Nonlinear dynamics near equilibrium points
for a Solar Sail
Ariadna Farrés
Àngel Jorba
[email protected]
[email protected]
Universitat de Barcelona
Departament de Matemàtica Aplicada i Anàlisi
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
y
Ecliptic plane
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.
~sail = β ms h~rs , ~ni2 ~n.
F
2
rps
• 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
~
F
Sail
~
F
E
~
F
S
t
µ
X
Sun
1−µ
WSIMS – p.5/36
Earth
Equations of Motion (RTBPS)
The equations of motion are:
ẍ
ÿ
z̈
=
=
=
2ẏ + x − (1 − µ)
−2ẋ + y −
−
x−µ
x+1−µ
1−µ
2
−
µ
+
β
h~
r
,
~
n
i
nx ,
s
3
3
2
rps
rpe
rps
µ
1−µ
+
3
3
rps
rpe
µ
1−µ
+
3
3
rps
rpe
z+β
y+β
1−µ
2
h~
r
,
~
n
i
ny ,
s
2
rps
1−µ
2
h~
r
,
~
n
i
nz ,
s
2
rps
where,
nx
=
cos(φ(x, y) + α) cos(ψ(x, y, z) + δ),
ny
=
sin(φ(x, y, z) + α) cos(ψ(x, y, z) + δ),
nz
=
sin(ψ(x, y, z) + δ),
with φ(x, y) and ψ(x, y, z) defining the Sun - Sail direction in spherical
coordinates (~rs = ~rps /rps ).
WSIMS – p.6/36
Equilibrium Points
• The RTBP has 5 equilibrium points (Li ). 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, S1
and S2 . It can be seen that S1 is diffeomorphic to a sphere and S2 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
0.2
0.01
0
0
y
y
0.4
-0.2
-0.01
-0.4
-0.02
-0.6
-0.03
-0.8
-1
-1.5
-1
-0.5
0
x
0.5
1
-0.04
-1.02
1.5
-1.01
-1
-0.99
-0.98
-0.97
-0.96
-0.95
x
Equilibrium points in the {x, y}- plane
0.2
0.015
0.15
0.01
0.1
0.005
0
z
z
0.05
-0.05
0
-0.005
-0.1
-0.01
-0.15
-0.2
-1.5
-1
-0.5
0
x
0.5
1
1.5
-0.015
-1.015
-1.01
-1.005
-1
x
-0.995
-0.99
-0.985
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
L1
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
L1
L1
Sun
Summer Solstice
Sun
S
Winter Solstice
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. Farrés and À. Jorba, “Solar Sail surfing along familes of equilibrium points.”, Acta
Astronautica Volume 63, Issues 1-4, July-August 2008, Pages 249-257.
• A. Farrés and À. 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
L2
= 0 and β = 0.051689.
Sun
Earth
L3
L1
SL2
SL1
SL3
• 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 SL1 .
WSIMS – p.11/36
Family of equilibrium points around SL1 for α
= 0 and β = 0.051689
0.015
0.01
z
0.005
L1
0
SL1
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)
=
A0 [cos(ω1 t + ψ1 )~v1 + sin(ω1 t + ψ1 )~
u1 ]
+
B0 [cos(ω2 t + ψ2 )~v2 + sin(ω2 t + ψ2 )~
u2 ]
+
C0 eλt~vλ + D0 e−λt~v−λ
Where,
◦ ±iω1 eigenvalues with ~v1 ± i~u1 as eigenvectors.
◦ ±iω2 eigenvalues with ~v2 ± i~u2 as eigenvectors.
◦ ±λ eigenvalues with ~vλ , ~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: A0 = γ and B0 = D0 = E0 = 0.
◦ Vertical family : B0 = γ and A0 = D0 = E0 = 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
CxS
SxS
0.01
z
0.005
0
-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
CxS
SxS
0.01
z
0.005
0
-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
CxS
SxS
0.01
z
0.005
0
-0.005
-0.01
-0.015
-0.992
-0.99
-0.988
-0.986
x
-0.984
-0.982
-0.98
WSIMS – p.17/36
Planar Family of Periodic Orbits
Periodic Orbits for δ
1
0.015
0.5
0.01
0
0.005
-0.5
z
-1
= 0.
0
-0.005
-1
0.04
-0.995
0.02
-0.99
-0.985
0
-0.98
-0.975
-0.02
y
-0.97
x
-0.04
-0.965
-0.96
-0.06
-0.955
-0.01
0.03
0.02
0.01
-0.015-1 -0.995
0
y
-0.01
-0.99-0.985-0.98
-0.02
-0.975-0.97-0.965
x
-0.03
0.06
Periodic Orbits for δ
= 0.01.
0.015
0.015
0.01
0.01
0.005
z
0.005
z
0
0
-0.005
-0.005
-0.01
-0.01
0.03
0.02
-0.015-1 -0.995-0.99
0 0.01 y
-0.01
-0.985
-0.02
x -0.98-0.975-0.97-0.965
-0.03
-0.015
0.04
0.03
0.02
0.010
-0.96
y
-0.01
-0.965
-0.97
-0.975
-0.02
-0.98
-0.985
-0.03
-0.99
-0.995
x
-0.04
-1
-1.005
WSIMS – p.18/36
Planar Family of Periodic Orbits
Familly for δ
=0
0.015
0.01
0.005
z
0
-0.005
-0.01
-0.015
0.05
0.04
0.03
0.01
0-0.03
y 0.02
-0.01
-0.955
-0.02
-0.96
-0.965
-0.975
-0.98
-0.04
-0.985
-0.99
-0.05 -1 -0.995
x -0.97
Familly for δ
= 0.01
0.015
0.01
0.005
z
0
-0.005
-0.01
-0.015
0.04
0.03
0.01
0
y0.02
-0.01
-0.96
-0.965
-0.02
-0.97
-0.98
-0.03
-0.985
-0.99
-0.04
x-0.975
-1.005 -1 -0.995
WSIMS – p.19/36
Vertical Family of Periodic Orbits
delta = 0
delta = 0.001
0.03
0.03
0.02
0.02
0.01
z
0.01
z
0
0
-0.01
-0.01
-0.02
-0.02
0.0008
0.0006
0.0004
-0.03
0.0002
-0.9818
-0.9816
0
-0.9814
-0.9812
-0.0002
y
-0.981
-0.9808
-0.0004
-0.9804
-0.0006
x-0.9806
-0.9802
-0.98
-0.0008
-0.9798
0.0008
0.0006
0.0004
-0.03
0.0002
-0.9818
-0.9816
0
-0.9814
-0.9812
-0.0002
y
-0.981
-0.9808
-0.0004
-0.9804
-0.0006
x-0.9806
-0.9802
-0.98
-0.0008
-0.9798
delta = 0.005
delta = 0.01
0.03
0.03
0.02
0.02
0.01
z
0
0.01
z
0
-0.01
-0.01
-0.02
-0.02
0.0008
0.0006
0.0004
-0.03
0.0002
-0.9818
-0.9816
0
-0.9814
-0.9812
-0.0002
y
-0.981
-0.9808
-0.0004
-0.9804
-0.0006
x-0.9806
-0.9802
-0.98
-0.0008
-0.9798
0.001
0.0008
0.0006
-0.03
0.0004
-0.9818
0.0002
-0.9816
0
-0.9814
-0.9812
-0.0002
y
-0.981
-0.0004
-0.9808
-0.9806
-0.0006
-0.9804
x
-0.0008
-0.9802
-0.98
-0.001
-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,
ẋ = Ax + f (x, y), x ∈ R4 ,
ẏ = By + g(x, y), y ∈ R2 ,
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
ẋ = Ax + f (x, v(x)).
WSIMS – p.21/36
Approximating the Centre Manifold
To find y
= v(x) we substitute this expression on the differential equations.
So v(x) must satisfy,
Dv(x)Ax − Bv(x) = g(x, v(x)) − Dv(x)f (x, v(x)).
(1)
We take,

v(x) = 
X
v1,k xk ,
|k|≥2
X
|k|≥2

v2,k xk  ,
k ∈ (N ∪ {0})4 ,
its expansion as power series.
The left hand side is a linear operator w.r.t
non-linear.
v(x) and the right hand side is
WSIMS – p.22/36
Approximating the Centre Manifold
The left hand side of equation (1),
L(x) = Dv(x)Ax − Bv(x),
diagonalizes if A and B are diagonal.
In particular, if A
= diag(iω1 , −iω1 , iω2 , −iω2 ) and B = diag(λ, −λ)
then,
 X
(iω1 k1 − iω1 k2 + iω2 k3 − iω2 k4 − λ)v1,k xk

 |k|≥2
L(x) =  X

(iω1 k1 − iω1 k2 + iω2 k3 − iω2 k4 + λ)v2,k xk
|k|≥2



.

WSIMS – p.23/36
Approximating the Centre Manifold
The right hand side of equation (1),
h(x) = g(x, v(x)) − Dv(x)f (x, v(x)),
can be expressed as,

h(x) = 
X
|k|≥2
k
h1,k x ,
X
|k|≥2
where hi,k depend on vi,j in a known way (i
T
h2,k xk  ,
= 1, 2).
• It can be seen that for a fixed degree |k| = n, the hi,k depend only on the
vi,j such that |j| < n.
WSIMS – p.24/36
Approximating the Centre Manifold
Now we can solve equation (1) in an iterative way, equalising the left and the
right hand side degree by degree. We have to solve a diagonal system at each
degree.
Notice:
• It is important to have a fast way to find the hi,k to get up to high degrees.
• We do not recommend to expand f (x, y) y g(x, y), and then compose
with y = v(x). One should find other alternative ways, faster in terms of
computational time.
• The matrixes A and B don’t have to be diagonal, but then one must solve
a larger linear system at each degree.
WSIMS – p.25/36
On the efficient computation of hi,j
We recall that the equations of motion for α
ẍ
ÿ
z̈
=
=
=
2ẏ + x − κs
−2ẋ + y −
−
= 0 are,
x−µ
z(x − µ)
x+1−µ
−
κ
+
κ
,
e
sail
3
3
3 r
rps
rpe
rps
2
κe
κs
+
3
3
rps
rpe
κs
κe
+
3
3
rps
rpe
z − κsail
y + κsail
zy
,
3 r
rps
2
r2
,
3
rps
where κs = (1 − µ)(1 − β cos3 α), κe = µ, κsail = β(1 − µ) cos2 α sin α.
• To expand the equations of motion we use the Legendre polynomials.
WSIMS – p.26/36
On the efficient computation of hi,j
For example:
• 1/rps can be expanded as,
X
cn Tn (x, y, z),
n≥0
where the Tn (x, y, z) are homogeneous polynomials of degree n that are
computed in a recurrent way.
n−1 2
2n − 1
xTn−1 −
(x + y 2 + z 2 )Tn−2 ,
Tn =
n
n
with
T0 = 1,
T1 = x.
WSIMS – p.27/36
On the efficient computation of hi,j
• The functions f (x̄, ȳ) and g(x̄, ȳ) can be computed in a reccurrent way
as they are found after applying a linear transformation to the expansion of
the system.
• Composing these recurrences with v(x̄) we can compute the expansions
of f (x̄, v(x̄)) and g(x̄, v(x̄)) in a recurrent way and so for the hi,j .
For example:
T0 = 1,
T1 = x(x̄, v(x̄)),
2n − 1
n−1
2
2
2
Tn =
x(x̄, v(x̄))Tn−1 −
x(x̄, v(x̄)) + y(x̄, v(x̄)) + z(x̄, v(x̄)) Tn−2 .
n
n
WSIMS – p.28/36
Validation Test
• Given an initial condition v0 , we denote v1 and ṽ1 to the integration at time
t = 0.1 of v0 on the centre manifold and the complete system respectively.
• The error behaves as: |ṽ1 − v1 | = chn+1 , where h is the distance to the
origin of v0 .
• If we consider the centre manifold up to degree 8:
h
|ṽ1 − v1 |
n+1
0.04 2.3643547906724647e − 15
0.08 1.2618898774811476e − 12 9.059923
0.16 6.9534006796827247e − 10 9.105988
0.32 3.9879163406855978e − 07 9.163700
WSIMS – p.29/36
Results for δ
=0
We have computed the reduction of to the centre manifold around Sub-L1 up
to degree 32. (it takes 17min of CPU time)
• After this reduction we are in a four dimensional phase space
(x1 , x2 , x3 , x4 ).
• We fix a Poincaré section x3 = 0 to reduce the system to a three
dimensional phase space.
• We have taken several initial conditions and computed their successive
images on the Poincaré section.
WSIMS – p.30/36
Results for δ
=0
x4 = 0.01
x4 = 0.05
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-1.5
-1
-0.5
0
0.5
1
1.5
-0.8
-1.5
-1
-0.5
x4 = 0.11
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-1
-0.5
0
0.5
1
1.5
0.5
1
1.5
x4 = 0.17
0.8
-0.8
-1.5
0
0.5
1
1.5
-0.8
-1.5
-1
-0.5
0
WSIMS – p.31/36
Results for δ
=0
x4 = 0.01
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0-1.5
-1
-0.5
0
0.5
x4 = 0.05
1
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
1.5-0.8
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0-1.5
-1
-0.5
0
x4 = 0.11
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0-1.5
-1
-0.5
0
0.5
0.5
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
1.5-0.8
1
0.8
0.6
0.4
0 0.2
-0.2
-0.4
-0.6
1.5-0.8
x4 = 0.17
1
0.8
0.6
0.4
0 0.2
-0.2
-0.4
-0.6
1.5-0.8
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05-1.5
-1
-0.5
0
0.5
WSIMS – p.32/36
Results for δ
= 0 (for a fixed energy level)
h = 0.09
h = 0.11
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
-1
-0.5
h = 0.13
0
0.5
1
0.5
1
h = 0.15
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
-1
-0.5
0
WSIMS – p.33/36
Results for δ
= 0.05
x4 = 0.28
x4 = 0.49
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1.5
-1
-0.5
0
x4 = 0.7
0.5
1
1.5
x4 = 0.84
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-1
-0.5
0
0.5
1
1.5
-1
-0.5
0
0.5
1
1.5
WSIMS – p.34/36
Results for δ
= 0.05
x4 = 0.28
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
-1.5
x4 = 0.49
1.5
1
0.5
-1
0
-0.5
0
-0.5
0.5
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
-1.5
-1
1
1.5
1
0.5
-1
0
-0.5
0
-1.5
1.5
x4 = 0.7
1
-1
-1.5
1.5
x4 = 0.84
1.1
1.2
1
1.1
1
0.9
0.9
0.8
0.8
0.7
0.6
0.5-1
-0.5
0.5
-0.5
0
0.5
1
-1
1.5 -1.5
0
-0.5
1
0.5
1.5
0.7
0.6
0.5-1
-0.5
0
0.5
1
0
-0.5
-1
1.5-1.5
1
0.5
1.5
WSIMS – p.35/36
The End
Thank You !!!
WSIMS – p.36/36