Trajectory analysis of spacecraft with propellantless propulsion systems

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Universit`

a di Pisa

Dottorato di Ricerca in Ingegneria Industriale

Curriculum in Ingegneria Aerospaziale Ciclo XXXI

Trajectory analysis of spacecraft with

propellantless propulsion systems

Author

Lorenzo Niccolai

Supervisors

Prof. Alessandro A. Quarta Prof. Giovanni Mengali

Coordinator of the PhD Program Prof. Giovanni Mengali

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Individual science fiction stories may seem as trivial as ever to the blinder critics and philosophers of today . . . but the core of sci-ence fiction, its esssci-ence, the concept around which it revolves, has become crucial to our salvation if we are to be saved at all.

Isaac Asimov (1920–1992) Asimov in Science Fiction (1981)

The expanse of space becomes a literal ana-logue to the open seas. If space is tomorrow’s ocean, then Earth’s surface is its shoreline.

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Abstract

The aim of this Thesis is to provide a systematic analysis of the different aspects involving the preliminary mission analysis and design for a spacecraft equipped with a propellantless propulsion system (solar sail or electric sail). These propul-sive systems are defined and described in the Introduction.

The first Part presents the available mathematical models that may be used to calculate the thrust generated by a solar sail or by an electric sail. These mathematical tools will be then used in the rest of the work. In particular, the solar sail thrust can be modeled assuming an ideal reflection, a non-perfect specular reflection, or, more realistically, through an optical reflection. As far as the electric sail thrust is concerned, different approaches are discussed, starting from the first and very simplified ones, up to the description of the recent and more accurate analytical model.

The second Part focuses on possible mission scenarios for solar sails and elec-tric sails, starting from a literature review. Then, two special mission scenarios for an electric sail are discussed in detail, whose analysis is not yet available or not accurate enough in the existing literature. The first case involves displaced non-Keplerian orbits (both circular and elliptic). The attitude and propulsive re-quirements for orbital maintenance are calculated, and the analysis also discusses some special situations, including the maintenance of an artificial Lagrangian point or of a heliostationary position. The second scenario involves the plasma brake, a derivation of the electric sail basic concept that may be usefully employed to perform an end-of-life deorbiting of a spacecraft in a Low Earth Orbit (LEO). An analytical approximate model capable of estimating the drag force generated by the plasma brake effect is presented and deeply discussed. A comparison with the results of an orbital propagator shows the accuracy of the approximation and the potentialities of such a technology.

The third Part deals with trajectory analysis. First, an approximate model, based on an asymptotic series expansion and on the hypothesis of small thrust magnitude, is used to derive an analytical formulation of the trajectory equation for a spacecraft propelled with a solar sail, an electric sail, or even a conventional electric thruster. Comparisons with the results of a numerical integration of the equations of motion show that these approximations are accurate, especially when a rectification procedure is included in the model. Then, the problem of determining the optimal trajectory is addressed, which usually amounts to finding the minimum-time transfer trajectory, due to the absence of propellant consumption. For a solar sail-equipped spacecraft, an analytical approximate version of the optimal steering law is found, based on the actual numerical values of the sail film optical coefficients, which allows the optimal trajectory to be

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found with a significantly smaller computational time than that required by a conventional approach. The optimal steering law is also derived in an exact form for the case of an electric sail.

The fourth Part analyzes the impact of solar activity fluctuations on the trajectory of a spacecraft propelled by a solar sail or by an electric sail. In the former case, the fluctuations are small, so that their effect is not so marked, even though they should be taken into account by means of a simple control system capable of adjusting the sail optical properties in response to a variation of the environmental conditions. The case of an electric sail is much more involved, since the fluctuations of the solar wind dynamic pressure are fast, chaotic, and have the same order of magnitude as the mean value. Therefore, the plasma dynamic pressure is modelled as a random variable with a gamma distribution and a its effect of the mission parameters is discussed with a statistical analysis. The simulation results highlight that the trajectory of an electric sail-equipped spacecraft under variable solar wind is very difficult to predict, and, for this reason, two possible control strategies are discussed to counteract this issue. Both control laws exploit a suitable adjustment of the electric sail grid voltage to modulate the thrust and compensate the effect of the solar wind fluctuations.

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Preface

This Thesis discusses the work and the main results obtained along my Ph.D. course, from November 2015 to October 2018, and carried out at the Department of Civil and Industrial Engineering of the University of Pisa.

Most of the contents of this Thesis may also be found in the following pub-lished papers:

[J.1] Niccolai L., Quarta A. A., Mengali G., “Electric Sail-Based Displaced Orbits with a Refined Thrust Model ”, Proceedings of the Institution of Mechanical Engineering - Part G: Journal of Aerospace Engineering, Vol. 232, Issue 3, pp. 423-432, March 2018, doi: 10.1177/0954410016679195;

[J.2] Niccolai L., Quarta A. A., Mengali G., “Electric Sail Elliptic Displaced Or-bits with Advanced Thrust Model ”, Acta Astronautica, Vol. 138, pp. 503-511, September 2017, doi: 10.1016/j.actaastro.2016.10.036;

[J.3] Orsini L., Niccolai L., Mengali G., Quarta A. A., “Plasma Brake Model for Preliminary Mission Analysis”, Acta Astronautica, Vol. 144, pp. 297-304, March 2018, doi: 10.1016/j.actaastro.2017.12.048;

[J.4] Niccolai L., Quarta A. A., Mengali G., “Solar Sail Trajectory Analysis with Asymptotic Expansion Method ”, Aerospace Science and Technology, Vol. 68, pp. 431-440, September 2017, doi: 10.1016/j.ast.2017.05.038;

[J.5] Niccolai L., Quarta A. A., Mengali G., “Two-dimensional Heliocentric Dy-namics Approximation of an Electric Sail with Fixed Attitude”, Aerospace Science and Technology, Vol. 71, pp. 441-446, December 2017,

doi: 10.1016/j.ast.2017.09.045;

[J.6] Niccolai L., Quarta A. A., Mengali G., “Orbital Motion Approximation with Constant Circumferential Acceleration”, Journal of Guidance, Control, and Dynamics, Vol. 41, Issue 8, pp. 1783-1789, 2018, doi: 10.2514/1.G003635; [J.7] Niccolai L., Quarta A. A., Mengali G., “Analytical Solution of the Optimal

Steering Law for Non-Ideal Solar Sail ”, Aerospace Science and Technology, Vol. 62, pp. 11-18, March 2017, doi: 10.1016/j.ast.2016.11.031;

[J.8] Niccolai L., Anderlini A., Mengali G., Quarta A. A., “Impact of Solar Wind Fluctuations on Electric Sail Mission Design”, Aerospace Science and Tech-nology. Vol. 82-83, pp. 38-45, November 2018, doi: 10.1016/j.ast.2018.08.032; and in the following conference papers:

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[C.1] Niccolai L., Quarta A. A., Mengali G., “Refined Analysis of Electric Sail-Based Displaced Orbits”, The 5th International Conference on Tethers in Space, 24-26 May, Ann Arbor (MI), USA, 2016;

[C.2] Niccolai L., Bassetto M., Quarta A. A., Mengali G., “Plasma Brake Ap-proximate Trajectory. Part I: Geocentric Motion”, 4th IAA Conference on University Satellite Missions & CubeSat Workshop, 4-7 December, Rome, Italy, 2017;

[C.3] Bassetto M., Niccolai L., Quarta A. A., Mengali G., “Plasma Brake Ap-proximate Trajectory. Part II: Relative Motion”, 4th IAA Conference on University Satellite Missions & CubeSat Workshop, 4-7 December, Rome, Italy, 2017;

[C.4] Niccolai L., Quarta A. A., Mengali G., “Solar Sail Trajectory Analysis with Asymptotic Expansion Method ”, The 4th International Symposium on Solar Sailing, 17-20 January, Kyoto, Japan, 2017;

[C.5] Niccolai L., Anderlini A., Mengali G., Quarta A. A., “Electric Sail Displaced Orbit Control With Solar Wind Uncertainties”, The 69th International As-tronautical Congress, 1-5 October, Bremen, Germany, 2018.

The contents of other work, developed in parallel with my Ph.D. project, may be found in the following journal paper:

[J.9] Bassetto M., Niccolai L., Quarta A. A., Mengali G., “Logarithmic Spiral Trajectories Generated by Solar Sails”, Celestial Mechanics and Dynamical Astronomy, Vol. 130, No. 2, February 2018, doi: 10.1007/s1056.

in the following submitted paper:

[J.10] Niccolai L., Bassetto M., Quarta A. A., Mengali G., “Solar Sailing with Femtosatellites: a Review of Smart Dust Architecture, Dynamics, and Mis-sion Applications”, submitted to Progress in Aerospace Sciences.

and in the following conference papers:

[C.6] Niccolai L., Quarta A. A., Mengali G., Petrucciani F., “Translunar Tra-jectories for Electric Sail Testing”, 10th IAA Symposium on the Future of Space Exploration: Towards the Moon Village and Beyond, 27-29 June, Turin, Italy, 2017

[C.7] Bassetto M., Niccolai L., Quarta A. A., Mengali G., “Notes on Logarithmic Spiral Trajectories Generated by Solar Sails”, The 7th International Meeting on Celestial Mechanics, 4-8 September, San Martino al Cimino (VT), Italy, 2017.

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Acknowledgements

I primi ringraziamenti sono ovvi ma assolutamente doverosi, e vanno ai miei supervisori, Prof. Alessandro Quarta e Prof. Giovanni Mengali, ai quali mi lega un rapporto di stima ed amicizia. Si può senz’altro affermare che senza il loro aiuto, i loro preziosi consigli ed il loro supporto non avrei mai portato a termine questo percorso. Ma si può anche dire che senza la loro serietà professionale e la passione da loro instillata per la meccanica del volo, non avrei nemmeno intrapreso la strada del dottorato.

Non sarei mai arrivato fin qua senza il supporto e all’amore dei miei genitori, che mi hanno permesso di coltivare la mia passione per lo studio. La tristezza per cui solo mamma potr`a assistere a questo traguardo `e in parte sollevata dalla consapevolezza che babbo ne sarebbe profondamente fiero. La persona che sono e i traguardi che raggiungo sono anche (molto) merito vostro. Grazie di cuore. Non posso non citare anche tutto il resto della mia famiglia, una parte di me anche se sparsa da Empoli a Siena.

Un grazie va anche a chi ha condiviso con me questo percorso, costellato di pranzi e pause caff`e. Grazie quindi ai “colleghi” Marco e Andrea, e ai “vicini” fluidodinamici Alessandron (n → ∞), Benedetto, Alessio.

Come chiunque mi conosca sapr`a, non sono molto bravo nell’esternare l’affetto, ma un grazie di cuore lo devo ai miei amici “extra-universitari”, sia i vicini Luigi, Francesca, Bob, Michel, Alessio, Alice, Letizia, Marco, Maddalena, Iacopo, Rob, Matteo2 . . . che i lontani Dario, Francesco e gli altri membri della Last Line Pio e Baru.

Infine, un grazie enorme, gigantesco, va alla persona con cui condivido un percorso ancora pi`u importante di quello del dottorato. Grazie per starmi accanto, supportarmi/sopportarmi, e rendermi ogni giorno una persona migliore. Grazie per la felicit`a reciproca che ci diamo. Grazie Ari, “meno male ti ho trovata”.

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Chapter

1

Introduction

In the history of space exploration, most of past missions were equipped with a chemical propulsion system, capable of providing a significant amount of thrust by consuming a large mass of reacting propellant. Typically, a chemical engine has a short firing time, such that its effect can be approximated with an impul-sive variation of the orbital velocity vector. However, the first pioneers of space propulsion, including Konstantin E. Tsiolkovsky and Robert Goddard, proposed the idea of generating thrust by accelerating charged particles with very small mass by means of electrical power [1–3]. Later, one of Tsiolkovsky’s scholars, Fridrickh A. Tsander, was the first to conceive the solar sail concept, suggesting the possibility of flying with enormous mirrors or very thin sheets [4].

Both electric propulsion systems and solar sails are capable of providing a low thrust, with the difference that the former consume a small amount propellant, unlike the latter that require no propellant at all. Both systems are able to operate for a long time, and provide a low continuous thrust. From the dawning of space era to the present, the progress in this field is emphasized by the theoretical research on low-thrust engines, by a number of studies on propelled trajectories, and is especially demonstrated by the past and currently operative space missions equipped with such propulsive systems. The electric propulsion was initially used in test missions (Sert 1 and Zond 2 ) or for station keeping purposes. The first satellite equipped with a low-thrust engine as its primary propulsion system was the Deep Space 1, launched by NASA on 26 October 1998 and directed towards the asteroid Braille and the comet Borrelly. Its Xenon-ion thruster was capable of providing a thrust of 0.092 N and fired for a total time of more than 14000 hours. Other recent or currently operative space missions equipped with electric ion thrusters are NASA’s Dawn, which visited Vesta and is now orbiting around Ceres, JAXA’s Hayabusa and Hayabusa 2, both directed towards Near-Earth Asteroids.

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2 1. Introduction

An even more recent trend in space propulsion research is focused on propel-lantless propulsion systems, such as the aforementioned solar sails. A solar sail basically consists of a thin reflective membrane that exchanges momentum with the impinging photons coming from the Sun, see Fig. 1.1. In essence, a solar sail

Figure 1.1: Artistic rendering of a square solar sail.

is propelled by the solar radiation pressure, so it does not require any propellant to work. Therefore, the potentialities of a solar sail are enormous, in particular for long mission durations, because it is capable of providing a low but continu-ous thrust for a theoretically infinite time. A very exhaustive discussion on solar sail concept, structure, technology, and possible applications may be found in Refs. [5–7].

Despite the action of solar radiation pressure is well known since the Mariner 10 mission, the feasibility of using solar sails as primary propulsive systems had been questioned for a long time. Indeed, before the last passage of Halley’s comet in 1986, NASA attempted to project a solar sail-equipped mission to perform a cometary rendezvous, but the project was eventually broken off due to the high associated risk. The first satellite that used a solar sail as a primary propulsion system and for attitude control purposes was JAXA’s IKAROS (Interplanetary Kite-craft Accelerated by Radiation Of the Sun) [8–10], launched on May 2010, which successfully performed a Venus flyby. More recently, on January 2011, NASA tested the deployment in a LEO of a small square solar sail (3.2 m-side length) with the NanoSail-D2 mission [11]. A private company, the Planetary So-ciety, launched the first private solar sail-equipped satellite, Lightsail-1 [12], which

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3

was equipped with a 32 m2 _{square sail and performed a fast deorbiting from a}

LEO thanks to the augmented atmospheric drag. The recent success of solar sails is also demonstrated by future space missions that will be equipped with such a propulsive system, including a JAXA’s mission towards the Trojan asteroids [13], the NASA’s NEA-Scout, which is going to perform a rendezvous with a Near-Earth asteroid [14], and the technological demonstrator mission Lightsail-2 from Planetary Society.

Other propellantless propulsive concepts exploits the solar wind momentum flux, and not the solar radiation pressure, to generate a propulsive acceleration. More precisely, the magnetic sail was proposed by Andrews and Zubrin [15], whereas the electric sail is an even more innovative concept, which was conceived by Dr. Pekka Janhunen in 2004 [16]. An Electric Solar Wind Sail (shortened in electric sail or E-sail) consists of a spinning grid of tethers, which are kept at a high (usually positive) potential. When immersed in the solar wind plasma, the tethers electrostatically interact with the ions coming from the Sun, and the momentum exchange generates a propulsive acceleration. A sketch of the basic structure of an E-sail is shown in Fig. 1.2.

solar wind electron gun body solar panel tether

Figure 1.2: Basic sketch of an E-sail structure.

The recent noteworthy interest shown by the scientific community and the space agencies about propellantless propulsive system constitutes the major mo-tivation of this Thesis. Its main objectives are to analyze possible mission sce-narios, to investigate the trajectory determination problem, and to estimate the impact of environmental uncertainties on the propelled trajectories generated by

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4 1. Introduction

solar sails and electric sails, to give a systematic analysis that should represent an useful guide for a preliminary mission design phase of a spacecraft equipped with such propulsive systems.

Part I discusses the available analytical models used to define the propulsive acceleration generated by a solar sail or by an electric sail. These models will then be used in the rest of the work. Then, Part II deals with possible mis-sion scenarios for propellantless propulmis-sion systems, starting from the fact that their peculiarities allow exotic mission scenarios to be envisaged. In particular, circular and elliptic displaced non-Keplerian orbits for electric sails are deeply discussed and the corresponding attitude and propulsive requirements are iden-tified. Moreover, an analytical model for analyzing the aforementioned plasma brake deorbiting strategy is presented, and its effectiveness is verified by com-parison with an orbital propagator. Part III focuses on the trajectory analysis of a propellantless propulsive system. First, an asymptotic series expansion is used to derive an analytical approximation of the trajectory equation for a solar sail, an E-sail, and also for a generic electric thruster. Then, the analysis concen-trates on the problem of the optimal trajectory, which in this case coincides with the minimum-time transfer trajectory. The optimal control law is expressed by means of an analytical approximate form for a solar sail. Finally, Part IV analyzes the impact of uncertainties due to solar activity fluctuations on the trajectories of a spacecraft equipped either with a solar- or an electric sail. In particular, the latter generates a thrust that depends on the very chaotic and unpredictable behaviour of the solar wind, so that a control system that adjusts the grid volt-age is required. Two possible strategies are presented and discussed, and their effectiveness is illustrated in potential mission scenarios.

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Part I

Thrust vector mathematical

model

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Chapter

2

Spacecraft motion with continuous low-thrust

This chapter concentrates on the formulation of the dynamics equations that describe the motion of a spacecraft propelled by a low continuous thrust. A brief review is given, with a distinction made between notable cases, such as radial, circumferential, or tangential thrust, for which the most important results available in the literature are summarized. The tools discussed in this chapter will be the basis of the mission analysis and trajectory design of a spacecraft equipped with propellantless propulsion systems, which will constitute the remainder of the work.

2.1 General model

Consider a spacecraft whose propulsive system generates a continuous thrust. To simplify the mathematical model, the spacecraft is assumed to orbit around a single primary with gravitational parameter µ and to be subjected to the grav-itational pull and the propulsive acceleration only, thus neglecting other per-turbation sources. Under these hypotheses, its motion can be described by the following equations, written in a reference frame whose origin is placed at the primary center of mass

˙r = v (2.1)

˙v = −µ

r3r + a (2.2)

where the dot symbol denotes the time derivative, r and v are the position and velocity vectors, respectively, r , krk is the primary-spacecraft distance, and a is the propulsive acceleration vector.

For a classical (either chemical or electric) propulsive system, the mathe-matical formulation given by Eqs. (2.1)–(2.2) must be completed with the mass

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8 2. Spacecraft motion with continuous low-thrust primary fixed direction q spacecraft r z plane P r

Figure 2.1: Polar reference frame TO(O; ρ, θ, z).

conservation equation dm dt = − m a g0Isp (2.3) where m is the total spacecraft mass, g0 is the standard gravity, a , kak, and

Isp is the engine specific impulse.

Note that Eqs. (2.1)–(2.2) may be decomposed into scalar equations by means of a cylindrical reference frame TO(O; ρ, θ, z), with origin O at the primary’s

center of mass and whose coordinate are defined as follows. First, the osculating orbital plane at the initial time instant t0 , 0 is denoted with P. Recall that

the coordinate r is the instantaneous primary-spacecraft distance r , krk = p

ρ2_{+ z}2_{. The radial coordinate ρ is the component of r on plane P. The angle}

θ is measured counterclockwise from a fixed direction on P to the projection of r on the plane P. Finally, the vertical coordinate z expresses the distance between the instantaneous spacecraft position and the plane P. The coordinates of TO

are illustrated in Fig. 2.1. Equations (2.1)–(2.2) can be written in TO as

˙ ρ = vρ (2.4) ˙ θ = vθ ρ (2.5) ˙ z = vz (2.6) ˙vρ= v2_θ ρ − µ (ρ2_{+ z}2₎3/2ρ + ar (2.7) ˙vθ = − vrvθ ρ + aθ (2.8) ˙vz = − µ (ρ2_{+ z}2₎3/2z + az (2.9)

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2.2 Special cases 9

where vρ, vθ, and vz(or aρ, aθ, and az) are the radial, circumferential, and vertical

components of the spacecraft velocity (or propulsive acceleration).

An usual simplification of the dynamical system given by Eqs. (2.4)–(2.9) is obtained by assuming that the thrust lies on the plane P throughout the whole motion. As a result, the osculating orbital plane does not change, so that the spacecraft dynamics is confined on the plane P and the trajectory is purely two-dimensional. Accordingly, ρ ≡ r, and the spacecraft dynamics can be described in a more compact form, viz.

¨

r − r ˙θ2= −µ

r2 + ar (2.10)

r ¨θ + 2 ˙r ˙θ = aθ (2.11)

which will be used in the succeeding discussion.

2.2 Special cases

The formulation of Eqs. (2.10)–(2.11) can be further simplified in special cases, as will now be discussed for some mission scenarios.

2.2.1 Radial thrust case

When the propulsive acceleration is radial, that is, is constantly aligned with the primary-spacecraft direction, Eqs. (2.10)–(2.11) can be simplified as

¨ r − r ˙θ2 = −µ r2 + a (2.12) 1 r dh dt = 0 (2.13)

where a > 0 (or a < 0) if a · r > 0 (or a · r < 0), and h , r2θ is the specific˙ angular momentum of the osculating orbit. Note that Eq. (2.13) implies that a radial thrust does not change the angular momentum h, and consequently the osculating semilatus rectum p = h2/µ is constant.

Tsien [17] studied the radial thrust problem assuming a constant value of the propulsive acceleration magnitude a and a circular parking orbit. He showed that the spacecraft may reach an escape trajectory only if the condition

a ≥ µ/(8 r2₀) (2.14)

is met. In this case, the escape radius resc can be calculated as

resc= r0 1 + µ 2 r2 0a (2.15) Note that the subscript 0 denotes the value of the variable at the initial time instant. The escape time tesc may be obtained by numerically solving a first-kind

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10 2. Spacecraft motion with continuous low-thrust

elliptic integral. The motion of a spacecraft subjected to radial thrust was also studied with a similar approach by Battin [18] and Boltz [19], who included in the analysis the mass loss due to the propellant consumption.

A different strategy for the analysis of the radial thrust case consists in study-ing the problem by an energetic point of view. This idea was proposed by Prussstudy-ing and Coverstone-Carroll [20] and is based on the expression of the osculating spe-cific mechanical energy E , which may be written as a function of its initial value, the primary-spacecraft distance r, and the propulsive acceleration magnitude a, viz. E = E₀+ a (r − r0) (2.16) with E , v 2 r 2 + v2_θ 2 − µ r = ˙r2 2 + µ r0 2 r2 − µ r (2.17)

where the condition r2θ =˙ √µ r0 has been used, see Eq. (2.13). Note that

Eqs. (2.16)–(2.17) are valid because the parking orbit is assumed to be circular. When the physical constraint ˙r2 _{≥ 0 is enforced, the (r/r}

0, E /E0) plane is divided

into a feasible and an unfeasible region. These results are in perfect accordance with those of Tsien, and prove again that motion is unbounded when the inequal-ity (2.14) is met, otherwise the spacecraft dynamics is confined into a bounded region around the primary.

2.2.2 Circumferential thrust case

A circumferential thrust is constantly aligned with the local horizon and or-thogonal to the primary-spacecraft direction. Accordingly, Eqs. (2.10)–(2.11) are rewritten as

¨

r − r ˙θ2 = −µ

r2 (2.18)

r ¨θ + 2 ˙r ˙θ = a (2.19)

where a can also have negative values if a · v < 0.

The trajectory determination of a spacecraft subjected to a circumferential propulsive acceleration is a classical problem of orbital mechanics, which has been deeply investigated since the pioneering work of Tsien [17], who, assuming a constant propulsive acceleration magnitude a, derived some interesting results, including the fact that the circumferential thrust is much more efficient for vary-ing the osculatvary-ing orbital elements when compared to the radial thrust case. Battin [18] proposed an interesting approximation for a simplified situation, that is, when the parking orbit is circular and the thrust magnitude is very low. Under

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2.2 Special cases 11

these assumptions, the time history of the primary-spacecraft distance r(t) is

r ' r0 1 − a t pµ/r0 !−2 ' r₀ 1 + 2 a t pµ/r0 ! (2.20)

Unlike the radial thrust case, an escape condition can always be reached using a circumferential thrust, if a > 0. The escape distance may be obtained as

resc=

r µ

2 a (2.21)

whereas the escape time is given by tesc = 1 a r µ r0 − r µ resc (2.22) where resc is taken from Eq. (2.21).

Quarta and Mengali [21] discussed the motion of a spacecraft propelled by a constant circumferential thrust with an alternative approach, based on the use of suitable dimensionless variables. They found the following approximate solution for the polar equation r(θ)

r ' s r0

1 − 4 a θ pµ/r2

0

(2.23)

as well as a set of analytical estimations of the escape conditions. The interested reader may find more details in Ref. [21].

2.2.3 Tangential thrust case

A tangential thrust is always aligned with the instantaneous orbital velocity vec-tor. Hence, Eqs. (2.10)–(2.11) may be written as

¨ r − r ˙θ2= −µ r2 + a ˙r p ˙r2_{+ r}2_θ˙2 (2.24) r ¨θ + 2 ˙r ˙θ = ap r ˙θ ˙r2_{+ r}2_θ˙2 (2.25) where, again, a < 0 if a · v < 0.

Battin [18] found interesting analytical results, assuming the parking orbit to be circular and the propulsive acceleration magnitude to be small when compared to the local gravity. In this case, for a > 0, the escape radius resc is

resc= √ µ r0 20 a2_r2 0 1/4 (2.26)

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12 2. Spacecraft motion with continuous low-thrust

while the escape time is

tesc = √ µ a√r0 " 1 − 20 a 2_r4 0 µ2 1/8# (2.27)

Zee [22] discussed an analytical spiral-shaped solution for the spacecraft trajectory by neglecting the oscillatory terms, whereas Boltz [23] accounted for the mass loss due to propellant consumption and derived some analytical approximate solutions for a thrust magnitude equal to a given fraction of the total spacecraft mass.

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Chapter

3

Propulsive acceleration models for

propellantless thrusters

This chapter discusses and analyzes the mathematical models available in the literature for calculating the propulsive acceleration vector of a spacecraft propelled by a solar sail or an electric sail, which are the propulsive technolo-gies used for the analysis in the remainder of the work. In the former case, the discussion starts from the ideal or η-perfect reflection model, and then focuses on the more realistic optical force model. As far as the electric sail case is concerned, the review covers the original simplified approach, a poly-nomial interpolation-based model, and the recent vectorial expression of the propulsive acceleration.

3.1 Introduction

For a solar sail or an electric sail-based spacecraft, the absence of propellant consumption reduces Eq. (2.3) to the trivial solution of constant spacecraft mass. Because the propulsive acceleration and the thrust are simply related by a direct proportionality, the objective of this section is confined to the definition of a mathematical model for the former, while the derivation of the thrust level is straightforward.

3.2 Solar sail

In this section, different possible thrust models for a solar sail are presented and analyzed. These models are characterized by different complexity and accuracy, but are all based on a fundamental assumption, that is, the sail surface is perfectly flat and any billowing effect under the applied load is neglected. This hypothesis is sufficiently realistic for a preliminary mission analysis phase, but a parametric model accounting for the sail deformation exists, and is thoroughly discussed in

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14 3. Propulsive acceleration models for propellantless thrusters

Ref. [6].

The magnitude of the propulsive acceleration generated by a solar sail is com-monly expressed as a function of a performance parameter. The most common choice is the lightness number β, which expresses the ratio between the maximum propulsive acceleration that a solar sail is able to generate at a given Sun-sail dis-tance r to the local Sun’s gravitational pull. It is worth noting that the maximum force generated by a solar sail is directly proportional to the solar radiation pres-sure P , which can be expressed as a function of the Sun-spacecraft distance r as P = P⊕ r_⊕ r 2 (3.1) where the subscript ⊕ denotes the value at a Sun-Earth distance, that is, at r⊕ , 1 au. Based on Eq. (3.1), it is possible to state that both the solar sail

propulsive acceleration and the Sun’s gravitational attraction scale with r−2, so that the lightness number β is a constant parameter. An alternative parameter for measuring the sail performance is the characteristic acceleration ac, which is

defined as the maximum propulsive acceleration that the solar sail can generate when the Sun-spacecraft distance is r⊕, 1 au.

Figure 3.1 shows the fundamental quantities that are used to define the spatial orientation of the thrust vector generated by a solar sail. In particular, the unit vector ˆr , r/r denotes the Sun-spacecraft direction, ˆn is the unit vector normal to the sail plane and directed away from the Sun, whereas ˆa , a/a identifies the propulsive acceleration direction. The pitch angle α is measured between the radial direction expressed by ˆr and the external normal to the sail ˆn, while the cone angle φ is the angle between ˆr and the propulsive acceleration unit vector ˆ

a.

3.2.1 Ideal force model

The simplest way to model the thrust generated by a solar sail consists in as-suming that all of the photons impinging the sail are specularly reflected. This condition is usually referred to as ideal force model, and in this case the sail propulsive acceleration vector a may be written as

a = 2 P⊕A m r ⊕ r 2 (ˆr · ˆn)2 nˆ (3.2)

where P⊕, 4.563 µN/m2 is the solar radiation pressure at a distance r = r⊕, A

is the sail area, and m is the total spacecraft mass. Note that for an ideal force model ˆa ≡ ˆn, so that α ≡ φ. The lightness number β at a Sun-Earth distance in a perfect reflection case reduces to

β = 2 P⊕A/m µ/r2⊕

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3.2 Solar sail 15

r

ˆ

n

ˆ

r

a

aˆ

solar sail

f

Sun

Figure 3.1: Fundamental angles and directions for a solar sail thrust model.

which, when substituted into Eq. (3.2), yields

a = β µ r2

(ˆr · ˆn)2 nˆ (3.4)

Note that Eq. (3.4) implies that the maximum magnitude of a is obtained for a Sun-facing configuration (i.e., ˆr ≡ ˆn), so that the characteristic acceleration is related to the lightness number as

ac= 2 P⊕A m = β µ r2 ⊕ (3.5)

The vector a given by Eq. (3.4) can be properly decomposed in the three-dimensional polar reference frame TO, if the sail attitude is determined through

the pitch angle α ∈ [0, π/2] and the clock angle δ ∈ [0, 2π]. As sketched in Fig. 3.2, the latter is the angle between the projection of ˆn onto the local horizon-tal plane and a fixed direction, which usually coincides with that of the osculating orbital angular momentum vector. Note that, in the special case ˆa ≡ ˆr ≡ ˆn, the clock angle is undefined, and a has only a radial component given by Eq. (3.4). Accordingly, Eq. (3.4) can be rewritten by means of three scalar equations

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fixed direction

ˆr

ˆ

n

local horizontal plane

a

d

sail center of mass

Figure 3.2: Pitch angle α and clock angle φ.

ar= β µ r2 cos3α (3.6) aθ= β µ r2

cos2α sin α sin δ (3.7)

az= β

µ

r2

cos2α sin α cos δ (3.8)

which could be substituted into Eqs.(2.7)–(2.9) to write the three-dimensional equations of motion for an ideal solar sail.

However, in many mission scenarios, the sail thrust lies on the osculating orbital plane along the whole motion path. In this case, there are only two possible values of the angle δ ∈ {π/2, 3π/2}, where the first (second) value corresponds to a positive (negative) circumferential thrust component, which tends to increase (decrease) the orbital energy. To simplify the analysis in a two-dimensional case, the range of variation of α can be extended to the negative values, that is, α ∈ [−π/2, π/2], where α < 0 denotes a negative circumferential component of the propulsive acceleration. Such a strategy allows the clock angle δ to be removed from the state variables. Thus, Eqs. (3.6) and (3.7) are rewritten as

ar= β µ r2 cos3α (3.9) aθ= β µ r2 cos2α sin α (3.10)

where the sail attitude is a function of α only. As is implied by Eqs. (3.6)–(3.8) or by Eqs. (3.9)–(3.10), when an ideal force model is adopted, the propulsive

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3.2 Solar sail 17

acceleration magnitude becomes a = β µ r2 cos2α (3.11)

for both the two-dimensional and the three-dimensional case.

3.2.2 Non-perfect reflection force model

A non-perfect specular reflection of impinging photons on the solar sail film can be modeled by slightly modifying the previously described ideal force model. In particular, the non-perfect reflection model assumes that the non-ideal reflective behaviour of the sail coating reduces the propulsive acceleration magnitude, but leaves its direction unchanged [24]. Consequently, Eq. (3.4) may be rewritten as

a = β µ r2

η (ˆr · ˆn)2 nˆ (3.12)

where the factor η accounts for the non-ideal reflection. Its value is usually taken close to η ' 0.85. This force model is often referred to as η-perfect reflection (η-PR) model. It is evident from Eq. (3.12) that the condition ˆn ≡ ˆa (or α ≡ φ) still holds. The characteristic acceleration becomes

ac= β µ r2 ⊕ η (3.13)

and the propulsive acceleration components for the two-dimensional case are ar = β µ r2 η cos3α (3.14) aθ = β µ r2 η cos2α sin α (3.15)

The generalization to the three-dimensional case is straightforward and is here omitted for the sake of conciseness. The magnitude of the propulsive acceleration can be calculated as a = β µ r2 η cos2α (3.16)

for both the two-dimensional and the three-dimensional case.

3.2.3 Optical force model

A general and more accurate mathematical model of the propulsive acceleration generated by a solar sail is the optical force model [5, 6, 25]. The latter accounts for photons that are specularly reflected, diffusely reflected, or emitted, neglecting the light transmission only. The propulsive acceleration is written as

a = 2 P⊕A m r ⊕ r 2 (ˆr · ˆn) {b1r + [bˆ 2(ˆr · ˆn) + b3] ˆn} (3.17)

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where b1, b2, and b3 are called force coefficients, and may be obtained from the

sail film optical properties as b1= 1 − crcs 2 (3.18) b2= crcs (3.19) b3= Bfcr(1 − cs) 2 + (1 − cr) (fBf − bBb) 2 (f + b) (3.20) where cris the reflection coefficient, csis the fraction of reflected photons that are

specularly reflected, Bf (or Bb) is the front (or back) sail surface non-Lambertian

coefficient, and f (or b) is the front (or back) sail surface emissivity coefficient.

An interesting consequence of Eq. (3.17) is that the propulsive acceleration for an optical sail is no longer parallel to ˆn, but still lies on the plane spanned by ˆr and ˆn. Hence, its direction is univocally determined by the cone angle φ and the clock angle δ, see Fig. 3.2.

Note that the ideal force model may be thought of as a special case of the optical one. Indeed, the assumption according to which all photons are specularly reflected corresponds to imposing cr = cs = 1, and therefore b1 = b3 = 0 and

b2 = 1. In this case, Eq. (3.17) reduces to Eq. (3.2). Experimental data for the

optical coefficients listed above were carried out in 1978 during the JPL mission project towards Halley’s comet [5], and were updated in 2015 by more refined measurements [26]. The previous and more recent results are reported in Tab 3.1 and compared with the ideal sail case.

model cr cs Bf Bb f b b1 b2 b3

ideal 1 1 2/3 2/3 0 0 0 1 0

JPL-1978 0.88 0.94 0.79 0.55 0.05 0.55 0.0864 0.8272 −0.0055 JPL-2015 0.91 0.94 0.79 0.67 0.025 0.27 0.0723 0.8554 −0.0030

Table 3.1: Optical and force coefficients for a flat solar sail with an ideal [6] and an optical [5, 26] force model.

It is evident from Eq. (3.17) that the maximum magnitude of the propulsive acceleration vector is obtained again for a Sun-facing attitude (ˆr ≡ ˆn). Hence, the value of the lightness number when an optical force model is adopted is

β = 2 P⊕A/m µ/r2⊕

(b1+ b2+ b3) (3.21)

whereas the characteristic acceleration can be expressed as ac= 2 P⊕A m (b1+ b2+ b3) = β µ r2 ⊕ (3.22)

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3.2 Solar sail 19

When Eq. (3.21) is substituted into Eq. (3.17), it is found that a = β µ r2 1 b1+ b2+ b3 (ˆr · ˆn) {b1r + [bˆ 2(ˆr · ˆn) + b3] ˆn} (3.23)

which expresses the propulsive acceleration vector as a function of the lightness number.

For a two-dimensional motion, the propulsive acceleration vector given by Eq. (3.23) can be thought of as the sum of a radial and a circumferential compo-nent, which are both functions of the pitch angle α ∈ [−π/2, π/2], viz.

ar= β µ r2 1 b1+ b2+ b3

(b1 cos α + b2 cos3α + b3 cos2α) (3.24)

aθ= β µ r2 1 b1+ b2+ b3

(b2 cos2α sin α + b3 cos α sin α) (3.25)

The acceleration vector a given by Eq. (3.17) may also be decomposed along two different orthogonal directions, defined by unit vectors ˆn (normal to the sail plane) and ˆt (parallel to the sail plane) viz.

a = a⊥n + aˆ kˆt (3.26)

where, taking into account Eq. (3.17), it is found that a⊥= 2 P⊕As m r_⊕ r 2

b1 cos2α + (b2cos α + b3) cos α

(3.27) ak = 2 P⊕As m r ⊕ r 2 b1 sin α cos α (3.28)

which can be rewritten as a⊥= β b1+ b2+ b3 µ r2

b1 cos2α + (b2cos α + b3) cos α

(3.29) ak= β b1+ b2+ b3 µ r2 b1 sin α cos α (3.30)

The derivation of three-dimensional components is possible by introducing the clock angle δ (see Fig. 3.2), but is omitted here for the sake of conciseness. The magnitude of the propulsive acceleration and the cone angle φ are simply calculated as a = q a2 r+ a2θ , φ = arctan aθ ar (3.31) where the expression of the components are given by Eqs. (3.24) and (3.25). Note that Eq. (3.31) assumes that the admissible range for the cone angle is extended to φ ∈ [−π/2, π/2], similarly to what is done for the pitch angle α.

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A further improvement of the optical force model may be introduced by con-sidering the degradation of the sail film material during the motion. However, for preliminary mission design phase, the described model has a sufficient accu-racy. The interested reader may find more details about available sail degradation models in Refs. [27, 28].

3.3 Electric solar wind sail

Existing literature on possible thrust models for an E-sail is significantly less extensive than that regarding solar sails, due to the novelty of the E-sail concept. A brief review on the available mathematical models is presented in this section. In all cases the E-sail is assumed to be axially-symmetric and all the grid tethers to be coplanar.

Unlike a solar sail, its thrust does not scale as r−2, so that the lightness number is not a constant. In fact, although the solar wind dynamic pressure p scales as r−2, similar to the solar radiation pressure, that is

p = p⊕

r_⊕ r

2

(3.32) the plasma Debye length1 increases with the distance from the Sun, so that a larger apparent tether width arises. Therefore, the most common performance parameter for an E-sail is its characteristic acceleration ac.

The direction of the E-sail propulsive acceleration vector is conveniently ex-pressed through angles that are very similar to those used for a solar sail, as can be seen from Fig. 3.3. In particular, the definitions of ˆr, ˆa, α, and φ are the same. Moreover, since all of the tethers are modeled as coplanar, their common plane is denoted as the E-sail nominal plane, and the unit vector ˆn for an E-sail is assumed to be orthogonal to the nominal plane and directed outward with respect to the Sun.

3.3.1 Original model

Preliminary simulations [29] suggested that the propulsive acceleration a gener-ated by an E-sail could be written as

a = ac r ⊕ r 7/6 ˆ a (3.33)

Note that, according to Eq. (3.33), the E-sail thrust magnitude does not depend on the sail attitude. The thrust direction can be expressed by the cone angle φ ∈ [0, π/2] as a function of the pitch angle α, viz.

φ = α

2 (3.34)

1_{In plasma physics, the Debye length is the distance within which any possible unbalance in} the charge distribution is not neutralized by the surrounding particles.

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3.3 Electric solar wind sail 21

r

Sun

ˆ

n _a

ˆ

r E-sail a f

E-sail nominal plane

ˆ

Figure 3.3: Fundamental angles and directions for an E-sail thrust model.

Numerical analyses also suggested the existence of a maximum value of α that, due to mechanical instability issues, should not exceed 60-70 deg. This constraint implies a maximum value of φ of about 30-35 deg.

The model discussed by Mengali et al. [29] was later updated by Janhunen [30]. In fact, numerical plasmadynamic simulations showed that the exponent of (r⊕/r)

in Eq. (3.33) should be adjusted as a = ac r_⊕ r ˆ a (3.35)

whereas the thrust direction was not further investigated.

3.3.2 Polynomial model

In 2013, Yamaguchi and Yamakawa [31] performed accurate simulations investi-gating the thrust vectors acting on each conducting tether of the E-sail grid, and derived two polynomial expressions for the cone angle φ and the propulsive accel-eration magnitude of the whole E-sail. In particular, a dimensionless parameter γ was defined as

γ , kak

ac(r⊕/r)

(3.36) which expresses the ratio of the actual to the maximum propulsive acceleration magnitude that the E-sail could generate for a given value of the pair {α, r}. The

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sixth-order polynomial interpolations of numerical results for φ and γ are φ = j6α6+ j5α5+ j4α4+ j3α3+ j2α2+ j1α + j0 (3.37)

γ = k6α6+ k5α5+ k4α4+ k3α3+ k2α2+ k1α + k0 (3.38)

where the values of the coefficients ji and ki (with i = 0, 1, ..., 6) are listed in

Tab. 3.2. Note that Eq. (3.37) gives a maximum value of the thrust angle smaller

i 0 1 2 3 4 5 6

ji 0 4.853 × 10−1 _{3.652 × 10}−3 _{−2.661 × 10}−4 _{6.322 × 10}−6 _{−8.295 × 10}−8 _{3.681 × 10}−10 ki 1.000 6.904 × 10−5 −1.271 × 10−4 _{7.027 × 10}−7 _{−1.261 × 10}−8 _{1.943 × 10}−10 _{−5.896 × 10}−13

Table 3.2: Best-fit interpolation coefficients of E-sail thrust parameters (φ and α in degrees). Data taken from Ref. [31].

than 20 deg.

3.3.3 Final model

Recently, Huo et al. [32] derived and discussed the following analytical expression for the propulsive acceleration generated by an E-sail with more than two identical tethers a = τ ac 2 r_⊕ r [ˆr + (ˆr · ˆn) ˆn] (3.39) where τ ∈ {0, 1} is a switching parameter that accounts for the possibility of switching on (τ = 1) or off (τ = 0) the electron gun. Equation (3.39) shows that the E-sail propulsive acceleration vector lies in the plane spanned by the Sun-spacecraft vector ˆr and the external normal to the E-sail nominal plane ˆn. Using the model proposed by Huo et al., the characteristic acceleration accan be

calculated as

ac=

N L σ⊕

m (3.40)

where N ≥ 2 is the number of tethers, L is the tether length, and the parameter σ⊕ is calculated as [33, 34]

σ⊕, 0.18 max (0, V − Vw)

√

0p⊕ (3.41)

In Eq. (3.41), V is the tether grid voltage (usually a few tens of kV), Vw is

the electric potential associated with the kinetic energy of solar wind particles (Vw' 1 kV), and 0 is the vacuum permittivity.

From Eq. (3.39), the magnitude of a can be expressed as a = τ ac 2 r ⊕ r p 1 + 3 cos2_α _(3.42)

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3.4 Final considerations 23

whereas the cone angle and the dimensionless factor γ are φ = arccos 1 + cos2α √ 1 + 3 cos2_α (3.43) γ = √ 1 + 3 cos2_α 2 (3.44)

These analytical expressions are in perfect accordance with the numerical results of Ref. [31]. Similarly to the solar sail case, in a two-dimensional scenario the possible values for the pitch and cone angles are extended to the negative cases, that is, α ∈ [−π/2, π/2] and φ ∈ [−π/2, π/2], so that Eq. (3.43) is rewritten as

φ = sign (α) arccos 1 + cos2α √ 1 + 3 cos2_α (3.45) whereas Eq. (3.44) is still valid. Note that Eq. (3.45) can be inverted as

α =                sign (φ) arccos   q 2 (3 cos2_{φ + cos φ}p 9 cos2_{φ − 8 − 2)} 2   sign (φ) arccos   q 2 (3 cos2_{φ − cos φ}p 9 cos2_{φ − 8 − 2)} 2   (3.46)

from which it is evident that two possible values of the pitch angle can generate the same cone angle, but the smaller one is better, since it gives a larger value of γ, see Eq. (3.44). Also note that the maximum value of the cone angle φmax

amounts to about 19.45 deg. The variations of φ and γ as functions of α are shown in Figs. 3.4(a) and 3.4(b).

The E-sail propulsive acceleration can be decomposed in a radial and a cir-cumferential component as ar= τ ac 4 r ⊕ r [3 + cos (2α)] (3.47) aθ= τ ac 4 r ⊕ r sin (2α) (3.48)

Equations (3.47)–(3.48) may be extended to the three-dimensional case. In fact, since ˆa is coplanar to ˆn and ˆr, as in solar sail case, the thrust direction is a function of the cone angle φ and the clock angle δ, see Fig. 3.2.

3.4 Final considerations

The thrust models presented for a solar sail and an electric sail allows some considerations to be discussed.

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24 3. Propulsive acceleration models for propellantless thrusters -90 -60 -30 0 30 60 90

,

[deg]

-20 -15 -10 -5 0 5 10 15 20 (a) φ(α) -90 -60 -30 0 30 60 90

,

[deg]

0.5 0.6 0.7 0.8 0.9 1 (b) γ(α)

Figure 3.4: Variation of the cone angle φ and the dimensionless acceleration γ as functions of the pitch angle α.

First, the thrust magnitude decreases with the Sun-spacecraft distance for both propulsive systems, but it scales as r−2 for a solar sail, and as r−1 for an E-sail. Therefore, the former seems a stronger candidate for missions directed to the inner Solar System, whereas the electric sail could guarantee a sufficient

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3.4 Final considerations 25

level of thrust at larger heliocentric distances. However, since this consideration does not take into account the different nature of the propulsive systems, it surely needs to be updated once that experimental data on the real performance level of an E-sail in deep space will be available.

The second remark is about the capability of generating a circumferential thrust component. This is important because it is the only component that al-lows the osculating orbital angular momentum to be modified, see Eq. (2.11). Figure 3.5 shows the possible values of the radial and circumferential compo-nents of the propulsive acceleration that are obtained with different pitch angles for an ideal solar sail, an optical solar sail, and an electric sail. The propulsive

ac-ideal solar sail

optical solar sail electric sail 0 0.2 0.4 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rad circ

Figure 3.5: Possible values of the dimensionless radial (“rad”) and circumferential (“circ”) propulsive acceleration components for different pitch angles.

celeration components are dimensionless, since they are divided by the maximum propulsive acceleration magnitude (corresponding to the radial thrust case). It is evident that both systems, and in particular the E-sail, have a limited capa-bility of generating a transverse propulsive acceleration. This could constitute a problem for mission scenarios requiring large variations of the osculating orbital angular momentum. Moreover, it is interesting to observe that, unlike the solar

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sail case, the E-sail thrust does not vanish for α = π/2. However, a spacecraft equipped with an E-sail still has the possibility of covering coasting arcs during its motion by simply switching off the electron gun, that is, setting τ = 0 in Eq. (3.39).

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Part II

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Chapter

4

Solar sail and E-sail mission applications

This chapter deals with possible mission scenarios that can be obtained with a solar sail or an electric sail-based spacecraft. First, a review discusses the potential applications of such propulsive systems. Then, the focus is put on two mission scenarios. The first one is constituted by the maintenance of displaced non-Keplerian orbits with an electric sail. This problem has been addressed in the literature only using the original E-sail thrust model and under the assumption of a circular orbit case. This work exploits the most recent thrust model to derive the attitude and propulsive requirements for both cases of circular and elliptic orbit. Some special mission scenarios are presented, including Type II displaced orbits, the generation of an artificial La-grangian point, the maintenance of a heliostationary position, and a displaced orbit constantly placed above (or below) a planetary pole. For the circular orbit case, a linear stability analysis is also performed. Finally, an analytical approximate method to calculate the Coulomb drag generated by a plasma brake tether is presented and applied to a LEO deorbiting case. The results discussed in this chapter have been published in [J.1], [J.2], and [J.3].

4.1 Introduction

The literature proposes a large number of potential mission scenarios for a solar sail. The first hypothesized solar sail mission scenario was a rendezvous with Halley’s comet [5]. Although the original project was eventually dismissed, a ren-dezvous with a comet, an asteroid [35], or a planet [36], even with the possibility of performing a sample return [37] has been discussed. Other conventional mission scenarios that could be achieved with a solar sail-equipped spacecraft consist of a phasing maneuver [38,39], a mission for monitoring the Earth magnetotail [40–44], and other planetocentric orbits [45]. The solar sail flexibility could be further in-creased by using electrochromic materials [46–48], which change their optical

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30 4. Solar sail and E-sail mission applications

properties when an electrical voltage is applied. These materials can be used to adjust the generated thrust and also to perform attitude maneuvers [49, 50], as was recently shown by JAXA’s IKAROS mission [51–53].

More exotic and fascinating mission scenarios are also possible for solar sail-equipped spacecraft, including the exploration of the outer Solar System [54], by means of quasi-linear trajectories [55, 56], a “solar photonic” assist [57, 58], or by reversing the angular momentum vector [59, 60]. Moreover, the gener-ation of non-Keplerian orbits by means of a solar sail is achievable thanks to its continuous thrust. Possible scenarios include heliostationary position mainte-nance [49,61–64], a pole-sitter mission [65,66], trajectories confined on cylindrical or spherical surfaces [67], formation flying on displaced orbits [68], and the main-tenance of a stationary position in the Sun-Earth gravitational field (artificial La-grangian point) in order to provide an early warning in case of solar events [69,70]. Finally, a very fascinating and recently proposed mission concept involves a sail accelerated by a high-energy laser and directed towards Alpha Centauri system for an interstellar flight [71–75].

A different application of the solar sail working principle has led to the de-velopment of femtosatellites (m < 100 g) with a very large area-to-mass ratio, named Smart Dusts, in analogy with similar very small units used for terrestrial applications [76]. Such a spacecraft is very sensible to the pressure force, includ-ing solar radiation pressure, so that it could be well suited for special applications such as diagnostic purposes, particle detection, and other scientific missions, both in a heliocentric [50, 77] and in a geocentric scenario [78–83]. Smart Dusts may also be used in a swarm [84], since their low launch cost allows many satellites to be inserted on orbit with a single launch. Moreover, Smart Dusts could be equipped with the previously described electrochromic materials, in order to per-form attitude maneuvers [85] and modify the thrust level.

The Electric sail has a less mature technology compared to the solar sail, and, so far, no operative missions have tested this propulsive technology yet. The first attempts of validation test involve the plasma brake concept, which is a derivation of the “classical” electric sail and will be treated in detail in Section 4.4, but no operative data are available yet. However, the peculiar nature of the E-sail allows several mission scenarios to be envisaged. Indeed, a spacecraft equipped with an E-sail could perform an orbital transfer to a planet [86, 87] or an asteroid [88, 89], and even more exotic scenarios, including the exploration of the outer Solar System [90], hazardous asteroid deflection [91,92], and displaced non-Keplerian orbits [93].

A displaced non-Keplerian orbit (DNKO) is characterized by the fact that the primary does not lie on the orbital plane, so that the orbital maintenance can

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4.2 Circular DNKOs 31

be achieved by means of a continuous thrust only. This concept can be tracked back to Forward [94], who proposed it as a possible application for a high per-formance solar sail, and has been deeply investigated since [95, 96], both for solar sails [97–100] and hybrid propulsion systems [66]. In this context, many possi-ble DNKO-related applications have been discussed, including the observation of planetary polar regions [65, 66], the maintenance of an out-of-plane geostationary orbit [101], the scientific study of high-latitude solar regions through DNKOs with an oscillating ecliptic shift [67], and the creation of a communication relay as a support for future Mars exploration [102]. The analysis conducted in Ref. [93] suggests DNKOs as a possible mission scenario for an electric sail-equipped space-craft. However, this analysis is based on the original E-sail thrust model, in which the thrust scales as r−7/6, see Eq. (3.33), and its magnitude is independent of the spacecraft attitude, as stated in Subsection 3.3.1. Moreover, the analysis was limited to circular DNKOs. Hence, this possible E-sail mission application has been deeply investigated by using the most recent thrust models in Sections 4.2 and 4.3.

An alternative application of the electric sail basic working principle, the plasma brake concept, will be the focus of Section 4.4.

4.2 Circular DNKOs

Consider a spacecraft equipped with an E-sail, whose center of mass S is covering a circular DNKO. Without loss of generality, assume that the orbital plane is parallel to the ecliptic plane. The Sun’s center of mass is located at point O, whose projection on the orbital plane is C. Let ˆρ be the unit vector from C to S, whereas ˆr is the radial unit vector. Hence, the scalar quantities ρ and r denote the C-S and O-S distance, respectively. The constant angular velocity of the spacecraft covering the DNKO is Ω, with Ω , Ω/ kΩk, and therefore the orbital period is T = 2 π/Ω. The elevation of the DNKO with respect to the ecliptic can be measured by the elevation angle ψ, defined as

ψ , arccos ρ

r

(4.1) or alternatively by the distance between the ecliptic plane and the DNKO plane Z = ρ tan ψ. The situation is sketched in Fig. 4.1.

The orbital maintenance of a circular DNKO is possible only by imposing the equilibrium of the gravitational, propulsive, and centrifugal accelerations acting on the spacecraft, see Fig. 4.1. For symmetry reasons, ˆa and ˆρ must belong to the plane containing ˆr and Ω. Recalling that ˆa lies in the plane spanned by ˆr and ˆn, the latter is therefore constrained to belong to the same plane. Note that the previous considerations involving orbital maintenance can be extended to the

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32 4. Solar sail and E-sail mission applications C O S

orbital plane

circular DNKO ecliptic plane orbital plane

ˆ

a

to Sun

ˆ

n

spacecraft S radial direction f a E-sail nominal plane

y r Z ˆ r W2 _r ˆ m - ₂ r r e ˆr r W

Figure 4.1: Sketch of a circular displaced non-Keplerian orbit.

general case of a DNKO whose orbital plane is not parallel to the ecliptic. The equilibrium condition of S can be conveniently expressed by means of two scalar equations, one for the component along ˆr and one for the direction orthogonal to ˆ r, viz. ac r ⊕ r γ cos φ = µ r2 − Ω 2_{ρ cos ψ} _(4.2) ac r ⊕ r γ sin φ = Ω2ρ sin ψ (4.3)

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4.2 Circular DNKOs 33

is not required. Equations (4.2) and (4.3) give the following requirements for the cone angle and the E-sail characteristic acceleration

φ = arctan (Ω/ΩK)2 tan ψ 1 + tan2ψ − (Ω/ΩK)2) (4.4) ac= µ r⊕r γ 1 − (Ω/ΩK) 2 1 + tan2_ψ s 1 + tan 2_ψ [(1 + tan2_ψ)/(Ω/Ω K)2− 1)]2 (4.5)

where ΩK ,pµ/r3 is the angular velocity of a reference Keplerian orbit with

radius r, viz.

ΩK,

p

µ/r3 (4.6)

The procedure for determining the conditions to be met for the maintenance of a given circular DNKO can be summarized as follows. The parameters r, ψ, and Ω are fixed when the DNKO is selected. The required value of the cone angle φ is found by means of Eq. (4.4), which gives the possible values of the required pitch angle α through Eq. (3.46). The corresponding value of the dimensionless propulsive acceleration γ is found by Eq. (3.44), from which it is evident that γ is a decreasing function of |α|, see Fig. 3.4(b). Hence, the first value of those reported in Eq. (3.46) is always preferable because the generated thrust is larger. Finally, the characteristic acceleration required for orbital maintenance is obtained from Eq. (4.5).

The obtained results are shown in Figs. 4.2(a) and 4.2(b), and highlight that a circular DNKO with large elevation angle and small heliocentric distance are very demanding in terms of propulsive requirements. The gray region in Figs. 4.2(a) and 4.2(b) identifies DNKOs that would require a thrust angle larger than φmax'

19.2 deg, see Eq. (3.45).

4.2.1 Type II circular DNKOs

A special mission application is constituted by a Type II DNKO, defined by McInnes [6] as a DNKO whose period is equal to that of a Keplerian orbit with radius r. A spacecraft placed on such a trajectory could be able to both observe the polar regions of a planet with a nearly-circular orbit and to easily commu-nicate with another satellite covering a Keplerian ecliptic orbit. For a Type II DNKO, the condition Ω = ΩK holds, and Eq. (4.4) reduces to

φ = π

2 − ψ (4.7)

meaning that, since φmax is about 20 deg, these orbits are physically feasible only

for large values of the elevation angle, i.e. approximately ψ > 70 deg.

Figure 4.3 shows the propulsive requirements for Type II DNKO maintenance as a function of the elevation angle and the Sun-spacecraft distance. The pitch

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34 4. Solar sail and E-sail mission applications 70 72 74 76 78 80 82 84 86 88 90 A [deg] 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 r [au] 3 4 5 8 10 15 0.6 0.7 0.8 0.9 1 1 2 2 / ( / ) c a me rÅ [deg] f max f forbidden region (a) r ≥ r⊕ (b) r ≤ r⊕

Figure 4.2: Required cone angle φ and characteristic acceleration acto maintain a

circular DNKO as functions of Sun-spacecraft distance r and elevation angle ψ.

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4.2 Circular DNKOs 35 a c [mm/s 2 ] 70 72 74 76 78 80 82 84 86 88 90 2 4 6 8 10 12 14 [deg] y 0.5 0.6 0.7 0.8 0.9 1 1.25 1.5 2 1.75 / r r Å

Figure 4.3: Characteristic acceleration ac required to maintain a circular Type II

DNKO as a function of the elevation angle ψ and Sun-spacecraft dis-tance r.

angle is selected taking the first value of those given by Eq. (3.46). From Fig. 4.3, it is evident that the characteristic accelerations needed for orbital maintenance are large, making this fascinating mission scenario quite beyond the current or near-term technology level. For instance, a circular DNKO with radius close to r⊕ is achievable only for an E-sail with ac' 6 mm/s2, which is at least six times

larger than the currently hypothesized maximum value of 1 mm/s2. Moreover, it is worth noting that the required value of acis a decreasing function of ψ, unlike

the results obtained by Mengali and Quarta [93] that showed an opposite trend. This happened because a larger value of the elevation angle implies a larger component of the gravitational attraction perpendicular to the DNKO plane, which cannot be balanced by the centrifugal force but only by the propulsive acceleration. However, if the dependance of the propulsive acceleration on the sail attitude is introduced, a larger value of ψ corresponds to a smaller value of the pitch angle α (see Eq. (4.7)), yielding a greater value of γ (see Eq. (3.44)). This effect is more significant with respect to the former one, as is illustrated in

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36 4. Solar sail and E-sail mission applications 70 72 74 76 78 80 82 84 86 88 90 y [deg] 3 4 5 6 7 8 9 10 11 12 ac [m m/s 2 ] 0.5 0.5 0.75 0.75 1 1 1.25 1.25 1.5 1.5 1.75 1.75 2 ₂ Å r / r

Figure 4.4: Propulsive requirements for a Type II DNKO as a function of ψ and r; original thrust model (dashed line) vs. recent thrust model (solid line).

Fig. 4.3.

Figure 4.4 shows a comparison between the results obtained in Ref. [93] and those obtained with the most recent E-sail thrust model. For r < r⊕, the new

results always provide more demanding propulsive requirements, whereas for r > r⊕ the opposite may happen, due to the different exponent in the propulsive

acceleration expression, see Eqs. (3.33) and (3.39).

4.2.2 Ecliptic DNKOs

An interesting special case of circular DNKO is obtained when the elevation angle ψ is zero, that is, when the DNKO lies on the ecliptic plane. This could be exploited to generate an artificial Lagrangian point [63, 103, 104] closer to the Sun than the real L1 by suitably adjusting the orbital period, as sketched in

Fig. 4.5, in which the equilibrium heliocentric distance is denoted with rL. For a

circular DNKO lying on the ecliptic plane, Eq. (4.4) gives φ = 0, corresponding to a Sun-facing attitude, i.e., α = 0 and γ = 1, see Eqs. (3.46) and (3.44). The

Trajectory analysis of spacecraft with propellantless propulsion systems

Universit`

a di Pisa

Trajectory analysis of spacecraft with

propellantless propulsion systems

Abstract

Preface

Acknowledgements

Contents

Chapter

1

Introduction

Part I

Thrust vector mathematical

model

Chapter

2

Spacecraft motion with continuous low-thrust

2.1

General model

2.2

Special cases

Chapter

3

Propulsive acceleration models for

propellantless thrusters

3.1

Introduction

3.2

Solar sail

r

ˆ

n

ˆ

r

a

aˆ

f

ˆr

ˆ

n

3.3

Electric solar wind sail

r

ˆ

ˆ

ˆ

3.4

Final considerations

,

[deg]

,

[deg]

Part II

Chapter

4

Solar sail and E-sail mission applications

4.1

Introduction

4.2

Circular DNKOs

orbital plane

ˆ

a

ˆ

n