Astrodynamics
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Astrodynamics Vol. 5, No. 2, 121{137, 2021 https://doi.org/10.1007/s42064-020-0093-6 Spiral trajectories induced by radial thrust with applications to generalized sails Marco Bassetto (B), Alessandro A. Quarta, Giovanni Mengali, and Vittorio Cipolla Department of Civil and Industrial Engineering, University of Pisa, Pisa I-56122, Italy ABSTRACT KEYWORDS In this study, new analytical solutions to the equations of motion of a propelled spacecraft radial thrust are investigated using a shape-based approach. There is an assumption that the spacecraft spiral trajectory travels a two-dimensional spiral trajectory in which the orbital radius is proportional shape-based approach to an assigned power of the spacecraft angular coordinate. The exact solution to the analytical solution equations of motion is obtained as a function of time in the case of a purely radial generalized sail thrust, and the propulsive acceleration magnitude necessary for the spacecraft to track the prescribed spiral trajectory is found in a closed form. The analytical results are then specialized to the case of a generalized sail, that is, a propulsion system capable of providing an outward radial propulsive acceleration, the magnitude of which depends on a given power of the Sun-spacecraft distance. In particular, the conditions for an outward radial thrust and the required sail performance are quanti ed and thoroughly discussed. It is worth noting that these propulsion systems provide a purely radial thrust when their orientation is Sun-facing. This is an important advantage from an engineering point of view because, depending on the particular propulsion system, a Sun-facing attitude Research Article can be stable or obtainable in a passive way. A case study is nally presented, where the Received: 27 April 2020 generalized sail is assumed to start the spiral trajectory from the Earth's heliocentric Accepted: 19 August 2020 orbit. The main outcome is that the required sail performance is in principle achievable © The Author(s) 2020 on the basis of many results available in the literature. 1 Introduction solution to the radial and the circumferential case, respec- tively. Since then, many other authors have discussed the Analytical solutions to the dierential equations that constant radial thrust problem. Prussing and Coverstone- govern the motion of an orbiting propelled spacecraft are Carroll addressed the problem of determining the escape available in few cases only [1{3]. Closed-form solutions conditions and the amplitude of the spacecraft radial represent a very useful tool in the preliminary phase of displacement (in the absence of an escape) by introduc- mission design as they signi cantly reduce the compu- ing the concept of potential energy well, but con ning tational cost that would otherwise be required by the their analysis to the case of a circular parking orbit [5]. numerical propagation of spacecraft dynamics. Later, Mengali and Quarta adopted the same approach in Analytical solutions to the equations of motion can order to extend the previous results to the more general be found by using two dierent approaches. The rst case of an elliptic parking orbit [6]. The explicit solu- possibility is to solve the equations of motion for an as- tion found by Battin in terms of elliptic integrals is also signed thrust pro le, such as the case of a spacecraft noteworthy [7], as well as the accurate approximations subjected to a constant radial or circumferential propul- of the spacecraft trajectory and the ight time found sive acceleration. This problem was rst investigated by Quarta and Mengali in terms of a Fourier series [8]. by Tsien [4], who found an explicit and an approximate Gonzalo and Bombardelli found alternative solutions that B marco.bassetto@ing.unipi.it 122 M. Bassetto, A. A. Quarta, G. Mengali, et al. Nomenclature a semimajor axis (km) a characteristic acceleration (mm/s ) a propulsive acceleration, with a , a r^ (mm/s ) r r r fc ; c g constants of integration, see Eq. (12) 1 2 e eccentricity vector, with e , kek h speci c angular momentum magnitude (km /s) O primary body center of mass p semilatus rectum (km) r orbital radius (km) r^ radial unit vector r reference distance 1 au S spacecraft center of mass t time (year) T polar reference frame v spacecraft velocity vector (km/s) v radial component of v (km/s) v transversal component of v (km/s) dimensionless spiral parameter, see Eq. (6) constant, see Eq. (61) dimensionless design parameter, see Eq. (38) polar angle (rad) transversal unit vector 3 2 gravitational parameter (km /s ) spacecraft true anomaly (rad) dimensionless auxiliary function, see Eq. (14) ! argument of periapsis (rad) Subscripts 0 initial, parking orbit max maximum the Earth the Sun Superscripts time derivative threshold value ? optimal involve asymptotic expansions [9], while Izzo and Biscani approach conceived by Bombardelli et al. [13]. Finally, showed that a solution relating the state variables to a Quarta et al. investigated the minimum-time trajecto- time parameter can be expressed in terms of Weierstrass ries from a circular parking orbit towards the apocenter elliptic and related functions [10]. of a rectilinear ellipse assuming that the spacecraft is After the pioneering work of Tsien [4], the constant subjected to a continuous circumferential thrust [14]. circumferential thrust problem has also been an object Benney was the rst author to propose approximate of study. For example, Battin proposed an approximate solutions to the motion of a spacecraft subjected to a solution under the assumptions of circular parking or- constant tangential thrust [15], while Boltz addressed bit, two-dimensional motion, and low thrust [7]. More the same problem assuming that the ratio of the thrust recently, Quarta and Mengali proposed an analytical to the local gravitational pull is constant [16]. Finally, approximation of the spacecraft escape conditions and it is worth mentioning the work of Roa et al. [17], who provided a simple expression for the escape distance [11], presented a new analytical solution to the motion of a while Niccolai et al. [12] analyzed the two-dimensional continuously accelerated spacecraft, wherein its thrust dynamics of a spacecraft with a constant and circumfe- magnitude is assumed to decrease with the square of the rential low propulsive acceleration using the perturbative spacecraft orbital radius. In particular, Roa et al. found Spiral trajectories induced by radial thrust with applications to generalized sails 123 out that the dynamical system admits two integrals of includes all the propellantless propulsion systems that motion resulting from the energy and angular momentum provide an outward radial propulsive acceleration with equations, and identi ed three subfamilies of spiral tra- a magnitude that scales as a certain power of the Sun{ jectories depending on the sign of a modi ed mechanical spacecraft distance. Electric solar wind sails (E-sails) [33{ energy [17]. 37], magnetic sails (magsails) [38{42], solar sails [20{22], The second possibility to determine the analytical so- and smart dusts (SDs) [43, 44] belong to this class of lutions to the equations of motion is represented by the propulsion systems. Each of them exploits a peculiar shape-based approach. In this case, the thrust pro le is form of energy coming from the Sun. In particular, E- derived such that a given trajectory may be followed by sails and magsails extract momentum from the charged the spacecraft. For example, the possibility of generating particles constituting the solar wind by using an arti cial a logarithmic spiral trajectory was rst investigated by electric and magnetic eld, respectively. Dierently, solar Bacon [18] and Tsu [19], the latter suggesting the use of sails and SDs take advantage of solar radiation pressure, solar sails [20{22]. In fact, a solar sail can travel along a which acts on a membrane made of re ective material, logarithmic spiral with a constant attitude with respect thus generating a propulsive acceleration. It is worth to the Sun{spacecraft line, as thoroughly discussed by noting that these propulsion systems provide a purely Bassetto et al. [23], who focused on the requirements to radial thrust when their orientation is Sun{facing, that be met to place a spacecraft in a logarithmic spiral trajec- is, when the sail nominal plane is orthogonal to the tory without any impulsive maneuver. Petropoulos and propagation direction of photons or solar wind particles. Longuski introduced an exponential sinusoid where four From an engineering point of view, this is a signi cant independent constants are used for describing the shape advantage because such an attitude is easy to maintain in of the spacecraft trajectory [24]. In particular, Ref. [24] most cases. For example, a Sun{facing attitude is shown derived closed-form relations for the angular rate and to be stable when using an E-sail [45], while SDs are able the required tangential acceleration. Then, Izzo used to passively obtain a Sun{facing orientation thanks to an the concept of the exponential sinusoid for solving the appropriate system architecture (which is composed of multi-revolution Lambert's problem [25], while Wall and a single plate with many surface etches called \facets") Conway proposed fth- and sixth-degree inverse polyno- that exploits solar radiation torques [46]. mial functions that are capable of providing near-optimal E-sail- and solar sail-based trajectories have already solutions to many mission scenarios [26]. Finally, Taheri been analyzed in an attempt to nd approximate analy- and Abdelkhalik developed a new exible and eective tical solutions to the equations of motion. For example, Quarta and Mengali proposed an analytical approxima- method, suitable for the preliminary design of dierent tion for the trajectory of a low-performance E-sail with scenarios, to approximate low-thrust trajectories using a constant thrust angle [47]. In a subsequent work [48], the nite Fourier series [27{29]. In the context of shape-based approaches, the aim of Quarta and Mengali oered an approximate expression this paper is to analyze the generation of two-dimensional of the E-sail heliocentric trajectory by reducing the pro- spiral trajectories in which the spacecraft orbital radius blem to the dynamics of an equivalent nonlinear oscillator is proportional to a given power of its angular coordinate. with a single degree of freedom, and assuming that the There is an assumption that the spacecraft is subjected E-sail provides a purely outward radial thrust. Moreover, Niccolai et al. analyzed the two-dimensional heliocentric to a purely radial propulsive acceleration of adjustable dynamics of an E-sail with a xed attitude, and found an magnitude during the entire transfer. It turns out that approximate solution to the equations of motion through an exact solution to the equations of motion exists, al- lowing all of the spacecraft state variables and the orbital the use of an asymptotic expansion procedure [49]. In the elements of the osculating orbit to be explicitly written eld of solar sails, Quarta and Mengali investigated ap- as a function of time. proximate solutions to circle-to-circle orbit transfers [50], The general solution, which can be applied to any and Niccolai et al. analyzed solar sail trajectories with propulsion system that can provide a purely radial thrust, an asymptotic expansion method [51], while Caruso et is specialized in a heliocentric framework to the class al. presented a procedure to generate an approximate of the generalized sails [30{32]. This type of thrusters optimal trajectory through a nite Fourier series [52]. 124 M. Bassetto, A. A. Quarta, G. Mengali, et al. In this study, the characteristic acceleration necessary for the generalized sail to track the desired heliocentric spiral trajectory is found analytically. The characteristic acceleration, which is a sail performance parameter, is here de ned as the magnitude of the radial propulsive acceleration at the reference distance of one astronomi- cal unit from the Sun. A continuous modulation of the characteristic acceleration is actually feasible for all the aforementioned sails. For example, the characteristic ac- celeration of an E-sail may be controlled by modulating the tether electrical voltage [45]. For a solar sail, such a control may be achieved by partially or totally refolding Fig. 1 Polar reference frame and spacecraft state variables. the re ective lm, by rotating parts of it, or by vary- ing the sail orientation over time to obtain a modulated radius r (i.e., the primary-spacecraft distance) and by the thrust that, on average, is purely radial [3, 53]. Another polar angle , the latter being measured counterclockwise option for controlling the characteristic acceleration of a on the parking orbit plane from a xed direction. Be- solar sail is to cover its re ective lm with electrochromic sides, let v , vr^ and v , v be the two components materials [54], which can change their re ectivity coe- of the spacecraft velocity vector v; see Fig. 1. At the cient through the application of an electrical voltage. initial time t = t , 0, the spacecraft angular position is The same strategy is commonly used for millimeter-scale , (t ), the orbital radius is r , r(t ), and the two 0 0 0 0 solar sails and SDs [43, 44]. Finally, the characteristic components of the spacecraft velocity are v , v (t ) r r 0 acceleration of a Magsail can be adjusted, in principle, by and v , v (t ). varying the electrical current owing in its ring [40, 41]. The spacecraft is equipped with a continuous-thrust The remainder of this paper is organized as follows. propulsion system that provides a purely radial propulsive Section 2 deals with the trajectory analysis and presents acceleration a = a r^ of adjustable magnitude ja j. As r r r the analytical solution to the equations of motion assum- a = 0, the speci c angular momentum h , rv is a ing a purely radial thrust and a direct proportionality constant of motion, and the transversal component of the between the spacecraft orbital radius and a given power of velocity vector may be written as v = r v =r. Hence, its angular coordinate. The magnitude of the propulsive the spacecraft dynamics is described by the following acceleration that is required for the spacecraft to track system of nonlinear dierential equations: the prescribed spiral trajectory is also derived in Section 2. Section 3 investigates the possibility of generating r _ = v (1) heliocentric spiral trajectories using a generalized sail r v = (2) and quanti es the conditions for an outward radial thrust 2 and the required sail performance. Section 4 discusses a 2 2 r v v _ = + + a (3) r r case study assuming that the sail enters the spiral trajec- 2 3 r r tory starting from the Earth's heliocentric orbit. Finally, r v v 0 r v _ = (4) Section 5 presents the concluding remarks. with initial conditions 2 Trajectory analysis r(t ) = r ; (t ) = ; v (t ) = v ; v (t ) = v (5) 0 0 0 0 r 0 r 0 0 0 With reference to Fig. 1, consider a spacecraft S that ini- There is an assumption that the spacecraft is able to tially covers a Keplerian parking orbit around a primary exactly track a generic spiral trajectory in the form body O of gravitational parameter , and introduce a polar reference frame T (O;r^;) with origin at O, where r^ r = r (6) ^ 0 and are the radial and transversal unit vectors, respec- tively. The spacecraft position is described by the orbital provided that the radial thrust magnitude may be suit- Spiral trajectories induced by radial thrust with applications to generalized sails 125 ably modulated. In Eq. (6), 2 R is a dimensionless as 6=0 parameter that characterizes the type of spiral. Typical v 1 1+2 = (13) examples are obtained with = 1 (the hyperbolic spi- v ral), = 1=2 (the lituus), = 1=2 (the Fermat's spiral), where and = 1 (the Archimedean spiral). Note that, when the (1 + 2)v t , 1 + (14) trajectory is in the form of Eq. (6), the spacecraft radial velocity v is constrained to the transversal velocity v . is a dimensionless auxiliary function of time. Substituting In fact, according to Eq. (6), the time derivative of r can Eq. (13) into Eq. (6), the -variation of the primary- be written in a compact form as spacecraft distance becomes r _ = (7) 1+2 r = r (15) Bearing in mind that v = r _ and v = r, Eqs. (6) and r while, with the aid of Eqs. (7) and (10), the two compo- (7) provide nents of the spacecraft velocity vector are r 1+ = (8) 1+2 1+2 v = v ; v = v (16) r r 0 0 so that the initial conditions in Eq. (5) must be selected Equation (15) provides a positive value of r as long as to satisfy the constraint (1 + 2)v = > 0. When, instead, (1 + 2)v = < 0, r r 0 0 the physical constraint r > 0 is met provided that v = (9) 0 r t < t , (17) max The time evolution of the spacecraft state variables (1 + 2)v fr; ; v ; v g can now be found. To that end, rst consider Finally, note that, when v > 0, r ! +1 as t ! +1 the time variation of the polar angle , (t), which may (or t ! t ) if < 1=2 or > 0 (or 1=2 < < 0). max be obtained by substituting Eq. (6) into Eq. (2) to get Conversely, r ! 0 as t ! t (or t ! +1) if < 1=2 max the following rst order dierential equation: or > 0 (or 1=2 < < 0). 2+1 The closed-form expressions of the state variables allow 0 2 = (10) r v 0 the orbital elements fa; e; !; g of the osculating orbit to be analytically determined as a function of time. To that which can be solved by separation of variables. The end, consider the eccentricity vector e, de ned as result is p r rv v 1+2 c [c + (1 + 2)t] ; if 6= f1=2; 0g ^ ^ 2 1 e , e r^ + e = r^ (18) = (11) c exp (c t); if = 1=2 2 1 2 2 where p p , r v = is the (constant) semilatus where fc ; c g are two constants of integration, given by 1 2 rectum of the spacecraft osculating orbit. From Eqs. (15) and (16), the radial and transversal components of e may ; if 6= f1=2; 0g be explicitly written as c = 2v : ; if = 1=2 r r v 0 0 8 1+2 e = 1 (19) 2 r > 1+2 >v r ; if 6= f1=2; 0g 1+ r v v 0 r r v 0 0 c = 0 r (12) 2 0 1+2 e = (20) : ; if = 1=2 2v Accordingly, the eccentricity e of the spacecraft oscu- The two cases of 6= f1=2; 0g and = 1=2 are now lating orbit is analyzed separately. 2 4 2 u r v 2r v 2 0 0 0 u 1+2 1+2 2.1 Case of 6=f1=2; 0g u 2 e = kek = (21) 2 2 2 t r v v 2+2 0 r 0 0 In this scenario, rearranging Eqs. (11) and (12), the time 1+2 +1 + variation of the spacecraft polar angle can be rewritten 126 M. Bassetto, A. A. Quarta, G. Mengali, et al. whereas the semimajor axis a is while the radial and transversal components of e are p r 2 0 0 r v 0 v t a = = 0 0 2 2+2 2 e = exp 1 (28) 1 e 2 2 r 1+2 1+2 1+2 2 r v r v 0 0 r 0 0 0 (22) r v v v t 0 r r 0 0 0 e = exp (29) Because the argument of periapsis is here de ned as the r angle between the initial primary-spacecraft line and the Accordingly, the osculating orbit eccentricity e is direction of e (see Fig. 2), the expression of ! 2 [0; 2) 2 4 2 rad is r v 2r v u 0 0 2v t v t r r 0 0 0 0 0 1 u exp exp r r 2 0 0 v 1+ u 1+2 u e = 2 2 2 B C r v v v 1 v 0 r 2v t 0 0 r 0 0 0 1+2 B C +1 + exp ! = ( 1) + arctan 2 r @ A 0 1+2 r v (30) (23) whereas the semimajor axis a becomes Finally, the spacecraft true anomaly 2 [0; 2) rad a = (31) along the osculating orbit is v t 2v t r r 0 2 0 2 exp r v exp 0 1 r r 0 0 v 1+ 1+2 2v t 2 0 B C r v exp 0 0 B C 0 = arctan (24) @ A 1+2 r v Finally, the angle ! is given by v 2v t 0 0 2.2 Case of =1=2 ! = 1 exp 2v r r 0 2 3 In this case, using again Eqs. (11) and (12), the function v v t r r 0 0 exp 6 7 , (t) can be written as v r 6 0 7 + arctan (32) 6 7 4 v t 5 v 2v t 0 0 exp = exp (25) 2 r r v 0 0 2v r 0 r 0 while the spacecraft true anomaly along the osculating Substituting Eq. (25) into Eq. (6) provides the time orbit is variation of the orbital radius, viz. 2 3 v v t r r v t 0 0 exp 6 7 r = r exp (26) v r 6 0 7 (t) = arctan (33) 6 7 4 v t 5 exp with r > 0 for any t > t . Note that now r ! +1 (or r r v 0 0 r ! 0) when v > 0 (or v < 0) as t ! +1. Moreover, r r 0 0 the two components of the spacecraft velocity vector are 2.3 Required propulsive acceleration v t v t r r 0 0 v = v exp ; v = v exp (27) r r 0 0 r r 0 0 Having shown that a spiral trajectory in the form of Eq. (6) is actually a particular solution to the equations of motion of a radially propelled spacecraft, the required propulsive acceleration can now be calculated. To that end, recall that a must satisfy Eq. (3) at any time t > t . r 0 Substituting Eqs. (1) and (2) into Eq. (3) yields _ a = r [ + ( 1) ] r 0 r + (34) Fig. 2 Eccentricity vector components and angle !. Spiral trajectories induced by radial thrust with applications to generalized sails 127 _ where f; g are obtained from Eqs. (13) and (25) as of a r^ at a Sun{spacecraft distance equal to r ), and v 2 > 0 is a dimensionless design parameter that de nes 1+2 if 6= f1=2; 0g the type of propulsion system. For example, a value = (35) v 2v t 0 0 > of = 1 models an E-sail [45, 55], = 4=3 describes : exp if = 1=2 r r 0 0 the heliocentric behavior of a magsail with a large loop 2v v 1+4 > r radius [56, 57], while = 2 is consistent with the thrust 0 0 > 1+2 if 6= f1=2; 0g 0 model of a solar sail [20] or a magsail with a huge loop 2v v 2v t > r r 0 0 0 radius [56, 57]. exp if = 1=2 r r In order to make the generalized sail-based spacecraft (36) capable of following the assigned spiral trajectory, the Accordingly, Eq. (34) can be rewritten using Eqs. (13), radial propulsive acceleration of Eq. (38) must comply (25), (35), (36) to obtain the expression of a , a (t) as r r with the required value of Eq. (37). To that end, it is a function of f; r ; v ; v g, viz. 0 r useful to rewrite Eq. (37) so that the dependence of a 0 0 r 2 on the Sun{spacecraft distance r is made explicit, viz. v 2+3 v > 1 + 3 2 > 0 0 1+2 1+2 1+2 + ; > 2+2 r r r 2 2 2 > 0 0 r v > (1 + )v r r 0 0 > 0 0 a = + (39) if 6= f1=2; 0g r > 2+3 2 3 r r " # r a = r v v 4v t v t r r r 0 0 0 > 0 where is the Sun's gravitational parameter. Note that > exp 1 exp > r 0 2 r v r 0 0 > r > Eq. (39) is valid for any 6= 0 and, therefore, it also 2v t > r includes the special case of = 1=2. From Eqs. (38) :+ exp ; if = 1=2 r r and (39), the required characteristic acceleration a can (37) now be written as a function of r as Equation (37) provides the spacecraft propulsive ac- 2+2 2 2 2 r v (1 + )v r r 0 0 0 0 celeration that is required to travel an assigned spiral a = + (40) 2+3 2 3 r r r r r r trajectory with r / . The next section analyzes the problem of creating a generic spiral trajectory using a The characteristic acceleration of a generalized sail generalized sail-based spacecraft [30{32]. must therefore be adjusted with continuity according to Eq. (40) to maintain the spacecraft along the assigned spiral trajectory of parameters f; r ; v ; v g. 0 r 0 0 3 Generalized sail-based spiral trajecto- ries 3.1 Condition for a generalized sail outward radial thrust The results obtained so far are general because they may be applied to any propulsive system that is able to provide A generalized sail can only generate an outward propul- a purely radial thrust. An interesting class of propulsion sive acceleration, which amounts to stating that its char- systems that may be employed for generating spiral tra- acteristic acceleration must be non-negative. For a given jectories is oered by the generalized sails [30{32]. The value of that de nes the spiral type, it is therefore nec- concept of generalized sail is useful for describing, in a essary to investigate the conditions under which a > 0 heliocentric mission scenario, the propulsive acceleration for t > t . of a propellantless propulsion system when the thrust First, consider the value of a when r = r . From c 0 vector is oriented along the outward radial direction and Eq. (40), one has its magnitude depends on a given power of the Sun{ 1 1 2 2 2 (1 + )v r r r v r 0 0 0 0 0 spacecraft distance r. Precisely, the expression of a r^ r a , a (r ) = + c c 0 r r r for a generalized is (41) a r^ = a (38) r c which is positive when where r , 1 au is a reference distance, a > 0 is the c (1 + )v v < ve , (42) 0 0 spacecraft characteristic acceleration (that is, the value 0 128 M. Bassetto, A. A. Quarta, G. Mengali, et al. e sin 0 0 while it is equal to zero when v = ve . In particular, 0 0 s (45) when < 1 or > 0, the inequality of Eq. (42) exists 1 + 1 + e cos e sin 0 0 0 only if jv j 6 (1 + ) =(r ). Figure 3 shows the r 0 maximum values of v that ensure a non-negative value Assuming e 2 (0; 1], Eq. (45) can now be solved with of a . The contour lines in Fig. 3 are reported in the respect to cos , and the result is plane (v ; r ) for = f1;1=2; 1=2; 1g, that is, for a hy- r 0 perbolic spiral, a lituus, a Fermat's (or parabolic) spiral, sign () + 4e (1 + ) cos = (46) and an Archimedean (or arithmetic) spiral, respectively. 2e Note that, when = 1, the initial transversal velocity where sign (2) is the signum function. When e > 1 component must be less than =r , regardless of v ; 0 r (hyperbolic parking orbit), Eq. (45) provides see Eq. (42) and Fig. 3(a). Instead, when = 1=2, the 2 2 > + 4e (1 + ) value of ve is greater than v for any r ; see Fig. 3(b). > 0 r 0 0 0 > 2e < 0 Finally, the grey areas in Figs. 3(c) and 3(d) correspond p if 2e (e e 1) < < 1 cos = 0 0 0 0 to the loci in which jv j > (1 + ) =(r ). r 0 > p > 2 2 > + + 4e (1 + ) Equation (41) allows the designer to nd the initial > 0 ; if 1 6 < 0 2e orbital parameters such that a = 0. To that end, the (47) initial radial velocity component may be written as which implies that no solutions exist if 6 2e (e 0 0 v = e sin (43) r 0 0 e 1) or > 0. When investigating generalized sail-based spiral tra- where e , e(t ) is the parking orbit eccentricity and , 0 0 0 jectories, it is reasonable to con ne the analysis to the (t ) is the initial spacecraft true anomaly. Substitute case of outward heliocentric spirals. In fact, an inward Eq. (43) into Eq. (42) to obtain spiral would entail a negative value of a in some parts 1 + of the propelled trajectory and, as such, it could not be 2 2 ve = 1 + e cos e sin (44) 0 0 0 0 0 0 followed by a generalized sail. The following analysis is also limited to the case of elliptic parking orbits. from which the tangent of the ight path angle when a = 0 is found as Assume now that the spacecraft starts a heliocentric spiral when a = 0. According to Eq. (46), the spacecraft e sin v 0 0 0 r initial true anomaly is given by 1 + e cos ve 0 0 (a) Hyperbolic spiral ( = 1). (b) Lituus ( = 1=2). (c) Fermat's spiral ( = 1=2). (d) Archimedean spiral ( = 1). Fig. 3 Variation of ve with fv ; r g for = f1=2; 1g. r 0 0 0 Spiral trajectories induced by radial thrust with applications to generalized sails 129 " # 2 2 sign () + 4e (1 + ) = e , arccos 0 0 2e (48) Note that a < 0 when 2 (0; e ), while a > 0 when c 0 0 c 0 0 2 (e ; rad). For example, Fig. 4 shows the variation 0 0 of e with for e = f0:1; 0:2; 0:3g. In particular, e < 0 0 0 (a) e = 0:1. =2 rad if 2 (1; 0), e = =2 rad if = 1, while e > =2 rad if < 1 or > 0. When = e , the value of a is non-negative along 0 0 c the generic spiral when 2= (1 + )v r 0 r r r 0 1 + 1 > 0 (49) r r where, bearing in mind Eq. (48), v and r are given by (b) e = 0:2. (p r ) 0 0 2 2 v = e (50) r 0 0 2 p r 2p r = (51) 2 2 sign () + 4e (1 + ) + 2 Equation (49) is met for any r > r when 6 2 or > 1. If, instead, 2 (2;1), Eq. (49) is valid as long as the orbital radius is less than a threshold value, (c) e = 0:3. beyond which the value of the required characteristic acceleration becomes negative. Fig. 5 Sign of a as a function of f; rg when a = 0 and c c e = f0:1; 0:2; 0:3g. In other terms, when 2 (2;1), the required charac- teristic acceleration increases until a maximum is reached, after which it starts decreasing and eventually takes neg- ative values. The value of r at which a becomes negative depends on and on the parking orbit characteristics, as shown in Fig. 5 for e = f0:1; 0:2; 0:3g. In particular, the grey areas in Fig. 5 correspond to the combinations of Fig. 6 Sign of a as a function of f; rg for a = 0 and c c e = e . and r when a < 0, while the contour lines represent the condition a = 0. Note that a < 0 occurs when r is several orders of magnitude greater than r . Thus, the required charac- teristic acceleration is in practise always non-negative, Fig. 4 Variation of e with for e = f0:1; 0:2; 0:3g. 0 0 130 M. Bassetto, A. A. Quarta, G. Mengali, et al. especially when small values of e are handled. For ex- ample, when e = e , the minimum value of r that would yield a negative a is more than 5 orders or magnitude greater than r , and it occurs when ' 1:04; see Fig. 6. 3.2 Required sail performance Having found the conditions under which the general- ized sail is able to provide an outward radial propulsive acceleration, the maximum required characteristic accel- eration can now be determined. First, it is necessary to nd the values of f; g such that a is upper limited, that is, it converges to a nite value as r ! +1. To that end, since no propellantless propulsion systems are known with > 2, assume that 6 2. In this case, the sum of the second and third term in the right-hand side of Eq. (40) reaches zero as r ! +1 when < 2, while it tends to =r as r ! +1 when = 2. There- Fig. 7 Limit of a as r ! +1 as a function of f; g. fore, bearing in mind Eq. (40), the required characteristic acceleration is upper limited as long as The solid black line that delimits the gray region in 2 + 3 the lower left part of Fig. 7 is described by the equation > 0 (52) = 3 + 2= with 2 [2;2=3]. If f; g belong to Figure 7 shows the limits of a as r ! +1 for 2 that locus, then the required value of the characteristic [2; 2] and 2 [0; 2]. When f; g belong to the grey acceleration becomes areas in Fig. 7, the inequality (52) is met and the required 2+2 propulsive acceleration tends to zero. In particular, when (1 + )v r r 0 a = + 1 (53) 2+3 2+3 2+ the limit is 0 , the required characteristic acceleration r r r r rst reaches a positive global maximum, then a negative from which global minimum and nally tends to zero. Instead, when r v > 0 the limit is 0 , a simply reaches the global maximum 0 c > > 1 ; if = 2 r 2 before decreasing to zero without ever having negative > 2+2 values. In the white region, a reaches a positive global (1 + )v r 0 0 lim a = maximum and then it decreases inde nitely. Note that c ; if 2 (2;2=3) r!+1 2+3 Fig. 7 con rms what was shown in the previous section > about the sign of a . In fact, when < 1, a actually v c c > : ; if = 2=3 reaches negative values, even if this occurs when r is 2r (54) several orders of magnitude greater than r . Finally, in the hatched area a ! +1. It is worth noting that the In particular, the sign of the second expression in the hatched area is crossed by a dotted line. When f; g lie right-hand side of Eq. (54) is below the dotted line, the required propulsive acceleration 2+2 (1 + )v r rst reaches a local maximum, then a local minimum and r 0 > 0 < 0; if 2 (2;1) > 2+3 nally tends to +1. Instead, when f; g lie above the > r dotted line, a is a monotonic increasing function of r. For 2+2 c > (1 + )v r r 0 example, when = 1=2, a has both a local maximum = 0; if = 1 (55) 2+3 and a local minimum as long as 6 1:6350. Moreover, r 2+2 when = f0; 1; 4=3g, a becomes a monotonic increasing > 2 (1 + )v r > r 0 > 0; if 2 (1;2=3) function of r when > f0:2395;0:3361;0:4000g, > 2+3 respectively. Spiral trajectories induced by radial thrust with applications to generalized sails 131 Another special case occurs when = 2 and the re- Equation (59) admits an analytical solution when = quired characteristic acceleration becomes f2;1; 2g. In those special cases the result is 2+2 (3 )r (1 + )v r p r 0 0 ; if = f2;1g a = + 1 (56) > c 2+ 2 2 " # 2 > r r r r > (3 )(1 )+ ? p r = 2 2 from which (3 ) (1 ) + 4(2 )(4 ) 8 > r v > 0 > 2(2 ) > > > 1 ; if = 2 : > 2 > r 2 if = 2 >1; if 2 (2;1) (60) lim a = ; if = 1 2 where r!+1 > 3v r +1; if 2 (1; 0) > 0 > , (61) > 2 : ; if 2 (0; 2] It is worth noting that the equation @a =@r = 0 can (57) also be analytically solved when = 1 (see Eq. (58)) Finally, note that the hatched and the white areas are for any value of . In this case, the necessary condition separated from each other by the vertical line of equation for a maximum (i.e., @a =@r = 0) provides = 1, where a ! 0 . In this case, which is consistent (3 )p with a hyperbolic spiral, Eq. (40) reduces to r = (62) a = 1 (58) 2 which is valid as long as < 2 because r must also r r r satisfy the constraint r > p . Substituting Eq. (62) into which is non-negative as long as r > p . 0 0 Eq. (58), the maximum value of a is given by The stationary points of a may be computed (provided (2 ) ? ? that they exist) as a function of f; g by solving the a (r ) = a , (63) 2 3 (3 ) r p equation @a =@r = 0 for a given value of . Assuming c 0 = e (i.e., when v = ve ), the result is Figure 8 shows the maximum value of the required 0 0 0 0 2 characteristic acceleration as a function of fp ; g when 2= (1 + )v r 3 + 2 r r 0 3 + = 1. It is worth noting that a decreases as p increases. + ( 2) + 3 = 0 (59) 4 Case study Consider a generalized sail-based spacecraft that leaves the Earth's sphere of in uence on a parabolic escape trajectory. Assume that the sail deployment occurs when the true anomaly on the heliocentric parking orbit (which therefore coincides with the Earth's orbit) is = e ; see 0 0 Eq. (48). In this scenario, the variation of e with is shown in Fig. 9 for 2 [2; 2], illustrating that, owing to the small Earth's orbit eccentricity, the value of e is very close to =2 rad when jj 2 [1=2; 1]. The ratios r =r that solve Eq. (59) are represented in Fig. 10 for 2 [1; 1], = f0; 4=3; 1; 1:635g, and e = e . Note that, when 2 (0:2395; 0), no stationary point exists for all the considered values of , which implies that Fig. 8 Maximum value of a as a function of fp ; g for a c 0 a always increases with r in that range; see also Fig. 7. hyperbolic spiral ( = 1). 132 M. Bassetto, A. A. Quarta, G. Mengali, et al. of when = f0; 4=3; 1g and 6 1=2 or > 0. This implies that, in those cases, the maximum required characteristic acceleration is substantially independent of the type of spiral that the spacecraft is travelling, but only depends on the speci c propulsion system. Therefore, Eq. (63) may be used to estimate a when =6 1. In the presented case study, p = p ' 0:9997208 au and e = e . Accordingly, Eq. (63) provides >a ' 0:8790 mm/s ; if = 0 < c (64) a ' 1:4829 mm/s ; if = 1 : 2 a ' 1:9320 mm/s ; if = 4=3 Figures 11{14 show the trajectories and the re- quired characteristic accelerations as a function of = Fig. 9 Variation of e with when e = e . 0 0 f1;1=2; 1=2; 1g, when p = p , e = e , and = e . 0 0 0 0 In all cases, as expected, a = 0. By comparing Moreover, consistently with Fig. 7, Fig. 10 shows that a Figs. 11(a), 12(a), 13(a), and 14(a), it is worth not- actually has two stationary points (that is, a positive local ing that the orbital radius grows faster when = 1=2, maximum and a positive local minimum) before going that is, when the spacecraft travels a lituus. Instead, the to +1 when 2 (1;0:2395). It is also worth noting slowest growth of r occurs when = 1=2, that is, when that the second stationary point (i.e., the positive local the spacecraft travels a Fermat's spiral. In fact, after minimum) is much greater than the rst one (i.e., the a 20 year-long journey, the spacecraft orbital radius is positive local maximum) when 2 (1;1=2] and = approximately equal to 8:1684 au when = 1=2, while f0; 1; 4=3g. This implies that the required characteristic it is only 1:7512 au when = 1=2. However, note that acceleration becomes greater than its local maximum the orbital radius grows even faster if 2 (1=2; 0). In after a such large time interval that, basically, the local fact, comparing Eqs. (15) and (26) it may be veri ed maximum may be used as a design parameter of the that, when 2 (1=2; 0): generalized sail. v t Figure 10 shows that the dimensionless distance r =r , 0 1+2 > exp (65) at which a local maximum occurs, is nearly independent (a) < 0 (b) > 0 Fig. 10 Ratio r =r that solves Eq. (59) as a function of f; g for a = 0 and e = e . 0 c 0 0 Spiral trajectories induced by radial thrust with applications to generalized sails 133 (a) Hyperbolic spiral trajectory from Earth's orbit (b) Required a as a function of ft; g. assuming = e = 90 deg. 0 0 Fig. 11 Generalized sail-based hyperbolic spiral. (a) Lituus trajectory from Earth's orbit assuming (b) Required a as a function of ft; g. = e ' 89:04 deg. 0 0 Fig. 12 Generalized sail-based lituus. for all t 2 (0; t ); see Eq. (17). Finally, Figs. 11(b), craft orbital radius is proportional to a given power of max 12(b), 13(b), and 14(b) con rm that the maximum value its angular coordinate. The expressions of the spacecraft of a essentially depends on only and prove that Eq. (63) state variables and the orbital elements of the osculating (which is exact when = 1) provides an accurate orbit have been derived in exact form as a function of estimate of a when 6= 1. time, and the required propulsive acceleration necessary for the spacecraft to track the prescribed spiral trajectory has been calculated a posteriori. 5 Conclusions The analytical results have been specialized to the case In this study, new analytical solutions to the equations of a generalized sail, a propulsion system in which the of motion of a radially propelled spacecraft have been magnitude scales with an assigned power of the Sun{ investigated, with the starting hypothesis that the space- spacecraft distance. The conditions for an outward radial 134 M. Bassetto, A. A. Quarta, G. Mengali, et al. (a) Fermat's spiral trajectory from Earth's orbit as- (b) Required a as a function of ft; g. suming = e ' 92:87 deg. 0 0 Fig. 13 Generalized sail-based Fermat's spiral. (a) Archimedean spiral trajectory from Earth's orbit (b) Required a as a function of ft; g. assuming = e ' 91:91 deg. 0 0 Fig. 14 Generalized sail-based Archimedean spiral. thrust and the required sail performance have been quan- liocentric orbit. In particular, it has been shown that ti ed and thoroughly discussed, demonstrating that the the maximum required characteristic acceleration is on maximum required characteristic acceleration (a sail per- the order of one millimeter per second squared when the formance parameter) is substantially independent of the spacecraft travels some reference spiral trajectories, a type of spiral, but it only depends on the features of feasible value on the basis of many results available in the speci c propulsion system. Moreover, an analytical the literature. approximation of the maximum required characteristic acceleration has been proposed, which is valid for any Funding note spiral and any type of generalized sail. A case study has nally been presented, in which the Open Access funding provided by University of Pisa spacecraft parking orbit coincides with the Earth's he- within the CRUICARE Agreement. Spiral trajectories induced by radial thrust with applications to generalized sails 135 References [16] Boltz, F. W. Orbital motion under continuous tangential thrust. Journal of Guidance, Control, and Dynamics, [1] Markopoulos, N. Analytically exact non-Keplerian mo- 1992, 15(6): 1503{1507. tion for orbital transfers. In: Proceedings of the Astro- [17] Roa, J., Pel aez, J., Senent, J. New analytic solution dynamics Conference, 1994: AIAA-94-3758-CP. with continuous thrust: Generalized logarithmic spirals. [2] Petropoulos, A. E., Sims, J. A. A review of some exact Journal of Guidance, Control, and Dynamics, 2016, solutions to the planar equations of motion of a thrusting 39(10): 2336{2351. spacecraft anastassios. International Symposium Low [18] Bacon, R. H. 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Use of magnetic sails for Mars for space-based geo-eEngineering. Journal of Guidance, exploration missions. In: Proceedings of the 25th Joint Control, and Dynamics, 2010, 33(3): 1017{1020. Propulsion Conference, 1989: AAAA-89-2861. [54] Aliasi, G., Mengali, G., Quarta, A. A. Arti cial Lagrange [39] Andrews, D. G., Zubrin, R. M. Progress in magnetic points for solar sail with electrochromic material panels. sails. In: Proceedings of the AIAA/ASME/SAE/ASEE Journal of Guidance, Control, and Dynamics, 2013, 26th Joint Propulsion Conference, 1990. 36(5): 1544{1550. [40] Andrews, D. G., Zubrin, R. M. Magnetic sails and in- [55] Huo, M. Y., Mengali, G., Quarta, A. A. Electric sail terstellar travel. Journal of the British Interplanetary thrust model from a geometrical perspective. Journal of Society, 1990, 43(6): 265{272. Guidance Control and Dynamics, 2017, 41(3): 735{741. [41] Zubrin, R. M., Andrews, D. G. Magnetic sails and in- [56] Quarta, A. A., Mengali, G., Aliasi, G. Optimal control terplanetary travel. Journal of Spacecraft and Rockets, laws for heliocentric transfers with a magnetic sail. Acta 1991, 28(2): 197{203. Astronautica, 2013, 89: 216{225. [42] Zubrin, R. The use of magnetic sails to escape from low [57] Bassetto, M., Quarta, A. A., Mengali, G. Magnetic sail- earth orbit. In: Proceedings of the 27th Joint Propulsion based displaced non-Keplerian orbits. Aerospace Science Conference, 1991: AIAA-91-3352. and Technology, 2019, 92: 363{372. [43] Niccolai, L., Bassetto, M., Quarta, A. A., Mengali, G. A review of smart dust architecture, dynamics, and mission Marco Bassetto graduated in applications. Progress in Aerospace Sciences, 2019, 106: aerospace engineering at the Uni- 1{14. versity of Pisa in 2016. In 2019, he [44] Quarta, A. A., Mengali, G., Niccolai, L. Smart dust received his Ph.D. degree in civil option for geomagnetic tail exploration. Astrodynamics, and industrial engineering at the 2019, 3(3): 217{230. Department of Civil and Industrial [45] Bassetto, M., Mengali, G., Quarta, A. A. Stability and Engineering of the University of control of spinning electric solar wind sail in heliostation- Pisa. He currently holds a post-doc ary orbit. Journal of Guidance, Control, and Dynamics, scholarship in the same department. His research activity 2019, 42(2): 425{431. focuses on trajectory design and attitude control of Spiral trajectories induced by radial thrust with applications to generalized sails 137 spacecraft propelled with low-thrust propulsion systems Vittorio Cipolla received his Ph.D. such as solar sails and electric solar wind sails. E-mail: degree discussing a thesis on high marco.bassetto@ing.unipi.it. altitude-long endurance UAVs pow- ered by solar energy. Between 2011 Alessandro A. Quarta received his and 2019 he has participated in several Ph.D. degree in aerospace engineering research projects, including \PARSI- from the University of Pisa in 2005, FAL" (PrandtlPlane Architecture for and is currently a professor of ight the Sustainable Improvement of Fu- mechanics at the Department of Civil ture Airplanes) and \PROSIB" (hybrid propulsion sys- and Industrial Engineering of the Uni- tems for xed and rotary wing aircraft). Since 2018 he is versity of Pisa. His main research a research fellow at the University of Pisa, where he also areas include space ight simulation, teaches applied aeroelasticity in M.Sc. course of aerospace spacecraft mission analysis and design, low-thrust tra- engineering. E-mail: vittorio.cipolla@unipi.it. jectory optimization, solar sail, and E-sail dynamics and control. E-mail: a.quarta@ing.unipi.it. Open Access This article is licensed under a Creative Com- mons Attribution 4.0 International License, which permits Giovanni Mengali received his use, sharing, adaptation, distribution and reproduction in any doctor of engineering degree in medium or format, as long as you give appropriate credit to aeronautical engineering in 1989 from the original author(s) and the source, provide a link to the the University of Pisa. Since 1990, Creative Commons licence, and indicate if changes were made. he has been with the Department The images or other third party material in this article are of Aerospace Engineering (now included in the article's Creative Commons licence, unless Department of Civil and Industrial indicated otherwise in a credit line to the material. If material Engineering) of the University of is not included in the article's Creative Commons licence and Pisa, rst as a Ph.D. student, then as an assistant and your intended use is not permitted by statutory regulation or an associate professor. Currently, he is a professor of exceeds the permitted use, you will need to obtain permission space ight mechanics. His main research areas include directly from the copyright holder. spacecraft mission analysis, trajectory optimization, To view a copy of this licence, visit http:// solar sails, electric sails, and aircraft ight dynamics and creativecommons.org/licenses/by/4.0/. control. E-mail: g.mengali@ing.unipi.it.

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