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On the Solar Sail periodic orbits near theEarth-Moon libration points
June 5, 2015
Ariadna Farrés, Àngel Jorba and Marc Jorba-Cuscóariadna.farres@maia.ub.es, angel@maia.ub.es, marc@maia.ub.es
XXIV CEDYACádiz
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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.
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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
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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.
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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)
⟩.
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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
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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.
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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
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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.
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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.
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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.
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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.
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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 .
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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.
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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 .
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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 .
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 .
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 .
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 .
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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 .
23
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.
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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.
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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.
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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.
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