Nonlinear dynamics near equilibrium points for a Solar Sail ` Ariadna Farr´ es Angel Jorba ari@maia.ub.es angel@maia.ub.es Universitat de Barcelona Departament de Matem` atica Aplicada i An` alisi WSIMS – p.1/36
Contents • Introduction to Solar Sails. • Families of Equilibria. • Periodic Motion around Equilibria. • Reduction to the Centre Manifold. WSIMS – p.2/36
What is a Solar Sail ? • It is a proposed form of spacecraft propulsion that uses large membrane mirrors. • The impact of the photons emitted by the Sun onto the surface of the sail and its further reflection produce momentum. • Solar Sails open a new wide range of possible mission that are not accessible for a traditional spacecraft. WSIMS – p.3/36
Some Definitions • The effectiveness of the sail is given by the dimensionless parameter β , the lightness number . • The sail orientation is given by the normal vector to the surface of the sail ( � n ), parametrised by two angles, α and δ , where α ∈ [ − π/ 2 , π/ 2] and δ ∈ [ − π/ 2 , π/ 2] . z � n δ Sun-line x α Ecliptic plane y WSIMS – p.4/36
Equations of Motion (RTBPS) • We consider that the sail is perfectly reflecting. So the force due to the sail is in the normal direction to the surface of the sail � n . F sail = β m s � n � 2 � � � r s ,� n. r 2 ps • We consider the gravitational attraction of Sun and Earth: we use the RTBP adding the radiation pressure to model the motion of the sail. Y � FSail � FE � FS t µ X Sun 1 − µ WSIMS – p.5/36 Earth
Equations of Motion (RTBPS) The equations of motion are: y + x − (1 − µ ) x − µ − µx + 1 − µ + β 1 − µ n � 2 n x , x ¨ = 2 ˙ � � r s ,� r 3 r 3 r 2 ps pe ps � 1 − µ � + µ y + β 1 − µ n � 2 n y , y ¨ = − 2 ˙ x + y − � � r s ,� r 3 r 3 r 2 ps pe ps � 1 − µ � + µ z + β 1 − µ n � 2 n z , z ¨ = − � � r s ,� r 3 r 3 r 2 ps pe ps where, n x = cos( φ ( x, y ) + α ) cos( ψ ( x, y, z ) + δ ) , n y = sin( φ ( x, y, z ) + α ) cos( ψ ( x, y, z ) + δ ) , n z = sin( ψ ( x, y, z ) + δ ) , with φ ( x, y ) and ψ ( x, y, z ) defining the Sun - Sail direction in spherical coordinates ( � r ps /r ps ). r s = � WSIMS – p.6/36
Equilibrium Points • The RTBP has 5 equilibrium points ( L i ). For small β , these 5 points are replaced by 5 continuous families of equilibria, parametrised by α and δ . • For a fixed and small β , these families have two disconnected surfaces, S 1 and S 2 . It can be seen that S 1 is diffeomorphic to a sphere and S 2 is diffeomorphic to a torus around the Sun. • All these families can be computed numerically by means of a continuation method. WSIMS – p.7/36
Equilibrium Points 1 0.04 0.8 0.03 0.6 0.02 0.4 0.01 0.2 0 0 y y -0.2 -0.01 -0.4 -0.02 -0.6 -0.03 -0.8 -1 -0.04 -1.5 -1 -0.5 0 0.5 1 1.5 -1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95 x x Equilibrium points in the { x, y } - plane 0.2 0.015 0.15 0.01 0.1 0.005 0.05 0 0 z z -0.05 -0.005 -0.1 -0.01 -0.15 -0.2 -0.015 -1.5 -1 -0.5 0 0.5 1 1.5 -1.015 -1.01 -1.005 -1 -0.995 -0.99 -0.985 x x Equilibrium points in the { x, z } - plane WSIMS – p.8/36
Some Interesting Missions • Observations of the Sun provide information of the geomagnetic storms, as in the Geostorm Warning Mission. z y CME x ACE Sun Earth L 1 0 . 01 AU Sail 0 . 02 AU • Observations of the Earth’s poles, as in the Polar Observer. N N Sail Sail z z Earth x Earth x L 1 L 1 Sun Sun Winter Solstice Summer Solstice S S WSIMS – p.9/36
• C. McInnes, “ Solar Sail: Technology, Dynamics and Mission Applications.”, Springer-Praxis , 1999. • D. Lawrence and S. Piggott, “ Solar Sailing trajectory control for Sub-L1 stationkeeping”, AIAA 2005-6173 . • J. Bookless and C. McInnes, “Control of Lagrange point orbits using Solar Sail propulsion.”, 56th Astronautical Conference 2005. • A. Farr´ es and ` A. Jorba, “Solar Sail surfing along familes of equilibrium points.”, Acta Astronautica Volume 63, Issues 1-4, July-August 2008, Pages 249-257. es and ` • A. Farr´ A. Jorba, “A dynamical System Approach for the Station Keeping of a Solar Sail.”, Journal of Astronautical Science . ( to apear in 2008 ) WSIMS – p.10/36
From now on ... We fix α = 0 and β = 0 . 051689 . Sun Earth L 2 L 3 L 1 SL 1 SL 3 SL 2 • Here, we have 3 families of equilibrium points on the { x, z } - plane parametrised by the angle δ . • The linear behaviour for all these equilibrium points is of the type centre × centre × saddle. • We want to study the families of periodic orbits that appear around these equilibrium points for a fixed δ . • For practical reasons we focus on the region around SL 1 . WSIMS – p.11/36
Family of equilibrium points around SL 1 for α = 0 and β = 0 . 051689 0.015 0.01 0.005 L1 SL1 0 z Earth -0.005 -0.01 -0.015 -1.005 -1 -0.995 -0.99 -0.985 -0.98 x WSIMS – p.12/36
Motion around the equilibrium points • As we have said, the linear behaviour around the fixed point is centre × centre × saddle. • So up to first order the solutions around the fixed point are: φ ( t ) = A 0 [cos( ω 1 t + ψ 1 ) � v 1 + sin( ω 1 t + ψ 1 ) � u 1 ] + B 0 [cos( ω 2 t + ψ 2 ) � v 2 + sin( ω 2 t + ψ 2 ) � u 2 ] C 0 e λt � v λ + D 0 e − λt � + v − λ Where, ◦ ± i ω 1 eigenvalues with � v 1 ± i � u 1 as eigenvectors. ◦ ± i ω 2 eigenvalues with � v 2 ± i � u 2 as eigenvectors. ◦ ± λ eigenvalues with � v λ , � v − λ as eigenvectors. WSIMS – p.13/36
Motion around the equilibrium points • We take the linear approximation to compute an initial periodic orbit for each family. We then use a continuation method to compute the rest of the family. ◦ Planar family: A 0 = γ and B 0 = D 0 = E 0 = 0 . ◦ Vertical family : B 0 = γ and A 0 = D 0 = E 0 = 0 . • We use a parallel shooting method to compute the periodic orbits. • We have done this for different values of δ . WSIMS – p.14/36
Planar Family of Periodic Orbits • We have computed the planar family for δ = 0 . (i.e. sail perpendicular to Sun). delta = 0 0.015 C x S S x S 0.01 0.005 0 z -0.005 -0.01 -0.015 -0.992 -0.99 -0.988 -0.986 -0.984 -0.982 -0.98 x WSIMS – p.15/36
Continuation of the Planar Family • We have computed the planar family for δ = 0 . 001 . delta =10^-3 0.015 C x S S x S 0.01 0.005 0 z -0.005 -0.01 -0.015 -0.992 -0.99 -0.988 -0.986 -0.984 -0.982 -0.98 x WSIMS – p.16/36
Continuation of the Planar Family 0.015 C x S S x S 0.01 0.005 0 z -0.005 -0.01 -0.015 -0.992 -0.99 -0.988 -0.986 -0.984 -0.982 -0.98 x WSIMS – p.17/36
Planar Family of Periodic Orbits Periodic Orbits for δ = 0 . 1 0.015 0.5 0.01 0 0.005 -0.5 z z 0 -1 -0.005 -0.01 0.03 0.06 0.02 -1 0.04 -0.01 0 0.01 -0.995 -0.015 -0.02 0 0.02 -0.99 -1 -0.995 y -0.985 -0.99 -0.985 -0.98 -0.02 -0.98 -0.975 -0.975 y x -0.97 -0.03 -0.97 -0.965 x -0.04 -0.965 -0.96 -0.955-0.06 Periodic Orbits for δ = 0 . 01 . 0.015 0.015 0.01 0.01 0.005 0.005 0 z z z z 0 -0.005 -0.005 -0.01 -0.01 -0.015 0.04 0.03 0.03 0.02 0.02 0.01 0.01 -0.015 0 -1 -0.995-0.99-0.985-0.98-0.975-0.97-0.965 0 y -0.01 -0.96 y -0.01 -0.965 -0.02 -0.97 -0.02 -0.975 x -0.03 -0.98 -0.985 -0.03 -0.99 -1.005-1-0.995 -0.04 x WSIMS – p.18/36
Planar Family of Periodic Orbits Familly for δ = 0 0.015 0.01 0.005 0 z z -0.005 -0.01 -0.015 0.05 0.04 0.03 0.02 0.01 0 y -0.01 -0.02 -0.955 -0.96 -0.03 -0.97 -0.965 -0.975 -0.04 -0.985 -0.98 -0.05 -0.99 -1-0.995 x Familly for δ = 0 . 01 0.015 0.01 0.005 0 z z -0.005 -0.01 -0.015 0.04 0.03 0.02 0.01 0 y -0.01 -0.96 -0.02 -0.965 -0.975 -0.97 -0.03 -0.98 -0.99 -0.985 -1.005-1-0.995 -0.04 x WSIMS – p.19/36
Vertical Family of Periodic Orbits delta = 0 delta = 0.001 0.03 0.03 0.02 0.02 0.01 0.01 z z z z 0 0 -0.01 -0.01 -0.02 -0.02 0.0008 0.0008 0.0006 0.0006 0.0004 0.0004 -0.03 -0.03 -0.9818 0.0002 -0.9818 0.0002 -0.9816 -0.9816 0 0 -0.9814 -0.9814 -0.9812 -0.9812 -0.0002 -0.0002 -0.981 y -0.981 y -0.9808 -0.9808 -0.0004 -0.0004 -0.9806 -0.9806 -0.9804 -0.9804 x -0.0006 x -0.0006 -0.9802 -0.9802 -0.98 -0.98 -0.0008 -0.0008 -0.9798 -0.9798 delta = 0.005 delta = 0.01 0.03 0.03 0.02 0.02 0.01 0.01 z z z z 0 0 -0.01 -0.01 -0.02 -0.02 0.0008 0.001 0.0008 0.0006 0.0006 0.0004 -0.03 -0.03 0.0004 -0.9818 0.0002 -0.9818 0.0002 -0.9816 -0.9816 -0.9814 0 -0.9814 0 -0.9812 -0.9812 -0.0002 -0.981 -0.0002 y -0.981 y -0.0004 -0.9808 -0.9808 -0.0004 -0.9806 -0.9806 -0.0006 -0.9804 -0.9804 x -0.0006 x -0.0008 -0.9802 -0.9802 -0.98 -0.0008 -0.98 -0.001 -0.9798 -0.9798 WSIMS – p.20/36
Reduction to the Centre Manifold Using an appropriate linear transformation, the equations around the fixed point can be written as, x ∈ R 4 , x ˙ = Ax + f ( x, y ) , y ∈ R 2 , y ˙ = By + g ( x, y ) , where A is an elliptic matrix and B an hyperbolic one, and f (0 , 0) = g (0 , 0) = 0 and Df (0 , 0) = Dg (0 , 0) = 0 . • We want to obtain y = v ( x ) , with v (0) = 0 , Dv (0) = 0 , the local expression of the centre manifold. • The flow restricted to the invariant manifold is x = Ax + f ( x, v ( x )) . ˙ WSIMS – p.21/36
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