Periodic orbits high above the ecliptic plane in the solar sail 3-body problem

(1)

Periodic orbits high above the ecliptic plane in

the solar sail 3-body problem

Thomas Waters

Department of Mechanical Engineering, University of Strathclyde, Glasgow.

4th October 2006

In conjunction with Prof. C.R. McInnes.

Research funded by EPSRC grant EP/D003822/1 - “Dynamics, Stability and Control of Highly Non-Keplerian Orbits”

(2)

Overview

1 Solar Sails

Sail design

2 3 Body Equations and Equilibria

Classical equilibria - the Lagrange points Sail equilibria

3 Periodic Solutions

Lindstedt-Poincaré method Families of periodic solutions

4 Applications

Polesitter

(3)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(4)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(5)

Sail design

A solar sail is a spacecraft without an engine, and therefore needs no fuel. It is pushed along by the pressure of

photons from the sun hitting the sail. Solar sails are typically large square sheets of a highly reflective film

(6)

Solar radiation pressure

r

1

n

Photons from the sun hitting the sail impart a small but constant radiation pressure.

The acceleration on the sail due to this pressure is given by

a = βMs

r_s2 (!rs.n)

2

n.

Here β is the “sail lightness number”, the ratio of the radiation pressure

(7)

Current designs

Solar sails are currently being built with a lightness number

β ∼ 0.2_.

For the purpose of this presentation we only consider

(8)

Possible missions

(9)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(10)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(11)

Classical equilibria - the Lagrange points

The CR3BP in a rotating coordinate frame:

µ

1₋_µ

x

y z

Earth, m2=µ

Sun, m1= 1−µ

r₁

r₂

r

(12)

Classical equilibria - the Lagrange points

In the inertial frame, the equations of motion of the third body are:

d2r

dt2 = −∇V, V = −

"₁ ₋

µ r1

+ µ

r2

#

.

However, in the rotating frame two additional forces are introduced: the Corioli force and the centrifugal force,

d2r dt2 +

$

2_θ × dr

dt + θ × (θ × r) %

= −∇_V_.

(13)

Classical equilibria - the Lagrange points

In the inertial frame, the equations of motion of the third body are:

d2r

dt2 = −∇V, V = −

"₁ ₋

µ r1

+ µ

r2

#

.

However, in the rotating frame two additional forces are introduced: the Corioli force and the centrifugal force,

d2r dt2 +

$

2_θ × dr

dt + θ × (θ × r)

%

= −∇_V_.

(14)

Classical equilibria - the Lagrange points

There are five equilibrium points in the rotating coordinate frame, called Lagrange points. All are in the plane of the primaries’ mutual orbit.

µ

1₋_µ

x

y z

Earth, m2=µ

Sun, m1= 1−µ

r₁

r₂

(15)

Classical equilibria - the Lagrange points

There are five equilibrium points in the rotating coordinate frame, called Lagrange points. All are in the plane of the primaries’ mutual orbit.

µ

1₋_µ

r1

r2

r

θ x

y z

Earth, m2=µ

Sun, m1= 1₋µ

L1

L2

L3

L4

(16)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(17)

Sail equilibria

In the rotating coordinate frame the equations of motion of the solar sail are

d2r

dt2 + 2θ × dr

dt = a − θ × (θ × r) − ∇V ≡ F,

where

a = β1 − µ

r₁2 (!r1.n)

2

n.

At equilibrium, r˙ and ¨r vanish, so an equilibrium point is a zero of F.

We seek equilibria in the x-z plane, thus we let

(18)

Sail equilibria

We find continuous surfaces of equilibria:

0.985 0.99 0.995 1 1.005 1.01 1.015

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

L₁ L₂

Earth β = 8 > > < > > :

0.1 0.2 0.3 0.5

xe

(19)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(20)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(21)

Periodic solutions

We use the Lindstedt-Poincaré method to find periodic solutions.

We Taylor expand around an equilibrium point





¨

x − 2_y_˙

¨

y + 2_x_˙

¨z



 = δra(∂_aF)|e +

1 2δr

a

δrb(∂_a∂_bF)|e

+1₆δraδrbδrc(∂_a∂_b∂_cF)|e + O(δr

4

)

At linear order

¨x − 2_y_˙ ₌ _ax ₊ _bz

¨y + 2_x˙ = _cy

(22)

Periodic solutions

There are two distinct regions,

depending on the eigenvalues of the linear system.

In region I, the linear spectrum is

I : * ± λ1i, ±λ2i,±λ_r

+

0.985 0.99 0.995 1 1.005 1.01 1.015 -0.015

-0.01 -0.005 0 0.005 0.01 0.015

II

I

L₁ L₂

Earth

Thus periodic solutions exist at linear order, by suppressing unwanted modes:

(23)

Periodic solutions

There are two distinct regions,

depending on the eigenvalues of the linear system.

In region I, the linear spectrum is

I : * ± λ1i, ±λ2i,±λ_r

+

0.985 0.99 0.995 1 1.005 1.01 1.015 -0.015

-0.01 -0.005 0 0.005 0.01 0.015

II

I

L₁ L₂

Earth

Thus periodic solutions exist at linear order, by suppressing unwanted modes:

(24)

Lindstedt-Poincaré method

We let ε be a small parameter (typically the amplitude of a periodic orbit), and we let

ω = 1 ₊ _εω₁ ₊ _ε2_ω₂ ₊ _ε3_ω₃ ₊ _O₍_ε4₎_.

The idea is, if the frequency of a periodic solution in the linear system is λ, then this method lets you find

approximations to periodic solutions in the nonlinear system with frequency ωλ.

We define a new time coordinate, τ = ωt, and let x_n = x_n(τ) etc. Then the following statements are equivalent:

The periodic solution has

frequency ωλ in t-seconds ⇔

(25)

Lindstedt-Poincaré method

We let ε be a small parameter (typically the amplitude of a periodic orbit), and we let

ω = 1 ₊ _εω₁ ₊ _ε2_ω₂ ₊ _ε3_ω₃ ₊ _O₍_ε4₎_.

The idea is, if the frequency of a periodic solution in the linear system is λ, then this method lets you find

approximations to periodic solutions in the nonlinear system with frequency ωλ.

We define a new time coordinate, τ = ωt, and let x_n = x_n(τ)

etc. Then the following statements are equivalent:

The periodic solution has

frequency ωλ in t-seconds ⇔

(26)

Lindstedt-Poincaré method

We find the solutions for x and z are trigonometric cosine series and y is a sine series. For example when n is odd

xn = pn3 cos(3T) + . . . + p_nn cos(nT),

and when n is even

x_n = p_n0 + p_n2 cos(2T) + . . . + p_nn cos(nT).

The problem reduces to solving systems of algebraic equations for the coefficients p_ni and ω_i.

For large amplitude periodic solutions high above the

ecliptic plane, we need to include up to the 7_{th term in the}

(27)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(28)

Families of periodic solutions

0.986 0.987 0.988 0.989 0.99 -0.001 0 0.001 0 0.002 0.004 0.006 -0.001 0

0.001 The Lindstedt-Poincaré series

provides our first guess for the initial data of a periodic

solution of the full nonlinear system.

We then use a differential

corrector to fine tune the initial data.

(29)

Families of periodic solutions

0.986 0.987 0.988 0.989 0.99 -0.001 0 0.001 0 0.002 0.004 0.006 -0.001 0

0.001 The Lindstedt-Poincaré series

provides our first guess for the initial data of a periodic

solution of the full nonlinear system.

We then use a differential

corrector to fine tune the initial data.

(30)

Stability - Control

The orbital stability of these periodic solutions is found by calculating the eigenvalues of the monodromy

matrix.

The spectrum is typically

*

1_, 1_, _λ_r_, 1_/_λ_r_,_λ_c_, _λ¯_c+

(31)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(32)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(33)

Polesitter

We may use these orbits to provide a constant view of one of the poles.

W intertime, N orthern Hemisphere Summertime, N orthern Hemisphere t= 0

(34)

View from the pole

By timing the orbit well we can narrow the angle subtended by the sail:

1 2 3 4 5 6

-20 20 40 60 E le va ti o n a b ov e th e h o ri zo n (d eg re es )

Time scale = 1 year

L₁ Equilibrium

(35)

Outline

1 Solar Sails

Sail design

4 Applications

Polesitter

(36)

Invariant manifolds

0.97 0.98 0.99 1 1.01 x

-0.01 -0.005 0 0.005 0.01

y 0 0.005 0.01 0.015 z 0 0.005 0.01 0.015

Associated with each periodic orbit are a set of invariant

manifolds.

We find these by integrating in the direction of the stable and unstable eigenvectors of the monodromy matrix.

These manifolds provide us with a mechanism to transfer between various periodic

(37)

Unusual orbits

1 1.002 1.004 1.006 1.008 x -0.001 0 0.001 y 0 0.001 0.002 0.003 0.004 z 0 0.001 0.002 0.003

Unexpected families of periodic solutions arise.

Interestingly, this particular orbit has monodromy matrix with spectrum