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Electric Solar Wind Sail Kinetic Energy Impactor for Asteroid Deflection Missions

  • Kouhei Yamaguchi
  • Hiroshi Yamakawa
Article

Abstract

An electric solar wind sail uses the natural solar wind stream to produce low but continuous thrust by interacting with a number of long thin charged tethers. It allows a spacecraft to generate a thrust without consuming any reaction mass. The aim of this paper is to investigate the use of a spacecraft with such a propulsion system to deflect an asteroid with a high relative velocity away from an Earth collision trajectory. To this end, we formulate a simulation model for the electric solar wind sail. By summing thrust vectors exerted on each tether, a dynamic model which gives the relation between the thrust and sail attitude is proposed. Orbital maneuvering by fixing the sail’s attitude and changing tether voltage is considered. A detailed study of the deflection of fictional asteroids, which are assumed to be identified 15 years before Earth impact, is also presented. Assuming a spacecraft characteristic acceleration of 0.5 mm/s 2, and a projectile mass of 1,000 kg, we show that the trajectory of asteroids with one million tons can be changed enough to avoid a collision with the Earth. Finally, the effectiveness of using this method of propulsion in an asteroid deflection mission is evaluated in comparison with using flat photonic solar sails.

Keywords

Electric solar wind sail Near Earth asteroid’s deflection Propellantless propulsion systems Kinetic energy impactor 

Introduction

It has long been said that Near Earth Asteroids (NEAs) could have a devastating impact on the Earth [1, 4, 25]. To represent such hazards, the Torino Impact Hazard Scale, which categorizes the risk of an asteroid’s approach to Earth on a 0-10 scale, was established in 1999 [2]. At present, the probability of an asteroid impact leading to severe damage for humanity is considered to be sufficiently low, but some known asteroids may be perturbed into orbits that will cross the Earth’s orbit, so the list of NEAs is continuously being updated. Thus the risk of collision is not zero. The most famous and severe example of an asteroid impact might be the one that caused the extinction of the dinosaurs at the Cretaceous Tertiary boundary. More recently, a 60- m asteroid exploded over Tunguska, Siberia in 1908. It released 10 Mt of energy and devastated about 2,000 m 2 of forest [5, 31]. Just from these two examples, it can be seen that the potential threat posed by an asteroid impact might be catastrophic and it is, therefore, important to prepare methods to deal with an asteroid that has an unacceptable risk of collision with the Earth. Though the simplest idea for this may be using some kind of nuclear warhead, serious political issues would inevitably arise. So various nonnuclear ideas have also been proposed and investigated. The method of pushing the asteroid using the gravitational coupling between an asteroid and a massive spacecraft was proposed by Lu and Love (18). This idea, called the gravity tractor, involves moving the asteroid without touching the asteroid surface. Another idea, the so-called Kinetic Energy Impactor (KEI), involves sending a spacecraft or projectile to collide with the asteroid, and is considered to be one of the most efficient methods for deflecting asteroids [8, 10].

The aim of this paper is to discuss the feasibility of a KEI with an electric solar wind sail (E-sail). The E-sail is a next-generation spacecraft propulsion system whose fundamental idea was conceived by Janhunen (11). It uses the electrostatic interaction between the natural solar wind plasma stream and an electric field generated by charged conducting tethers [11, 12, 14]. The tethers are deployed by spinning the spacecraft body and held at a high positive potential through the loss of electrons caused by onboard electron guns [21], see Fig. 1.
Fig. 1

A conceptual sketch of an E-sail based spacecraft

The experiments to perform the deployment of the tether and measurement of the sail force have been planned. The ESTCube-1 was launched in 2013 to deploy 10 m tether and measure the sail force [7]. While the E-sail tether deployment failed, the overview of the mission was summarized and the lessons learned were discussed in detail [28]. In addition, the E-sail experiment onboard Aalto-1 [17], which is intended to deploy the 100 m tether and to measure the electrostatic drag force, will provide significant information about this innovative thruster. The tether factory capable of producing 1 km has been presented and the study concluded that larger scale production of E-sail tether is possible and practical. Moreover, because the top level feasibility study has shown that the E-sail can accelerate the spacecraft more effective than any other current propulsion technologies, NASA Marshall Space Flight Center studied the E-sail for rapid heliopause mission1. E-sail trajectories for missions toward the inner or outer Solar System have been calculated [21, 22] and an investigation has shown that the E-sail would allow a spacecraft to rendezvous with 67 % of currently known potentially hazardous asteroids within 1 year [23]. Performance of the E-sail to complete nodal flyby missions toward NEAs have been studied and the results showed that most NEAs can be reached within a mission time of 300 days, and over 60 % of NEAs within 100 days [20]. In addition, a sample return campaign with E-sail targetting the asteroid named 1998 KY 26 was performed, and the analysis of the mission end date as a function of scientific mission time and start date offered a flexibility in the launch window [24]. Though a remote unit placed at the tip of each tether has a thruster unit for controlling the main tether’s angular velocity, the E-sail allows the spacecraft to perform orbit transfer with a small consumption of reaction mass. By making use of this characteristic, the E-sail could also be an attractive system for producing the high impact velocity that is required by a KEI.

In this paper, we analyze the performance of an E-sail KEI as follows. Section “Fictional Asteroid Deflection Mission Scenario” describes a fictional asteroid deflection scenario. The b-plane concept, which models the change in the orbit of the asteroid is also discussed. A simulation model to calculate the trajectory of an E-sail is formulated in Section “Simulation Model for E-sail”. A relation between sail attitude and its thrust, and equations of motion are given here. In addition, orbital maneuvers using active tether voltage control are also described. Section “Numerical Simulations” summarizes the results of simulations to deflect three virtual Earth impactors. Optimized trajectories, impact velocity of projectiles, and the resulting deflection distance are provided, and a comparison with a flat photonic solar sail is made. The final conclusions and brief discussion of the results are presented in Section “Conclusion”.

Fictional Asteroid Deflection Mission Scenario

To investigate the performance of the E-sail KEI proposed in this paper, a fictional asteroid mitigation scenario is investigated. Figure 2 shows an overview of this scenario, that consists of an asteroid detection and identification on 1 January 2015. The fictional asteroid is assumed to be found 15 years before Earth impact on 1 January 2030. After spacecraft launch, the flight time until asteroid impact is t f; the time between this impact and the predicted Earth impact is t s. Assuming a circular orbit with 1 AU radius for the motion of the Earth, a necessary condition for an asteroid impact with the Earth is that the asteroid has an orbit with a perihelion smaller than 1 AU and an aphelion larger than 1 AU.
Fig. 2

Mission time-line for deflecting a fictional asteroid

We considered three fictional asteroids (FAs). Figure 3 shows the orbits of these FAs.
Fig. 3

Orbits of three fictional asteroids (FAs)

Orbital elements for the FAs were selected from catalogued NEAs and modified to yield Earth impacts at the same predicted time (1 January 2030). They are summarized in Table 1. To study the E-sail KEI concept, three different FAs are considered. The three FAs have the same properties except for orbital elements. The orbit type of FA I and FA II is Apollo group, which has a semimajor axis larger than 1 AU, and that of FA III is Aten, which has a semimajor axis smaller than 1 AU. In each case we assumed a spherical asteroid with a diameter of 100 m. As summaryzed in Ref. [3], a density of an asteroid depends on several parameters. Here, a typical density of 2 g/cm 3, which is a little rough but reasonable value for investigating the performance of the E-sail KEI, is assumed. The mass of each FA m ast is about one million tons; although this is relatively small, such asteroids could cause serious regional damage.
Table 1

Orbital elements of three fictional asteroids

 

a, AU

e

i, deg

ω, deg

Ω, deg

M, deg

FA I

2.01677

0.521434

0.455074

153.941

280.261

262.934

FA II

1.54241

0.502447

8.41607

287.732

100.261

236.254

FA III

0.739846

0.433178

6.71126

337.0785

280.261

120.629

Notes: a is the semimajor axis, e is the eccentricity, i is the inclination, ω is the argument of perigee, Ω is the ascending node, and M is the mean anomaly at MJD=57,023.

The change in asteroid orbit is expressed using the b-plane concept. The b-plane is oriented normal to the unperturbed geocentric velocity vector of an asteroid U, as shown in Fig. 4a. To express the asteroid path in terms of the b-plane, a planetocentric coordinate system (ξ, η, ζ) is introduced. The η-axis is along U, the ζ-axis points in the opposite direction to the projection of the heliocentric Earth velocity vector v E on the b-plane. The ξ-axis is chosen to comply with the right-hand rule. Changes in the asteroid’s orbit are given by Δξ, Δζ. The geometry of the velocity vectors are shown in Fig. 4b. A change in the asteroid’s approach distance arises from two different sources. The first is a phase shift, which is caused by a change of orbital period, and will accumulate over the time t s. The second is a change in the shape of the orbit, which does not cause any secular effect. Since the first contribution is dominant for large t s [9], we disregard the second contribution and, therefore, we take Δξ = 0. Then, the following analytical relation [9] between the KEI strategy and the change in asteroid path along the ζ-axis can be used.
$$ \Delta \zeta =K\frac{3a v_{E} \sin{\theta_{aE}}}{\mu} \frac{m_{sc}}{m_{sc}+m_{ast}}t_{s} \boldmath{V}_{\text{sa}} \cdot \boldmath{v}_{\text{ast}} $$
(1)
where v E is the magnitude of the heliocentric velocity of the Earth, 𝜃 aE is the angle between the heliocentric velocity of the Earth and the relative velocity vector of the asteroid U at encounter (Fig. 4b), μ is the gravitational parameter of the Sun, m sc is the mass of the projectile, V sa is the relative encounter velocity of the asteroid and the spacecraft, and v ast is the asteroid heliocentric velocity vector. The parameter K represents the kinetic momentum transportation factor which largely accounts for the composition of the asteroid and the structure of the projectile. The determination of the upper limit of V sa requires further study on the relation between the disruption and kinetic energy delivered by the projectile. The required energy to fragment an asteroid depends on its composition and structure. For example, the disruption energy could be largely different depending on whether the asteroid is composed largely of ice or silicate [32]. To allow for an imperfectly inelastic impact, the quite conservative value of K = 1 [9] is assumed here. The deflection distance Δζ is used as the objective function in the trajectory optimization for the E-sail. Though the deflection distance Δζ is not the true change of the distance of approach, we can judge whether the asteroid will collide with the Earth or not by scaling the Earth radius to the b-plane. The scaled Earth radius R b is given by [9]
$$ R_{b} = R_{\mathrm{E}} f_{\infty} $$
(2)
$$ f_{\infty}=\sqrt{1+\frac{2 \mu_{\mathrm{E}}}{R_{\mathrm{E}} U^{2}}} $$
(3)
where R E=6,378km is the Earth’s equatorial radius, f is the focusing factor, and μ E is the gravitational parameter of the Earth. An asteroid deflection mission usually comes down to making Δζ greater than R b at the predicted Earth collision date. In Table 2, f , R b, and |U| for the three FAs are given.
Fig. 4

Definition of the b-plane and planetocentric coordinate system (ξ, η, ζ)

Table 2

Orbital deflection descriptions for the three FAs

 

f

R b , km

|U|, km/s

FA I

1.656

10,560

8.459

FA II

1.272

8,123

14.20

FA III

1.515

9,661

9.810

Simulation Model for E-sail

In this section, some points concerning the trajectory of an E-sail are considered. The first problem addressed here is the development of a simulation model for calculating the heliocentric trajectory of the E-sail. A method for orbital maneuvering appropriate for an E-sail is also explained.

Natural Solar Wind Dynamic Pressure Force Model

Here, we define the E-sail thrust vector taking into account its structure and attitude. For simplicity, we assume that every tether of the E-sail confined to the same plane by a sufficiently large centrifugal force. To express the E-sail thrust acceleration a sw and attitude, the sail plane’s normal direction, inclination angle α, clock angle δ, and thrust direction angle α T are introduced, as shown in Fig. 5.
Fig. 5

Sail plane’s inclination angle α, thrust direction α T, and clock angle δ

In Fig. 5, \(\hat {\mathbf {r}}\) is a unit vector along the Sun-spacecraft line and \(\hat {\mathbf {p}}\) is a unit vector directs to \(\hat {\mathbf {r}}\times \hat {\mathbf {v}}\), where \(\hat {\boldmath {v}}\) is the normalized heliocentric velocity vector of the spacecraft. Using these definitions, the thrust direction \(\hat {\mathbf {a}}_{\mathbf {sw}}\) is
$$ \hat{\mathbf{a}}_{\mathbf{sw}}=\cos{\alpha_{\mathrm{T}}}\hat{\mathbf{r}}+\sin{\alpha_{\mathrm{T}}}\sin{\delta}\hat{\mathbf{p}}+\sin{\alpha_{\mathrm{T}}}\cos{\delta} \hat{\mathbf{p}}\times\hat{\mathbf{r}}. $$
(4)
In Refs. [21, 22, 23], the corresponding thrust angle α T was assumed to be equal to one-half the sail plane’s inclination angle α. Moreover, the magnitude of the propulsive acceleration was not defined as a function of α, but depended only on the distance from the Sun. In this study, a more detailed analysis was performed. Here, we assume a spinning body with 100 conducting tethers, which was assumed in our previous work [34]. Since the E-sail mass budget model [16] provides the required number of the conducting tether, we can calculate the relation between the angle α and the thrust vector with the same way. Using an equation, which expresses the force exerted on a conducting tether kept at a given voltage [14], thrust vectors for every tether were calculated. By summing all of these vectors varying the sail plane’s inclination angle α in steps of 0.1 , it was found that the E-sail thrust direction α T and thrust acceleration magnitude γ, which is a normalized a sw so as to be 1 when the angle α = 0, depend on α as shown in Fig. 6. Note that we used a constant and uniform value of 400 km/s for the solar wind velocity. We also used 7.3 × 10 6 m −3 as the solar wind electron density, and 12 eV for its temperature which are average values at 1 AU from the sun [26, 27].
Fig. 6

Relations between E-sail plane’s inclination angle α and thrust vector direction [34]

Figure 6a shows that the relation between α and α T assumed in previous works [21, 22, 23] is roughly correct within the range 0α≤30. However, for α≥30, α T behaves non-linearly. After reaching a maximum value when α is about 55 , α T decreases as α grows. Looking at the relation between α and the thrust acceleration magnitude γ shown in Fig. 6b, one sees that γ also depends on the sail plane’s inclination angle α, varying from 0.5 to 1 as α changes. Note that the effect of the overlapping of the electric potentials generated around the tethers is not considered here, because a model of how the thrust behaves inside the overlap region is not available yet [16]. Sixth-order polynomial equations fitted to the calculated results are as follows.
$$\begin{array}{@{}rcl@{}} \alpha_{\mathrm{T}}(\alpha)=3.681\times 10^{-10}\alpha^{6}-8.295\times 10^{-8}\alpha^{5} +6.322 \times 10^{4} \alpha^{4} \\ -2.661 \times 10^{-4} \alpha^{3}+3.652 \times 10^{-3}\alpha^{2}+4.853 \alpha \end{array} $$
(5)
$$\begin{array}{@{}rcl@{}} \gamma(\alpha)=-5.896 \times 10^{-13}\alpha^{6}+1.943 \times 10^{-10} \alpha^{5} - 1.261 \times 10^{-8} \alpha^{4} \\ +7.027 \times 10^{-7}\alpha^{3} -1.271 \times 10^{-4} \alpha^{2} + 6.904 \times 10^{-5} \alpha +1.000 \end{array} $$
(6)
Note that the unit of the angle α and α T is in degree, here. These analytic forms enable us to approximate the thrust angle α T and thrust magnitude γ as a function of the angle α.

Formulation of the Equations of Motion

To calculate the trajectory of the E-sail, we start from the heliocentric equations of motion in polar inertial coordinates (r, 𝜃, φ) (see Fig. 7)
$$\begin{array}{@{}rcl@{}} \dot{v}_{\mathrm{r}}&=&\frac{\mathrm{d}v_{\mathrm{r}}}{\mathrm{d}t}+\frac{1}{r} \left( {v_{\phi}}^{2}+{v_{\mathrm{\theta}}}^{2}\right)+\frac{\mu}{r^{3}}+S \end{array} $$
(7)
$$\begin{array}{@{}rcl@{}} \dot{v}_{\mathrm{\theta}}&=&\frac{1}{r} ({v_{\mathrm{\theta}}}v_{\mathrm{\phi}}\tan{\phi}-{v_{\mathrm{r}}}v_{\mathrm{\theta}})+T \end{array} $$
(8)
$$\begin{array}{@{}rcl@{}} \dot{v}_{\mathrm{\phi}}&=&-\frac{1}{r} \left( v_{\mathrm{r}}v_{\mathrm{\phi}}+v_{\mathrm{\theta}}^{2} \tan{\phi}\right)+W \end{array} $$
(9)
Fig. 7

Reference frame and thrust components

where r = (r, 𝜃, φ) is the position of the spacecraft, v = (v r ,v 𝜃 ,v φ ) is the velocity, μ is the Sun’s gravitational parameter, and a s w =(S, T, W) is the propulsive acceleration. In a previous study [14], the Coulomb drag force produced by a conducting tether kept at a high potential (on the order of kilovolts) with respect to the solar wind was studied using 1D and 2D particle in cell simulations, and a simplified propulsion model depending on the spacecraft’s distance from the Sun r was proposed. The force exerted on the tether increases ∝(1/r) η as the sail moves closer to the Sun. Results in published earlier said that the value of η was 7/6 [14], and some investigations on the performance of the E-sail were performed based on this result [15, 21, 22, 23, 34]. A strict analysis focusing on the removal of the trapped electrons by spacecraft induced orbit scattering and subsequent tether collisions was performed, and η = 1 became more general [13]. In this paper we assume η = 1 and the difference between two values of η is briefly explained later to show that the orbital maenuvering technique proposed in our previous work [34] can also be applied for η = 1 model. E-sail propulsion is produced by the interaction between the solar wind natural plasma stream and the artificial electric shield. Thus we can change the thruster’s ON/OFF state by switching the onboard electron gun. This ON/OFF state is usually modeled by a parameter τ∈{0,1}. Now, by using Eq. 6, we can write the magnitude of the spacecraft’s propulsive acceleration a sw as
$$ a_{\text{sw}}=\tau a_{0} \left( \frac{r_{0}}{r}\right)^{\eta} \gamma(\alpha) $$
(10)
where r 0=1 AU is the astronomical unit, and a 0 is the spacecraft characteristic acceleration, which is produced when the sail plane is oriented normal to the solar wind (α = 0) and tethers are kept at their maximum voltage. Then using Eqs. 5 and 10, the radial, transverse, and normal acceleration components S, T, and W can be written as
$$\begin{array}{@{}rcl@{}} S&=&\tau a_{0} \left( \frac{r_{0}}{r}\right)^{\eta} \gamma(\alpha) \cos{\alpha_{\mathrm{T}}(\alpha)} \end{array} $$
(11)
$$\begin{array}{@{}rcl@{}} T&=&\tau a_{0} \left( \frac{r_{0}}{r}\right)^{\eta} \gamma(\alpha) \sin{\alpha_{\mathrm{T}}(\alpha)}\sin{\delta} \end{array} $$
(12)
$$\begin{array}{@{}rcl@{}} W&=&\tau a_{0} \left( \frac{r_{0}}{r}\right)^{\eta} \gamma(\alpha) \sin{\alpha_{\mathrm{T}}(\alpha)}\cos{\delta} \end{array} $$
(13)

Introducing these components into the heliocentric equations of motion, a numerical simulation model for the E-sail is formulated.

Orbital Maneuvering by Changing the Thrust Acceleration Magnitude

Here, we describe an orbital maneuvering strategy for the E-sail. Previous works [21, 22, 23], which investigated the E-sail’s mission applicability, used the active change of the sail plane’s inclination angle α to direct thrust in the desired direction. In addition, by turning off the electron gun, a coasting arc could be used. To reduce the required energy for changing the E-sail attitude, some mission analyses were performed in which the sail angular rate was restricted to less than 1 deg/day. The method to control the sail angle with respect to the Sun direction by modulating the voltage of each tether separately was proposed and investigated [29]. Here, we propose another type of orbital maneuvering strategy in which the sail’s attitude is not actively changed. Instead the attitude is fixed in the inertial frame and only the electric potential of conducting tethers is changed. By turning the electron gun on and off with optimal timing, the E-sail’s trajectory can be controlled, with the thrust direction and magnitude depend on the position of the E-sail. To investigate the feasibility of this method, local switching laws (LSLs) that increase some specific osculating orbital elements with a maximum rate were generated. With the LSLs, the electron gun is turned on when the thrust is along the optimal direction defined by Lagrange’s planetary equations. Figure 8 shows an example of a locally optimal trajectory generated with the LSL for increasing eccentricity with η = 1. Note that a characteristic acceleration a 0=0.5 mm/s 2 and initial sail attitude (α 0,δ 0) = (0,90) were assumed. E-sail thrust was mainly ON in the first and fourth quadrant, and eccentricity gradually increased.
Fig. 8

An example of a locally optimal trajectory of an E-sail using the LSL for increasing eccentricity

For comparative purpose, the difference between the two values of η is briefly investigated. The LSL for increasing eccentricity is applied for two values of η, and the time histories of semimajor axis and eccentricity are summarized in Fig. 9. The solid lines are the results with η = 1 and dashed lines are with η = 7/6. Looking at Fig. 9, one see that the difference of two models is small. Change in the semi-major axis with η = 1, shown in Fig. 9a, matches to that of η = 7/6 to within 6 %, and the eccentricity in (b) matches to within 1 %. This results mean that the LSL can be used for η = 1 model which is adopted here.
Fig. 9

Time histories of change in orbital elements with two different values of η

As in Ref. [14], the force per unit length of conducting tether can be expressed as a function of its voltage, which it suggests that a desired thrust can be obtained by changing tether voltage. We, therefore, assume that the active modulation of thrust acceleration magnitude can be achieved by controlling the electric potential of the conducting tethers. In Ref. [14], the rise time of the electric potential from zero to intended value was set to 5 msec. Under this condition, force exerting on a tether was saturated about several tens of milliseconds. Therefore, we can consider that the E-sail thrust is changed instantaneously as long as the interval of changing the electric potential of the tethers are sufficiently long compared with the millisecond time scale. To represent the active modulation of thrust acceleration magnitude, we redefine the discrete switching parameter τ∈{0,1} of Eq. 10 to be a continuous coefficient τ∈[0,1]. A previous study investigated the efficacy of this maneuver as an orbital control method [34]. By optimizing the initial trajectory generated by the LSL with a direct optimization method, circle to circle rendezvous missions with an E-sail were achieved. This shows that the maneuver strategy can be applied to more complex orbital transfer problems. Hence, we assume that this maneuvering strategy is a practical option for designing the E-sail trajectory for a KEI asteroid deflection mission.

Numerical Simulations

In this section, the feasibility of an E-sail-based KEI is investigated through numerical simulations. Using the b-plane concept, equations of motion, and orbital maneuvering which were discussed in the previous sections, an E-sail trajectory was optimized for deflecting each of three FAs. The E-sail KEI is also compared with a solar sail KEI. In all simulations, the differential equations were converted into non-dimensional form and numerically integrated at double precision by a fourth-order Runge-Kutta scheme with absolute erros of 10−12.

Deflection Mission Planning

We assume a spinning E-sail with a characteristic acceleration of a 0=0.5 mm/s 2 (the probability of E-sail KEI with smaller values of a 0 is also investigated and briefly explained for the case of FA I). The mass of the projectile is set to 1,000 kg, which is consistent with the mass budget models in Ref. [16]. By assuming the projectile is a payload mass, the total mass of the E-sail spacecraft becomes approximately 2,700 kg [16]. The E-sail’s attitude is fixed to the inertial frame, and the trajectory is controlled by changing the thrust acceleration magnitude through τ. At the spacecraft’s departure from the Earth, the E-sail’s state is assumed to be
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} (r, \theta, \varphi) &=& (r_{0}, 0, 0)\\ (v_{r}, v_{\theta}, v_{\varphi}) &=& (0, \sqrt{\mu/r_{0}}, 0) \end{array} \end{array} $$
(14)
The E-sail’s attitude at Earth departure is a parameter to be optimized. Let the spacecraft’s flight time t f be divided into 100 segments of equal length. Each of these segments can have their own switching parameter τ i (i = 1,⋅⋅⋅,100). Then, the optimization variables to express this problem are departure date, flight time t f, initial attitude of the E-sail (α 0,δ 0), and the switching parameters τ i (i = 1,⋅⋅⋅,100). To optimize the E-sail trajectory, we use the following objective function and constraints.
$$\begin{array}{@{}rcl@{}} \text{Maximize:} & |\Delta \zeta| \end{array} $$
(15)
$$\begin{array}{@{}rcl@{}} \text{subject \hspace{2mm} to:}&\left\{ \begin{array}{r c l} d_{\text{sa}}&=&0\\ 0\leq &\tau_{i}& \leq 1 \hspace{2mm} (i=1, \cdot \cdot \cdot, 100)\\ -90^{\circ} \leq &\alpha_{0}& \leq 90^{\circ} \\ 0^{\circ}\leq &\delta_{0}& \leq 180^{\circ} \end{array} \right. \end{array} $$
(16)
where d sa is a final boundary constraint which expresses the distance between the spacecraft and the asteroid at the end of the flight. As a baseline for the mission, initial values of τ i were generated with the LSL for increasing eccentricity [34]. The problem was solved using the successive quadratic programming (SQP) method and the spacecraft trajectory (i.e., from the spacecraft’s departure to the projectile’s impact) was optimized for Eqs. 15 and 16. The projectile’s impact with the asteroid is usually planned to occur near the asteroid’s perihelion in order to achieve maximum deflection [30]. In this paper, however, the impact position is left free.

Trajectory Optimization for Deflecting Three Fictional Asteroids

The optimal trajectories to deflect three fictional asteroids are described here. Table 3 gives results for the three missions including pertinent mission parameters. Note that the final boundary constraint d sa was satisfied to less than 1,000 km, for every result.
Table 3

Optimization results for three fictional asteroid impact scenarios

 

FA I

FA II

FA III

Departure date

3 January 2015

3 January 2015

14 February 2015

Impact date

15 July 2021

17 December 2018

30 September 2018

Flight time t f, days

2,385

1,445

1,324

Deflection distance Δζ, km

17,070

17,300

2,936

Initial attitude α 0, deg

7.114

32.97

18.23

Initial attitude δ 0, deg

47.57

27.61

21.18

Impact velocity, km/s

23.84

21.27

8.509

The optimal trajectory for FA I is shown in Fig. 10. A two-dimensional plot (Fig. 10a), the time history of the switching parameter (b), and time histories of the propulsive components normalized with the characteristic acceleration S/a 0, T/a 0, and W/a 0 (c) are shown. The E-sail is launched on 3 January 2015 and impacts FA I on 15 July 2021, after a flight time of 2,385 days. At the impact point of the projectile, the true anomaly of FA I is around -60 . We can say that the impact point is relatively near the perihelion of FA I, and this is preferred for increasing the resulting deflection distance Δζ. The impact velocity is 23.84 km/s and this yields a deflection distance Δζ = 17,070 km on the b-plane at the predicted Earth impact date. For FA I, the scaled Earth radius on the b-plane R b is 10,560 km and thus the resulting Δζ corresponds to 162 % of R b. The deflection achieved is well above the value required to deflect FA I from the collision trajectory. The impact geometry |V sav ast|= 472.0 km 2/s 2, and this represents the change in the asteroid’s orbital energy. Figure 10b describes the time history of the switching parameter τ. Areas filled with gray indicate the regions in which the E-sail transverse thrust component T becomes negative when τ is not zero. The switching parameter τ is changed every 24 days to guide the spacecraft to the impact point. The time histories of the thrust components shown in Fig. 10c show that the radial thrust S makes the largest contribution to the trajectory.
Fig. 10

The optimal trajectory to deflect FA I

We also solve the problem with lower values of spacecraft characteristic acceleration a 0, which are 0.1 mm/s 2 and 0.25 mm/s 2. When we consider a 0 of 0.1 mm/s 2, flight time becomes 2,391 days, impact geometry 235.4 km 2/s 2, and deflection distance 8,493 km. In the second case, we consider a 0 of 0.25 mm/s 2, flight time becomes 1,489 days, impact geometry 262.3 km 2/s 2, and deflection distance 9,256 km. In both of two cases, the deflection distances are reduced as the value of impact geometry decreases. On the other hand, to increase such impact geometry and deflection distance, flight time tends to be lengthened. From these two results, we can say that it is also possible to design asteroid deflection missions with smaller a 0; while, some considerable attention to deal with long flight time and strategies to increase the deflection distance would be required.

The next case is the optimized trajectory for deflecting FA II, which has an inclined orbit. The results are summarized in Fig. 11. The E-sail is launched on 3 January 2015 and will impact FA II on 17 December 2018, after a flight time of 1,445 days. The perihelion becomes smaller than about 0.7 AU, and this maneuver increases the eccentricity to 0.4824. In Fig. 11b, the time history of τ is given. The dotted line in Fig. 11b indicates the ON/OFF switching pattern of the LSL for increasing eccentricity, which is in good agreement with the optimal τ. The velocity of the E-sail increases with the eccentricity so this is an efficient method for increasing impact velocity. As the E-sail thrust becomes greater when the spacecraft is closer to the Sun, the magnitudes of thrust components are larger than those for the case of FA I (see Fig. 11c). As a result, the impact velocity becomes 21.27 km/s and this yields a deflection distance of Δζ = 17,300 km on the b-plane. The impact geometry, which indicates the change in the asteroid orbit energy, is |V sav ast| = 322.4 km 2/s 2. Because the scaled Earth radius on the b-plane was R b=8,123 km for FA II, the deflection achieved is well above the value required to deflect FA II from the collision trajectory. In this case, the out-of-plane motion of the E-sail, is also significant (Fig. 12). The optimized initial sail angles are (32.97 , 27.01 ) and a thrust component normal to the orbital plane W is generated, as shown in Fig. 11c. A three-dimensional representation of the orbits tells a better story of this mission: in Fig. 12, the optimal trajectory, the asteroid’s orbit, and the normalized thrust vectors are illustrated.
Fig. 11

The optimal trajectory to deflect FA II

Fig. 12

Three-dimensional orbit diagram for deflecting FA II

Note that to enhance the clarity of the figure, the z-axis is scaled to be about 5 times larger than the x and y-axes. In Fig. 12, thrust vectors mainly orient toward the positive z direction, and the orbital inclination is increased to 6.1 . Looking at the last minutes before the projectile’s impact, FA II is travelling in the + z direction, and the E-sail is travelling downward. At the impact, E-sail and asteroid’s z velocities are -1.457 km/s and 3.539 km/s respectively and, looking only at motion along the z-axis, it can be regarded as a head-on collision. To deflect FA II, increasing both eccentricity and inclination are important for an E-sail KEI. In Table 4, we also summarize some other optimal mission scenarios for deflecting FA II for launches after 2015. We can select appropriate mission design, considering situations like different required deflection distance or remaining time for the mission. These solutions also indicate that it is possible to obtain various solutions for other scenarios.
Table 4

Possible mission designs to deflect FA II

Departure

Impact

Deflection Δζ, km

3 January 2016

13 November 2020

12,100

22 April 2017

12 August 2020

7,540

9 January 2018

8 October 2022

9,973

12 June 2019

17 July 2022

8,308

9 February 2020

14 September 2024

5,452

13 March 2021

13 August 2024

6,034

Here, we describe the KEI strategy for FA III, which belongs to the Aten asteroids group. Figure 13 shows the optimized results. The E-sail is launched on 14 February 2015 and will impact on 30 September 2018, after a flight time of 1,324 days. As shown in Fig. 13a, the impact point is quite near the aphelion of FA III and the axis of the impact trajectory is almost parallel to the asteroid’s orbit. Aten asteroids have a semi major axis smaller than 1 AU, and the perihelion of FA III is 0.42 AU from the sun. To achieve the preferred impact near the perihelion, the E-sail needs to be strongly decelerated by using a negative value of the transverse thrust T. Since FA III’s orbit is inclined at 6.7 deg from the ecliptic plane, a thrust component W normal to the orbital plane is also needed (see Fig. 13c). However, the thrust direction and magnitude are constrained by the position of the E-sail, so it is a more practical option for this case to impact near the aphelion of FA III which is 1.1 AU from the sun. The impact velocity of 8.509 km/s and impact geometry |V s a v a s t | = 155.0 km 2/s 2 yield an achievable b-plane deflection of Δζ = 2,936 km, which is 30.39 % of R b. To illustrate the significance of the z motion of the E-sail, a three-dimensional orbit representation is provided in Fig. 14.
Fig. 13

The optimal trajectory to deflect FA III

Fig. 14

Three-dimensional orbit diagram for deflecting FA III

For clarity, the z-axis is scaled to be about ten times larger than the x and y-axes. Throughout the mission, thrust vectors are oriented in the out-of-plane direction and the orbital inclination gradually increases to 3.8 . At the impact point, FA III is travelling downward and the E-sail is travelling up so as to achieve a z-axis head-on collision.

Compared with the cases of FA I and FA II, making the deflection distance of FA III greater than the scaled Earth radius R b is clearly more challenging. However, E-sail KEIs may become useful by designing an improved threat mitigation campaign. For instance, multiple impacts using several E-sail spacecraft could increase the deflection distance. Its usefulness also depends on the asteroid’s distance of approach to the Earth. If the asteroid doesn’t pass through the center of the Earth, the required deflection would be smaller than R b. In addition, to deflect an asteroid away from a gravitational keyhole, the order of magnitude of the deflection distance would be a few kilometers, which would be achievable with an E-sail KEI. In Table 4, optimal mission designs with small Δζ are shown for launches after 2015. These solutions show the feasibility of performing such practical missions with an E-sail KEI (Table 5).
Table 5

Possible mission designs to deflect FA III

Departure

Impact

Deflection Δζ, km

29 December 2015

5 August 2020

2,341

2 February 2017

22 May 2019

2,466

28 January 2018

22 May 2019

2,508

19 February 2019

8 May 2021

1,600

3 March 2020

20 April 2023

679.5

23 January 2021

25 June 2024

1,307

In three hypothetical mission cases, the KEI strategy produced inclined orbits, resulting in a head-on collision along the z-axis, by increasing the orbital inclination. The results showed the existence of an optimal mission strategy in three fictional asteroid deflection scenarios and proved the feasibility of using E-sail KEIs to perform practical asteroid deflection .

Comparison Between E-sail and Solar Sail KEIs

The performance of the E-sail KEI is compared with that of a photonic solar sail KEI. A solar sail uses solar radiation pressure to produce an acceleration. Because it also allows a spacecraft to perform high-energy orbit transfers [19], a solar sail can be regarded as a viable option for accelerating a projectile. Inserting a few hundred kilograms of projectile into a retrograde orbit with a cranking maneuver has been proposed as an advanced strategy for a solar sail KEI. A head-on collision, which is produced by the retrograde orbit, would achieve high relative impact velocities up to 80 km/s. This strategy has been optimized for the same asteroid impact scenarios and the results showed that it might be the best solution for increasing impact velocity and deflection distance. However, a highly advanced solar sail and a spacecraft system with a long lifetime in the region close to the Sun would be required. In addition, the retrograde orbit will increase the projectile’s velocity relative to the asteroid, which will make it difficult to ensure the projectile collides with the asteroid. Hence, we consider only a prograde orbit in this paper. For the sake of conciseness, we assumed a perfectly reflecting, flat, and square solar sail. Perfectly reflecting sail contains an ideal solar radiation pressure force model in which the thrust force is always along the sail normal direction. This model provides a simple analytical expression of solar sail. We also assumed that the solar sail is moving only under the influence of the Sun’s gravity and solar radiation pressure and that the solar sail’s attitude can be changed instantaneously. The optimization parameters are the Earth departure date, flight time, and sail attitudes, which are defined in Ref. [19] as sail cone and clock angle and which are absent in E-sail maneuvering. The mass of projectile was set to 1,000 kg. By using the same projectile mass for both E-sail and solar sail KEI, a reasonable comparison of the performances of these two propulsion systems can be performed. We assume three sizes of solar sail: 40 m × 40 m, 80 m × 80 m, and 160 m × 160 m. The mass properties of these solar sails are provided in Refs. [6, 32]. Now, our main aim is to conduct a parametric comparison to analyze the difference between an E-sail and different sizes of solar sail.

Nine optimal solutions are provided in Fig. 15. As is clearly shown in Fig. 15, the performance of a solar sail as a KEI strongly depends on the size of the sail, and the type of asteroid. The results for FA I and FA II, belonging to the Apollo group, show that an E-sail with a 0=0.5 mm/s 2 is comparable in performance to a 160 m × 160 m solar sail, whose characteristic acceleration is only about 0.20 mm/s 2. Since it is easier to change the thrust direction for a solar sail than for an E-sail [33], when their characteristic accelerations are of the same order, the optimal solar sail solutions are better than those of the E-sail in terms of achievable deflection. On the other hand, when deflecting the Aten asteroid FA III, the 160 m × 160 m solar sail achieved a much higher deflection distance of 6,018 km on the b-plane. This solar sail achieves an impact at a distance of 0.4961 AU from the Sun, and yields the impact geometry |V s a v a s t |= 485.4 km 2/s 2. In this case, it can also be said that the performance of the 80 m × 80 m solar sail is comparable to that of the E-sail considered in this paper. The characteristic acceleration of this solar sail is only about 0.056 mm/s 2.
Fig. 15

Optimal solutions for deflecting fictional asteroids with a square solar sail KEI

These intriguing results emphasize the difference in the relations between the magnitude of the thrust acceleration and distance from the Sun. The solar sail’s thrust varies as the inverse of the squared distance from the Sun, while the E-sail’s thrust varies as the inverse of the distance. Thus, when the spacecraft spirals outward, E-sail thrust experiences a smaller reduction of thrust than the solar sail, and when the spacecraft spirals inward, solar sail thrust has a larger increase. For the preferred projectile impact on FA III, the spacecraft should be decelerated and approach near the perihelion, which is about 0.42 AU from the Sun. As a result, the solar sail achieves a better result when the FA III is the target. When FA I and FA II are considered as targets, the projectile’s impacts occurred at around 1 AU from the Sun. In this region, an E-sail can make the best use of its acceleration performance. Thus the benefit of using an E-sail as the propulsion system for a KEI can be gained when Apollo asteroids are the intended target.

Conclusion

In this paper, the feasibility of an E-sail KEI was investigated. To evaluate the relation between the E-sail thrust and attitude, thrust vectors for every conducting tether were calculated. By fitting the obtained results with polynomial equations, analytic expressions for the thrust direction and magnitude were provided. In addition, to reduce the energy required for changing the direction of the thrust, an orbital maneuver fixing the E-sail attitude to the inertial frame and changing only the tether voltage was proposed.

The parameters for three fictional asteroid deflection missions were found by maximizing the deflection distance at the predicted Earth collision date. For two Apollo asteroids, the E-sail KEI achieved a deflection distance comparable to the Earth radius. Though the achievable deflections for an Aten asteroid were relatively small compared with the Earth radius, they would be practical for multiple impact or gravitational keyhole deflection missions.

Furthermore, a comparison between KEIs using the E-sail and various sizes of solar sail was performed. An E-sail with a characteristic acceleration of 0.5 mm/s 2 was able to achieve an asteroid deflection comparable that of a 160 m × 160 m perfectly reflecting solar sail when the target is an Apollo asteroid. On the other hand, since the E-sail thrust increases by a smaller amount than that of a solar sail when the spacecraft gets closer to the sun, the asteroid deflection achievable by an E-sail is comparable with that of a 80 m × 80 m solar sail for Aten asteroids. The results obtained here show that it is feasible to use an E-sail KEI to deflect asteroids threatening to collide with the Earth, especially asteroids of the Apollo type.

One problem that must be considered for the E-sail KEI is the terminal guidance with impulsive thrusts. Since the condition of the solar wind changes dynamically during the mission, the terminal guidance to cancel the accumulated position and velocity error has to be performed prior to the impact.

Footnotes

Notes

Acknowledgements

The present study was supported by JSPS KAKENHI Grant Number 15J08268.

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Copyright information

© American Astronautical Society 2016

Authors and Affiliations

  1. 1.Division of Creative Research and Development of Humanosphere, Research Institute for Sustainable HumanosphereKyoto UniversityGokasho, Uji-cityJapan

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