Nonlinear dynamics near equilibrium points

for a Solar Sail

Ariadna Farres Angel Jorba

ari@maia.ub.es angel@maia.ub.es

Universitat de Barcelona

Departament de Matematica Aplicada i Analisi

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].

α

δ Sun-line

~n

x

y

z

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.

~Fsail = βms

r2ps

〈~rs, ~n〉2~n.

• We consider the gravitational attraction of Sun and Earth: we use the

RTBP adding the radiation pressure to model the motion of the sail.

1 − µ

µ

~FS

~FSail~FE

Sun

Earth

Y

Xt

WSIMS – p.5/36

Equations of Motion (RTBPS)

The equations of motion are:

x = 2y + x− (1 − µ)x− µ

r3ps

− µx+ 1 − µ

r3pe

+ β1 − µ

r2ps

〈~rs, ~n〉2nx,

y = −2x+ y −

(

1 − µ

r3ps

+µ

r3pe

)

y + β1 − µ

r2ps

〈~rs, ~n〉2ny,

z = −

(

1 − µ

r3ps

+µ

r3pe

)

z + β1 − µ

r2ps

〈~rs, ~n〉2nz,

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

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1 1.5

y

x

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-1.02 -1.01 -1 -0.99 -0.98 -0.97 -0.96 -0.95

y

x

Equilibrium points in the {x, y}- plane

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1.5 -1 -0.5 0 0.5 1 1.5

z

x

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-1.015 -1.01 -1.005 -1 -0.995 -0.99 -0.985

z

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.

SunEarth

x

y

z

0.01 AU

0.02 AU

L1

ACE

Sail

CME

• Observations of the Earth’s poles, as in the Polar Observer.

Sun

Earthx

z

L1

N

SSummer Solstice

Sail

Sun

Earthx

z

L1

N

SWinter Solstice

Sail

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. Farres and A. Jorba, “Solar Sail surfing along familes of equilibrium points.”, Acta

Astronautica Volume 63, Issues 1-4, July-August 2008, Pages 249-257.

• A. Farres and 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.

Earth SunL1

L2 L3

SL1SL2SL3

• 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

-0.005

0

0.005

0.01

0.015

-1.005 -1 -0.995 -0.99 -0.985 -0.98

z

x

Earth

L1 SL1

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(ω1t+ ψ1)~v1 + sin(ω1t+ ψ1)~u1]

+ B0[cos(ω2t+ ψ2)~v2 + sin(ω2t+ ψ2)~u2]

+ C0eλt~vλ +D0e

−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).

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-0.992 -0.99 -0.988 -0.986 -0.984 -0.982 -0.98

z

x

delta = 0

C x SS x S

WSIMS – p.15/36

Continuation of the Planar Family

• We have computed the planar family for δ = 0.001.

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-0.992 -0.99 -0.988 -0.986 -0.984 -0.982 -0.98

z

x

delta =10^-3

C x SS x S

WSIMS – p.16/36

Continuation of the Planar Family

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

-0.992 -0.99 -0.988 -0.986 -0.984 -0.982 -0.98

z

x

C x SS x S

WSIMS – p.17/36

Planar Family of Periodic Orbits

Periodic Orbits for δ = 0.

-1-0.995-0.99-0.985-0.98-0.975-0.97-0.965-0.96-0.955-0.06-0.04

-0.02 0

0.02 0.04

0.06

-1

-0.5

0

0.5

1

x y-1 -0.995-0.99-0.985-0.98-0.975-0.97-0.965-0.03-0.02-0.01 0 0.01 0.02 0.03

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

z

xy

z

Periodic Orbits for δ = 0.01.

-1 -0.995-0.99-0.985-0.98-0.975-0.97-0.965-0.03-0.02-0.01 0 0.01 0.02 0.03

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

z

xy

z

-1.005-1-0.995-0.99-0.985-0.98-0.975-0.97-0.965-0.96-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04-0.015

-0.01

-0.005

0

0.005

0.01

0.015

z

xy

z

WSIMS – p.18/36

Planar Family of Periodic Orbits

Familly for δ = 0

-1-0.995-0.99-0.985-0.98-0.975-0.97-0.965-0.96-0.955-0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05-0.015

-0.01

-0.005

0

0.005

0.01

0.015

z

xy

z

Familly for δ = 0.01

-1.005-1-0.995-0.99-0.985-0.98-0.975-0.97-0.965-0.96-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04-0.015

-0.01

-0.005

0

0.005

0.01

0.015

z

xy

z

WSIMS – p.19/36

Vertical Family of Periodic Orbits

-0.9818-0.9816-0.9814-0.9812-0.981-0.9808-0.9806-0.9804-0.9802-0.98-0.9798-0.0008-0.0006-0.0004-0.0002 0 0.0002 0.0004 0.0006 0.0008

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

z

delta = 0

xy

z

-0.9818-0.9816-0.9814-0.9812-0.981-0.9808-0.9806-0.9804-0.9802-0.98-0.9798-0.0008-0.0006-0.0004-0.0002 0 0.0002 0.0004 0.0006 0.0008

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

z

delta = 0.001

xy

z

-0.9818-0.9816-0.9814-0.9812-0.981-0.9808-0.9806-0.9804-0.9802-0.98-0.9798-0.0008-0.0006-0.0004-0.0002 0 0.0002 0.0004 0.0006 0.0008

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

z

delta = 0.005

xy

z

-0.9818-0.9816-0.9814-0.9812-0.981-0.9808-0.9806-0.9804-0.9802-0.98-0.9798-0.001-0.0008-0.0006-0.0004-0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001-0.03

-0.02

-0.01

0

0.01

0.02

0.03

z

delta = 0.01

xy

z

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 = Ax + f(x, y), x ∈ R4,

y = 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

x = 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) =

∑

|k|≥2

v1,kxk,

∑

|k|≥2

v2,kxk

, k ∈ (N ∪ {0})4,

its expansion as power series.

The left hand side is a linear operator w.r.t v(x) and the right hand side isnon-linear.

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,

L(x) =

∑

|k|≥2

(iω1k1 − iω1k2 + iω2k3 − iω2k4 − λ)v1,kxk

∑

|k|≥2

(iω1k1 − iω1k2 + iω2k3 − iω2k4 + λ)v2,kxk

.

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) =

∑

|k|≥2

h1,kxk ,

∑

|k|≥2

h2,kxk

T

,

where hi,k depend on vi,j in a known way (i = 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 = 0 are,

x = 2y + x− κs

x− µ

r3ps

− κe

x+ 1 − µ

r3pe

+ κsail

z(x− µ)

r3psr2,

y = −2x+ y −

(

κs

r3ps

+κe

r3pe

)

y + κsail

zy

r3psr2,

z = −

(

κs

r3ps

+κe

r3pe

)

z − κsail

r2r3ps

,

where κs = (1 − µ)(1 − β cos3 α), κe = µ, κsail = β(1 − µ) cos2 α sinα.

• 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,∑

n≥0

cnTn(x, y, z),

where the Tn(x, y, z) are homogeneous polynomials of degree n that are

computed in a recurrent way.

Tn =2n − 1

nxTn−1 −

n − 1

n(x2 + y2 + z2)Tn−2,

with T0 = 1, T1 = x.

WSIMS – p.27/36

On the efficient computation of hi,j

• The functions f(x, y) and g(x, y) 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(x) we can compute the expansions

of f(x, v(x)) and g(x, v(x)) in a recurrent way and so for the hi,j .

For example:

T0 = 1, T1 = x(x, v(x)),

Tn =2n− 1

nx(x, v(x))Tn−1 −

n− 1

n

(

x(x, v(x))2 + y(x, v(x))2 + z(x, v(x))2)

Tn−2.

WSIMS – p.28/36

Validation Test

• Given an initial condition v0, we denote v1 and v1 to the integration at time

t = 0.1 of v0 on the centre manifold and the complete system respectively.

• The error behaves as: |v1 − v1| = chn+1, where h is the distance to the

origin of v0.

• If we consider the centre manifold up to degree 8:

h |v1 − 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 Poincare 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 Poincare section.

WSIMS – p.30/36

Results for δ = 0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.5 -1 -0.5 0 0.5 1 1.5

x4 = 0.01

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.5 -1 -0.5 0 0.5 1 1.5

x4 = 0.05

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.5 -1 -0.5 0 0.5 1 1.5

x4 = 0.11

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.5 -1 -0.5 0 0.5 1 1.5

x4 = 0.17

WSIMS – p.31/36

Results for δ = 0

-1.5 -1 -0.5 0 0.5 1 1.5-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

x4 = 0.01

-1.5 -1 -0.5 0 0.5 1 1.5-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

x4 = 0.05

-1.5 -1 -0.5 0 0.5 1 1.5-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

x4 = 0.11

-1.5 -1 -0.5 0 0.5 1 1.5-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

0.55

x4 = 0.17

WSIMS – p.32/36

Results for δ = 0 (for a fixed energy level)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

h = 0.09

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

h = 0.11

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

h = 0.13

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

h = 0.15

WSIMS – p.33/36

Results for δ = 0.05

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

x4 = 0.28

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

x4 = 0.49

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5

x4 = 0.7

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5

x4 = 0.84

WSIMS – p.34/36

Results for δ = 0.05

-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1

-0.5 0

0.5 1

1.5

0.2 0.25

0.3 0.35

0.4 0.45

0.5 0.55

0.6 0.65

0.7 0.75

x4 = 0.28

-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1

-0.5 0

0.5 1

1.5

0.45 0.5

0.55 0.6

0.65 0.7

0.75 0.8

0.85

x4 = 0.49

-1 -0.5 0 0.5 1 1.5-1.5-1

-0.5 0

0.5 1

1.5

0.5

0.6

0.7

0.8

0.9

1

1.1

x4 = 0.7

-1 -0.5 0 0.5 1 1.5-1.5-1

-0.5 0

0.5 1

1.5

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

x4 = 0.84

WSIMS – p.35/36

The End

Thank You !!!

WSIMS – p.36/36