conditions, the long-term orbital behaviour is assessed asymptotically by means of the two-variable expansion procedure. An implicit expression for the eccentricity is derived and explicit results are established by an iteration scheme. The other orbital elements can be expressed in terms of the eccentricity and their asymptotic series for near-circular initial orbits are also obtained.
While equations for the higher-order contributions as well as the periodic parts of their solutions can be formulated readily, their secular terms are determined only for a circular initial orbit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- a :
-
semi-major axis
- a e :
-
semi-major axis of Earth's orbit, 1 A.U.=1.496×108 km
- a j nk ,b j nk :
-
slowly varying Fourier coefficients, Appendix II
- c j nk ,d j nk :
-
slowly varying Fourier coefficients, Appendix II
- c s :
-
constant in spiral trajectory −u′(ν)/u(ν)
- c t :
-
constant, Equation (9)
- e :
-
eccentricity
- e f ,e b :
-
emissivities of front, and back side of surface element
- e p :
-
modified eccentricity\(\left[ {e^2 _{00} + 2\varepsilon _s p_{00} R + \varepsilon _s ^2 R^2 } \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} /\left( {1 - \varepsilon _s R} \right)\)
- h :
-
angular momentum (per unit mass) vector,r×v
- i :
-
inclination of orbital plane with respect to ecliptic
- l :
-
semi-latus rectum
- l p :
-
modified semi-latus rectum,l 00/(1-ε s R)
- m :
-
mass of satellite
- n :
-
number of illuminated surface components
- p :
-
auxiliary element,e cos\(\tilde \omega \)
- q :
-
auxiliary element,e sin\(\tilde \omega \)
- r:
-
radius vector, pointing from origin to satellite, Figute 1
- t :
-
time (non-dimensional)
- u :
-
inverse radius, 1/r
- u n k :
-
(u n kx ,u n ky ,u n kz ), unit vector normal to surface componentA k with components along local coordinate axes
- u s :
-
(u s x ,u s y ,u s z ), unit vector along direction of radiation with components along local coordinate axes
- w :
-
auxiliary element, 1−(1−e 2)1/2
- A :
-
total effective illuminated surface area of the satellite, cross-sectional area for spherical satellite
- \(A_0 (\bar v),B_0 (\bar v)\) :
-
auxiliary functions in expression forM 0, Equation (21)
- \(A_j (\bar v),B_j (\bar v)\) :
-
(j=2, 3, 4, 5) auxiliary functions in second-order results, Equation (41)
- A k :
-
k-th surface component (non-dimensional)
- A nk ,B nk :
-
(n=1, 2, ...;k=0, 1, 2, ...) integrals, defined and evaluated in Appendix I
- B :
-
auxiliary constant,c s T/(2S)=ε s T/C
- C :
-
constant in spiral trajectory,u(ν)l(ν)
- \(C_0 (\bar v)\) :
-
auxiliary function of\(\bar v\), (A 2 0+B 2 0)1/2
- F=(F x ,F y ,F z ):
-
solar radiation force with components along local coordinate axes
- K :
-
component ofK along orbit-normal
- K k :
-
K(K k )
- K :
-
unit vector alongZ-axis with components along local coordinate axes
- L :
-
component ofK along local horizontal
- M :
-
component ofK along local vertical
- M k :
-
M(v k )
- R :
-
component ofR along local vertical
- R=(R, S, T):
-
functions of rotation angles α and β, and material properties, Equation (4) with components along local reference frame
- S :
-
component ofR along local horizontal
- S c :
-
solar constant, 1.35 kW m−2
- S′:
-
solar radiation pressure,S c /(velocity of light), 4.51×10−6 N m−2
- T :
-
component ofR along orbit-normal
- T f ,T b :
-
temperature of front, and back side of surface element
- W :
-
rotation vector,w r+\(\dot v\)
- X, Y, Z :
-
inertial reference axes, Figure 1
- α=(α, β, γ):
-
Eulerian control angles defining orientation of surface element with respect to orbital plane, Figure 1
- α k , β k :
-
Eulerian control angles for surface componentA k
- α s :
-
spiral angle, arctan (c s )
- ε s :
-
ratio of solar radiation and gravity forces for heliocentric orbits, 2S′(A/m)a 2 e/μ s =1.52×10−3(A/m)
- κ:
-
material parameter (e f T f 4−e b T b 4)/(e f T f 4+e b T b 4)
- μ s :
-
Sun's gravitational parameter, 1.326×1020 m3 s−2
- ν:
-
quasi-angle in osculating plane\(\dot v = \dot \phi \), ν(0)=0, employed as independent variable
- \(\bar v\) :
-
slow independent variable,ε s v
- v k :
-
(k=0, 1, 2, ...) abbreviation forv k =kπ/(1+B 2)1/2
- ξ0,η0,ζ0 :
-
reference axes, fixed to osculating plane in heliocentric orbits, ξ0 along the local vertical, η0 along the local horizontal and ζ0 along the orbit-normal, Figure 1
- ξ1,η1,ζ1 :
-
intermediate frame of reference after rotation of solar sail by α, Figure 1
- ξ, η, ζ:
-
reference frame fixed to solar sail after rotations by α and β, Figure 1
- ρ:
-
material parameter characterizing specular reflectivity of surface component, ρ1ρ2
- ρ1 :
-
portion of incident photons which are reflected
- ρ2 :
-
portion of reflected photons which are reflected specularly
- ρ b ,ρ f :
-
specular reflectivity for back and front side of surface element, respectively
- ρ k :
-
specular reflectivity for surface componentA k
- σ:
-
material parameter for homogeneous flat plate, σ1+σ2+ρ, or homogeneous sphere, (1-τ)/2+2σ2/3
- σ1, σ2 :
-
material parameters σ1=(1-ρ-τ)/2 and σ2=[ρ1(1-ρ2)+κ(1-ρ1-τ)]/3
- σ1k ,σ12 :
-
σ1 and σ2 for surface componentA k
- τ:
-
material parameter denoting transmissivity of surface element
- ϕ:
-
argument of latitude, i.e. position angle of satellite as measured from the line of nodes
- ψ:
-
angle characterizing shift of orbital plane, ν−ϕ
- ω:
-
argument of the perihelion with respect to the line of nodes
- \(\tilde \omega \) :
-
position of the perihelion measured in osculating plane from axis ν=2πk
- \(\tilde \omega \) p :
-
modified position of the perihelion, arctan [q 00/p 00+ε s R)]
- Ω:
-
longitude of ascending node, measured from the autumnal equinox, Figure 1
References
Kiefer, J. W.: 1965,Proceedings of the 15th International Astronautical Congress of the I.A.F., Gauthier-Villars, pp. 383–416.
London, H. S.: 1960,ARS Journal 30, 198–200.
Modi, V. J., Pande, K. C., and Nicks, G. W.: 1973,Proceedings of the 10th International Symposium on Space Technology and Science, AGNE Publishing Company, pp. 375–382.
Nayfeh, A. J.: 1973,Perturbation Methods, Wiley, Ch. 6.
Pozzi, A. and De Socio, L.: 1961,ARS Journal 31, 422–427.
Tsu, T. C.: 1959,ARS Journal 29, 422–427.
Van der Ha, J. C. and Modi, V. J.: 1977,Acta Astronautica 4, 813–831
Wesseling, P.: 1967,Astronautica Acta 13, 431–440.
Author information
Authors and Affiliations
Additional information
Single subscripts refer to the order of the perturbation terms; 00 indicates initial conditions: dots and primes refer to differentiation with respect to time and ν, respectively.
Rights and permissions
About this article
Cite this article
Van Der Ha, J.C., Modi, V.J. Long-term evaluation of three-dimensional heliocentric solar sail trajectories with arbitrary fixed sail setting. Celestial Mechanics 19, 113–138 (1979). https://doi.org/10.1007/BF01796085
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01796085