Mission analysis of spacecraft equipped with advanced propulsion systems (Analisi di missione di satelliti equipaggiati con sistemi propulsivi avanzati)

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Universit`

a di Pisa

Dottorato di Ricerca in Ingegneria Industriale

Curriculum in Ingegneria Aerospaziale Ciclo XXXII

Mission analysis of spacecraft equipped with

advanced propulsion systems

Author

Marco Bassetto

Supervisors

Dr. Alessandro A. Quarta Prof. Giovanni Mengali

Coordinator of the PhD Program Prof. Giovanni Mengali

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Marco Bassetto

Dipartimento di Ingegneria Civile e Industriale Universit`a di Pisa

Via G. Caruso, 8 I-56122 Pisa, Italy

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If we knew what it was we were doing, it would not be called research, would it?

Albert Einstein (1879–1955)

Behind every problem there is always an op-portunity.

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Abstract

The aim of this Thesis is to provide an in-depth analysis concerning low-thrust propellantless propulsion systems for space navigation and attitude control. The performance of Electric Solar Wind Sails (E-sails), Magnetic sails (Magsails), and solar sails is investigated in the context of various mis-sion scenarios, either in a heliocentric or geocentric framework. The main features of these fascinating propulsion systems are described in Chap-ter 1, which also offers a brief overview of all the missions that have so far exploited these advanced concepts.

The first Part of the Thesis deals with the mathematical model that describes both the thrust and torque vectors of a spinning E-sail, showing how its performance vary as a function of the tether shape. An approxi-mation of the E-sail equilibrium shape is then obtained in analytical form assuming that the E-sail nominal plane is orthogonal to the solar wind ion stream. The corresponding tether equilibrium shape is accurately de-scribed by a natural logarithmic arc, an important information that allows the thrust and torque vectors to be expressed in closed-form as a function of the E-sail attitude. The differential equations of the orbital and attitude dynamics of an E-sail-based spacecraft are also introduced.

The second Part analyzes the opportunity of exploiting a tether voltage modulation in order to control (or simply maintain) the E-sail attitude. The proposed analytical control law is easy to implement and offers good performance in terms of attitude maneuver time. The presented solution, which requires a very small modulation of the tether electrical voltage, is validated through a set of numerical simulations.

The third Part focuses on some of the possible mission scenarios involv-ing an E-sail. The problem is addressed usinvolv-ing a locally-optimal formula-tion, in which a linear combination of the time derivatives of the osculating orbital elements is optimized at any time. The simulation results show the effectiveness of the proposed method and its practical usefulness in case of very complex mission scenarios, such as the rendez-vous with comet 1P/Halley, or the generation of Earth-following orbits. The stability and control of an E-sail in a heliostationary equilibrium point is also thoroughly investigated. Finally, the performance of an E-sail as a de-orbiting device (the so-called “plasma brake”) is evaluated using an analytical approach, in which the spacecraft trajectory is calculated starting from the linearized

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Hill-Clohessy-Wiltshire equations.

The fourth and last Part deals with the analysis of other mission scenar-ios with different propellantless propulsion systems. The Magsail concept is used to address the problem of determining the requirements for maintain-ing circular displaced non-Keplerian orbits around the Sun. The analysis is performed using a recent mathematical model in which the Magsail thrust vector is expressed as a function of the Sun-spacecraft distance and the sail attitude. A linear stability analysis is also carried out in order to rec-ognize the unstable displaced orbits when an error in the orbital insertion is assumed. Moreover, the fourth Part provides a systematic study regard-ing the generation of logarithmic spiral trajectories usregard-ing a solar sail. The analysis takes into consideration both an ideal and an optical force model, and underlines the differences between the two models in terms of offered performance. The required conditions to be satisfied in terms of attitude angle, thrust level, and initial conditions are thoroughly discussed. The closed-form variations of the osculating orbital parameters are explored, and the obtained analytical results are then used for the design of a phas-ing maneuver along an elliptic heliocentric orbit.

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Preface

This Thesis discusses the work and the main results obtained along my Ph.D. course, from November 2016 to October 2019, 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 journal papers:

[J.1] Bassetto M., Mengali G., Quarta A. A., “E-sail attitude control with tether voltage modulation”, Acta Astronautica, Vol. 166, pp. 350– 357, January 2020, DOI:10.1016/j.actaastro.2019.10.023.

[J.2] Bassetto M., Quarta A. A., Mengali G., “Magnetic sail-based dis-placed non-Keplerian orbits”, Aerospace Science and Technology, Vol. 92, pp. 363–372, September 2019, DOI:10.1016/j.ast.2019.06.018. [J.3] Bassetto M., Mengali G., Quarta A. A., “Attitude dynamics of an

electric sail model with a realistic shape”, Acta Astronautica, Vol. 159, pp. 250–257, June 2019, DOI:10.1016/j.actaastro.2019.03.064. [J.4] Bassetto M., Mengali G., Quarta A. A., “Stability and control of

spinning electric solar wind sail in heliostationary orbit ”, Journal of Guidance, Control, and Dynamics, Vol. 42, Issue 2, pp. 425–431, February 2019, DOI:10.2514/1.G003788.

[J.5] Bassetto M., Quarta A. A., Mengali G., “Locally-optimal electric sail transfer ”, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, Vol. 233, Issue 1, pp. 166–179, January 2019, DOI:10.1177/0954410017728975.

[J.6] Bassetto M., Mengali G., Quarta A. A., “Thrust and torque vector characteristics of axially-symmetric E-sail ”, Acta Astronautica, Vol. 146, pp. 134–143, May 2018, DOI:10.1016/j.actaastro.2018.02.035. [J.7] Bassetto M., Niccolai L., Quarta A. A., Mengali G., “Logarithmic

spiral trajectories generated by solar sails”, Celestial Mechanics and Dynamical Astronomy, Vol. 130, Issue 2, Article number 18, Febru-ary 2018, DOI:10.1007/s10569-017-9812-6.

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[J.8] Bassetto M., Niccolai L., Quarta A. A., Mengali G., “Plasma brake approximate trajectory. Part II: Relative motion”, Advances in the Astronautical Sciences, Vol. 163, pp. 249–259, 2018.

and in the following conference papers:

[C.1] Bassetto M., ´Alvarez Mera L. S., Mengali G., Quarta A. A., “E-sail attitude control with tether voltage modulation”, 6th International Conference on Tethers in Space, 12–14 June 2019, Madrid, Spain. [C.2] Bassetto M., Mengali G., Quarta A. A., “Attitude dynamics of an

electric sail model with a realistic shape”, 69th International Astro-nautical Congress, 1–5 October 2018, Bremen, Germany.

[C.3] Bassetto M., Niccolai L., Quarta A. A., Mengali G., “Plasma brake approximate trajectory. Part II: Relative motion”, 4th IAA Con-ference on University Satellite Mission & CubeSat Workshop, 4–7 December 2017, Rome, Italy.

[C.4] Bassetto M., Niccolai L., Quarta A. A., Mengali G., “Notes on loga-rithmic spiral trajectories generated by solar sails”, 7th International Meeting on Celestial Mechanics, 4–8 September 2017, San Martino al Cimino (VT), Italy.

[C.5] Bassetto M., Quarta A. A., Mengali G., “Locally-optimal electric sail transfer ”, 10th IAA Symposium on the Future of Space Exploration: Towards the Moon Village and Beyond, 27–29 June 2017, Turin, Italy.

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

[J.9] Quarta A. A., Mengali G., Bassetto M., “Optimal solar sail trans-fers to circular Earth-synchronous displaced orbits”, Astrodynamics, DOI:10.1007/s42064-019-0057-x.

[J.10] Niccolai L., Bassetto M., Quarta A. A., Mengali G., “A review of Smart Dust architecture, dynamics, and mission applications”, Progress in Aerospace Sciences, Vol. 106, pp. 1-14, April 2019, DOI:10.1016/j.paerosci.2019.01.003.

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v

[J.11] Niccolai L., Bassetto M., Quarta A. A., Mengali G., “Plasma brake approximate trajectory. Part I: Geocentric motion”, Advances in the Astronautical Sciences, Vol. 163, pp. 235–247, 2018.

in the following article in press:

[J.12] Bassetto M., Caruso A., Quarta A. A., Mengali G., “Optimal steering law of refractive sail ”, Advances in Space Research, DOI:

10.1016/j.asr.2019.10.033. in the following submitted papers:

[J.13] Caruso A., Bassetto M., Mengali G., Quarta A. A., “Optimal solar sail trajectory approximation with finite Fourier series”, submitted to Advances in Space Research.

[J.14] Bassetto M., Boni L., Mengali G., Quarta A. A., “Electric sail phas-ing maneuvers with radial thrust ”, submitted to Acta Astronautica. [J.15] Boni L., Bassetto M., Mengali G., Quarta A. A., “Static structural analysis of an electric sail with Finite Element approach”, submitted to Acta Astronautica.

and in the following conference papers:

[C.6] Bassetto M., Quarta A. A., Mengali G., “Optimal steering law for refractive sails”, 5th International Symposium on Solar Sailing, 30 July – 2 August 2019, Aachen, Germany.

[C.7] Bassetto M., Boni L., Mengali G., Quarta A. A., “Electric sail phas-ing maneuvers with radial thrust ”, 11th IAA Symposium on the Fu-ture of Space Exploration. Moon, Mars and beyond: Becoming an Interplanetary Civilization, 17–19 June 2019, Turin, Italy.

[C.8] Boni L., Bassetto M., Mengali G., Quarta A. A., “Static structural analysis of an electric sail with Finite Element approach”, 11th IAA Symposium on the Future of Space Exploration. Moon, Mars and beyond: Becoming an Interplanetary Civilization, 17–19 June 2019, Turin, Italy.

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[C.9] Niccolai L., Bassetto M., Quarta A. A., Mengali G., “Plasma brake approximate trajectory. Part I: Geocentric motion”, 4th IAA Con-ference on University Satellite Mission & CubeSat Workshop, 4–7 December 2017, Rome, Italy.

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Acknowledgements

Vorrei innanzitutto ringraziare i professori Giovanni Mengali e Alessandro A. Quarta, che in questi tre anni hanno guidato il mio lavoro con grande professionalit`a e dedizione. A loro sono rivolti non solo la mia stima, ma anche il mio affetto e la mia gratitudine.

Non avrei potuto raggiungere questo obiettivo senza l’instancabile sup-porto dei miei genitori e di Martina. La loro fiducia nei miei confronti `e sempre stata il segreto dei miei successi. Grazie di cuore.

Ho compiuto la seconda met`a di questo percorso con Silvia, che ringrazio per tutto l’amore e il sostegno che non mi ha mai fatto mancare. Con lei `

e stato tutto pi`u facile e pi`u bello.

Grazie infine a tutti coloro che hanno condiviso con me questa espe-rienza. Ai colleghi, amici e compagni di trasferta Lorenzo e Andrea, ai compagni di pranzo del Laboratorio di Fluidodinamica, e a tutte le per-sone con le quali ho trascorso momenti indimenticabili.

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Chapter

1

Introduction

This Thesis collects most of the results obtained during my Ph.D. course, whose research activity has been addressed to the analysis of advanced propulsion concepts for space navigation. The choice of this topic has been motivated by the growing interest shown by academies and space agencies towards propellantless propulsion systems. In fact, due to the finite amount of available propellant, the use of reaction engines limits the spacecraft maneuvering capabilities to such an extent that, sometimes, they may become ineffective. For this reason, propellantless propulsion systems represent a fascinating and interesting option, especially when high levels of delta-v are required, or long-term deep-space missions must be performed without resorting to gravity assist maneuvers. The aim of this Thesis is to investigate the effectiveness of electric sails, magnetic sails, and solar sails in the context of some heliocentric and geocentric mission scenarios.

The Electric Solar Wind Sail

Most of the Thesis focuses on the Electric Solar Wind Sail (E-sail), an inno-vative propellantless propulsion system proposed by Dr. Pekka Janhunen in 2004 [1]. An E-sail exploits the solar wind particle momentum to gener-ate a continuous thrust in the interplanetary space [1–3]. In particular, the solar wind ions interact with an artificial electric field generated through a spinning grid of charged tethers [4–6], which are kept at a voltage level on the order of a few tens of kilovolts by means of an electron emitter [2, 3]. An artistic impression of the E-sail concept is shown in Fig. 1.1. The same

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physical principle is also used by the plasma brake, which is a promising option for reducing the decay time of satellites in low Earth orbits at their end-of-life [4, 7, 8]. A number of space missions have already been designed to demonstrate the feasibility of the plasma brake in the geocentric context. In particular, the Estonian one-unit CubeSat ESTCube-1 [9,10], which was designed by the University of Tartu and launched on 7 May 2013, was the first-ever satellite to have attempted the use of an E-sail. ESTCube-1 was a 10 × 10 × 11.35-centimeter cube, with a total mass of 1.048 kilograms. Among its mission targets, ESTCube-1 was designed to deploy a 10-meter-long Heytether, positively charged to 500 volts by means of two electron emitters. Other mission requirements included the spinning rate control of the spacecraft and its de-orbiting phase with a plasma brake-based system. However, the tether deployment failed, possibly due to a stuck reel, and the mission ended on 17 February 2015. Aalto-1 [11] is a Finnish research nanosatellite, designed by Aalto University and launched on 23 June 2017. Aalto-1 has a total mass of 3.9 kilograms, and incorporates a plasma brake device, which is designed to perform attitude maneuvers and eventually to de-orbit the satellite at the end of its operational lifespan, with the intent of avoiding the creation of space junk [12]. The plasma brake payload consists of a 100-meter-long conductive tether, a reel mechanism for tether storage, a high voltage source, and electron emitters to maintain the tether charge [13]. Finally, ESTCube-2 [14] is a three-unit CubeSat, which is de-signed to test the Coulomb drag propulsion. The launch is scheduled in 2019, and ESTCube-2 will deploy a 300-meter-long tether, which will be charged negatively to 1 kilovolt. The charged tether will be used to slow down the satellite, and it is expected that ESTCube-2 may reduce its alti-tude from 700 to 500 kilometers in half a year. According to conservative estimates, the mass of a 300-meter-long tether is about 30 grams, which makes the plasma brake a cost-effective de-orbiting system. On the other hand, the E-sail concept has not been applied in the context of heliocentric mission scenarios, yet.

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Figure 1.1: Artistic impression of an Electric Solar Wind Sail. Courtesy of Alexandre Szames, Antigravit´e (Paris).

The Magnetic sail

Another propellantless propulsion system for deep space navigation is rep-resented by the Magnetic sail (Magsail), which was conceived by Andrews and Zubrin [15] in 1990; see Fig. 1.2. The Magsail extracts momentum from the solar wind, whose particles interact with an artificial magnetic field generated on board through an electrical current that crosses a large loop of conducting material.

The solar sail

Finally, solar sailing is one of the most promising innovations among low thrust propulsion systems. Fridrickh A. Tsander was the first to conceive the solar sail concept, suggesting the possibility of navigating in space with huge mirrors or very thin sheets [16]. In essence, a solar sail consists of a thin reflective film that exchanges momentum with the impinging photons coming from the Sun. As such, a solar sail does not require any propellant to work, nor electrical power unlike E-sails or Magsails. An artistic ren-dering of a square solar sail is shown in Fig. 1.3. A comprehensive review about solar sail concept, design, and feasible applications can be found in Refs. [17–19]. Until today, various missions have already been launched

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Magsail

solar wind

Figure 1.2: Magsail conceptual scheme.

to demonstrate the potentialities of such a fascinating propulsion system, and to evaluate its in-space performance. The JAXA’s Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) was the first success-ful interplanetary solar sail demonstration mission [20,21], which succeeded in deploying a 196-square-meter solar sail in 2010. Currently, JAXA is also developing a solar sail aimed at propelling a large-size spacecraft towards Jupiter and the Trojan asteroids [22]. The estimated propulsion system is a so-called solar power sail, which is a square 2500-square-meter thin mem-brane exposed to sunlight that should guarantee the required propulsive acceleration and supply the electrical power necessary to operate an ion engine. In 2010 NASA launched the NanoSail-D2 [23], which was a three-unit CubeSat intended to study the deployment mechanism of a 10-square-meter solar sail. The Near-Earth Asteroid Scout (NEA Scout) [24–26] is another NASA project, scheduled to launch in 2020, whose aim is to fly a six-unit CubeSat towards near-Earth asteroids using a solar sail with an area of 86 square meters. The Planetary Society has recently developed two three-unit CubeSats, the LightSail 1 [27, 28] and the LightSail 2 [29]. These CubeSats were launched in 2015 and 2019, respectively, to test the solar sailing in a low Earth orbit using a 32-square-meter solar sail.

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Figure 1.3: Artistic impression of a square solar sail.

The Thesis is organized as follows. Part I deals with the thrust and torque vector model of an axially-symmetric E-sail, describing how its per-formance varies as a function of the tether shape. An approximation of the E-sail equilibrium shape is then derived in analytical form, thus allowing the thrust and torque vectors to be expressed in closed-form as a function of the E-sail attitude. Moreover, the differential equations that describe both the orbital and the attitude motions of an E-sail-based spacecraft are introduced. Part II analyzes the possibility of maintaining and controlling the E-sail attitude with a simple but effective control law based on a mod-ulation of the tether electrical voltage. Such a control law is provided in analytical form, in which the electrical voltage of each tether is expressed as a function of the time and the current sail attitude. Part III focuses on some of the possible mission scenarios involving an E-sail. The problem is addressed using a locally-optimal formulation, in which a scalar perfor-mance index that depends on the time derivatives of the osculating orbital elements is minimized at any time. The simulation results show the effec-tiveness of the proposed approach and its practical usefulness in case of

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complex mission scenarios, such as the rendez-vous with comet 1P/Halley, or the generation of Earth-following orbits. The stability and control of an E-sail in heliostationary orbit is also investigated, whereas the performance of the plasma brake device is evaluated using an iterative process, in which the altitude loss is computed starting from the linearized Hill-Clohessy-Wiltshire equations. Finally, Part IV deals with mission applications with other propellantless propulsion systems, such as Magsails and solar sails. In particular, the problem of determining the requirements for maintain-ing circular displaced non-Keplerian orbits around the Sun is addressed using a Magsail. In this context, a linear stability analysis is carried out in order to identify the marginally stable displaced orbits assuming an er-ror in the orbital insertion. Finally, Part IV provides a systematic study concerning the possibility of inserting a solar sail-based spacecraft into a heliocentric logarithmic spiral trajectory without using impulsive maneu-vers. The obtained analytical results are then used for investigating the phasing maneuver of a solar sail along an elliptic heliocentric orbit.

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Part I

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Chapter

2

Thrust and torque vector models

This chapter deals with the mathematical expressions of the thrust and torque vectors of an axially-symmetric E-sail as a function of its shape and orientation in the space. It is also shown that an approximate solu-tion to the tether equilibrium shape can be found under the assumpsolu-tion that the E-sail nominal plane is orthogonal to the solar wind velocity. The corresponding tether equilibrium shape is accurately described by a natural logarithmic arc, whose geometrical characteristics are related to the combined effects of centrifugal and solar wind-induced forces. Such a result allows the thrust and torque vectors to be expressed in closed-form as a function of the E-sail attitude, thus improving the understanding of the E-sail behavior in the interplanetary space. The results discussed in this chapter have been published in [J.6] and [J.4].

2.1 Introduction

The E-sail is an innovative propellantless propulsion system that extracts momentum from the solar wind flow, which interacts electrostatically with long conducting tethers maintained charged by an electron emitter [1–3]; see Fig. 2.1. The tethers are deployed and maintained stretched by spin-ning the spacecraft about a symmetry axis [4–6] so that, in a simplified model, the E-sail can be assumed to take the shape of a spoked wheel [30]. Along with the more typical solar sail, the E-sail is a promising propellant-less propulsion systems, even though it needs electrical power to produce the required electric field. However, unlike solar sails, whose propulsive acceleration varies as the inverse-square distance from the Sun, a very

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in-Sun E-sail solar wind charged tether electric field main body

spin

axis

Figure 2.1: Spinning E-sail conceptual sketch.

teresting property of the E-sail is that its thrust magnitude is inversely proportional to the heliocentric distance [31].

In order to get preliminary simulation results, the E-sail thrust vector is often modelled in a simplified way assuming that the tether arrange-ment resembles that of a rigid disc of given radius [30, 32–37]. In such an ideal configuration, Huo et al. [38] have recently obtained an analyt-ical description of the E-sail thrust vector using a geometranalyt-ical approach, and assuming that all the tethers have the same uniform electrical voltage. However, such an approximation may be inaccurate as the actual shape of each tether depends on the chaotic interaction between the solar wind dynamic pressure and the centrifugal force due to the spacecraft rotation. Accordingly, the tethers are not perfectly straight, and the uncertainty as-sociated with their actual shape makes it difficult to obtain an estimate of the thrust and torque vectors, which are necessary information for both trajectory analysis and attitude control design. A non-negligible portion of the current research is intended to investigate how the geometrical fea-tures of such a propulsion system may affect its in-flight performance in terms of thrust and torque vectors [39–41]. In this context, Toivanen and Janhunen [41] have studied the shape of a rotating E-sail using a numer-ical approach, showing that the tethers form a cone near the spacecraft, whereas they are flattened by the centrifugal force near the tip region.

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2.2 Geometrical preliminaries 11

Nevertheless, in a preliminary phase of mission design, the mathematical model adopted to describe the sail shape must be simple enough to be suc-cessfully implemented within a simulation code, especially when optimal trajectories are investigated [30, 42]. In the latter case, indeed, a consid-erable number of transfer trajectories need to be simulated to minimize a scalar performance index, such as the flight time [43–46]. The new mathe-matical relations discussed in this chapter represent a useful improvement over existing models, as they allow the influence of the tether arrangement on the E-sail performance to be quantified without the use of numerical algorithms.

2.2 Geometrical preliminaries

Consider a spacecraft, whose primary propulsion system is an E-sail, and assume the vehicle to be modelled as an axially-symmetric rigid body spinning about its symmetry axis ˆn at an angular velocity ω , ω ˆn of constant magnitude ω. The E-sail consists of N ≥ 2 tethers, which are modelled as planar cables belonging to the plane (ˆik, ˆn), where ˆik (with k ∈ {0, 1, . . . , N − 1}) is orthogonal to ˆn; see Fig. 2.2.

ˆ B i ˆ B j ˆ B k ˆ n B x B y B z 1 tetherst ( +1)-th tetherk iˆk k x k z sail nominal plane ˆ ˆ ( , ) planei k_k _B S N-th tether

Figure 2.2: E-sail geometrical arrangement.

The displacement of the generic tether with respect to the spacecraft main body can be evaluated by introducing a principal body reference

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frame TB(S; xB, yB, zB) with the origin S at the spacecraft center of mass and unit vectors {ˆiB, ˆjB, ˆkB} defined as

ˆ

kB , ˆn , îB , î0 , ˆjB , ˆn × î0 (2.1) Note that the plane (îB, ˆkB) contains the first tether (labelled with k = 0), whereas the unit vector îk can be written as

ˆ_i_k_{= cos ζ}_kˆ_i_B_{+ sin ζ}_kˆ_j

B (2.2)

where ζk is the angle, measured counterclockwise from the direction of ˆ_i_B_{, between x}_k _{and x}_B_{; see Fig. 2.2. Likewise, ζ}_k _{is the angle between} the plane containing the k-th tether and the plane identified by the unit vectors ˆiB and ˆkB. Assume now that the shape of the generic tether can be described, in the plane (ˆik, ˆkB), through a continuously differentiable function fk = fk(xk) : [xrk, xtk] → R, where xrk ≥ 0 (or xtk > 0) is the

distance of the tether root (or tip) from the spacecraft spin axis zB; see Fig. 2.3. ˆ B k ˆ k i S k r x k x k t x k f ds_k root tip ( +1)-th tetherk spin axis k d

Figure 2.3: Generic tether displacement.

The position vector dk of an infinitesimal arc-length dsk of the k-th conducting tether is given by

dk= xkˆik+ fkˆkB (2.3) with dsk = q 1 + (f_k0)2_dx k (2.4)

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2.3 Total force and torque acting on the E-sail 13

where f_k0 _{, df}k/dxk is the local tether slope. From Eqs. (2.3) and (2.4), the expression of the unit vector ˆsk tangent to the generic tether at point (xk, fk) is ˆ sk, ddk dsk = dxk ˆ_i_k_{+ df}_kˆ_k_B p1 + (f0 k)2dxk = ˆ_i_k_{+ f}0 kkˆB p1 + (f0 k)2 (2.5) which can be rewritten using Eq. (2.2) as a function of {ˆiB, ˆjB, ˆkB} as

ˆ sk= cos ζkˆiB+ sin ζkˆjB+ fk0kˆB p1 + (f0 k)2 (2.6)

2.3 Total force and torque acting on the E-sail

The aim of this section is to obtain an analytical expression of both the thrust and torque vectors generated by a spinning E-sail of a given three-dimensional shape, under the main assumption that each tether belongs to a plane containing the spacecraft spin axis zB. Both the total force and the total torque generated by the E-sail can be computed starting from the elementary force dFkdue to the solar wind dynamic pressure acting on an infinitesimal arc-length dsk of the generic conducting tether. According to the recent works by Janhunen and Toivanen [4, 39, 41], when the Sun-spacecraft distance r is on the order of r⊕ , 1 au, the thrust dFk gained by dsk is

dFk= σku⊥kdsk (2.7)

where u⊥k is the component of the solar wind velocity u perpendicular to ˆ sk, whereas σk , σ⊕ r_⊕ r with σ⊕, 0.18 max(0, Vk− Vw) √ 0mpn⊕ (2.8)

in which Vk (ranging in the interval [20, 40] kV) is the tether voltage, Vw is the electric potential of the solar wind ions (with a typical value of about 1 kV [4]), 0 is the vacuum permittivity, mp is the proton mass, and n⊕ = 5 × 106m−3 is the average solar wind number density at r = r⊕.

Assuming a purely radial solar wind stream, that is, u , u ˆr, where ˆr is the Sun-spacecraft unit vector and u ' 400 km/s is the solar wind speed, the term u⊥k in Eq. (2.7) becomes

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Moreover, according to Fig. 2.4, the Sun-spacecraft unit vector ˆr can be written as a function of {ˆiB, ˆjB, ˆkB} as

ˆ

r = sin αn cos δnˆiB+ sin αn sin δnˆjB+ cos αnkˆB (2.10) where

αn, arccos(ˆr · ˆkB) ∈ [0, π] rad (2.11) referred to as pitch angle, is the angle between ˆr and ˆkB, while

δn,            arccos r · ˆˆ iB ||ˆr × ˆkB|| ! if (ˆr · ˆj_B) ≥ 0 2π − arccos ˆr · ˆiB ||ˆr × ˆkB|| ! otherwise ∈ [0, 2π] rad (2.12) is the clock angle, measured counterclockwise starting from ˆiB, between xB and the projection of ˆr on the plane (xB, yB); see Fig. 2.4. Note that δnis undefined when ˆkBis parallel to ˆr, that is, when αn= 0. Taking into

ˆ B i ˆ B j ˆ B k B x B y B z 1 tetherst ˆ k i k x k z sail nominal plane S Sun ˆr n d n a

Figure 2.4: Pitch (αn) and clock (δn) angles.

account Eqs. (2.6) and (2.10), the dot product ˆr · ˆsk in Eq. (2.9) can be rearranged as ˆ r · ˆsk = ˆ r · ˆik+ fk0r · ˆˆ kB p1 + (f0 k)2 = cos(δn− ζk) sin αn+ f 0 k cos αn p1 + (f0 k)2 (2.13)

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2.3 Total force and torque acting on the E-sail 15

Therefore, with the aid of Eqs. (2.4), (2.6), (2.9), and (2.13), the thrust dFk given by Eq. (2.7) can be rewritten as

dFk= σku ˆ r −cos(δn− ζk) sin αn+ f 0 k cos αn 1 + (f_k0)2 cos ζkˆiB+ + sin ζkˆjB+ f 0 kˆkB i q 1 + (f_k0)2_dx k (2.14) 2.3.1 Total force

Starting from Eq. (2.14), the total force dFk acting on the infinitesimal arc-length dsk of the k-th tether is

dFk= dAkr + dBˆ kˆiB+ dCkˆjB+ dDkˆkB (2.15) where dAk, σku q 1 + (f_k0)2_dx k (2.16) dBk, −σku cos ζk

cos(δn− ζk) sin αn+ fk0 cos αn p1 + (f0

k)2

dxk (2.17)

dCk, −σku sin ζk

cos(δn− ζk) sin αn+ f_k0 cos αn p1 + (f0

k)2

dxk (2.18)

dDk, −σku fk0

cos(δn− ζk) sin αn+ fk0 cos αn p1 + (f0

k)2

dxk (2.19)

Therefore, the force Fk acting on the generic conducting tether is Fk= Z x_tk x_rk dFk = Akr + Bˆ kˆiB+ CkˆjB+ DkˆkB (2.20) with A_k _, Z x_tk x_rk dAk, Bk, Z x_tk x_rk dBk, Ck , Z x_tk x_rk dCk, Dk , Z x_tk x_rk dDk (2.21) whereas the total force F acting on the E-sail (composed of N ≥ 2 tethers) is given by F = N −1 X k=0 Fk= A ˆr + B ˆiB+ C ˆjB+ D ˆkB (2.22) where A , N −1 X k=0 A_k , B , N −1 X k=0 B_k , C , N −1 X k=0 C_k , D , N −1 X k=0 D_k (2.23)

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2.3.2 Total torque

The torque dTk given by an infinitesimal arc-length dsk of the k-th con-ducting tether is

dTk = dk× dFk (2.24)

where the symbol × denotes the cross product. Taking into account the expressions of dkand dFkgiven by Eqs. (2.3) and (2.15), respectively, and using Eq. (2.10), dTk can be written as a function of {ˆiB, ˆjB, ˆkB} as

dTk= dEkˆiB+ dFkˆjB+ dGkkˆB (2.25) where dEk, ( xk sin ζk " σku cos αn−

f_k0σku (sin αn cos(δn− ζk) + fk0 cos αn) 1 + f_k02

#

+

+fk sin ζk

σku (sin αn cos(δn− ζk) + f_k0 cos αn)

1 + f_k02 + −f_kσku sin αn sin δn} q 1 + f_k02 dxk (2.26) dFk , ( −x_k cos ζk " σku cos αn−

f_k0 σku (sin αn cos(δn− ζk) + fk0 cos αn) 1 + f_k02

#

+

−f_k cos ζk

σku (sin αn cos(δn− ζk) + f_k0 cos αn)

1 + f_k02 + +fkσku sin αn cos δn} q 1 + f_k02 dxk (2.27) dGk, σku xk sin αn sin(δn− ζk) q 1 + f_k02 dxk (2.28)

Therefore, the torque Tk acting on the generic tether is Tk= Z x_tk x_rk dTk = EkˆiB+ FkˆjB+ GkkˆB (2.29) with E_k_, Z x_tk x_rk dEk , Fk, Z x_tk x_rk dFk , Gk , Z x_tk x_rk dGk (2.30)

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2.4 Case of a Sun-facing E-sail 17

whereas the total torque T acting on the E-sail is

T = N −1 X k=0 Tk= E ˆiB+ F ˆjB+ G ˆkB (2.31) where E , N −1 X k=0 E_k , F , N −1 X k=0 F_k , G , N −1 X k=0 G_k (2.32)

Equations (2.22) and (2.31) are the expressions of the total force and torque acting on an E-sail with a given tether shape, length, and angular sepa-ration between tethers. These results will be applied to the noteworthy case of a Sun-facing E-sail [47, 48], thus obtaining a set of closed-form rela-tions. Some simplifying assumptions need to be introduced to get a more tractable form of both F and T , as is thoroughly discussed in the next section.

2.4 Case of a Sun-facing E-sail

The previous general results are now specialized to the noteworthy case of a Sun-facing E-sail [47, 48], corresponding to the situation in which the spacecraft spin axis zBcoincides with the Sun-spacecraft line (i.e., ˆkB≡ ˆr). In this case, the pitch angle αn is zero by construction, whereas δn can be set to zero without loss of generality, because ˆr is orthogonal to the plane (xB, yB), viz.

αn= δn= 0 (2.33)

Assuming all the tethers to have the same length L and the same voltage Vk (that is, the same value of σk; see Eq. (2.8)), the E-sail may reasonably be assumed to have a cylindrical symmetry around the zB-axis. The notation can be therefore simplified by dropping the subscript k in the variables {x_k, fk, xrk, xtk, σk}. Accordingly, all the tethers have the same shape

(i.e., their shape is described by the same mathematical function f = f (x)), and are arranged at the same angle apart from each other, that is

ζk = 2π N k (2.34) where k = {0, 1, . . . , N − 1}.

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Taking into account Eqs. (2.33) and (2.34), and bearing in mind that ˆ

kB≡ ˆr, from Eq. (2.22) the total force F becomes

F = N σ u Z xt xr 1 p1 + (f0₎2 dx ! ˆ r+ − σ u Z xt xr f0 p1 + (f0₎2dx " ˆ_i_B N −1 X k=0 cos 2π N k + +ˆj_B N −1 X k=0 sin 2π N k # (2.35)

whereas, starting from Eq. (2.31), the total torque is

T = σ u Z xt xr x + f f0 p1 + (f0₎2 dx " ˆ_i_B N −1 X k=0 sin 2π N k + −ˆj_B N −1 X k=0 cos 2π N k # (2.36)

Note that the single tether length L can be written as a function of {x_r, xt, f0} as L = Z xt xr p 1 + (f0)2_dx _(2.37)

According to Ref. [38], when N ≥ 2 the summations in Eqs. (2.35) and (2.36) give N −1 X k=0 sin 2π N k = N −1 X k=0 cos 2π N k = 0 (2.38)

and, therefore, the final form of the total force and torque given by a Sun-facing E-sail reduces to

F = N σ u Z xt xr 1 p1 + (f0₎2 dx ! ˆ r , T = 0 (2.39)

Note that the result T = 0 is consistent with the assumption of a Sun-facing E-sail with a cylindrical symmetry with respect to the spin axis, whereas the actual expression of the total force F (that is, the E-sail thrust)

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2.4 Case of a Sun-facing E-sail 19

depends on the tether shape via the analytical function f0 = df /dx. Some noteworthy cases are now discussed to better investigate the impact of the tether shape on the E-sail total force F .

2.4.1 Flat shape

When all the tethers are arranged on a flat surface that coincides with the E-sail nominal plane (xB, yB), the condition f0 = 0 is to be enforced in Eq. (2.39). The total force becomes

F = N L σ u ˆr (2.40)

where L = (xt− xr); see Eq. (2.37). In particular, Eq. (2.40) is consistent with the result discussed in Ref. [38] for a Sun-facing E-sail (i.e., when αn= 0). However, the case of a purely flat E-sail is only a first approximation of the actual sail shape. In fact, all tethers tend to move away from the E-sail nominal plane (xB, yB), and to take a three-dimensional arrangement. 2.4.2 Conic shape

An interesting approximation of the actual E-sail three-dimensional ar-rangement is given by a conic shape. In that case, each tether may be analytically described as f (x) = bcxr x xr − 1 with x ∈ [xr, xt] (2.41) where bc > 0 is a dimensionless constant, whose value depends on the aperture angle β of the right circular cone through the formula

β , π − 2 arctan(bc) (2.42)

Since f0(x) = bc, from Eq. (2.39) the total force is F = N L σ u

p1 + b2 c

ˆ

r (2.43)

where L = (xt− xr) p1 + b2c is the tether length. Equation (2.43) is similar to Eq. (2.40), where a sort of “effective” tether length (equal to L/p1 + b2

c) is considered in place of the actual length. In particular, note that L/p1 + b2

c is the length of the tether projection on the E-sail nominal plane (xB, yB).

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2.4.3 Parabolic shape

A simple way to take the tether curvature into account is to consider a parabolic shape. In this case, the shape of each tether may be modelled as

f (x) = bpxr x xr − 1 2 with x ∈ [xr, xt] (2.44) where the dimensionless constant bp > 0 depends on the tether curvature. In this case f0(x) = 2 bp x xr − 1 (2.45) and, bearing in mind Eq. (2.39), the total force becomes

F = N σ u xr 2 bp arcsinh 2 bp xt xr − 1 ˆ r (2.46)

where {xr, xt, bp} are related to the tether length L according to

L = xr 4 bp arcsinh 2 bp xt xr − 1 + + bpxr xt xr − 1 s xt xr − 1 2 + 1 4 b2 p (2.47) 2.4.4 Logarithmic shape

An interesting case is obtained when the tether shape is described through a logarithmic function of the distance x. Therefore, let the shape function be f (x) = blxt ln x + xt xr+ xt with x ∈ [xr, xt] (2.48) from which f0(x) = blxt x + xt (2.49) where the dimensionless coefficient bl> 0 is a given parameter. Substitut-ing Eq. (2.49) into Eq. (2.39), the resultant force vector becomes

F = N σ u xt   q 4 + b2 l − s b2 l + xr xt + 1 2   ˆr (2.50)

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2.5 Tether equilibrium shape of a Sun-facing E-sail 21

where {xr, xt, bl} are related to the tether length L through the equation

L = xt q 4 + b2 l − blarcsinh bl 2 + − s b2 l + xr xt + 1 2 + blarcsinh blxt xr+ xt   (2.51)

The result represented by Eq. (2.50) is very useful from a practical point of view. Indeed, the next section will show that, when the spin rate ω is sufficiently large, the tether equilibrium shape of a Sun-facing E-sail follows a natural logarithmic arc.

2.5 Tether equilibrium shape of a Sun-facing E-sail

The problem of describing the actual E-sail equilibrium shape has a sub-stantial simplification when the spacecraft spin axis is aligned with the solar wind velocity vector. In that case, each tether can be thought of as being aligned with the force field and belonging to a plane containing the spacecraft spin axis. An estimation of the tether equilibrium shape of a Sun-facing E-sail can be obtained using the approach discussed in Ref. [41]. Assuming a spinning E-sail, Toivanen and Janhunen [41] de-scribe the tether equilibrium shape with an integral equation, which is solved numerically. In particular, using an analytical approximation of the tether shape, Toivanen and Janhunen [41] also obtain closed-form expres-sions for both the thrust and torque arising from the solar wind momentum transfer. Their results essentially state that the tethers form a cone near the spacecraft, while they are substantially flattened around the tip region by the centrifugal force. A new approximation of the tether equilibrium shape will be achieved in the next section using an analytical procedure.

In this section, instead, it is shown that the tether tip slope may be found in closed-form with some simplifying hypotheses. The procedure to perform such a calculation starts by assuming that the equilibrium shape of each tether results from a combination between the centrifugal force due to the spacecraft spin about ˆn and the electrical interaction of the solar wind particles with the conducting tether. Since each tether is assumed to be at equilibrium, the sum between the centrifugal and the electrical forces is balanced by the constraint reaction at the root section of the cable. It

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is worth noting that this simplified model neglects the inertial forces due to the E-sail acceleration. Actually, the tether shape also depends on the instantaneous acceleration of the spacecraft, which, in its turn, is affected by the current shape of the E-sail. However, such a problem is not solvable with an analytical approach, as its solution requires an iterative numerical procedure, which is very expensive from a computational point of view. The removal of the inertial forces in the presented model is equivalent to assume that the E-sail velocity is constant, a reasonable approximation within sufficiently small time intervals. Also note that at a Sun-sail dis-tance r = r⊕, the ratio of the electrical to the gravitational force per unit

length is equal to

σ⊕u

ρ µ/r2⊕

' 6.2737 (2.52)

where ρ is the tether linear mass density (approximately equal to 10−5kg/m for a µm-diameter aluminum tether), whereas σ⊕ ' 9.3 × 10−13kg/m/s

when V = 20 kV; see Eq. (2.8). Therefore, from Eq. (2.52), the gravita-tional effects on the E-sail are small when compared to the electrical forces acting on the tethers.

The elementary centrifugal force dFωk acting on dsk can be written

recalling that xk is the distance of dsk from the spacecraft spin axis zB (see Fig. 2.3), that is

dFωk = ρ ω 2_x kdskîk= ρ ω2xk q 1 + (f_k0)2_dx kîk (2.53) where ρ is the (uniform) tether linear mass density, whereas îk is given by Eq. (2.2) as a function of {îB, ˆjB}. Enforcing the Sun-facing condition αn = δn = 0 into Eqs. (2.15)–(2.19), the sum between dFk and dFωk

becomes dFsk , dFk+ dFωk = σku ρ ω2_x k σku − f 0 k 1 + (f_k0)2 ˆ_i_k₊ + 1 1 + (f_k0)2 ˆkB _q 1 + (f_k0)2_dx k (2.54) where ˆik, given by Eq. (2.2), is the unit vector obtained from the projection of dFsk on the E-sail nominal plane (xB, yB). Without loss of generality,

the notation may be simplified by dropping the subscript k in the variables {xk, fk0, σk, ˆik} of Eq. (2.54).

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2.5 Tether equilibrium shape of a Sun-facing E-sail 23

Assume now the generic tether to have no bending stiffness, so that only an internal tension acts tangentially to its neutral axis. In this case, according to Toivanen and Janhunen [41], the direction of the vector tan-gent to the tether at a generic point P of abscissa x ∈ [xr, xt] is parallel to the direction of the integral of dFs from x to xt (i.e., the integral of the total force from P to the tether tip). Therefore, from Eq. (2.54), the tether slope f0at point P is the solution of the following integro-differential equation f0(x) = σ u Z xt x dξ p1 + (f0₎2 ρ ω2 Z xt x ξp1 + (f0₎2_{dξ − σ u} Z xt x f0dξ p1 + (f0₎2 (2.55)

where the numerator (or denominator) in the right-hand side is the com-ponent along the zB-axis (or xB-axis) of the resultant force acting on the tether arc between P and the tip, that is

Fx(x) , ρ ω2 Z xt x ξp1 + (f0)2_{dξ − σ u} Z xt x f0dξ p1 + (f0₎2 (2.56) Fz(x) , σ u Z xt x dξ p1 + (f0₎2 (2.57)

Introduce now the dimensionless abscissa h , x/xt, with h ∈ [hr, 1], where hr , xr/xt ≥ 0 is the value at the root section. Equation (2.55) can be conveniently rewritten as f0(h) = Z 1 h dξ p1 + (f0₎2 K Z 1 h ξp1 + (f0₎2_{dξ −} Z 1 h f0dξ p1 + (f0₎2 (2.58)

where K > 0 is a dimensionless shaping parameter, defined as

K , ρ ω 2_x

t

σ u (2.59)

which relates the tether equilibrium shape of a Sun-facing E-sail to the ratio between the centrifugal (ρ ω2_x

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tether slope at the tip, that is, the exact value of f_t0 _{, f}0(h = 1) can be obtained from Eq. (2.58) using a limiting procedure, viz.

f_t0 = lim h→1f 0_{(h) =} 1 K [1 + (f_t0)2_{] − f}0 t (2.60) Equation (2.60) can be rewritten as

f_t0− 1 K 1 + (f0 t)2 = 0 (2.61)

whose only real solution is

f_t0 = 1 K = σ u ρ ω2_x t (2.62) As expected, the tether slope at the tip sharply reduces as the E-sail spin rate increases. The variation of f_t0 with {xt, ω} is shown in Fig. 2.5 for ρ = 10−5kg/m and V = 20 kV. In particular, f_t0 ≤ 0.1 (or K ≥ 10) when ω ≥ 5 rph (with 1 rph ' 1.7453 × 10−3rad/s) and xt≥ 5 km, which implies a tether slope at the tip less than 6 deg.

Having obtained the exact value of f_t0, it is now possible to calculate the function f0(x) (or f0(h)). To that end, a recursive procedure is necessary, which, starting from the tether tip and backward proceeding towards the root, numerically solves Eq. (2.58) for a given value of K. The results of such a procedure are summarized in Fig. 2.6 for some values of the shaping parameter K. In particular, Fig. 2.6 shows that f_t0 = 1/K, in agreement with Eq. (2.62). Note also that the tether slope at the root becomes f0(0) ' 2 f_t0 = 2/K when the shaping parameter is sufficiently large (that is, when K ≥ 5), and hr= 0. In particular, the latter condition amounts to neglecting the main body width and to assuming the root section to be attached to the zB-axis. In that case, 1/K ≤ f0 ≤ 2/K or, equivalently K2+ 1 K2 ≤ 1 + (f 0 )2 ≤ K 2_{+ 4} K2 (2.63) which implies 1 + (f0)2 ' 1 (2.64)

The tether shape may be obtained by means of a numerical integration, and the results are summarized in Fig. 2.7 assuming hr = 0. Notably, an accurate analytical approximation may also be obtained, as is discussed in the next section.

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2.5 Tether equilibrium shape of a Sun-facing E-sail 25 0 5 10 15 20 10-3 10-2 10-1 100 101 102 xt [km] f 0 t 10 3 2 1 0.5 [rph] w 5

Figure 2.5: Tip slope f_t0 as a function of the spin rate ω and the spin axis-tip distance xt; see Eq. (2.62).

2.5.1 Analytical approximation of the tether shape

An accurate analytical approximation of the tether shape can be obtained for a sufficiently large value of the shaping parameter, for example when K ≥ 5. In that case, substituting Eq. (2.64) into Eq. (2.58), the result is

f0(h) ' Z 1 h dξ K Z 1 h ξ dξ − Z 1 h f0_dξ = 2 (1 − h)/K 1 − h2_{− 2}[ft− f (h)] K xt (2.65)

Since max{2 [ft− f (h)]/(K xt)} ' 0.11 (see Fig. 2.7), the last relation may be further simplified as

f0(h) ' 2

K (1 + h) (2.66)

Notably, the approximation of Eq. (2.66) gives the exact value at tether tip (i.e., f_t0 = 1/K), and also captures the approximate value at tether root

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0 0.2 0.4 0.6 0.8 1 10-2 10-1 100 101 h f 0 30 20 10 3 2 1 K 5 50 100

Figure 2.6: Tether slope f0 as a function of the dimensionless abscissa h , x/xtand the shaping parameter K; see Eq. (2.59).

(i.e., f0(0) = 2/K), in agreement with the estimate obtained in the pre-vious section. Figure 2.8 compares the analytical approximation given by Eq. (2.66) (dashed line) with the numerical solution (solid line), showing that the two results are nearly coincident when K ≥ 5. Accordingly, an ac-curate analytical solution of the tether shape can be found from Eq. (2.66). Indeed, using a variable separation and integrating both sides, it may be verified that f (h) = 2 xt K ln 1 + h 1 + hr with h ∈ [hr, 1] (2.67) or, using Eq. (2.59)

f (x) = 2 σ u ρ ω2 ln x + xt xr+ xt with x ∈ [xr, xt] (2.68)

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2.5 Tether equilibrium shape of a Sun-facing E-sail 27 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 h fx/ t 100 30 20 10 5 K 50

Figure 2.7: Tether shape as a function of h , x/xt and K obtained

through numerical integration.

It is worth noting that the latter coincides with Eq. (2.48) when bl= 2 σ u ρ ω2_x t = 2 K (2.69)

Equation (2.68) proves the importance of the logarithmic function for de-scribing the equilibrium shape of a Sun-facing E-sail. Its actual accuracy is better appreciated with the aid of Fig. 2.9, which plots Eq. (2.67) with hr= 0.01. The obtained results are nearly coincident with those reported in Fig. 2.7, which correspond to a numerical integration of the actual tether slope.

This result is qualitatively in accordance with the numerical simulations by Toivanen and Janhunen [41], which show that the tethers form a cone near the spacecraft, whereas they are flattened by the centrifugal force near

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0 0.2 0.4 0.6 0.8 1 10-2 10-1 100 101 h f 0 100 50 30 20 10 54 3 2 1 K

Figure 2.8: Tether slope f0 as a function of h and K: numerical (solid line) vs. analytical approximation (dashed line).

the tip region. Actually, the analytical approximation given in Ref. [41] estimates a parabolic shape of the tethers, with the effect of a null slope at their tips. The discrepancy between the two models is consistent with the assumption that in Ref. [41] the tips of the main tethers host remote units connected to an external rim in order to provide mechanical stability to the sail. In the presented case, instead, the remote units are not included in the model with the aim of decreasing the stress at tether root. As such, a nonzero tip slope is an expected result.

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2.6 Tether root force 29 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 h fx/ t 10050 30 20 10 5 K 0.3

Figure 2.9: Tether approximate shape as a function of the dimensionless abscissa h , x/xtand K when hr= 0.01; see Eq. (2.67).

2.6 Tether root force

Due to the E-sail rotation, each tether experiences a tension force τ with a maximum value τr that occurs at the root section x = xr (or h = hr). The value of τr is obtained by imposing the equilibrium condition of all forces acting on the tether at the root, that is τr = pFx2r + F

2

zr, where

Fxr , Fx(xr) and Fzr , Fz(xr); see Eqs. (2.56) and (2.57). The tension at

the root section is therefore τr = p1 + (f0 r)2 f0 r Fzr (2.70) where Fzr = σ u Z xt xr dξ p1 + (f0₎2 (2.71)

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while f_r0 _{, f}0(xr) = Fzr/Fxr is the tether slope at the root section.

Equa-tion (2.70) can be rewritten in a dimensionless form as τr σ u xt = p1 + (f 0 r)2 f0 r Z 1 hr dξ p1 + (f0₎2 (2.72)

whose numerical solution is obtained (for a given value of K) using the function f0(h) calculated in the last section. For example, the dimension-less value of τr is shown in Fig. 2.10 as a function of K assuming hr= 0. Note that τr/(σ u xt) has a nearly linear dependence on K with an an-gular coefficient equal to 0.5. Such a result will now be confirmed by an analytical approximation. 0 20 40 60 80 100 0 5 10 15 20 25 30 35 40 45 50

K

=

r

/

(<

ux

t

)

Figure 2.10: Dimensionless tension at tether root as a function of K when hr= 0; see Eq. (2.67).

2.6.1 Analytical approximation of τr

Assuming a shaping parameter K ≥ 5, the tether slope is well approxi-mated by Eq. (2.66), therefore

Z 1 hr dξ p1 + (f0₎2 = 2√K2_{+ 1 −} q K2_{(1 + h} r)2+ 4 K (2.73)

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2.6 Tether root force 31

Substituting this last relationship into Eq. (2.72), and observing that f_r0 ' 2/K/(1 + hr) (see Eq. (2.66)), the result is

τr σ u xt = q K2 _{(1 + h} r)2+ 4 2 K 2pK2_{+ 1 −} q K2_{(1 + h} r)2+ 4 (2.74)

In the limit as hr → 0, the last relation may be further simplified taking into account that K2 1. The solution is

τr σ u xt = K 2 = ρ ω2xt 2 σ u (2.75)

which is in agreement with the plot shown in Fig. 2.10. This last relation allows the value of τr to be related with the tether length L. Indeed, assuming xr = 0 and substituting Eq. (2.75) into Eq. (2.69) and then into Eq. (2.51), it is possible to verify that

L xt = s 4 + σ u xt τr 2 − s 1 + σ u xt τr 2 + + σ u xt τr arcsinh σ u xt τr − arcsinh σ u xt 2 τr (2.76)

which is drawn in Fig. 2.11 when K ∈ [5, 100]. The tension at the root section can be expressed as a function of the pair {ω, L} by combining Eqs. (2.75) and (2.76). Its maximum value cannot exceed the tether yield strength, which is about 0.1275 N for a µm-diameter aluminum tether with ρ ' 10−5kg/m [49].

For example, assuming V = 20 kV [49], Fig. 2.12 shows how the tension τr varies with the tether length L and spin rate ω when xr= 0. Note that each level curve breaks down when the yield strength τmaxis achieved (i.e., when τr= τmax). According to Fig. 2.12, the tension τr roughly exhibits a parabolic behaviour with the spacecraft spin rate ω for a given value of L. In particular, the figure shows that the maximum allowable spin rate for a baseline tether length of 20 km is about ω = 4.57 rph, whereas the value of xt is 19.983 km. In this case, K ' 34 and the dimensionless root tether is τmax/(σ u xt) ' 17, in agreement with the numerical results shown in Fig. 2.10.

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1 1.01 1.02 1.03 1.04 1.05 0 5 10 15 20 25 30 35 40 45 50 L x/ t =r /( < ux t ) 5 K <

Figure 2.11: Dimensionless tether tension at root as a function of the di-mensionless tether length when hr= 0.

2.7 Thrust and torque vector model

This section concludes this chapter by presenting a simplified model for describing the thrust and torque vectors of an E-sail as a function of its attitude. The tethers are assumed to be uniformly distributed about the zB-axis, and to be maintained at a uniform electrical voltage V . Moreover, the assumption is made that the E-sail maintains the equilibrium shape found in the Sun-facing configuration, a reasonable hypothesis when the sail pith angle αn is sufficiently small. As such, all the tethers have the same two-dimensional shape, which can be described through a suitable differentiable function f = f (x), where the x-axis is orthogonal to zB, and (x, zB) defines the plane where the generic tether lies; see Fig. 2.3. Using the general mathematical model discussed in Section 2.3, the thrust (F ) and torque (T ) vectors acting on an axially-symmetric E-sail of given shape can be expressed in analytical form as a function of the E-sail attitude. In

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2.7 Thrust and torque vector model 33 w [rph] =r [N] 0 2 4 6 8 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 max t tether failure 5 K < [km] L 15 20 10 7 5 3

Figure 2.12: Root tension τr as a function of L and ω when ρ =

10−5kg/m, xr= 0, and τmax= 0.1275 N. Data taken from

Ref. [49].

fact, assuming r ' r⊕, the vectors F and T become

F = 1 2N L σ u h (2 − P) ˆr + (3 P − 2)r · ˆˆ kB ˆkB i (2.77) T = 1 2M N L 2_{σ u}ˆ_k B× ˆr (2.78) where P ∈ [0, 1] and M ∈ [0, 1] are two dimensionless coefficients related to the tether shape f through the equations

P , 1 L Z xt 0 dx p1 + (f0₎2 (2.79) M , 1 L2 Z xt 0 f 1 + 2 (f0)2 + x f0 p1 + (f0₎2 dx (2.80)

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where L is given by L = Z xt 0 p 1 + (f0₎2_dx _(2.81)

and the abscissa of the root section is neglected (i.e., xr= 0). It is worth noting that T induces a pitch oscillation resembling that of a spherical pendulum.

In the special case of a flat shape, that is, when f = f0 = 0 and all tethers belong to the (xB, yB) plane, Eqs. (2.79)–(2.81) give P = 1, M = 0, and L = xt. In that case, the torque is zero independent of the E-sail attitude, whereas Eq. (2.77) reduces to

F = 1 2N L σ u h ˆ r +r · ˆˆ kB ˆkB i (2.82) consistently with the results presented in Ref. [38]. The magnitude of F when f = f0= 0 is kF k = 1 2N L σ u p 1 + 3 cos2_α n (2.83)

which depends on the Sun-spacecraft distance r through the parameter σ defined in Eq. (2.8).

2.7.1 Case of logarithmic tether shape

Section 2.5 introduces the dimensionless shape coefficient K, which relates the tether equilibrium shape of a Sun-facing E-sail to the ratio between the centrifugal (ρ ω2xt) and the electrical (σ u) effects; see Eq. (2.59). Section 2.5.1 provides an analytical approximation of the tether shape, which is valid as long as K ≥ 5, or

ω ≥ ωmin ,

r 5 σ u ρ xt

(2.84) which means that the E-sail spin rate ω must be sufficiently high. The analytical expression of the tether equilibrium shape found in Section 2.5.1 (see Eq. (2.68)) is here reported assuming xr= 0, viz.

f (x) = blxt ln 1 + x xt with x ∈ [0, xt] (2.85) where bl is given by Eq. (2.69). However, the spin rate cannot exceed a maximum value ωmax related to the tether yield strength τmax, that is [50]

ω ≤ ωmax, s

2 τmax

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2.7 Thrust and torque vector model 35

Using a µm-diameter aluminum tether [49] with ρ ' 10−5kg/m and τmax = 0.1275 N, the allowable pairs {ω, xt} are shown in Fig. 2.13 when xt ∈ [1, 10] km. For example, assuming ω = 10 rph, Fig. 2.13 shows that the maximum value of xt is about 9 km. In that case, Fig. 2.14 shows the variation of {L, P, M} with xt∈ [1, 9] km according to Eqs. (2.79)–(2.81).

1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 xt [km] w [rph] max w min w

Figure 2.13: Allowable spin rates as a function of xt for a µm-diameter

aluminum tether [49].

Even though the coefficients P and M must be calculated numerically, they can be also accurately estimated with an analytical approximation. As a result, closed-form expressions of the E-sail propulsive characteristics can be easily derived, which are very useful for both trajectory simulation, and preliminary mission analysis purposes. In fact, the condition ω ≥ ωmin, with ω taken from Eq. (2.69) and ωmin from Eq. (2.84), implies bl < 0.4. Observing that

f0 = blxt xt+ x

≤ b_l (2.87)

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1 2 3 4 5 6 7 8 9 1 1.005 1.01 1.015

L/x

t 1 2 3 4 5 6 7 8 9 0.97 0.98 0.99 1

P

1 2 3 4 5 6 7 8 9 0 0.05 0.1 0.15 0.2

xt

[km]

M

Figure 2.14: Variation of {L, P, M} with xt when ω = 10 rph.

result is

L ' xt , P ' 1 , M ' ln (2) bl (2.88) in accordance with the graphs of Fig. 2.14, and consistently with Eq. (2.64). Substituting now Eqs. (2.88) into Eqs. (2.77) and (2.78), the approximate expression of the thrust vector reduces to Eq. (2.82) (with the magnitude given by Eq. (2.83)), whereas the torque vector becomes

T = 1 2 ln (2) blN L 2_{σ u} ˆ_k B× ˆr (2.89)

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2.8 Conclusions 37 whose magnitude is kT k = 1 2 ln (2) blN L 2_{σ u sin α} n (2.90)

Finally, from Eqs. (2.88) the function f (x) describing the shape of each tether can by simplified as

f (x) ' 2 σ u ρ ω2 ln 1 + x L with x ∈ [0, L] (2.91)

2.8 Conclusions

This chapter has addressed the problem of determining the thrust and torque vectors provided by an E-sail of given shape as a function of the spacecraft attitude. The general expressions of the thrust and torque vec-tors have been then specialized to the case of a Sun-facing E-sail, showing that the equilibrium shape of each tether is close to a logarithmic arc when the spin rate is sufficiently high. With the assumption that the E-sail maintains the logarithmic shape found in the Sun-facing configuration, analytical expressions of the thrust and torque vectors have finally been derived as a function of the sail orientation.

The new mathematical model allows the performance of an E-sail to be quantified in closed-form and, as such, it is easy to implement in a simulation routine in order to obtain preliminary mission results.

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Chapter

3

E-sail dynamics

This chapter introduces the differential equations that describe the motion of a spacecraft under the action of the propulsive force and torque provided by a rigid E-sail with a uniform electrical voltage. The orbital motion and the attitude dynamics are studied separately, thanks to the marked separation between their characteristic timescales. The mathematical model used to describe the thrust and torque vectors is taken from Chapter 2. The results discussed in this chapter have been published in [J.5] and [J.4].

3.1 Orbital dynamics

Consider an E-sail-based spacecraft that covers a heliocentric parking orbit of given characteristics. The spacecraft is modelled as a point mass sub-jected to the gravitational force of the Sun and to the E-sail thrust. The spacecraft state is defined by a set of non-singular Modified Equinoctial Orbital Elements [51, 52] (MEOEs) {p, f, g, h, k, L}, which are related to the classical orbital elements {a, e, i, Ω, ω, ν} of the osculating orbit by the following relationships

p = a 1 − e2

, f = e cos(Ω + ω) , g = e sin(Ω + ω) ,

h = tan (i/2) cos Ω , k = tan (i/2) sin Ω , L = ν + Ω + ω (3.1) where a is the semimajor axis, e is the orbital eccentricity, i is the orbital inclination, Ω is the right ascension of the ascending node, ω is the argu-ment of perihelion, and ν is the true anomaly. The spacecraft heliocentric

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motion is described through the vectorial differential equation [53] ˙

x = A [a]TRTN+ c (3.2)

where x , [p, f, g, h, k, L]T

is the state vector, A ∈ R6×3 is the state matrix, defined as A , r p µ                        0 2 p q 0 sin L (q + 1) cos L + f q g (k cos L − h sin L) q − cos L (q + 1) sin L + g q f (h sin L − k cos L) q 0 0 (1 + h 2_{+ k}2_{) cos L} 2 q 0 0 (1 + h 2_{+ k}2_{) sin L} 2 q 0 0 h sin L − k cos L q                        (3.3)

where q , (1 + f cos L + g sin L), whereas c ∈ R6×1 is given by c ,r µ

p3 q

2_{[0 0 0 0 0 1]}T

(3.4) in which µ ' 1.327 × 1011km3/s2 is the Sun’s gravitational parameter.

In Eq. (3.2), a is the spacecraft propulsive acceleration vector, whose com-ponents must be expressed in the radial-transverse-normal reference frame T_RTN, centered at the spacecraft center of mass S, of unit vectors

ˆ

er, ˆr , eˆn,

r × v

||r × v|| , eˆt, ˆen× ˆer (3.5) where r is the Sun-sail position vector, while v is the spacecraft absolute velocity; see Fig. 3.1. Note that Eq. (3.2) is free from singularities, since q ≡ p/r > 0, being p the semilatus rectum of the spacecraft osculating orbit. The spacecraft propulsive acceleration vector a can be obtained from Eq. (2.82) by computing the ratio of the thrust vector F to the spacecraft mass m, viz.

a , F m = ac 2 r_⊕ r [ˆer+ (ˆer· ˆn) ˆn] (3.6)

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3.1 Orbital dynamics 41 n a d ˆ_n e ˆ n ˆ_r e a a ˆ_r e Sun _osculating orbit E-sail ˆ_n e ˆ_t e r ˆ ˆ ( , ) planee e_t _n

Figure 3.1: Reference frame and E-sail characteristic angles.

where, consistently with the nomenclature adopted in Section 2.2, ˆn is the unit vector perpendicular to the E-sail plane pointing in the opposite direction to the Sun, while

ac,

N L σ⊕u

m (3.7)

is the characteristic acceleration of the E-sail, that is, the maximum feasible propulsive acceleration when r = r⊕. In particular, such a performance

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Indeed, when ˆn ≡ ˆr and r = r⊕, then

a = aceˆr (3.8)

With reference to Fig. 3.1, when αn 6= 0, a belongs to the plane (ˆer, ˆn), and the angle α between ˆerand a, referred to as thrust cone angle, is given by α = arccos 1 + cos2αn √ 1 + 3 cos2_α n (3.9) whose maximum value, reached when αn ' 54.74 deg, is approximately equal to 19.47 deg. It is also interesting to evaluate the dependence of the propulsive acceleration magnitude on αn. This can be done by computing the dimensionless propulsive acceleration γ, defined as

γ , ||a|| r acr⊕ = √ 1 + 3 cos2_α n 2 (3.10)

Figure 3.2 shows γ and α as a function of αn.

0 15 30 45 60 75 90 0.5 0.6 0.7 0.8 0.9 1 . 0 15 30 45 60 75 90 ,n [deg] 0 5 10 15 20 , [deg]

Figure 3.2: Dimensionless propulsive acceleration and cone angle as a func-tion of αn.

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3.2 Attitude dynamics 43

Finally, the components of a in TRTN are

[a]TRTN ,   ar at an   (3.11) with ar, a · ˆer= ac 2 r ⊕ r 1 + cos2αn (3.12) at, a · ˆet= − ac 2 r_⊕ r

cos αn sin αn sin δ (3.13)

an, a · ˆen= ac 2 r_⊕ r

cos αn sin αn cos δ (3.14) where δ ∈ [0, 2π] rad is the angle (measured counterclockwise) between the projection of ˆn on the local horizontal plane (i.e., the plane (êt, ên) perpendicular to the Sun-spacecraft line) and ên; see Fig. 3.1.

3.2 Attitude dynamics

This section deals with the problem of analyzing the attitude dynamics of a spinning E-sail-based spacecraft. Indeed, when the spacecraft attitude is perturbed from the Sun-facing configuration, that is, when the zB-axis slightly differs from the Sun-spacecraft direction (ˆkB 6= ˆr), the spacecraft experiences a non-zero propulsive torque whose approximate expression is given by Eq. (2.89). The attitude motion of the spacecraft is here studied without any type of control. Conversely, the next part of the Thesis will address the problem of investigating a strategy capable of maintaining and controlling the E-sail attitude by generating a suitable control torque. In particular, it will be shown that the attitude control can be performed through a suitable modulation of the electrical voltage of each tether.

Under the assumption that the spacecraft (including the E-sail) behaves like a rigid body, the dimensionless coefficient M (see Eq. (2.80)) and the spacecraft inertia tensor are both constant. The effects of the torque due to the tether inflection on the spacecraft dynamics can therefore be analyzed by a numerical integration of the Euler’s equations. To that end, the components of T in the body reference frame are written as a function of the three Euler’s angles {φ, θ, ψ}, which define the orientation of TB with respect to an inertial reference frame TI(S; xI, yI, zI) of unit vectors

Mission analysis of spacecraft equipped with advanced propulsion systems (Analisi di missione di satelliti equipaggiati con sistemi propulsivi avanzati)

Universit`

a di Pisa

Mission analysis of spacecraft equipped with

advanced propulsion systems

Abstract

Preface

Acknowledgements

Contents

Chapter

1

Introduction

Magsail

solar wind

Part I

Chapter

2

Thrust and torque vector models

2.1

Introduction

spin

axis

2.2

Geometrical preliminaries

2.3

Total force and torque acting on the E-sail

2.4

Case of a Sun-facing E-sail

2.5

Tether equilibrium shape of a Sun-facing E-sail

2.6

Tether root force

K

=

/

(<

ux

)

2.7

Thrust and torque vector model

L/x

P

xt

[km]

M

2.8

Conclusions

Chapter

3

E-sail dynamics

3.1

Orbital dynamics

3.2

Attitude dynamics