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On the Solar Sail periodic orbits near the Earth-Moon libration points June 5, 2015 Ariadna Farrés, Àngel Jorba and Marc Jorba-Cuscó , , XXIV CEDYA Cádiz

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Page 1: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

On the Solar Sail periodic orbits near theEarth-Moon libration points

June 5, 2015

Ariadna Farrés, Àngel Jorba and Marc Jorba-Cuscó[email protected], [email protected], [email protected]

XXIV CEDYACádiz

Page 2: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

23

SS PO near the EM LP

M. Jorba-Cuscó et al.

1Introduction

The Model

Background

Results

Conclusions andfuture work

MAiA

What is a Solar Sail

I A Solar sail is a recent proposed form of spacecraftpropulsion. It consists of large, light and highly reflectingmembrane mirrors used to take advantadge on the effectof solar radiation pressure (SRP).

I This technology opens a wide new range of possiblemission applications that cannot be achieved by traditionalspacecraft.

Page 3: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

2Introduction

The Model

Background

Results

Conclusions andfuture work

MAiA

Motivation

Solar sails in space:I IKAROS (JAXA) in 2010. First spacecraft to use solar

sailing as the main propulsion.

I NanoSail-D2 (NASA) in 2011. Solar Sail deployed in aLEO.

Planned missions: LightSail-1 (Planetary Society) 2016, LunarFlashlight (NASA) 2018. Why the Earth-Moon System?

I Has been studied by several authors1 but not deeply.

1Simo et al. 2009, Wawrzyniak et al. 2011, Heiligers et al. 2014

Page 4: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

23

SS PO near the EM LP

M. Jorba-Cuscó et al.

3Introduction

The Model

Background

Results

Conclusions andfuture work

MAiA

Modeling the SRP effect

We have considered the sail to be flat and perfectly reflecting.

~a = βmS

r2PS〈~rPS, ~n〉2~n,

The sail lightness numbermeasures the area-to-massratio of the spacecraft. Thenormal direction ~n givesthe orientation of the sail.Call α the angle between~rPSand ~n .

reflecte

d r

ad

iation

aref

incoming radiationaabs

incoming radiation

Sail normal

Sail

Sail

The sail is parallel if α = ±π/2. The sail is perpendicular ifα = 0.

Page 5: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

4The Model

Background

Results

Conclusions andfuture work

MAiA

Solar Sails in the Earth-Moon system

We have taken the planar Bicircular problem (BCP) and addedthe solar radiation pressure (BCPS).

I Two coupled planar RTBP.I Take units so that the distance between Earth and Moon is

one, the sum of their masses is one and their period is 2π.I Take the SRP acceleration to be constant.

µ = 0.01215 mS = 328900.56ωS = 0.925196 aS = 388.811143

The dynamics in positions-momenta coordinates is given bythe Hamiltonian function:

H =12

(p2x + p2

y ) + ypx − xpy −1− µrPE

− µ

rPM− mS

rPS(1)

− mS

a2S

(y sin θ − x cos θ) + βmS

a2S

cos2 α⟨R(α)s(θ), (x , y)

⟩.

Page 6: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

5Background

Results

Conclusions andfuture work

MAiA

Known facts

We can split H in three parts:

H = HRTBP + HS(t) + HSS(α, β, t).

I If α = ±π/2 or β = 0, we have HSS(α, β) = 0 and ourmodel is reduced to the Bicircular Problem (BCP).

I RTBP: There exist five equilibrium points: Li fori = 1, . . . ,5.

I BCP: The equilibrium points are replaced2 by periodic or-bits with the same period as Sun. The equivalents L1,L2 and L3 are just small unstable periodic orbits that canbe uniquely continued from the corresponding equilibriumpoint. The equivalents of L4 (and L5) are three periodicorbits, one of them small and unstable and the other twolarger and stable.

2Simó et al. 1995

Page 7: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

23

SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

6Background

Results

Conclusions andfuture work

MAiA

Dynamical Equivalents

LemmaLet x ∈ Rn, t ∈ R. Assume f ∈ C1(Rn,Rn) andg ∈ C1(Rn × T,Rn).{

x = f (x) + εg(x , t),x(t0) = x0.

(2)

Suppose that the following hypotheses are fulfilled:I The point x̄ ∈ Rn is a zero of f .I For each k ∈ Z we have ik /∈ Spec {Dx f (x̄)}.

then, there is ε0 > 0 for which, for each ε < ε0, (2) has exactlyone 2π-periodic solution xε such that xε → x̄ when ε→ 0.We call these solutions the dynamical equivalents (DE) of thefixed point. Stress that, if ε is large enough, the existence anduniqueness of DE may be compromised.

Page 8: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

7Background

Results

Conclusions andfuture work

MAiA

The case of the Bicircular Problem

-1

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5

L1L2 L3

L4

L5

EM

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Page 9: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

8Background

Results

Conclusions andfuture work

MAiA

Periodic orbits near L4.

0

0.2

0.4

0.6

0.8

1

-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

Continuation of L4 as a periodic orbit. The red part stands forthe stable orbits, blue for unstable ones. Horizontal axis: x . Thevertical axis represents an additional parameter ε multiplying themass of Sun. When ε = 0, the model is the RTBP, when ε = 1the model is the BCP.

Page 10: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

9Background

Results

Conclusions andfuture work

MAiA

Stroboscopic map

I Consider the map given by the evaluation of the flow attime TS = 2π

ωS, we call it Stroboscopic map (SM).

I The TS-periodic orbits of the original system appear asfixed points of the SM.

I There exist surfaces of fixed points (of the SM)parametrized by α and β. We fix β = β0 and get curves offixed points parametrized by α.

I We trace numerically these curves in order to study howthe initial fixed points change with respect to the parame-ters. Due to the large instability near the collinear points(and even near L4 for some values of α) it is necessary toimplement a multiple shooting method

GOAL: To study how SRP change the periodic orbits of theBCP.

Page 11: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

10Results

Conclusions andfuture work

MAiA

Notation

I We name the dynamical equivalents of the Lagrangianpoints in the BCP: x̃1, x̃2, x̃3, x̃ i

4, x̃ i5; i = 1,2,3, which are

fixed points of the Stroboscopic map.

I We fix the values β1 = 0.01, β2 = 0.02, β3 = 0.03 andβ4 = 0.04.

I Given a fixed β, we call

ψjβ : {−π/2, π/2} 7→ R4, j = 1 . . . 5

the continuation curve that starts from x̃j for j = 1,2,3 andx̃1

j for j = 4,5.

Page 12: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

11Results

Conclusions andfuture work

MAiA

Continuations near L1 and L2

I For each i = 1, . . . ,4, the curves ψ1βi

start from x̃1 atα = −π/2.

I The fixed points (of the SM) along these curves neverabandon its condition of linearly unstable fixed points.

I These curves are closed, the continuation always return tox̃1 when it reaches the homotopy level {α = π/2}.

I For each i = 1, . . . ,4, the curves ψ2βi

behave as in the latercase.

I In the case of L2 the cuves of fixed points grow larger.

Page 13: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

12Results

Conclusions andfuture work

MAiA

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

-0.8376 -0.8374 -0.8372 -0.837 -0.8368 -0.8366 -0.8364 -0.8362

β=0.01β=0.02β=0.03β=0.04

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-1.15 -1.145 -1.14 -1.135 -1.13 -1.125 -1.12 -1.115 -1.11 -1.105 -1.1

β=0.01β=0.02β=0.03β=0.01

Left: Projections of the curves ψ1βi

for i = 1, . . . ,4. Right:Projections of the curves ψ2

βifor i = 1, . . . ,4. Horizontal axis: x ,

vertical axis: y .

Page 14: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

13Results

Conclusions andfuture work

MAiA

Continuations near L3, L4, L5

I The curves ψ4βi

change its stability several times.

I Different types of eigenvalue collisions take place.

I There exists a value β∗ near 0.03732 . . . for which thecurves ψ3

β , ψ4β and ψ5

β collide.

I New connections between x̃15 , x̃1

4 and x̃3 appear.

Page 15: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

14Results

Conclusions andfuture work

MAiA

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Continuation curves ψ3β , ψ4

β and ψ5β for β = 0.01. Red stands

for stable and blue for unstable fixed points. Horizontal axis, α.Vertical axis y .

Page 16: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

15Results

Conclusions andfuture work

MAiA

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Continuation curves ψ3β , ψ4

β and ψ5β for β = 0.02. Red stands

for stable and blue for unstable fixed points. Horizontal axis, α.Vertical axis y .

Page 17: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

23

SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

16Results

Conclusions andfuture work

MAiA

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Continuation curves ψ3β , ψ4

β and ψ5β for β = 0.03. Red stands

for stable and blue for unstable fixed points. Horizontal axis, α.Vertical axis y .

Page 18: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

23

SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

17Results

Conclusions andfuture work

MAiA

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Continuation curves ψ3β , ψ4

β and ψ5β for β = 0.0373. Red stands

for stable and blue for unstable fixed points. Horizontal axis, α.Vertical axis y .

Page 19: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

23

SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

18Results

Conclusions andfuture work

MAiA

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Continuation curves ψ43β , ψ54

β and ψ35β for β = 0.0374. Red

stands for stable and blue for unstable fixed points. Horizontalaxis, α. Vertical axis y .

Page 20: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

19Results

Conclusions andfuture work

MAiA

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Continuation curves ψ43β , ψ54

β and ψ35β for β = 0.04. Red stands

for stable and blue for unstable fixed points. Horizontal axis, α.Vertical axis y .

Page 21: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

20Results

Conclusions andfuture work

MAiA

Eigenvalue collisions

1. (Type I) A collision of conjugated eigenvalues at 1. Thefixed point goes from elliptic to hyperbolic, or viceversa. Itmay branch a new family of fixed points.

2. (Type II) A collision of conjugated eigenvalues at the point−1. The fixed point goes from elliptic to hyperbolic, orviceversa. From this collision may branch a new family of2-periodic points.

3. (Type III) A collision of reciprocal eigenvalues at the pointλ = eiη. After the collision the eigenvalues abandon theunit circle (Complex instability). If η = p

q ∈ Q a branchingof a family of q-periodic fixed points can be produced, butgenerically η is irrational and usually Diophantine, in thiscase a family of invariant curves may branch.

Page 22: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

21Results

Conclusions andfuture work

MAiA

Eigenvalue collisions II

β = 0.02 β = 0.04α Type α Type

-1.429686 I -1.460538 I-1.711267 I -1.681738 I-1.126725 II -1.240388 II-0.874575 II -1.049999 II

-0.659631 III-0.375200 III

Eigenvalue collisions of the curves ψ4β2

and ψ4β4

, for α < 0. Theremaining ones can be obtained by symmetry.

Page 23: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

Results

22Conclusions andfuture work

MAiA

Conclusions

We have studied the effect of SRP upon a Solar Sail.

I Near L1 and L2, the dominance of the gravitational fields ofEarth and Moon do not allow to change remarkably thedynamical substitutes of these Lagrangian points.

I We have seen how the change of orientation of the sailcan make a notable difference in the shape and the size ofthe trajectories near L3, L4 and L5.

I If the sail lightness number is big enough, a connectionbetween L3, L4 and L5 appears. This could allow tonavigate between these equilibrium points.

Page 24: On the Solar Sail periodic orbits near the Earth-Moon libration pointsmarc/talks/cedya2015.pdf · 2015. 9. 14. · What is a Solar Sail I A Solar sail is a recent proposed form of

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SS PO near the EM LP

M. Jorba-Cuscó et al.

Introduction

The Model

Background

Results

23Conclusions andfuture work

MAiA

Future work

I Compute the branchings associated to the eigenvaluecollisions.

I Study the non-Hamiltonian case.

I Study how these results can be applyied to real missions.