Dynamics, Geometry and Solar SailsI

Ariadna Farresa, Angel Jorbab

aInstitut de Matematica, Universitat de Barcelona,Gran Via 585, 08007 Barcelona, Spain

bDepartament de Matematiques i Informatica, Universitat de Barcelona,Gran Via 585, 08007 Barcelona, Spain

Abstract

This note focuses on some dynamical aspects of a solar sail. The first part ofthe paper is a survey of the use of dynamical systems tools to control a solarsail near an unstable equilibrium point of the Earth-Sun system. The secondpart focuses on new results on the dynamics near an equilibrium point of a sailnear an asteroid. The main tool is a reduction to the centre manifold to focuson the bounded motions. In both cases, the role of the geometrical structuresof the phase space is highlighted.

Keywords: Periodic orbits, centre manifolds, Halo orbits2010 MSC: 37J15, 65P30, 70F07

1. Introduction

Dynamical systems have proven to be a useful tool for the design of spacemissions. For instance, the use of invariant manifolds is now common to derivecontrol and transfer strategies. In this note we focus on a specific kind of lowthrust propulsion, known as solar sailing. Solar sailing is based on the useof large membrane mirrors to take advantage of the solar radiation pressureto propel the spacecraft. Although the acceleration produced is smaller thanthe one achieved by a traditional chemical thruster, solar radiation pressureacts continuously and and it is unlimited in time. This makes some long termmissions more accessible, and opens a wide new range of possible applicationsthat cannot be achieved by a traditional spacecraft.

Up to now, three solar sails have been succesfully deployed in space: IKAROS,NanoSail-D2 and LightSail-A. IKAROS (Interplanetary Kite-craft Acceleratedby Radiation Of the Sun) is a Japan Aerospace Exploration Agency experimen-tal spacecraft with a 1414 m2 sail. The spacecraft was launched on May 21st2010, together with Akatsuki (Venus Climate Orbiter). On December 8th 2010,IKAROS passed by Venus at about 80.800 km. NanoSail-D2 is a small solar

IWork supported by the grants MTM2015-67724-P and 2014 SGR 1145.

Email addresses: ari@maia.ub.es (Ariadna Farres), angel@maia.ub.es (Angel Jorba)

1

~FEarth

~FSun

Sail~n

X

Y

Z

EarthSun -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

L1L2 L3

L5

L4

SE

Figure 1: Left: Scheme of the forces acting on the sail. Right: The five equilibrium points ofthe Restricted Three-Body Problem.

sail (10 m2, 4kg) deployed by NASA on January 2011 in a low Earth orbit, thatreentered the atmosphere on September 17th 2011. LightSail-A is a small testspacecraft (32 m2) of the Planetary Society, that was launched on May 20th2015 and deployed its solar sail on June 7th 2015. It reentered the atmosphereon June 14th 2015.

In this paper we will focus on the dynamics of a solar sail in a couple of situ-ations. We will introduce this problem focusing on a solar sail in the Earth-Sunsystem. In this case, the model used will be the Restricted Three Body Problem(RTBP for short) plus solar radiation pressure (see Figure 1, left). The effect ofthe solar radiation pressure on the RTBP produces a 2D family of artificialequilibria, coming from the well known equilibria of the RTBP (see Figure 1,right or [25] for more details). This new equilibria can be parametrised by theorientation of the sail. We will describe the dynamics around some of these ar-tificial equilibrium points. We note that, due to the solar radiation pressure,the system is Hamiltonian only for two cases: when the sail is perpendicular tothe Sun - sail line; and when the sail is aligned with the Sun - sail line (i.e., nosail effect). The main tool used to understand the dynamics is the computationof centre manifolds, for both the Hamiltonian and non-Hamiltonian cases.

The second example is the dynamics of a solar sail close to an asteroid. Notethat, in this case, the effect of the sail becomes very relevant due to the lowmass of the asteroid. We will use, as a model, a modified Hill problem thatincludes the effect of the solar radiation pressure, to describe some aspects ofthe natural dynamics of the sail.

The paper is organised as follows. Section 2 is a short introduction to thedynamics of a solar sail and its applications. Section 3 is a summary of someprevious work of the authors on the use of the geometry of the phase space tocontrol a sail. Section 4 introduces a modification of the classical Hill problemto model a solar sail close to an asteroid. Section 5 explains the so-calledreduction to the centre manifold for the previous model, and finally Section 6uses the centre manifold to describe the phase space.

2

reflecte

d r

ad

iation

aref

incoming radiationaabs

incoming radiation

Sail normal

Sail

Sail

reflecte

d r

ad

iation

aref

incoming radiationaabs

incoming radiation

Sail normal

Sail

Sail

Figure 2: The effect of light on a solar sail. Left: Force produced by the reflected light. Right:Force produced by the absorbed light.

2. Solar sail models

Here, a solar sail is modelled as a flat surface that reflects a large portion ofthe sunlight, while a small portion is absorbed. The reflected photons producean impulse in the normal direction of the sail, while the absorbed photonsproduce an impulse in the opposite direction of the Sun, see Figure 2. We notethat the impulse produced by a reflected photon is the sum of two impulses: theimpulse produced by its absorption and the impulse produced by its emission.The sail orientation is given by the normal vector to the surface of the sail, ~nand it is parametrised by two angles, and . The acceleration can be writtenas

~a = ~rs, ~n( ~rs, ~n~n+

1

2(1 )~rs

), (1)

where denotes the reflectivity coefficient ( = 0 corresponds to a perfectsolar panel that absorbs all the photons, and = 1 corresponds to a perfectlyreflecting solar sail). The scalar factor depends on the size of the sail, itsdistance to the Sun, the total mass of the satellite and the units used (see [5]for more details).

In this section, to simplify the discussion, we will assume that the sail isperfectly reflecting, that is, = 1 (the model with < 1 will be used in the casedescribed in Section 4).

2.1. A dynamical model

Here we use the Restricted Three Body Problem (RTBP) taking the Sunand Earth as primaries and including the solar radiation pressure. In this case,writing = msr2ps

, we have that the acceleration of the sail is given by

~a = msr2ps~rs, ~n2 ~n,

where now is a constant, that can be seen as the ratio of the solar radiationpressure in terms of the solar gravitational attraction ( = 1 means that, if the

3

sail is perpendicular to Sun direction, the effect of the solar radiation pressure onthe sail equals the gravitational attraction of the Sun). With current technology,it is considered reasonable to take 0.05 ([20]). This means that a spacecraftof 100 kg has a sail of 58 58 m2.

The equations of motion are:

x = 2y + x (1 )x r3ps

x+ 1 r3pe

+ 1 r2ps~rs, ~n2nx,

y = 2x+ y (

1 r3ps

+

r3pe

)y +

1 r2ps~rs, ~n2ny, (2)

z = (

1 r3ps

+

r3pe

)z +

1 r2ps~rs, ~n2nz,

where ~n = (nx, ny, nz) is the normal to the surface of the sail with

nx = cos((x, y) + ) cos((x, y, z) + ),ny = sin((x, y, z) + ) cos((x, y, z) + ),nz = sin((x, y, z) + ).

Here, ~rs = (x , y, z)/rps is the Sun - sail direction and the angles and refer to the position of the probe w.r.t. the Sun, in spherical coordinates (see [7]for details).

The properties of this system depend on the values of the parameters. If = = 0 the system is Hamiltonian for any value of . If = 0 and 6= 0,the system is not Hamiltonian but it is still reversible, under the symmetryR : (x, y, z, x, y, z, t) (x,y, z,x, y,z,t). For general values of the pa-rameters, the flow of (2) preserves volume but is neither Hamiltonian nor re-versible. Of course, if is small, it is close to a Hamiltonian system.

It is well known that the RTBP has 5 equilibrium points (Li, i = 1, . . . , 5,see Figure 1, right). For small , these 5 points are replaced by 5 continuousfamilies of equilibria, parametrised by and . For a small value of , we have5 disconnected families of equilibria near the classical Li. For larger values of, these families merge into each other. We end up having two disconnectedsurfaces, S1 and S2, where S1 is like a sphere and S2 is like a torus around theSun ([21, 7, 9]).

2.2. Interesting missions applications

There are some proposed missions where the capabilities of solar sails arefundamental ([20]). One of the first is the so-called Geostorm Warning Mission,whose goal is the continuous observation of the Sun to provide informationand early warnings of geomagnetic storms. The proposed location is near thepoint L1 of the Earth-Sun system. In this case, due to the effect of the sail,the equilibrium point L1 is closer to the Sun, which is better for this mission.Moreover, if the sail is not orthogonal to the Sun direction, the equilibriumpoint is displaced away from the Earth-Sun line (see Figure 3) which makesradio communications possible: if the probe were sitting on the Earth-Sun line

4

SunEarth

x

y

z

0.01 AU

0.02 AU

L1

ACE

Sail

CME

Figure 3: The Geostorm Warning Mission.

then, as seen from Earth, it would be on the middle of the solar disk and itsradio signals would be