Heliocentric Orbital Repositioning of a Sun-Facing Diffractive Sail with Controlled Binary Metamaterial Arrayed Grating

Abstract

This paper investigates the performance of a spacecraft equipped with a diffractive sail in a heliocentric mission scenario that requires phasing along a prescribed elliptical orbit. The diffractive sail represents an evolution of the more traditional reflective solar sail, which converts solar radiation pressure into thrust using a large reflective surface typically coated with a thin metallic film. In contrast, the diffractive sail proposed by Swartzlander leverages the properties of an advanced metamaterial-based film to generate a net transverse thrust even when the sail is Sun-facing, i.e., in a configuration that can be passively maintained by a suitably designed spacecraft. Specifically, this study considers a sail membrane covered with a set of electro-optically controlled diffractive panels. These panels employ a (controlled) binary metamaterial arrayed grating to steer the direction of photons exiting the diffractive film. This control technique has recently been applied to achieve a circle-to-circle interplanetary transfer using a Sun-facing diffractive sail. In this work, an optimal control law is employed to execute a rapid phasing maneuver along an elliptical heliocentric orbit with specified characteristics, such as those of Earth and Mercury. The analysis also includes a limiting case involving a circular heliocentric orbit. For this latter scenario, a simplified and elegant control law is proposed based on a linearized form of the equations of motion to describe the heliocentric dynamics of the diffractive sail-based spacecraft during the phasing maneuver.

1. Introduction

The diffractive sail [1,2] can be considered a sort of evolution of the more classical reflective solar sail (RSS), which, as is well known, converts the Sun’s solar radiation pressure into a propulsive thrust through the aid of a large (and sufficiently light) reflective surface, thus making a spacecraft propelled by an RSS resemble a large orbiting mirror that moves in space without consuming propellant [3]. In the context of an RSS and other currently available propellantless propulsion systems as, for example, the Electric Solar Wind Sail [4,5] or the Magnetic Sail [6,7], the recent review article by Berthet et al. [8] offers an interesting analysis of the technologies available today and a comprehensive discussion of their possible advances in the near future. The useful review articles by Fu et al. [9] or by Gong and Macdonald [10], while slightly older than Ref. [8], instead focus exclusively on the potentialities of RSSs, allowing the interested reader to also appreciate the evolution of this fascinating space propulsion concept over the decades, starting from the pioneering works of Tsiolkovsky [11] and Tsander [12] to arrive at the recent space missions that have effectively tested this type of spacecraft thruster in space [13,14]. A historical perspective on the development of that propellantless propulsion system can also be found in Wright’s now classic book [15], which places particular emphasis on the rapid evolution of solar sail technology due to JPL’s mid-1970s project (unfortunately, only at the proposal stage) to launch an RSS-based mission to Halley’s comet.

Unlike a typical and well-known RSS, the diffractive sail, proposed by Swartzlander [2] in his seminal 2017 work, exploits the behavior of an advanced metamaterial that constitutes the film covering the membrane of the sail to obtain a thrust vector whose properties cannot be obtained from a classical RSS, which in fact passively reflects the photons coming from the Sun [16]. In this context, the recent work of Thompson et al. [17] compares the flight performance and the design of a (more general) transmissive sail with that of a typical RSS under realistic conditions. The original mathematical model proposed by Swartzlander in 2017 [2] has recently been refined in a 2022 paper by the same author [18], which analyzes in detail the momentum transfer efficiency, obtainable by a diffractive sail that maintains a fixed attitude in an orbital reference frame, as a function of the characteristics of the sail film’s prism grating. Moreover, a simple thrust vector model for a diffractive sail is also illustrated in [1], while the work by Srivastava et al. [19] extends the diffractive sail concept by studying an advanced (and passively stable) laser-driven sail. This latter solution, which can actually be considered an evolution of the diffractive sail concept, could constitute a valid alternative to more conventional propulsion systems for the design of advanced scientific missions such as those, for example, requiring reaching the boundaries of the Solar System [20] or even traveling into interstellar space [21]. In this regard, the works of Srivastava et al. [22] and Chu et al. [23] provide a recent snapshot of the current state of research in this rather advanced propulsion system concept, while the paper by Davoyan et al. [24] analyzes the required characteristics of the material of the sail’s membrane in a set of high-energy mission scenarios.

In the field of a diffractive sail-propelled spacecraft (in the rest of the paper indicated as “diffractive sailcraft”), as widely discussed in Refs. [2,18], among the attractive properties of such propulsion system, one of the most interesting is the possibility of generating a transverse (i.e., an off-radial component of the) thrust even when the nominal plane of the membrane is perpendicular to the line joining the Sun to the space vehicle. In that case, the transverse component of the thrust vector has a fixed direction in a body reference frame. This specific configuration, commonly called “Sun-facing”, can be maintained during heliocentric flight in a passive manner. Indeed, according to McInnes [25], this useful characteristic can be obtained, appropriately designing the shape of the sail structure and carefully choosing the distribution of masses in order to have a suitable position of the center of mass relative to the center of pressure, i.e., designing a slightly conical sail with the apex directed towards the Sun. The presence of a transverse component of the thrust vector when the attitude is Sun-facing is a peculiarity of the diffractive sailcraft, since an RSS produces exclusively an outward radial thrust in this configuration. The RSS, for example, exploits a Sun-facing attitude in a heliocentric scenario to reduce the local value of the Sun’s gravitational acceleration and, therefore, to generate non-Keplerian orbits, which can however be described still through classical conic sections [25]. On the other hand, an RSS must vary the orientation of its nominal plane with respect to an orbital reference frame and, therefore, must perform an appropriate attitude maneuver [26], both to generate a transverse thrust component of appreciable magnitude and to direct the sail-induced thrust vector towards a specific direction, which is usually obtained through the solution of a suitable optimal control problem [27]. A Sun-facing diffractive sail, instead, is able to change the azimuthal direction of the transverse component of the local thrust vector by simply rotating the sail nominal plane around the Sun-sailcraft (radial) direction. This interesting property has been exploited by Dubill et al. [28] to investigate the transfer performance of a diffractive sailcraft in obtaining a large change in the inclination of a given heliocentric orbit, and by Quarta et al. [29] to obtain the optimal flight time in a typical interplanetary mission that requires the transfer between two assigned Keplerian orbits.

However, in the context of a two-dimensional mission scenario involving a diffractive sailcraft, Swartzlander [2] proposed another simple and elegant strategy to modify (i.e., reverse in this two-dimensional case) the verse of the transverse component of the thrust vector without rotating the sail nominal plane along the radial line. More precisely, the method illustrated in section 3.B of [2] is based on the employment of a set of electro-optically controlled diffractive panels, which use a binary metamaterial arrayed grating to vary the direction of photons leaving the membrane’s film. This solution allows for rapidly “switching” between two opposite, but equal in modulus, diffraction orders [2]. From a trajectory design point of view, this type of binary characteristic of the (controlled) diffractive film can be used to change the direction of the transverse component of the sail-induced thrust by $180\mathrm{deg}$, simply by applying an appropriate voltage to the diffractive panels. In broad terms, this ingenious design concept recalls that used in the Japanese solar sail IKAROS [30], that is, the first RSS deployed in an interplanetary environment, to achieve attitude control via a set of electrochromic panels installed on the edge of the sail’s square membrane. Starting from the work of Swartzlander [2], the interesting paper by Chu et al. [31] extends the concept of controlling the transverse component of the diffractive sailcraft’s thrust using a set of liquid crystal optical phased arrays, which are also able to generate suitable (and adjustable) torques without employing other devices. The model proposed in [31], although more complex than the one originally discussed in [2], is elegant in its analytical form and can be potentially used to design the trajectory of future generations of advanced, electro-optically controlled diffractive sailcraft.

In the context of the Swartzlander’s original idea [2] and assuming a two-dimensional heliocentric scenario, the in-flight control of the transverse direction of the thrust vector occurs by simply changing the electrical voltage applied to the panels, that is, by using a single (binary) scalar control term. This specific control technique was employed 2 years ago by the author [32] to investigate the performance of a Sun-facing electro-optically controlled diffractive sailcraft in a classical circle-to-circle orbit transfer. The results illustrated in [32], which also proposes an implicit and compact form of the optimal control law to design the heliocentric transfer trajectory, demonstrate the effectiveness of a solution that combines the concept of a diffractive sailcraft with that of electro-optically controlled panels. Furthermore, Ref. [32] also discussed an analytical model of the thrust vector as a function of the binary control parameter, which is general and can be used to describe the heliocentric dynamics of the diffractive sailcraft in a wide range of mission applications. The latter analytical thrust model is used in this work to study the performance of a diffractive sailcraft with a Sun-facing attitude in a heliocentric mission scenario different from the one analyzed in [32].

More precisely, the aim of this paper is to investigate the performance of such an advanced propulsion system in a mission scenario involving the orbital repositioning of a diffractive sailcraft along an elliptical heliocentric orbit of assigned characteristics, that is, in a mission scenario that is consistent with a heliocentric phasing maneuver, as described in the textbook of Curtis [33]. Such a scenario is of scientific interest and has already been investigated in the literature, because it allows, for example, for moving spacecraft on the same heliocentric orbit of the Earth in a different angular position with respect to that of the planet, in such a way as to observe the Sun simultaneously from different observation points. This concept of phasing along the Earth’s heliocentric orbit was used, for example, in NASA’s STEREO mission [34] launched in 2006 to obtain a stereoscopic image of the Sun [35] through two space probes moving ahead of and behind our planet with a phase angle increasing with time, thanks also to a suitable succession of lunar gravity assist maneuvers. On the other hand, a phasing maneuver along the Earth’s heliocentric orbit can be used in a preliminary mission design to approximate the transfer performance towards the Sun–Earth triangular Lagrange points [36] that are located approximately $60\mathrm{deg}$ ahead or behind the planet. For this reason, the author has already studied the performance of a spacecraft propelled by a propellantless propulsion system [37,38] or a CubeSat equipped with a solar electric thruster [39] in a scenario that includes the azimuthal repositioning of a deep space probe along a circular orbit of radius equal to one astronomical unit. The latter, in fact, approximates the actual heliocentric orbit of the Earth and allows for simplifying the mathematical model for the study of the phasing maneuver.

In this paper, instead, the performances of a Sun-facing diffractive sailcraft, equipped with electro-optically controlled panels, are studied, considering a generic heliocentric elliptical orbit, so that the mathematical model presented in Section 2 is general and can be applied to a closed (heliocentric) Keplerian orbit of assigned semimajor axis and eccentricity. In particular, following the usual approach in the preliminary design of missions involving propellantless thrusters [40], the method presented in the next section determines the transfer performance of the diffractive sailcraft in an optimization framework that allows for minimizing the total flight time [41]. The results of the model are then applied to two different scenarios in which the orbital parameters of the reference elliptical orbit are consistent with those of the heliocentric orbit of the Earth and Mercury, as reported in Section 3. Furthermore, the limiting case of a circular reference orbit is also studied in order to obtain a set of numerical results in this important special case of a closed Keplerian orbit that can be used to rapidly approximate the Earth-based mission scenario. Finally, this last circular case is used to define a simplified, non-optimal control law, which allows for completing the phasing maneuver with an analytical form of the panels’ switching strategy. The effectiveness of the analytical control law is investigated in Section 4, using a linearized formulation of the heliocentric dynamics of the spacecraft [42], which enables the flight time to be computed in an open loop as a function of the design parameters of the diffractive sailcraft as its characteristic acceleration.

Consequently, the innovative contribution of this paper to the literature on Sun-facing diffractive sailcraft is essentially twofold. First, it investigates the performance of such sailcraft in the relevant mission scenario of orbital phasing along a generic elliptical orbit within a heliocentric framework. Second, it introduces a simplified control law that enables a rapid estimation of mission performance—specifically, the flight time—in cases where a linearized spacecraft’s heliocentric dynamics around a circular orbit are adopted. The proposed analysis is confined to a Sun-facing attitude; however, an extension to the more general case, in which the sailcraft’s attitude varies during interplanetary flight, is straightforward. The results related to these two aspects of mission planning are further discussed in Section 5, which presents the final remarks and concludes the paper.

2. Thrust Vector Description and Mathematical Model

This section is substantially divided into three parts. The first one concerns the description of the thrust vector of the Sun-facing diffractive sailcraft and the definition of the single (binary) scalar control variable. In this context, the paper resorts to the results of the literature limiting the discussion to the description of the vectorial expression that allows for writing the propulsive acceleration of a diffractive sailcraft equipped with a set of electro-optically controlled panels. Additional comments regarding the origin of that simplified thrust model can be found in the recent work of the author [32] and in the seminal paper by Swartzlander [2].

The obtained thrust vector’s expression is then used in the second part of the section to write the nonlinear equations of motion of the Sun-facing diffractive sailcraft in a two-dimensional, heliocentric, mission scenario. In this case, a classical polar reference frame was used, while the boundary conditions were calculated considering the non-zero eccentricity of the elliptical reference orbit. The mathematical model used in that part of the section to describe the heliocentric dynamics of the diffractive sailcraft is consistent with the approach commonly employed in the preliminary design of interplanetary missions [43]. In particular, this method allows the designer to explore a wide range of possible transfer trajectories with reduced computational effort. Naturally, a subsequent and more refined mission analysis requires a detailed description of the spacecraft dynamics, in which, for example, both perturbations and possible uncertainties in the thrust vector are taken into account.

The third and last part of this section instead describes the mathematical model used to determine the optimal control law (in implicit form) and the transfer trajectory in an assigned phasing maneuver. In that context, indeed, a typical optimization model has been employed, as usually happens in problems of this type, in order to determine the nominal performances (i.e., in the absence of uncertainties and perturbations) of the Sun-facing diffractive sailcraft. Additionally, in this case, the discussion is reduced to the essential part of the model in order to allow the reader to have the entire mathematical model available for the reproduction of the numerical results described in Section 3. Being an optimization approach well known in the literature [44] and largely validated in many mission scenarios [45], the last part of this section omits specific comments on the meaning of some steps, referring to recent bibliographical references [32].

The proposed approach, in which the sailcraft orientation is maintained Sun-facing throughout the entire interplanetary transfer, can be extended, with minor modifications to the mathematical model, to the more general case of a piecewise constant steering law, where the sail’s attitude assumes a finite set of discrete orientations. In this context, the recent work by Bai et al. [46] analyzes the optimal transfer of a spacecraft propelled by a conventional continuous-thrust propulsion system, employing an elegant analytical method that enables the sensitivity matrix to be derived in closed form, thereby reducing computational effort during the trajectory optimization.

2.1. Analytical Thrust Vector Model of Sun-Facing Diffractive Sailcraft

Consider a two-dimensional heliocentric mission scenario in which the Sun-facing diffractive sailcraft moves in the plane of the reference elliptical orbit. The vehicle orbits in the interplanetary space under the effect of the gravitational attraction of the Sun and the thrust provided by the propellantless propulsion system. The latter has an array of electro-optically controlled panels, formed from a grid of binary metamaterial. In this case, paralleling the approach proposed by Swartzlander [2] and using the analytical results obtained in [32], the propulsive acceleration vector $\mathit{a}$ can be expressed as a function of the solar distance r, which is appropriately dimensionless by means of the reference distance $r_{\oplus }\triangleq 1\mathrm{AU}$; the Sun-sailcraft radial unit vector $\widehat{\mathit{r}}$; the transverse unit vector $\widehat{\mathit{t}}$ in the direction of the sailcraft’s heliocentric velocity vector $\mathit{v}$, i.e., $\mathit{v}·\widehat{\mathit{t}}>0$ during all the interplanetary flight; the dimensionless binary control term $\tau \in \{-1,1\}$, which allows the verse of the transverse component of the thrust to be changed; and the performance parameter $a_c$ commonly indicating as “characteristic acceleration”, in analogy with the case of an RSS-based spacecraft. In particular, $a_c$ is the maximum value of $\parallel \mathit{a}\parallel$ when the sail solar distance is equal to $r_{\oplus }$. Therefore, according to [32], the compact form of the propulsive acceleration vector $\mathit{a}$ of a Sun-facing diffractive sailcraft is

$$\mathit{a}={\displaystyle \frac{a_c}{\sqrt{2}}}{\left({\displaystyle \frac{r_{\oplus }}{r}}\right)}^2\left(\widehat{\mathit{r}}-\tau \widehat{\mathit{t}}\right)$$

(1)

The latter equation is valid when the grating momentum unit vector, which is fixed in a body reference frame, is aligned with the direction of $\widehat{\mathit{t}}$. In this case, during the interplanetary flight, the direction of the vector $\mathit{a}$ remains in the plane of the reference (parking) elliptical orbit, and the motion of the Sun-facing diffractive sailcraft is two-dimensional. Note that ${\tau }^2\equiv 1$ so that when $r=r_{\oplus }$, one has $\parallel \mathit{a}\parallel =a_c$, as expected. In particular, $\tau =1$ indicates that the transverse component of the thrust is opposite to the direction of the vehicle’s heliocentric velocity vector $\mathit{v}$, while the case of $\tau =-1$ is consistent with an increasing in the sailcraft’s kinetic energy. However, as highlighted in [32], the choice of the sign of the binary control variable $\tau$ is rather arbitrary, since it depends on the direction of the grating momentum unit vector whose position in the membrane plane is fixed. The important aspect that emerges from Equation (1) is that the binary variable $\tau$ is actually able to vary the verse of the transverse component of the thrust vector, as sketched in Figure 1.

Furthermore, Equation (1) indicates that the magnitude of $\mathit{a}$ depends linearly on the characteristic acceleration and varies inversely quadratically with the solar distance. In other words, once the value of $a_c$ and r have been assigned, the magnitude of the propulsive acceleration vector [$\parallel \mathit{a}\parallel =a_c{\left(r_{\oplus }/r\right)}^2$] is fixed as are its radial [$\mathit{a}·\widehat{\mathit{r}}=(a_c/\sqrt{2}){\left(r_{\oplus }/r\right)}^2$] and transverse [$\mathit{a}·\widehat{\mathit{t}}=-\tau (a_c/\sqrt{2}){\left(r_{\oplus }/r\right)}^2$] components. In particular, the verse of the transverse component of $\mathit{a}$ depends on the value of $\tau$. This aspect clearly indicates a reduced maneuverability of the Sun-facing diffractive sailcraft ultimately linked to the choice of having a fixed orientation during flight and, therefore, to exploit the possibility of passively maintaining this specific attitude. A possible improvement of the maneuverability could be obtained by abandoning the Sun-facing attitude hypothesis and, perhaps, employing a particular model of diffractive sail with a Littrow transmission grating [2], whose potential in an orbit-to-orbit interplanetary transfer has been recently investigated in [47].

As for the value of the characteristic acceleration, in this work, we consider a low-performance diffractive sailcraft, so the value of this important performance parameter is assumed to vary in the range $a_c\in [0.06,0.12]\mathrm{mm}/{\mathrm{s}}^2$. This specific interval was chosen, taking into account the actual performances of two RSS-propelled interplanetary spacecraft recently designed or even already launched. In fact, the value of $0.06\mathrm{mm}/{\mathrm{s}}^2$ is consistent with the characteristic acceleration of the (ill-fated) NASA’s scientific mission Near-Earth Asteroid Scout (NEA Scout) [48], which was lost shortly after launch in November 2022. The upper value of the range, $0.12\mathrm{mm}/{\mathrm{s}}^2$, refers instead to the NASA’s Solar Cruiser demonstration mission [49], which was recently canceled at an advanced stage of planning. It should be noted, however, that these performance values refer to a more classic RSS and therefore consider the current state of the technology required for the construction of a reflective sail. A diffractive sailcraft could require different construction solutions that would imply, for the same exposed surface, a membrane of different mass compared with an RSS. A discussion in this sense is certainly beyond the scope of this work and is currently analyzed in the context of dedicated studies [50].

Considering this range of variation for the characteristic acceleration, and taking a solar distance $r\in [0.3,1.2]\mathrm{AU}$, Equation (1) gives the graph reported in Figure 2, which shows the variation of the propulsive acceleration magnitude with r and $a_c$. In particular, the red (or green) area in that figure indicates the range of solar distance that is obtained if one considers a Sun-facing diffractive sailcraft moving in the vicinity of the elliptical orbit of Mercury (or of the Earth), that is, of the two reference elliptical orbits that will be considered in the numerical simulations illustrated in Section 3. For example, Figure 2 shows that a low-performance diffractive sailcraft with $a_c\simeq 0.6\mathrm{mm}/{\mathrm{s}}^2$ gives a propulsive acceleration magnitude of roughly $0.3\mathrm{mm}/{\mathrm{s}}^2-0.6\mathrm{mm}/{\mathrm{s}}^2$ in a Mercury-based mission scenario.

2.2. Heliocentric Dynamics of a Sun-Facing Diffractive Sailcraft

The heliocentric two-dimensional motion of the Sun-facing diffractive sailcraft can be conveniently studied by introducing a typical polar reference frame $\mathcal{T}$ of unit vectors $\{\widehat{\mathit{r}},\widehat{\mathit{t}}\}$, in which the polar angle $\theta$ is measured from the direction of the eccentricity vector of the reference elliptical orbit, that is, from the line connecting the Sun and the perihelion point of that orbit. The latter is a Keplerian (closed) trajectory with a given value of both the semimajor axis $a_0$ and the eccentricity $e_0$ so that the semilatus rectum is $p_0=a_0(1-e_0^2)$. In the special case of circular reference orbit (i.e., when $e_0=0$ and $p_0\equiv a_0$), the polar angle is measured from the line connecting the Sun and the sailcraft at the beginning of the flight, i.e., at the initial time $t=t_0\triangleq 0$. At that time, the state of the space vehicle is given by the following four scalar terms:

$$\begin{array}{c}r\left(t_0\right)={\displaystyle \frac{p_0}{1+e_{0}cos{\nu }_0}},\theta \left(t_0\right)={\nu }_0,v_r\left(t_0\right)=\sqrt{{\displaystyle \frac{{\mu }_{\odot }}{p_0}}}{e}_{0}sin{\nu }_0,\hfill \\ \hfill \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{v}_t\left(t_0\right)=\sqrt{{\displaystyle \frac{{\mu }_{\odot }}{p_0}}}\left(1+e_{0}cos{\nu }_0\right)\end{array}$$

(2)

where ${\nu }_0$ is the true anomaly of the sailcraft along the reference orbit at time $t_0$, $v_r$ (or $v_t$) is the radial (or transverse) component of the heliocentric velocity vector $\mathit{v}$, and ${\mu }_{\odot }$ is the gravitational parameter of the primary body (i.e., the Sun). The initial conditions reported in the previous equation and the spacecraft dynamics can be more conveniently expressed in a dimensionless form by introducing the following terms:

$$\tilde{r}\triangleq {\displaystyle \frac{r}{p_0}},{\tilde{v}}_r\triangleq {\displaystyle \frac{v_r}{\sqrt{{\mu }_{\odot }/p_0}}},{\tilde{v}}_t\triangleq {\displaystyle \frac{v_t}{\sqrt{{\mu }_{\odot }/p_0}}},\tilde{t}\triangleq {\displaystyle \frac{t}{\sqrt{p_0^3/{\mu }_{\odot }}}},{\tilde{a}}_c\triangleq {\displaystyle \frac{a_c}{{\mu }_{\odot }/p_0^2}}$$

(3)

so that Equation (2) becomes

$$\tilde{r}\left({\tilde{t}}_0\right)={\displaystyle \frac{1}{1+e_{0}cos{\nu }_0}},\theta \left({\tilde{t}}_0\right)={\nu }_0,{\tilde{v}}_r\left({\tilde{t}}_0\right)=e_{0}sin{\nu }_0,{\tilde{v}}_t\left({\tilde{t}}_0\right)=1+e_{0}cos{\nu }_0$$

(4)

Using the terms in Equation (3) and taking Equation (1) into account, the dimensionless form of the propulsive acceleration vector $\tilde{\mathit{a}}$ is simply

$$\tilde{\mathit{a}}={\displaystyle \frac{{\tilde{a}}_c}{\sqrt{2}}}{\left({\displaystyle \frac{{\tilde{r}}_{\oplus }}{\tilde{r}}}\right)}^2\left(\widehat{\mathit{r}}-\tau \widehat{\mathit{t}}\right)$$

(5)

where ${\tilde{r}}_{\oplus }\triangleq r_{\oplus }/p_0$. The expression of $\tilde{\mathit{a}}$ given by the previous equation is used to write the equations of motion of the Sun-facing diffractive sailcraft in $\mathcal{T}$, and the result is

$${\tilde{r}}^{\prime }={\tilde{v}}_r,{\theta }^{\prime }={\displaystyle \frac{{\tilde{v}}_t}{\tilde{r}}},{\tilde{v}}_r^{\prime }={\displaystyle \frac{{\tilde{v}}_t^2}{\tilde{r}}}+{\displaystyle \frac{{\tilde{a}}_c{\tilde{r}}_{\oplus }^2-\sqrt{2}}{\sqrt{2}{\tilde{r}}^2}},{\tilde{v}}_t^{\prime }=-{\displaystyle \frac{{\tilde{v}}_r{\tilde{v}}_t}{\tilde{r}}}-{\displaystyle \frac{\tau {\tilde{a}}_c{\tilde{r}}_{\oplus }^2}{\sqrt{2}{\tilde{r}}^2}}$$

(6)

where the apex indicates the derivative with respect to the dimensionless time $\tilde{t}$ defined in Equation (3).

In the previous equation, the single control term is the binary parameter $\tau$, whose variation with $\tilde{t}$ is chosen in order to obtain an assigned phasing angle $\mathrm{\Delta }\varphi \ne 0$ with the minimum flight time $\mathrm{\Delta }\tilde{t}={\tilde{t}}_f-{\tilde{t}}_0\equiv {\tilde{t}}_f$, where ${\tilde{t}}_f$ is the final time. In particular, the phasing angle $\mathrm{\Delta }\varphi$ indicates the variation of the angular position along the elliptical reference orbit of the sailcraft, at the end of the maneuver, with respect to a virtual point whose initial position coincides with that of the sailcraft and which continues to travel along the reference (Keplerian) orbit in the time interval $\mathrm{\Delta }\tilde{t}$. More precisely, a value $\mathrm{\Delta }\varphi >0$ (or $\mathrm{\Delta }\varphi <0$) refers to a phasing ahead (or behind) the virtual point. The latter may represent, for example, a celestial body such as a planet, or another spacecraft with respect to which the diffractive sailcraft changes angular position along the same orbit. In particular, the polar angle ${\theta }_{p_f}$ of the virtual point at the end of the phasing maneuver (i.e., at time $\tilde{t}={\tilde{t}}_f$) can be obtained by solving a classical Kepler problem [51] or by integrating the equations of motion (6) enforcing the condition ${\tilde{a}}_c=0$ to obtain a Keplerian heliocentric motion.

Bearing in mind that, (1) at the end of the phasing maneuver, the osculating orbit of the diffractive sailcraft coincides (again) with the reference elliptical orbit, and (2) the polar angle $\theta$ is measured from the (fixed) apse line of the reference orbit, the dimensionless boundary conditions at time $\tilde{t}={\tilde{t}}_f$ are

$$\tilde{r}\left({\tilde{t}}_f\right)={\displaystyle \frac{1}{1+e_{0}cos{\nu }_f}},\theta \left({\tilde{t}}_f\right)={\theta }_{p_f}+\mathrm{\Delta }\varphi ,{\tilde{v}}_r\left({\tilde{t}}_f\right)=e_{0}sin{\nu }_f,{\tilde{v}}_t\left({\tilde{t}}_f\right)=1+e_{0}cos{\nu }_f$$

(7)

where ${\nu }_f=\mathrm{mod}\left[\theta \left({\tilde{t}}_f\right),2\pi \right]$ is the true anomaly of the Sun-facing diffractive sailcraft along the elliptical reference orbit at the end of the phasing maneuver. The value of ${\nu }_f\in [0,2\pi ]\mathrm{rad}$ is calculated through the modulo operation. The (dimensionless) temporal variation of the binary control term $\tau$ is obtained through the optimization approach briefly described in the next subsection.

2.3. Notes on the Control Law and the Sailcraft’s Trajectory Design

For an assigned value of the dimensionless pair $\{{\tilde{a}}_c,\mathrm{\Delta }t\}$, the control law $\tau =\tau \left(\tilde{t}\right)$ is selected to minimize the flight time $\mathrm{\Delta }\tilde{t}$ required to reach the final boundary conditions reported in Equation (7). In this regard, the minimization of the time of flight (or, equivalently, the maximization of the dimensionless performance index $J\triangleq -{\tilde{t}}_f$) has been obtained using an indirect method [52] and following the general approach illustrated by Colasurdo and Casalino [44], to which the interested reader is referred for useful comments on the procedure illustrated below. In particular, the method used in this part of the section is general, in the sense that it can be applied to any elliptical orbit with a prescribed eccentricity $e_0$. Within this framework, the special case of a circular reference orbit (i.e., $e_0=0$) allows for a simplified form of the control law, as discussed later in this paper.

The employed method, based on the calculus of variations, introduces the dimensionless variables $\{{\lambda }_{\tilde{r}},{\lambda }_{\theta },{\lambda }_{{\tilde{v}}_r},{\lambda }_{{\tilde{v}}_t}\}$ adjoint to the states $\{\tilde{r},\theta ,{\tilde{v}}_r,{\tilde{v}}_t\}$ to write the dimensionless Hamiltonian function $\mathcal{H}$ defined as

$$\mathcal{H}={\lambda }_{\tilde{r}}{\tilde{v}}_r+{\displaystyle \frac{{\lambda }_{\theta }{\tilde{v}}_t}{\tilde{r}}}+{\displaystyle \frac{{\lambda }_{{\tilde{v}}_r}{\tilde{v}}_t^2}{\tilde{r}}}+{\displaystyle \frac{{\lambda }_{{\tilde{v}}_r}({\tilde{a}}_c{\tilde{r}}_{\oplus }^2-\sqrt{2})}{\sqrt{2}{\tilde{r}}^2}}-{\displaystyle \frac{{\lambda }_{{\tilde{v}}_t}{\tilde{v}}_r{\tilde{v}}_t}{\tilde{r}}}-{\displaystyle \frac{\tau {\lambda }_{{\tilde{v}}_t}{\tilde{a}}_c{\tilde{r}}_{\oplus }^2}{\sqrt{2}{\tilde{r}}^2}}$$

(8)

which gives the Euler–Lagrange equations

$${\lambda }_{\tilde{r}}^{\prime }=-{\displaystyle \frac{\partial \mathcal{H}}{\partial \tilde{r}}},{\lambda }_{\theta }^{\prime }=-{\displaystyle \frac{\partial \mathcal{H}}{\partial \theta }},{\lambda }_{{\tilde{v}}_r}^{\prime }=-{\displaystyle \frac{\partial \mathcal{H}}{\partial {\tilde{v}}_r}},{\lambda }_{{\tilde{v}}_t}^{\prime }=-{\displaystyle \frac{\partial \mathcal{H}}{\partial {\tilde{v}}_t}}$$

(9)

whose explicit form is

$$\begin{array}{cc}{\lambda }_{\tilde{r}}^{\prime }\hfill & ={\displaystyle \frac{{\tilde{v}}_t({\lambda }_{\theta }-{\lambda }_{{\tilde{v}}_t}{\tilde{v}}_r+{\lambda }_{{\tilde{v}}_r}{\tilde{v}}_t)}{{\tilde{r}}^2}}-{\displaystyle \frac{2{\lambda }_{{\tilde{v}}_r}-\sqrt{2}{\tilde{a}}_c{\lambda }_{{\tilde{v}}_r}{\tilde{r}}_{\oplus }^2+\sqrt{2}\tau {\tilde{a}}_c{\lambda }_{{\tilde{v}}_t}{\tilde{r}}_{\oplus }^2}{{\tilde{r}}^3}}\hfill \end{array}$$

(10)

$$\begin{array}{cc}{\lambda }_{\theta }^{\prime }\hfill & =0\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\lambda }_{{\tilde{v}}_r}^{\prime }& ={\displaystyle \frac{{\lambda }_{{\tilde{v}}_t}{\tilde{v}}_t}{\tilde{r}}}-{\lambda }_{\tilde{r}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\lambda }_{{\tilde{v}}_t}^{\prime }& =-{\displaystyle \frac{{\lambda }_{\theta }-{\lambda }_{{\tilde{v}}_t}{\tilde{v}}_r+2{\lambda }_{{\tilde{v}}_r}{\tilde{v}}_t}{\tilde{r}}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hfill \end{array}$$

(13)

The expression of $\mathcal{H}$ is used to obtain the implicit form of the optimal control law through the Pontryagin maximum principle (PMP) [53], and the result is a very simple relation that is consistent with the discussion in [32], i.e.,

$$\tau =-\mathrm{sign}\left({\lambda }_{{\tilde{v}}_t}\right)$$

(14)

where $\mathrm{sign}\left(\square \right)$ is the signum function. The latter equation, in fact, gives the value of the binary control term that maximizes the local value of the Hamiltonian function. Note that Equation (10) indicates that ${\lambda }_{\theta }$ is a constant of motion, so keeping in mind that the Euler–Lagrange equations are invariant under a scaling factor, the constant value of ${\lambda }_{\theta }$ is taken as a scaling factor, i.e., $|{\lambda }_{\theta }|=1$. The minimum flight time ${\tilde{t}}_f$ and the initial value of the remaining three adjoint variables $\{{\lambda }_{\tilde{r}}\left({\tilde{t}}_0\right),{\lambda }_{{\tilde{v}}_r}\left({\tilde{t}}_0\right),{\lambda }_{{\tilde{v}}_t}\left({\tilde{t}}_0\right)\}$ are obtained by enforcing the four final boundary conditions summarized in Equation (7). In this regard, a typical single-shooting procedure has been employed with a trial-and-error procedure to estimate the first guess solution, because the analytical method recently proposed in [54] fails in this specific scenario.

3. Results of Numerical Simulations and Case Studies

The mathematical model illustrated in Section 2 is employed to evaluate the phasing maneuver’s performance of a Sun-facing diffractive sailcraft of assigned characteristic acceleration. In this section, three mission scenarios are investigated, namely, (1) a Mercury-based case in which the reference elliptical orbit has a semimajor axis $a_0=0.3870\mathrm{AU}$ and an eccentricity $e_0=0.2056$, so that the semilatus rectum is $p_0=0.3706\mathrm{AU}$; (2) an Earth-based case in which $a_0=1\mathrm{AU}$, $e_0=0.0167$, and $p_0=0.9997\mathrm{AU}$; and (3) a circular orbit case where $e_0=0$ and $a_0\equiv p_0=1\mathrm{AU}$. The latter case is consistent, for example, with a simplified form of the Earth-based mission scenario.

3.1. Mercury-Based Scenario

This specific mission scenario, thanks to the non-negligible value of the eccentricity $e_0$, allows the reader to appreciate the impact of the initial true anomaly ${\nu }_0$, i.e., the angular position of the Sun-facing diffractive sailcraft along the reference elliptical orbit at the beginning of the phasing maneuver, on the minimum value of the flight time $\mathrm{\Delta }t$.

In this regard, Figure 3 shows the value of $\mathrm{\Delta }t$ as a function of ${\nu }_0$, when $\mathrm{\Delta }\varphi \in [-8,8]\mathrm{deg}$, and the value of the characteristic acceleration is $0.1\mathrm{mm}/{\mathrm{s}}^2$, i.e., a value in the range of $a_c$ considered in this work; see Section 2.1. The figure clearly shows how, for a fixed value of $\mathrm{\Delta }\varphi$, there is a notable dependence of the minimum flight time on the initial true anomaly ${\nu }_0$. This characteristic is evident even for small values of the phasing angle since, for example, there are differences in flight time of about $20\mathrm{days}$ when $\mathrm{\Delta }\varphi =5\mathrm{deg}$ starting from different points of the reference elliptical orbit; see Figure 3a. On the other hand, as expected, for a given value of $\left|\mathrm{\Delta }\varphi \right|$, a phasing ahead requires a longer flight time than a phasing behind, since, in the second case, the displacement of the virtual point on its reference (elliptical) Keplerian orbit is exploited.

The optimal transfer strategy in a phasing ahead is different from that in a phasing behind, as shown in Figure 4, which reports the trajectory when $\mathrm{\Delta }\varphi =\{-8,8\}\mathrm{deg}$, ${\nu }_0=90\mathrm{deg}$, and $a_c=0.1\mathrm{mm}/{\mathrm{s}}^2$. This difference in the transfer topology is also reflected in the control law $\tau =\tau \left(t\right)$, as can be observed in Figure 5, which shows the temporal variation of the binary control term.

Finally, Figure 6 shows the minimum flight time for a phasing ahead in which $\mathrm{\Delta }\varphi \in (0,60]\mathrm{deg}$ and $a_c=0.1\mathrm{mm}/{\mathrm{s}}^2$, as a function of the initial angular position. In particular, this figure highlights how the value of the pair $\{{\nu }_0,\mathrm{\Delta }\varphi \}$ influences the performance when the value of the eccentricity $e_0$ is non-negligible.

3.2. Earth-Based Scenario

In this case, the small value of $e_0$ makes the results less sensitive to the value of ${\nu }_0$. This aspect is evident in Figure 7.

In fact, Figure 7 indicates that, for a given value of $\mathrm{\Delta }\varphi$, the generic function $\mathrm{\Delta }t=\mathrm{\Delta }t\left({\nu }_0\right)$ has a substantially constant value with oscillations of a few days, even when the phasing angle reaches high values. This aspect is confirmed by Figure 8, which shows the curve levels of the function $\mathrm{\Delta }t=\mathrm{\Delta }t({\nu }_0,\mathrm{\Delta }\varphi )$. In particular, the curve levels in the figure essentially collapse into a single line that indicates the variation of the minimum flight time with the phasing angle.

For example, using the graph reported in Figure 8, one obtains that a phasing maneuver in which $\mathrm{\Delta }\varphi =60\mathrm{deg}$ (or $\mathrm{\Delta }\varphi =-60\mathrm{deg}$) requires a minimum flight time of about $670\mathrm{days}$ (or $600\mathrm{days}$) when the characteristic acceleration of the Sun-facing diffractive sailcraft is $a_c=0.1\mathrm{mm}/{\mathrm{s}}^2$. This is an interesting result because such a phasing maneuver models a simplified version of an interplanetary transfer towards the Sun–Earth triangular Lagrange points [36]. In these cases of a transfer towards $L_4$ or $L_5$, the trajectory is reported in Figure 9 when the starting true anomaly is ${\nu }_0=90\mathrm{deg}$ and $a_c=0.1\mathrm{mm}/{\mathrm{s}}^2$, while Figure 10 shows the corresponding optimal control law.

The last part of this subsection analyzes the dependence of the flight time on the value of the characteristic acceleration. The value of $a_c$ has been selected in the interval $[0.06,0.12]\mathrm{mm}/{\mathrm{s}}^2$. In this case, the initial true anomaly has been set to ${\nu }_0=0\mathrm{deg}$ (i.e., a starting point at the perihelion has been selected) to reduce the computational effort. Note, however, that the previous analysis has highlighted that the flight time is substantially insensitive to the value of ${\nu }_0$. The results of this parametric study are reported in Figure 11, which shows that the dependence of the minimum flight time on the value of the characteristic acceleration becomes more marked as the value of the phasing angle $\mathrm{\Delta }\varphi$ increases. For example, in the case of $\mathrm{\Delta }\varphi =30\mathrm{deg}$, the flight time of a sailcraft with $a_c=0.06\mathrm{mm}/{\mathrm{s}}^2$ is about 120 days greater than that obtainable with $a_c=0.12\mathrm{mm}/{\mathrm{s}}^2$. On the other hand, if $\mathrm{\Delta }\varphi =5\mathrm{deg}$, the difference in flight time is about 40 days. The plots in Figure 11 allow the reader to quickly estimate the value of $\mathrm{\Delta }t$ for a low-performance diffractive sailcraft in an Earth-based mission scenario, when the phasing angle range is $\mathrm{\Delta }\varphi \in [-30,30]\mathrm{deg}$. However, in this case, the small value of $e_0$ suggests the use of a simplified mathematical model where the reference orbit is circular. This limiting case is discussed in the next subsection.

3.3. Case of a Circular Reference Orbit

In this last part of Section 3, we analyze the limiting case of a circular reference orbit. Accordingly, consider the scenario in which $e_0=0$ and the semilatus rectum is equal to $1\mathrm{AU}$. In this case, Equation (4) simplifies as

$$\tilde{r}\left({\tilde{t}}_0\right)=1,\theta \left({\tilde{t}}_0\right)=0,{\tilde{v}}_r\left({\tilde{t}}_0\right)=0,{\tilde{v}}_t\left({\tilde{t}}_0\right)=1$$

(15)

where the initial true anomaly ${\nu }_0$ was set to zero for symmetry reasons (the initial orbit is circular). On the other hand, the final constraints given by Equation (7) now become

$$\tilde{r}\left({\tilde{t}}_f\right)=1,\theta \left({\tilde{t}}_f\right)={\tilde{t}}_f+\mathrm{\Delta }\varphi ,{\tilde{v}}_r\left({\tilde{t}}_f\right)=0,{\tilde{v}}_t\left({\tilde{t}}_f\right)=1$$

(16)

The dimensionless angular velocity of the virtual point along the (circular) reference orbit is a constant of motion with a value equal to 1 so that the polar angle ${\theta }_{p_f}$ of the virtual point at the (dimensionless) final time is simply equal to ${\tilde{t}}_f$.

Using the initial and final conditions given by Equations (15) and (16), respectively, the (dimensionless) optimization problem modeled by the equations of motion (6) and the Euler–Lagrange Equation (9) can be solved with a standard method as a function of the characteristic acceleration ${\tilde{a}}_c$ and the phasing angle $\mathrm{\Delta }\varphi$. For example, assuming $a_c=0.1\mathrm{mm}/{\mathrm{s}}^2$, i.e., ${\tilde{a}}_c\simeq 0.0169$, the variation of the minimum flight time with the phasing angle $\mathrm{\Delta }\varphi \in [-60,60]\mathrm{deg}$ is reported in Figure 12. This figure shows, as expected, clear similarities with the graphs reported in Figure 8.

This aspect confirms that a preliminary study of the performances in the phasing maneuver in the case of an Earth-based scenario can be carried out using the simplified circular orbit model. The transfer strategy in the simplified circular case is similar to that obtained in the Earth-based mission scenario. For example, assuming again $a_c=0.1\mathrm{mm}/{\mathrm{s}}^2$ and $\mathrm{\Delta }\varphi =\pm 60\mathrm{deg}$, the optimal trajectories for the sailcraft reported in Figure 13 are obtained. The corresponding optimal control law $\tau =\tau \left(t\right)$ is shown in Figure 14. In particular, the specific form of the curve drawn in Figure 14a suggests a simplified form of the control law, which originates from the fundamental work by McInnes [42]. This simplified, non-optimal form of the control law is illustrated in the next section.

4. Simplified Form of the Control Law

The problem of studying the heliocentric phasing maneuver, in the case where the reference orbit is circular and the spacecraft is equipped with a low-performance RSS, was studied in 2003 by McInnes [42] using an elegant mathematical model that describes the linearized dynamics of the sailcraft in the case where its solar distance remains, during the flight, close to the radius of the circular reference orbit. This linearized model, which is essentially a sort of extension of the classical Clohessy–Wiltshire model, approximates well the actual heliocentric dynamics of the sailcraft even when its phasing angle at the virtual point is large. Therefore, for this reason, it is well suited to quickly study the phasing performance in a heliocentric circular case.

In this context, assume that $e_0=0$, $p_0=r_{\oplus }=1\mathrm{AU}$, and observe that the angle $\varphi$ between the Sun-sailcraft line and the Sun-virtual point line is given by

$$\varphi =\theta -\sqrt{{\displaystyle \frac{{\mu }_{\odot }}{r_{\oplus }^3}}}t\equiv \theta -\tilde{t}$$

(17)

with $\theta \left({\tilde{t}}_0\right)=0\mathrm{deg}$, and introduce a dimensionless auxiliary variable $\rho$ defined as

$$\rho \triangleq {\displaystyle \frac{r}{p_0}}-1\equiv \tilde{r}-1$$

(18)

Keeping in mind that the dimensionless form of the propulsive acceleration vector is given by Equation (5), and in parallel with the procedure described in [42], it is possible to obtain the following set of linear differential equations describing the (linearized) dynamics of the Sun-facing diffractive sailcraft in the vicinity of the reference circular orbit, as follows:

$${\rho }^{\prime \prime }=2{\varphi }^{\prime }+3\rho +{\displaystyle \frac{{\tilde{a}}_c}{\sqrt{2}}},{\varphi }^{\prime \prime }=-2{\rho }^{\prime }-\tau {\displaystyle \frac{{\tilde{a}}_c}{\sqrt{2}}}$$

(19)

with zero initial conditions, viz.

$$\rho \left({\tilde{t}}_0\right)=0,{\rho }^{\prime }\left({\tilde{t}}_0\right)=0,\varphi \left({\tilde{t}}_0\right)=0,{\varphi }^{\prime }\left({\tilde{t}}_0\right)=0$$

(20)

For a given temporal variation of the binary control parameter $\tau$, the linear differential system (19) with the initial conditions (20) can be solved using a standard method to obtain the function $\varphi =\varphi \left(\tilde{t}\right)$, which gives the temporal variation of the angle between the position vectors of the sailcraft and the virtual point. However, the final constraints of the phasing maneuver require that the diffractive sail return to the reference circular orbit at the end of the transfer, i.e., when $\tilde{t}={\tilde{t}}_f$. This corresponds to obtaining, at $\tilde{t}={\tilde{t}}_f$, the following conditions:

$$\rho \left({\tilde{t}}_f\right)=0,{\rho }^{\prime }\left({\tilde{t}}_f\right)=0,{\varphi }^{\prime }\left({\tilde{t}}_f\right)=0$$

(21)

while the value $\varphi \left({\tilde{t}}_f\right)$ gives the actual phasing angle obtained through the maneuver, that is, $\varphi \left({\tilde{t}}_f\right)\equiv \mathrm{\Delta }\varphi$.

In his original work [42] on an RSS-based scenario with a piecewise-constant attitude, McInnes suggests using a very simple and effective control law that allows for naturally satisfying the final constraints given by Equation (21). In particular, McInnes indicates that a control law in which the pitch angle of the RSS is kept constant for the first half of the flight (i.e., for $\tilde{t}\in [{\tilde{t}}_0,{\tilde{t}}_f/2]$) and then reversed for the remaining second half (i.e., for $\tilde{t}\in [{\tilde{t}}_f/2,{\tilde{t}}_f]$) allows the sailcraft to meet the final constraints of Equation (21) if ${\tilde{t}}_f=2m\pi$, where m is a positive integer.

Starting from the elegant solution of McInnes [42], in this case of a Sun-facing diffractive sailcraft in which the transverse direction of the propulsive thrust can be modified by the parameter $\tau$, a simple control law has been hypothesized in the following form:

$$\tau \left(\tilde{t}\right)=\pm \mathrm{sign}\left(sin\left({\displaystyle \frac{2\pi N\tilde{t}}{{\tilde{t}}_f}}\right)\right)\mathrm{where}{\tilde{t}}_f=2m\pi$$

(22)

in which {m, N} are two positive integers. The resulting control law, in essence, resembles a classical square wave.

The effectiveness of the simplified control law given by Equation (22) was assessed by numerical simulation integrating the linear differential Equations (19) considering $m\in \{1,2,3\}$ and $N\in \{1,2,\dots ,20\}$, and an assigned value of the characteristic acceleration. For each pair of parameters $\{m,N\}$, the satisfaction of the final conditions given by Equation (21) was verified and, if not, the obtained value of $\mathrm{\Delta }\varphi$ was discarded.

The results of the (open loop) simulations are summarized in Figure 15 for two values of the diffractive sailcraft performance, namely, $a_c=0.1\mathrm{mm}/{\mathrm{s}}^2$ and $a_c=0.06\mathrm{mm}/{\mathrm{s}}^2$. According to this figure, and remembering that the sailcraft dynamics are that described by the linearized model (19), the “square wave“-type control law allows the phasing maneuver to be performed with a sufficiently large set of possible angles $\mathrm{\Delta }\varphi$. In this regard, it is interesting to observe the case of negative sign on the right side of Equation (22), $m=2$, and $N=1$. In that case, the flight time is 2 years, the control law is $\tau =1$ for the first half of the transfer, and $\tau =-1$ for the remaining year, while Figure 15 indicates that a phasing angle sightly above $\mathrm{\Delta }\varphi \simeq 60\mathrm{deg}$ is reached. This result and the corresponding control law are consistent with the actual performance described in Figure 14a.

Of course, the simple control law given in Equation (22) is not optimal in a general mission scenario; moreover, the linearized dynamical system given in Equation (19) can be studied using an optimization approach to evaluate the optimal time-of-flight performance. This aspect seems a useful extension of this work and is left to future research.

5. Conclusions

In this paper, the performance of an interplanetary spacecraft equipped with a diffractive sail in a phasing maneuver along an elliptical orbit with an assigned semimajor axis and eccentricity has been studied. The mission scenario hypothesized is two-dimensional, and it was assumed that the diffractive sail nominal plane during the flight was kept perpendicular to the Sun–spacecraft line. The sail membrane is covered by a set of electro-optically controlled panels that allow the transverse component of the thrust vector to be reversed simply by applying an adequate electrical voltage. This specific aspect reduces the control terms to a single binary variable and, consequently, facilitates the design of the control law, which, in this work, has been obtained by solving an appropriate optimization problem that allows for minimizing the flight time for a given phasing angle. The optimal control law used is simply related to that of the adjoint variable and is consistent with the results of recent literature regarding this specific type of Sun-facing diffractive sails.

Numerical simulations were performed considering a low-performance sail, i.e., assuming a characteristic acceleration one order of magnitude lower than the value used in recent literature results. This value of the characteristic acceleration, which is consistent with the value used in the design of the most recent interplanetary spacecraft equipped with a reflective solar sail, allows us to have an idea of the performance achievable by such an advanced propulsion system in this specific mission scenario considering the technology level hopefully achievable in the near future. In this regard, considering the heliocentric orbit of the Earth, simulation results indicate that a low-performance diffractive sail is able to complete a phasing maneuver with a phasing angle of the order of a few tens of degrees with a flight time of a few hundred days, with a substantial reduction in time in case the phasing is performed behind instead of ahead of the planet.

The proposed procedure has also been applied to a Mercury-based mission scenario, where the reference heliocentric orbit has a non-negligible eccentricity, and to the limiting (but important) case of a circular orbit. This last scenario allowed for designing an extremely simple control law, whose performances have been determined by resorting to linearized dynamics of the spacecraft. Although not optimal, such a simplified control law allows for determining potential phasing performances (for circular orbit) using a periodic time variation of the single binary control variable. The concept underlying this simplified control law could be the starting point of a first possible extension of the model discussed in this paper. Such a possible extension could consider linearized dynamics of the spacecraft around an elliptical heliocentric orbit, in which the vehicle’s trajectory around the Sun during the maneuver remains close to the reference elliptical orbit.

A second, potential continuation of this work, instead, would concern the use of the thrust model recently proposed in the literature capable of modeling the behavior of a diffractive sail with liquid crystal optical arrays. In that case, not only the direction of the transverse thrust component could be changed by varying the electric voltage, but also the thrust vector module itself could be controlled, within certain limits, in order to expand the steering capabilities of a diffractive sail with a Sun-facing attitude. This interesting innovative feature could improve the performance of the diffractive sail in a wide range of mission scenarios.