Optimal Earth Gravity-Assist Maneuvers with an Electric Solar Wind Sail

Abstract

Propellantless propulsive systems such as Electric Solar Wind Sails are capable of accelerating a deep-space probe, only requiring a small amount of propellant for attitude and spin-rate control. However, the generated thrust magnitude is usually small when compared with the local Sun’s gravitational attraction. Therefore, the total velocity change necessary for the mission is often obtained at the expense of long flight times. A possible strategy to overcome this issue is offered by an Earth gravity-assist maneuver, in which a spacecraft departs from the Earth’s sphere of influence, moves in the interplanetary space, and then re-encounters the Earth with an increased hyperbolic excess velocity with respect to the starting planet. An Electric Solar Wind Sail could effectively drive the spacecraft in the interplanetary space to perform such a particular maneuver, taking advantage of an augmented thrust magnitude in the vicinity of the Sun due to the increased solar wind ion density. This work analyzes Earth gravity-assist maneuvers performed with an Electric Solar Wind Sail based probe within an optimal framework, in which the final hyperbolic excess velocity with respect to the Earth is maximized for a given interplanetary flight time. Numerical simulations highlight the effectiveness of this maneuver in obtaining a final heliocentric orbit with high energy.

1. Introduction

An Electric Solar Wind Sail (or E-sail) is a propellantless propulsion system, first proposed by Dr. Pekka Janhunen [1] about twenty years ago, which generates thrust by exploiting the electrostatic interaction between solar wind ions and a charged tether grid [2]. Similarly to other propellantless propulsive systems, such as the more conventional photonic solar sails [3,4,5,6,7] and magnetic sails [8,9,10], the E-sail is able to operate without requiring any propellant stored onboard to propel the spacecraft in interplanetary space. This feature makes an E-sail a very interesting option for missions requiring a continuous propulsive acceleration, including the generation of displaced non-Keplerian orbits [11,12] and the maintenance of artificial equilibrium points in the Sun–Earth gravitational field [13,14].

Since the magnitude of the propulsive acceleration generated by an E-sail is typically small [15] when compared with that generated by conventional (chemical) thrusters, long flight times are usually necessary to meet the velocity changes required by the scientific mission. Examples of long-lasting missions with E-sail-based thrusts are available in the literature. For example, Quarta et al. [16] discussed the possibility of performing an E-sail-propelled cometary rendezvous and Janhunen et al. [17] analyzed an advanced mission towards Uranus, while other more exotic and fascinating scenarios are represented by in situ studies of the outer regions of the Solar System [18,19,20]. These mission concepts require the spacecraft to significantly increase its heliocentric velocity in order to reach the target within a reasonable flight time. To that end, a possible option is given by the so-called solar wind assist maneuver [21,22], in which the spacecraft first approaches the Sun with the aim of increasing the E-sail thrust magnitude due to the higher density of solar wind ions, and then moves towards the outer Solar System regions with a very large inertial velocity magnitude. This strategy is similar to the well-known solar photonic assist [23], when a solar sail approaches the Sun to increase the solar radiation pressure acting on the reflective membrane.

A different method for increasing the spacecraft heliocentric velocity (without using any propellant) is offered by the so-called $\mathsf{\Delta }V$-Earth Gravity-Assist ($\mathsf{\Delta }V$-EGA) maneuver [24]. To perform such a maneuver, a spacecraft first escapes from the gravitational field of the launch planet and, after a heliocentric phase, re-encounters the starting planet with an increased hyperbolic excess velocity. This allows high-energy final heliocentric orbits to be obtained without requiring an increase in the propellant consumption of the launch vehicle. In its original concept, suitable impulsive maneuvers (provided by chemical thrusters) and intermediate flybys were used for the spacecraft to obtain a desired increase in excess velocity relative to the Earth. The application of $\mathsf{\Delta }V$-EGA maneuvers has also been proposed for spacecrafts equipped with low-thrust propulsion systems [25,26,27], such as electric thrusters [28], solar sails [29,30], or a combination of both [31]. After a $\mathsf{\Delta }V$-EGA maneuver and the subsequent re-encounter with Earth, the spacecraft can move towards other planets [32,33] and moons [34], or correct possible orbital injection errors [35]. A further possible application of $\mathsf{\Delta }V$-EGA maneuvers consists in obtaining a final heliocentric orbit with a very high orbital energy, which allows the outer Solar System region to be reached in a reasonable time interval. This kind of mission is especially suited for a propellantless propulsive system [36], since a $\mathsf{\Delta }V$-EGA maneuver with a Sun approach would be able to combine its effect with the thrust increase induced by the solar wind assist maneuver (or solar photonic assist maneuver in the case of a solar sail spacecraft).

The aim of this work is to analyze the performance a $\mathsf{\Delta }V$-EGA maneuver with a spacecraft propelled by an E-sail, in order to quantify its effectiveness in obtaining high-energy heliocentric orbits. The problem is addressed within an optimal framework [37], where the hyperbolic excess velocity relative to the Earth after a given flight time is maximized. The optimal control problem is solved by means of an indirect multiple shooting technique [38], which allows the time histories of the E-sail attitude and the grid voltage to be obtained. To simplify the analysis, the Earth’s heliocentric orbit is assumed to be circular, the spacecraft motion is two-dimensional, and the dimensions of Earth’s sphere of influence are neglected.

The remainder of the paper is structured as follows. Section 2 describes the mathematical model and formulates the optimal control problem. Section 3 applies the optimal steering law to some potential mission scenarios. Section 4 highlights the main outcomes of the work and suggests possible future developments.

2. Mathematical Model

The analysis of an E-sail-based $\mathsf{\Delta }V$-EGA maneuver is made within a two-body heliocentric framework, in which the Earth is assumed to describe a circular orbit around the Sun with radius $r_{\oplus }\triangleq 1\mathrm{au}$. The spacecraft leaves the Earth’s sphere of influence at $t_0\triangleq 0$ with a hyperbolic excess velocity $V_{{\infty }_0}$ when the E-sail is deployed. The dimensions of the Earth’s sphere of influence are neglected here, so that the initial Sun–spacecraft distance is approximately equal to $r_{\oplus }$. Without loss of generality, the Sun–spacecraft line at the initial time $t_0$ is chosen as the reference (fixed) direction. The spacecraft motion for $t\ge t_0$ is described with the aid of a heliocentric polar reference frame $\mathcal{T}(O;r,\theta )$, where O is the Sun’s center of mass, r is the Sun–spacecraft distance, and $\theta$ is a polar angle measured counterclockwise from the reference direction to the Sun–spacecraft line; see Figure 1. The radial and circumferential unit vectors are denoted as ${\widehat{\mathit{i}}}_r$ and ${\widehat{\mathit{i}}}_{\theta }$, respectively.

To establish the system dynamics for $t\ge t_0$, a mathematical model that defines the propulsive acceleration vector $\mathit{a}$ generated by the E-sail is necessary. According to Huo et al. [39], the magnitude of $\mathit{a}$ is inversely proportional to r and depends on the E-sail attitude, defined by the orientation of the unit vector $\widehat{\mathit{n}}$ normal to the E-sail nominal plane in the direction opposite to the Sun; see Figure 1. In vectorial terms, the E-sail-induced propulsive acceleration is modeled as

$$\mathit{a}=\tau {\displaystyle \frac{{\mathit{a}}_c}{2}}\left({\displaystyle \frac{{\mathit{r}}_{\oplus }}{\mathit{r}}}\right)\left[{\widehat{\mathit{i}}}_{\mathit{r}}+({\widehat{\mathit{i}}}_{\mathit{r}}·\widehat{\mathit{n}})\widehat{\mathit{n}}\right]$$

(1)

where $\tau \in \{0,1\}$ is a dimensionless parameter that accounts for the possibility of switching either on ($\tau =1$) or off ($\tau =0$) the electron gun that maintains the E-sail electric voltage, and $a_c$ is the spacecraft characteristic acceleration, that is, the classical sail performance parameter defined as the maximum propulsive acceleration at a Sun–spacecraft distance equal to $r_{\oplus }$. The projection of the propulsive acceleration vector given by Equation (1) along the radial and circumferential directions yields

$$\begin{array}{cc}\hfill a_r& \triangleq \mathit{a}·{\widehat{\mathit{i}}}_r=\tau {\displaystyle \frac{a_c}{4}}\left({\displaystyle \frac{r_{\oplus }}{r}}\right)[3+cos\left(2{\alpha }_n\right)]\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill a_{\theta }& \triangleq \mathit{a}·{\widehat{\mathit{i}}}_{\theta }=\tau {\displaystyle \frac{a_c}{4}}\left({\displaystyle \frac{r_{\oplus }}{r}}\right)[sin\left(2{\alpha }_n\right)]\hfill \end{array}$$

(3)

where ${\alpha }_n\in [-\pi /2,\pi /2]\mathrm{rad}$ is the E-sail pitch angle, that is, the angle between $\widehat{\mathit{n}}$ and ${\widehat{\mathit{i}}}_r$, as sketched in Figure 1. In this work, the switching parameter $\tau$ and the pitch angle ${\alpha }_n$, which may be varied by the onboard control system [40], are the two control variables of the control problem to be discussed in the next section. Here, we assume that the E-sail can freely vary its attitude with continuity. A more detailed analysis of E-sail attitude dynamics is beyond the scope of this paper, but the interested reader may refer to refs. [41,42,43].

Using the expressions of $\{a_r,a_{\theta }\}$ given by Equations (2) and (3), the spacecraft dynamics in $\mathcal{T}$ are written as

$$\begin{array}{cc}\hfill \dot{r}& =u\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \dot{\theta }& =v/r\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \dot{u}& =-{\displaystyle \frac{{\mu }_{\odot }}{r^2}}+{\displaystyle \frac{v^2}{r}}+\tau {\displaystyle \frac{a_c}{4}}\left({\displaystyle \frac{r_{\oplus }}{r}}\right)[3+cos\left(2{\alpha }_n\right)]\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \dot{v}& =-{\displaystyle \frac{uv}{r}}+\tau {\displaystyle \frac{a_c}{4}}\left({\displaystyle \frac{r_{\oplus }}{r}}\right)sin\left(2{\alpha }_n\right)\hfill \end{array}$$

(7)

where ${\mu }_{\odot }$ is the Sun’s gravitational parameter, and u (or v) is the radial (or circumferential) component of the spacecraft heliocentric velocity vector. The dynamical system expressed by Equations (4)–(7) is completed by four initial conditions (subscript 0)

$$\begin{array}{cc}\hfill r\left(t_0\right)& \triangleq r_0=r_{\oplus }\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \theta \left(t_0\right)& \triangleq {\theta }_0=0\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill u\left(t_0\right)& \triangleq u_0=V_{{\infty }_0}cos{\varphi }_0\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill v\left(t_0\right)& \triangleq v_0=\sqrt{{\displaystyle \frac{{\mu }_{\odot }}{r_{\oplus }}}}+V_{{\infty }_0}sin{\varphi }_0\hfill \end{array}$$

(11)

where the auxiliary angle ${\varphi }_0\in (-\pi ,\pi ]\mathrm{rad}$ is measured counterclockwise from the radial unit vector ${\widehat{\mathit{i}}}_r$ to the geocentric velocity vector ${\mathit{V}}_{{\infty }_0}$ at $t_0$; see Figure 1. Note that Equations (10) and (11) may be rewritten, in a compact form, as

$$u_0^2+{\left[v_0-\sqrt{{\displaystyle \frac{{\mu }_{\odot }}{r_{\oplus }}}}\right]}^2=V_{{\infty }_0}^2$$

(12)

Spacecraft Trajectory Optimization

The objective of a $\mathsf{\Delta }V$-EGA maneuver is to increase the Earth-relative hyperbolic excess velocity (with respect to its value at the launch phase) when the spacecraft re-encounters the Earth at a given time instant $t=t_f$. The optimal control problem is therefore formulated by maximizing the cost function

$$J\triangleq \sqrt{u_f^2+{\left[v_f-\sqrt{{\displaystyle \frac{{\mu }_{\odot }}{r_{\oplus }}}}\right]}^2}$$

(13)

where the subscript f denotes the values at $t_f$, that is, $u_f\triangleq u\left(t_f\right)$ and $v_f\triangleq v\left(t_f\right)$. Note that the cost function J represents the hyperbolic excess speed relative to the Earth at the end of the $\mathsf{\Delta }V$-EGA maneuver, that is, when the spacecraft re-enters the Earth’s sphere of influence. The constraints on the spacecraft state variables at time $t_f$ are therefore

$$\begin{array}{cc}\hfill r\left(t_f\right)& \triangleq r_f=r_{\oplus }\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \theta \left(t_f\right)& \triangleq {\theta }_f={\theta }_0+\sqrt{{\displaystyle \frac{{\mu }_{\odot }}{r_{\oplus }^3}}{t}_f\equiv \sqrt{{\displaystyle \frac{{\mu }_{\odot }}{r_{\oplus }^3}}}{t}_f}\hfill \end{array}$$

(15)

The optimal control problem aimed at maximizing the value of J in Equation (13) is solved by means of an indirect method [37]. To this end, first introduce a set of costate variables $\{{\lambda }_r,{\lambda }_{\theta },{\lambda }_u,{\lambda }_v\}$ associated with the physical state variables $\{r,\theta ,u,v\}$, the dynamics of which are given by Equations (4)–(7). Then, define the control Hamiltonian function $\mathcal{H}$ as [44]

$$\mathcal{H}={\lambda }_{r}u+{\displaystyle \frac{{\lambda }_{\theta }v}{r}}+{\lambda }_u\left({\displaystyle \frac{v^2}{r}}-{\displaystyle \frac{{\mu }_{\odot }}{r^2}}\right)-{\displaystyle \frac{{\lambda }_{v}uv}{r}}+{\lambda }_u{a}_r+{\lambda }_v{a}_{\theta }$$

(16)

where $\{a_r,a_{\theta }\}$ are given by Equations (2) and (3). The time variation of the costates is described by the Euler–Lagrange equations, viz.

$$\begin{array}{cc}\hfill {\dot{\lambda }}_r& =-{\displaystyle \frac{\partial H}{\partial r}}={\displaystyle \frac{v{\lambda }_{\theta }}{r^2}}-{\lambda }_u\left({\displaystyle \frac{2{\mu }_{\odot }}{r^3}}-{\displaystyle \frac{v^2}{r^2}}-{\displaystyle \frac{a_r}{r}}\right)-{\lambda }_v\left({\displaystyle \frac{uv}{r^2}}-{\displaystyle \frac{a_{\theta }}{r}}\right)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_{\theta }& =-{\displaystyle \frac{\partial H}{\partial \theta }}=0\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_u& =-{\displaystyle \frac{\partial H}{\partial u}}={\displaystyle \frac{v{\lambda }_v}{r}}-{\lambda }_r\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\dot{\lambda }}_v& =-{\displaystyle \frac{\partial H}{\partial v}}={\displaystyle \frac{u{\lambda }_v-2v{\lambda }_u-{\lambda }_{\theta }}{r}}\hfill \end{array}$$

(20)

In particular, Equation (18) states that ${\lambda }_{\theta }$ is a constant of motion.

The trajectory that maximizes J may be found by enforcing the optimal steering law $\{{\alpha }_n={\alpha }_n^{\star }\left(t\right),\tau ={\tau }^{\star }\left(t\right)\}$, which defines the optimal time variations of the two control parameters. According to Pontryagin’s maximum principle, the optimal steering law maximizes, at any time, the Hamiltonian function given by Equation (16). Starting from the results of ref. [39], the optimal steering law may be written as

$$\begin{array}{cc}\hfill {\alpha }_n& \equiv {\alpha }_n^{\star }={\displaystyle \frac{{\alpha }_p}{2}}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \tau & \equiv {\tau }^{\star }={\displaystyle \frac{1+\mathrm{sign}\left(1+3cos{\alpha }_p\right)}{2}}\hfill \end{array}$$

(22)

where $sign\left(\square \right)$ denotes the signum function, and the auxiliary angle ${\alpha }_p\in [-\pi /2,\pi /2]\mathrm{rad}$ is defined as

$${\alpha }_p\triangleq arctan\left({\displaystyle \frac{{\lambda }_v}{{\lambda }_u}}\right)$$

(23)

Having defined the optimal steering law through Equations (21) and (22), the spacecraft trajectory that maximizes the cost function J is obtained by solving the associated two-point boundary value problem (TPBVP). To this end, the dynamical Equations (4)–(7) and the Euler–Lagrange Equations (17)–(20) are numerically integrated in double precision using a variable-order Adams–Bashforth–Moulton solver scheme [45,46], with initial conditions given by Equations (8), (9) and (12), and terminal conditions expressed by Equations (14) and (15).

The set of boundary conditions is completed by the transversality condition [44], to be enforced both at $t=t_0$ and $t=t_f$, that is

$$\begin{array}{cc}\hfill {\lambda }_{u_0}& ={\lambda }_{v_0}cot{\varphi }_0\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\lambda }_{u_f}& ={\displaystyle \frac{u_f}{J}}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\lambda }_{v_f}& ={\displaystyle \frac{v_f-\sqrt{{\mu }_{\odot }/r_{\oplus }}}{J}}\hfill \end{array}$$

(26)

where ${\lambda }_{u_0}\triangleq {\lambda }_u\left(t_0\right)$, ${\lambda }_{v_0}\triangleq {\lambda }_v\left(t_0\right)$, ${\lambda }_{u_f}\triangleq {\lambda }_u\left(t_f\right)$, and ${\lambda }_{v_f}\triangleq {\lambda }_v\left(t_f\right)$, while J is given by Equation (13). Note that, according to Equation (24), the initial values of ${\lambda }_u$ and ${\lambda }_v$ are not independent but are related through the auxiliary angle ${\varphi }_0$.

Equations (24)–(26) complete the set of eight boundary conditions necessary to fully define the TPBVP. A multiple shooting procedure is applied to obtain the values of the set $\{{\varphi }_0,{\lambda }_{r_0},{\lambda }_{{\theta }_0},{\lambda }_{v_0}\}$. Note that Equation (24) assumes that ${\varphi }_0\notin \{0,\pi \}\mathrm{rad}$, that is, the spacecraft velocity relative to the Earth at $t_0$ is not aligned with the Sun–spacecraft line; see Figure 1. In the special case when ${\varphi }_0\in \{0,\pi \}\mathrm{rad}$, the condition that relates ${\lambda }_{u_0}$ and ${\lambda }_{v_0}$ is no longer valid, and the multiple shooting procedure gives the set of initial values of the costates $\{{\lambda }_{r_0},{\lambda }_{{\theta }_0},{\lambda }_{u_0},{\lambda }_{v_0}\}$ as outputs.

3. Case Study

The previously described methodology to find the optimal $\mathsf{\Delta }V$-EGA trajectory is now simulated in some potential mission applications. In the following analysis, in analogy with previous works [29,30,31,36], it is assumed that the spacecraft leaves the Earth’s sphere of influence with hyperbolic excess velocity in the set $V_{{\infty }_0}=\{0.5,1,1.5,2,2.5,3\}\mathrm{km}/\mathrm{s}$. At $t_0$, the spacecraft deploys an E-sail with a characteristic acceleration in the set $a_c\in \{0.1,0.2,0.5,1\}\mathrm{mm}/{\mathrm{s}}^2$. These values of $a_c$ are compatible with a near- and mid-term technology level [15,47].

A $\mathsf{\Delta }V$-EGA maneuver is then started, with the aim of maximizing the hyperbolic excess speed relative to the Earth at its re-encounter, after a flight time $t_f=1\mathrm{year}$. The performance parameter P, defined as

$$P\triangleq {\displaystyle \frac{J-V_{{\infty }_0}}{V_{{\infty }_0}}},$$

(27)

is used to quantify the relative increase in hyperbolic excess velocity obtained from the $\mathsf{\Delta }V$-EGA maneuver. In particular, note that $P=0$ corresponds to when the $\mathsf{\Delta }V$-EGA maneuver gives no increase in the hyperbolic excess velocity, while $P=1$ indicates that the final excess velocity is twice its initial value.

Figure 2 shows the values of P obtained after a flight time of $1\mathrm{year}$ as functions of the initial hyperbolic excess velocity $V_{{\infty }_0}$ and the E-sail characteristic acceleration $a_c$. It is evident that, regardless of the E-sail performance level, the $\mathsf{\Delta }V$-EGA maneuver is more effective in increasing the hyperbolic excess velocity when its initial value is small. In those cases, indeed, the final values are several times larger than the initial values. However, the $\mathsf{\Delta }V$-EGA contribution is remarkable (on the order of some kilometers per second), even for large values of $V_{{\infty }_0}$.

Figure 3, Figure 4, Figure 5 and Figure 6 show some examples of the optimal $\mathsf{\Delta }V$-EGA maneuvers with $t_f=1\mathrm{year}$, $V_{{\infty }_0}=1\mathrm{km}/\mathrm{s}$, and different $a_c$ and show the spacecraft trajectory and the corresponding optimal steering law of the pitch angle. The steering law of the switching parameter is not shown because ${\tau }^{\star }\equiv 1$ for the whole flight in all of the considered scenarios. This is not surprising, because the aim of the $\mathsf{\Delta }V$-EGA maneuver is to obtain a large value of J, which requires the E-sail to be always switched on during the approach phase towards the Sun.

4. Conclusions

This paper analyzed the performance of an Earth gravity-assist maneuver for a spacecraft propelled by an Electric Solar Wind Sail. To this end, an optimal control problem was formulated with the aim of finding the steering law that maximizes the spacecraft hyperbolic excess velocity when it re-encounters the Earth. The results, obtained with an indirect multiple shooting technique, highlight that an Earth gravity-assist maneuver is able to provide a significant increment in the hyperbolic excess velocity relative to the Earth, and the relative increase is larger when the initial excess velocity is smaller. Accordingly, this strategy could allow high-energy heliocentric orbits to be obtained, even when the initial excess velocity provided by the launcher is small, thus opening the possibility of reaching large heliocentric distances in reasonable flight times. Future work will concentrate on a refinement of the model discussed here by accounting for the Earth’s gravity and the eccentricity and mutual inclination of the planetary heliocentric orbits.