Astrophys Space Sci (2015) 355:213–223 DOI 10.1007/s10509-014-2165-7

O R I G I N A L A RT I C L E

Equilibria near asteroids for solar sails with reflection control devices Shengping Gong · Junfeng Li

Received: 29 April 2014 / Accepted: 16 October 2014 / Published online: 5 December 2014 © Springer Science+Business Media Dordrecht 2014

Abstract Solar sails are well-suited for long-term, multipleasteroid missions. The dynamics of solar sails near an asteroid have not yet been studied in detail. In this paper, outof-plane artificial equilibria in a Sun-asteroid rotating frame and hovering points in a body-fixed rotating frame are studied (using a solar sail equipped with reflection control devices). First, the dynamics and the stability of out-of-plane artificial equilibria are studied as an elliptical restricted three body problem. Next, the body-fixed hovering problem is discussed as a two-body problem. Hovering flight is only possible for certain values of the latitude of the asteroid’s orbit. In addition, the feasible range of latitudes is determined for each landmark on the asteroid’s surface. The influence of the sail lightness number on the feasible range is also illustrated. Several special families of hovering points are discussed. These points include points above the equator and poles and points with an altitude equal to the radius of the synchronous orbit. In both of these types of problems, the solar sail (equipped with reflection control devices) can equilibrate over a large range of locations. Keywords Solar sail · Reflection Control Device · Asteroid · Artificial equilibrium · Body-fixed hovering 1 Introduction Solar sails were initially proposed as long ago as the 1920s to use solar radiation pressure (SRP) to propel a spaceS. Gong (B) · J. Li School of Aerospace, Tsinghua University, Beijing, 100084, China e-mail: [email protected] J. Li e-mail: [email protected]

craft by means of a large, deployable reflective membrane. However, the concept has only become practical recently because of advancements in micro-technologies and thin membranes (Curtis et al. 2000). After having being studied for several decades, solar sails were proven to be useful for some mission applications (Heiligers et al. 2012; McInnes et al. 2001; Macdonald et al. 2006). One such application is for multiple-objective missions. Several previous studies have investigated the use of solar sails for multipleasteroid exploration. One such mission proposed by the National Aeronautics and Space Administration (NASA) is the Prospecting Asteroid Mission (PAM). PAM uses swarms of very small spacecraft equipped with large segmented solar sails to explore asteroids in the main belt (Curtis et al. 2003). The designs may lead to flight times from Earth to the main belt on the order of 2.5 years. Another similar mission is the Multiple Near Earth Objects Rendezvous Mission (MNEORM) proposed by the European Space Administration (ESA) and the German Aerospace Center (DRL) using Gossamer Roadmap Technology (Dachwald et al. 2013). It takes less than 10 years to rendezvous with three NEOs using a solar sail with ac = 0.3 mm/s2 . Interplanetary transfer trajectories of solar sails have been studied extensively (Zhukov and Lebedev 1964; Dachwald 2004; Gong et al. 2011). However, the behavior of solar sails near asteroids has not been studied in detail. There are two main categories of operations of solar sails near asteroids. The first is to equilibrate at artificial Lagrange points in a Sunasteroid restricted three body problem (RTBP). The second is to hover above a point on the asteroid’s surface in a bodyfixed reference frame. Morrow et al. (2001) and Williams and Abate (2009) studied artificial equilibria near the asteroid in a circular restricted three body problem (CRTB). The out-of-plane artificial equilibria in the elliptic restricted three body problem

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(ERTBP) were rarely studied. Baoyin and McInnes (2006) studied the planar artificial equilibria in the ERTBP. The results indicated that a fixed-area solar sail was incapable of out-of-plane equilibria because three free parameters are required to achieve time-varying control force in the out-ofplane direction. However, a solar sail with a fixed area can only provide two free parameters. Many asteroids have considerable eccentricities and the ERTBP is necessary to study the dynamics of a spacecraft near the asteroids. The out-ofplane equilibria of a solar sail will be studied in this paper. Rather than maintaining a fixed position in the Sunasteroid rotating system, the hovering problem requires the solar sail to maintain a fixed position relative to a landmark on the surface of a rotating asteroid. This motion can prolong the observation of a specific area on the surface. Similarly, a body-fixed hovering operation cannot be achieved by a solar sail with a fixed area. Hovering flight near an asteroid has been studied for a spacecraft equipped with a continuous thrust propulsion system (Sawai et al. 2002; Broschart and Scheeres 2005). Recently, Williams and Abate (2009) used a furlable solar sail whose area can be adjusted to study the hovering flight problem. However, the sunlight vector was assumed to be constant in the body-fixed rotating frame (Williams and Abate 2009). In reality, the sunlight vector will change as the asteroid rotates around the Sun. Therefore, their constant-vector assumption makes their conclusions invalid. In reality, permanent body-fixed hovering flight is impossible. Even for a furlable solar sail, only short-term body-fixed hovering flight is possible at certain positions of the asteroid with respect to the Sun. This will be illustrated in this paper. For out-of-plane equilibria in the ERTBP and in the bodyfixed hovering problem, the solar sail should provide control forces with three degrees of freedom. One way to accomplish this is to change the sail area, which is difficult in engineering practice. In this paper, a solar sail equipped with a reflection control device (RCD) is used to achieve this purpose. The successful use of an RCD was demonstrated during a mission of the Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) (Tsuda et al. 2011). The RCD was mounted on the edge of the sail membrane to generate a torque by changing the induced force on each small element’s surface by switching between on and off states (Mimasu et al. 2011). The idea of using the RCD for orbit control and attitude-orbit control was discussed in the GeoSail Mission (Mu et al. 2013, 2014) and the Artificial Lagrange Point Mission (Aliasi et al. 2013; Gong and Li 2014b). In this paper, the artificial equilibria and the body-fixed hovering problem of a solar sail are revisited. For the artificial equilibria, the equation of motion of a solar sail with RCDs is derived in the Sun-asteroid Hill ERTBP. The boundaries of the allowed region (for the artificial equilib-

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ria) are identified for different reflectivities. The sail lightness numbers required by different equilibria in the allowed region are given for different sized asteroids. For the bodyfixed hovering problem, a two-body model is used to derive the equation of motion. The body-fixed hovering problem is discussed using a time-varying sunlight vector. In addition, the feasible range of latitudes of the asteroid’s orbit is obtained analytically based on restriction of the direction of the SRP force. In reality, the magnitude of the SRP force is also restricted because a fixed-area solar sail is considered. The feasible region is obtained numerically, while considering the restrictions on the magnitude and the direction of the SRP force. The influence of the sail lightness number on the boundary of the feasible range is studied. In addition, the solar sail parameters that are required for feasible hovering are also calculated. The results indicate that a hovering altitude that is equal to the synchronous orbit radius is advantageous over other altitudes. This finding is observed because, in that case, the solar sail attitude is independent of the rotation of the asteroid.

2 Equilibria in a non-uniformly rotating frame 2.1 Equation of motion in the elliptic RTBP Consider a Sun-asteroid system and the motion of a massless solar sail. The asteroid revolves in an elliptical orbit around the Sun (under the gravitational attraction of the Sun). The motion of the asteroid in an inertial frame system is described by μS d2 R =− 3R (1) 2 dt R where μS is the gravitational constant of the Sun, and R is the position vector of the asteroid with respect to the Sun. The orbit of the asteroid is a Keplerian orbit because only a two-body Newtonian gravitational force is present. The orbit of the asteroid can be described by its distance from the Sun. R=

a(1 − e2 ) 1 + e cos f

(2)

where a is the semi-major axis of the orbit; e is the eccentricity; f is the true anomaly. The position vectors of the solar sail with respect to the Sun and the asteroid are denoted by R 1 and r, respectively. Then, the equation of motion of the solar sail (in the inertial frame) can be written as μS μP d2 R 1 = − 2 R1 − 3 r + aS 2 dt r R1

(3)

where μP is the gravitational constant of the asteroid and a S is the SRP acceleration.

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Thus, the equation of motion of the solar sail around the asteroid can be obtained (in the inertial frame) as

The SRP accelerations of a 1 and a 2 due to the sail areas of A1 and A–A1 can be written as

d2 r μS μS μP = − 2 R1 + 2 R − 3 r + aS dt 2 R r R1

P A1 (nS · n)nS (8) m P (A − A1 )  a2 = (1 − ρS )(nS · n)nS + 2ρS (nS · n)2 n (9) m where P is the SRP; n is the unit vector along the sail normal direction; and nS is the unit vector along the direction of the sunlight. The magnitude of the SRP force exerted on the sail can be represented by the lightness number, β, which is defined as the ratio of the ideal maximum solar radiation force to the gravitational force of the Sun. The ideal maximum SRP force is calculated with the assumption that all of the sunlight is perfectly reflected and the sail normal is along the direction of the sunlight. The lightness number depends directly on the total mass per unit area of the spacecraft. If the mass-to-area ratio is denoted by σ (g/m2 ), the lightness number can be given by (McInnes 1999)

(4)

This equation of motion will be described in a non-uniformly rotating coordinate system. This is defined as follows: the origin is the mass center of the Sun; the x axis points to the asteroid; the z axis is along the direction of the angular momentum of the asteroid’s orbit; and the y axis forms a right-handed triad with the x and z axes. The xy coordinateplane rotates with a variable angular velocity, in such a way that the asteroid is always on the x axis. We assume that the solar sail is always in the vicinity of the asteroid and that the distance between the solar sail and the asteroid is very small compared to the distance between the Sun and the asteroid. Then, Eq. (4) can be linearized (in the non-uniformly rotating frame) as r¨ + 2ω × r˙ + ω × (ω × r) + ω˙ × r   μS μP μS = − 3 I3 + 3 5 RR T r − 3 r + a S R R r

(5)

where Ik is an identity matrix of k × k and ω is the angular velocity of the asteroid around the Sun. We denote differentiation with respect to f by a prime and define new dimensionless coordinates as x˜ = Rx , y˜ = y z R , z˜ = R . Equation (5) can then be rewritten as (Szebehely 1967)   ⎧ μ 1 ⎪ ⎪ x˜  − 2y˜  = 3 x ˜ − x ˜ + a ˜ ⎪ x ⎪ 1 + e cos f r˜ 3 ⎪ ⎪ ⎪   ⎨ μ 1 − 3 y˜ + a˜ y y˜  + 2x˜  = (6) ⎪ 1 + e cos f r˜ ⎪ ⎪   ⎪ ⎪ μ 1 ⎪ ⎪ ⎩ z˜  + z˜ = − 3 z˜ + a˜ z 1 + e cos f r˜ where μ = μμPS is the dimensionless gravitational constant of the asteroid; a˜ x , a˜ y , a˜ z are the dimensionless SRP accelerations along the x, y, and z axes, respectively. Their expressions are defined below. The SRP force model of a solar sail with an RCD (Gong and Li 2014a) is used to derive the SRP force exerted on the solar sail. The RCD has two states, and all of the sunlight is absorbed in state one. For state two, part of the sunlight is specularly reflected, and the ratio of the reflected sunlight is ρS . The remaining sunlight is absorbed, and the corresponding ratio is 1 − ρS . The RCD is either in state one or state two, with no intermediate state. Therefore, the state switch can be modeled simply as power on or power off. Consider a solar sail with a total area A and mass m. At some instant, A1 of the area is in state one, and the remaining area is in state two. The SRP acceleration exerted on the solar sail can be written as aS = a1 + a2

(7)

a1 =

1.53 (10) σ Using the lightness number, the SRP acceleration can be rewritten as μS β(nS · n) aS = R2  1 × (1 − ρS + υρS )nS /2 + (1 − υ)ρS (nS · n)n (11) β=

where υ is the ratio of the area in state one divided by the total area (υ = A1 /A) and is called the reflectivity rate. Because the distance from the solar sail to the asteroid is very small compared to the distance from the asteroid to the Sun, the distance from the solar sail to the Sun can be approximated by the distance from the asteroid to the Sun, R. In addition, the sunlight vector can be approximated by the direction of the Sun-asteroid line (x axis of the nonuniformly rotating frame). Then, the SRP acceleration in the non-uniformly rotating frame can be written as ⎡ ⎡ ⎤ 1 μS β cos α ⎣ 1 − ρS + υρS ⎣ ⎦ aS = 0 2 R2 0 ⎡ ⎤⎤ cos α (12) + (1 − υ)ρS cos α ⎣ sin α cos γ ⎦⎦ sin α sin γ where the pitch angle α is measured from the Sun-asteroid line to the sail normal vector, and the clock angle γ is the angle between the y axis and the projection of the sail normal vector in the yz plane. In the dimensionless equations, the dimensionless distance between the Sun and the asteroid and the dimensionless gravitational constant of the Sun become unitary. Thus,

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the dimensionless SRP acceleration components along the x, y, and z axes in the non-uniformly rotating frame are ⎧  2 ⎪ ⎨ a˜ x = β cos α (1 − ρS + υρS )/2 + (1 − υ)ρS cos α (13) a˜ y = β(1 − υ)ρS cos2 α sin α cos γ ⎪ ⎩ a˜ z = β(1 − υ)ρS cos2 α sin α sin γ Equation (6) can be written in a vector form as r˜  + 2J r˜  =

1 (∇Ω + a˜ S ) 1 + e cos f

(14)

where the pseudopotential of the problem can be written as μ 1 2 (15) + 3x˜ − (1 + e cos f )˜z2 r˜ 2 The dimensionless distance from the sail to the asteroid is given by  r˜ = x˜ 2 + y˜ 2 + z˜ 2 (16)

Ω=

The coefficient matrix J is given by ⎡ ⎤ 0 −1 0 J = ⎣1 0 0⎦ 0 0 0

anomaly. The reflectivity rate is restricted to vary between 0 and υmax (whose value is determined by the maximum area equipped with the RCD). Therefore, the lightness number should be chosen so as to ensure that the reflectivity rate is within the allowed range when the true anomaly varies from 0 to 2π . To achieve this, the lightness number is determined using the average method. Integrating (18) with respect to f and averaging it, Eq. (18) becomes ⎧ ⎪ ∂ Ω¯ ⎪ ⎪ + a˜ x = 0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎨ ∂ Ω¯ (19) + a˜ y = 0 ⎪ ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂ Ω¯ ⎪ ⎪ ⎩ + a˜ z = 0 ∂z where the average pseudopotential of the problem is  μ 1 2 (20) + 3x˜ − z˜ 2 r˜ 2 For out-of-plane equilibria in the xz plane, Eq. (19) can be written in a scalar form as ⎧   1 ⎪ 2 ⎪ ⎪ β cos α (1 − ρS + υ0 ρS ) + (1 − υ0 )ρS cos α ⎪ ⎪ 2 ⎪ ⎨ μ (21) = 3 x˜ − 3x˜ ⎪ ⎪ r˜ ⎪ ⎪ ⎪ μ ⎪ ⎩ β cos2 α(1 − υ0 )ρS sin α = z˜ + z˜ r˜ 3

Ω¯ = (17)

2.2 Generation of equilibrium points in the non-uniformly rotating frame This paper focuses on the out-of-plane equilibria. An out-ofplane equilibrium point (in the xz plane) has been proposed for planetary pole observation. This cannot be achieved by an equilibrium point in the xy plane. Morrow et al. (2001) studied the out-of-plane equilibria of a fixed-area solar sail in the CRTBP. However, out-of-plane equilibria do not exist for a fixed-area solar sail in the ERTBP. The generation of these out-of-plane equilibria will be studied for a solar sail with RCDs. To obtain an out-of-plane equilibrium point, Eq. (6) must be solved under the condition x˜  = y˜  = z˜  = x˜  = y˜  = z˜  = 0. ⎧ ∂Ω ⎪ ⎪ + a˜ x = 0 ⎪ ⎪ ⎪ ∂ x˜ ⎪ ⎪ ⎨ ∂Ω + a˜ y = 0 (18) ⎪ ∂ y˜ ⎪ ⎪ ⎪ ⎪ ∂Ω ⎪ ⎪ ⎩ + a˜ z = 0 ∂ z˜ Three free variables (of the solar sail) are required to make the solar sail equilibrate at an out-of-plane equilibrium point. This is because the force balance condition in the z direction is time-varying. It has been assumed that the lightness number is constant because changing it is difficult in practice. For a given equilibrium point, the attitude angles and the reflectivity rate can be varied periodically to balance the time-varying force induced by the variation of the true

where υ0 = υmax /2 is the middle value of the reflectivity rate. The analytical solutions are given by   1 k α = sin−1 + tan−1 k (22) √ 2 (1 − υ0 )ρS 1 + k 2   μ Ω¯ z z˜ β = 3 +1 where k = . r˜ cos2 α(1 − υ0 )ρS sin α Ω¯ x (23) The solar sail parameters can be determined step by step over one orbit of the asteroid around the Sun. First, the middle value of the reflectivity rate, υ0 = υmax /2, is used to calculate the lightness number using Eq. (19). Then, the lightness number is fixed to calculate the reflectivity rate and the attitude angles for different true anomalies using Eq. (18). The three algebraic equations can be solved by a process of iteration with the solution of Eq. (19) as the initial value. This numerical method can be used to determine the exact boundary of the allowed region (where out-of-plane equilibrium exists). The solar sail parameters are calculated over one orbit of the asteroid around the Sun. If the pitch angle is greater than 90 degrees, or the reflectivity rate is out of its allowed range, the equilibrium is out of the allowed region. Otherwise, it located in the allowed region. The results indicate that the boundary determined from the average Eq. (19)

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Table 1 Parameters of two asteroids ϕe (deg)

Semi-major axis (AU)

Eccentricity

Inclination (deg)

Argument of perigee (deg)

315

74

2.7668

0.07580

10.593

80.3276

99

−3

1.2485

0.3249

15.3249

8.9666

Name

λe (deg)

Ceres 1992 SK

RAAN (deg)

Mass (kg)

Rotation Period (hr)

72.2921

9.4300e20

9.0742

233.5510

2.1700e12

7.3190

Fig. 1 Boundaries in the xz plane for different reflectivity rates

Fig. 2 Pitch angle and reflectivity over one orbit of the primary

provides a very good approximation to the exact boundary. The variations of the solar sail parameters induced by the true anomaly variation are very small. This means that a point is usually within the allowed region determined by Eq. (18) if it is in the allowed region determined by Eq. (19). Only those points that are very close to the boundary that is determined by Eq. (19) may be out of the allowed region. Such points usually require a pitch angle that is close to 90 degrees. Therefore, the average equation is used to evaluate the boundary of the allowed region. It can be seen from Eq. (19) that the boundary depends on the allowed maximum value of the reflectivity rate and is independent of the eccentricity of the asteroid’s orbit. A main-belt asteroid and a near-Earth asteroid are used to illustrate the boundaries of the allowed regions in the xz plane. The parameters of two selected asteroids are obtained from the JPL Small-Body Database Browser. These are listed in Table 1. The parameters include the latitude φe and the longitude λe of the spin axis of the asteroid, the classical orbital elements and the spin period of the asteroid. New variables, xP = xh ˜ s and zP = z˜ hs , are defined to illustrate the problem more clearly. hs is the synchronous orbit

radius that is defined by hs = ( μωP2 )1/3 , where ωa is the spin a rate of the asteroid. Figure 1 presents the boundaries of the allowed regions for different reflectivity rates. The boundary is determined by the restriction of the direction of the SRP force. For an ideally reflective solar sail, the direction of the SRP force may change in half of the space. If absorption is considered, the direction will be restricted in a cone. As the reflectivity rate increases, the SRP force component in the sail plane increases, and the cone-apex angle will decrease. Therefore, the allowed region will shrink, as shown in Fig. 1. To verify that the boundary is a good approximation to the exact one, the solar sail parameters are calculated (over one orbit of the asteroid) for a point close to the boundary. This is shown in Fig. 2. The variations of the pitch angle and the reflectivity rate (induced by variation of the true anomaly) depend on the eccentricity of the asteroid’s orbit. The variations are very small for a small eccentricity, as shown in Fig. 2(b). Even for a considerably large eccentricity of 0.3249 (1992SK), the variations are still not too large, as shown in Fig. 2(a). The value of the reflectivity rate is always in the allowed range (0 ∼ υmax ). Therefore, the

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Fig. 3 Contours of the lightness numbers in the allowed regions

boundary based on the average method may be employed for mission design. This is because the points very close to the boundary are seldom used because the required pitch angles are close to 90 degrees. In addition, the boundary also depends on the mass and the semi-major axis of the asteroid. As shown in Fig. 1, the allowed region of Ceres is much larger than that of the 1992SK asteroid. The lightness numbers for different artificial equilibria are represented in Fig. 3. The lightness number decreases as the equilibrium point approaches the classical equilibrium point where no SRP force is required. The lightness number also decreases as the gravitational constant of the asteroid decreases. For a small asteroid, a very small lightness number is enough to achieve the equilibrium point. Considering the 1992SK asteroid for example, a solar sail with a lightness number of approximately 1e-5 can equilibrate at certain points near the asteroid (less than 100 kilometers) in the allowed region. Based on Eq. (10), the corresponding mass-to-area ratio for a lightness number equal to 1e-5 is approximately 153 kg/m2 . This is feasible for a deep space explorer. 2.3 Stability of equilibria in the non-uniformly rotating frame The linear stability characteristics of the equilibria can be studied in the usual manner by linearizing the dynamical equations around the equilibria. A perturbation S = [d T d  T ]T is added to the equilibria. Then, the variational equation of Eq. (14) can be obtained as   ∂(∇Ω) ∂ a˜ S ∂ a˜ S 1   d = −2J d + + d+ δn 1 + e cos f ∂ r˜ ∂ r˜ ∂n (24) The variation of the sail normal vector, δn, is independent of the state variables, and it is determined by the attitude control strategy. Different attitude control strategies lead to different behaviors of the stability. Several simple attitude control strategies were used to study the stability of relative motion around displaced solar orbits (Gong et al. 2007). The first one is that the sail normal vector follows its reference

value in the non-uniformly rotating and pulsating coordinate system, i.e., δn = 0. The second one is that the attitude angles follow their reference values. In this case, the variation of the normal vector can be approximated in a feedback form. δn =

∂n d ∂ r˜

(25)

For both cases, the variational equation is rewritten in a unified form as S  = N(f ) · S

(26)

The coefficient matrices of the two cases are given by  N(f ) =  N(f ) =

03×3 ∂(∇Ω) ∂ r˜

+

I3 ∂ a˜ S ∂ r˜



I3

03 ∂(∇Ω) ∂ r˜

+

∂ a˜ S ∂ r˜

(27)

−2J +

∂ a˜ S ∂n ∂n ∂ r˜



−2J

(28)

N (f ) is evaluated at the periodic orbit, where 03 is a zero matrix of 3×3. Because matrix N(f ) is time-dependent and periodic, the problem becomes a linear system with periodic coefficients. Floquet theory shows that the fundamental solution of such a system can be given by S(f ) = Φ(f, f0 )S(f0 )

(29)

Recasting the variational equations in terms of the state transition matrix and substituting Eq. (29) into Eq. (26), we have Φ  (f, f0 ) = N(f )Φ(f, f0 ),

Φ(f0 , f0 ) = I6

(30)

Φ(f, f0 ) is the state transition matrix, and the eigenvalues of Φ(2π, 0) tell us about the linear orbital stability of the equilibrium point. Numerical analysis is used to calculate the eigenvalues of Φ(2π, 0). The numerical results indicate that the periodic orbits are unstable for both cases. Active control of artificial equilibria has been studied in the SunEarth system (Gong and Li 2014b). The equilibrium can be stabilized using either two attitude angles or the reflectivity rate as the control input.

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3 Equilibria in a body-fixed rotating frame The hovering problem requires the solar sail to maintain a fixed position relative to a point on the surface of a rotating asteroid. A body-fixed frame is required to describe the body-fixed hovering problem. The origin of this frame is located at the mass center of the asteroid. The zb axis is along the spin axis of the asteroid, and the xb and yb axes are along the equator of the asteroid. Hovering is only possible at those points that lie above the asteroid equator at the synchronous orbit radius (Broschart and Scheeres 2005) and if no extra forces other than the gravitational force are exerted on the spacecraft. Williams and Abate (2009) studied the body-fixed hovering problem using a furlable solar sail. However, the literature assumes that the sunlight vector is constant in the body-fixed rotating frame. In fact, the sunlight vector changes periodically in the body-fixed rotating frame as the asteroid rotates the Sun. In this section, the body-fixed hovering problem is revisited by considering a time-varying sunlight vector. 3.1 Feasible regions for the body-fixed hovering problem The hovering point is usually very close to the asteroid, and the solar gravitational force is negligible. Only the gravitational force of the asteroid and the SRP force are considered. The equation of motion in the body-fixed frame are given by μP (31) r¨ + 2ωa J r˙ + ωa × (ωa × r) = − 3 r + a S r where ωa is the angular velocity vector of the asteroid’s rotation; J is given by Eq. (17); and the SRP acceleration a S is given by Eq. (12). The SRP acceleration required to generate an equilibrium point in the body-fixed frame is given by μP (32) a R = ωa × (ωa × r) + 3 r r Equation (32) can be projected in the non-uniformly rotating frame   T  μP a Ro = ωa × (ωa × r) + 3 r = aRx aRy aRz r o (33) The subscript ‘o’ denotes the projection of a vector in the non-uniformly rotating frame. The SRP force is restricted and cannot be directed toward the Sun. The component of the SRP force along the Sun-asteroid line is always positive. Therefore, the condition for feasible hovering flight can be given by aRx > 0

(34)

Two coordinate transition matrices are defined to obtain the expression for aRx . The first matrix is for the transition from the body-fixed frame to the inertial frame, and the second

one is for the transition from the inertial frame to the nonuniformly rotating frame. The spin vector of the asteroid (in the ecliptic inertial frame) is described by two angles, the latitude φe and the longitude λe . Consider the sequence C 3 (−θ ) ← C 2 (φe − π/2) ← C 3 (−λe ) from the body-fixed reference frame to the inertial frame. The corresponding rotation matrix is Ab2I = C 3 (−λe )C 2 (φe − 90◦ )C 3 (−θ ) ⎡ cos λe sin φe cos θ − sin λe sin θ = ⎣ sin λe sin φe cos θ + cos λe sin θ − cos φe cos θ − cos λe sin φe sin θ − sin λe cos θ − sin λe sin φe sin θ + cos λe cos θ cos φe sin θ

⎤ cos λe cos φe sin λe cos φe ⎦ sin φe (35)

where θ = ωa t is determined by the self-rotation of the asteroid. Similarly, the rotation matrix from the non-uniformly rotating frame to the inertial frame can be obtained by considering the rotation sequence C3(u) ← C1(i) ← C3(Ωe ). AI 2o = C 3 (u)C 1 (i)C 3 (Ωe ) ⎡ cos u cos Ωe − sin u cos i sin Ωe = ⎣ − sin u cos Ωe − cos u cos i sin Ωe sin i sin Ωe cos u sin Ωe + sin u cos i cos Ωe − sin u sin Ωe + cos u cos i cos Ωe − sin i cos Ωe

⎤ sin u sin i cos u sin i ⎦ cos i (36)

where u, i, and Ωe are the latitude, the inclination, and the right ascension of the ascending node of the asteroid’s orbit around the Sun. Then, the transition matrix from the body-fixed frame to the non-uniformly frame can be written as ⎡ ⎤ a11 a12 a13 Ab2o = AI 2o Ab2I = ⎣ a21 a22 a23 ⎦ (37) a31 a32 a33 The expressions for the elements of the matrix can be obtained by substituting Eqs. (35) and (36) into Eq. (37). The hovering point above a fixed surface point on the asteroid can be described by a hovering altitude h and the desired latitude φ0 and longitude λ0 . The xb axis of the bodyfixed frame is defined so that the hovering point is located on the xb axis. T  r b = h cos φ0 0 sin φ0 (38) The subscript ‘b’ denotes the projection of a vector in the body-fixed frame. Using these definitions, the hovering

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point in the non-uniformly rotating frame can be represented by ⎤ ⎡ a11 cos φ0 + a13 sin φ0 r o = Ab2o r b = h ⎣ a21 cos φ0 + a23 sin φ0 ⎦ (39) a31 cos φ0 + a33 sin φ0 The x-component of the vector can be obtained by substituting Eqs. (37) and (38) into Eq. (39). Similarly, ωa × (ωa × r) can be represented in the bodyfixed frame as   T ωa × (ωa × r) b = −ωa2 h cos φ0 0 0 (40) This can be projected in the non-uniformly rotating frame as  T  ωa × (ωa × r) o = −ωa2 hAb2o cos φ0 0 0  T = −ωa2 h cos φ0 a11 a12 a13 (41) Substitute Eqs. (39) and (41) into Eq. (33) and rearrange it. Then, one has  aRx = hωh2 (c1 cos u + c2 sin u) cos θ + (c3 cos u + c4 sin u) sin θ + c5 cos u + c6 sin u  where ωh = μP / h3 . (42) The coefficients are determined by the spin vector of the asteroid, the orbital elements of the asteroid’s orbit, and the latitude of the hovering point.   ωa2 c1 = 1 − 2 cos φ0 sin φe cos Ω1 (43) ωh   ω2 c2 = − 1 − a2 [cos φ0 cos i sin φe sin Ω1 ωh + cos φ0 sin i cos φe ]   ω2 c3 = 1 − a2 cos φ0 sin Ω1 ωh   ωa2 c4 = 1 − 2 cos φ0 cos i cos Ω1 ωh c5 = sin φ0 cos φe cos Ω1

(44)

c6 = −(sin φ0 cos i cos φe sin Ω1 − sin φ0 sin i sin φe )

(48)

Ω1 = Ωe − λe

(49)

Based on Eq. (42), the solution of aRx > 0 can be obtained as ⎧ 2 2 ⎪ ⎨ (c3 cos u + c4 sin u) + (c1 cos u + c2 sin u) (50) < (c5 cos u + c6 sin u)2 ⎪ ⎩ c5 cos u + c6 sin u > 0 The solution set can be represented by the intersection of two sets. Π(u) = Π1 (u) ∩ Π2 (u) The two solution sets are given by    Π1 (u) = u  c22 + c42 − c62 tan2 u + 2(c1 c2 + c3 c4 − c5 c6 ) tan u    + c12 + c32 − c52 < 0 Π2 (u) = {u | c5 cos u + c6 sin u > 0}

(46) (47)

Next, we need to solve the inequality aRx > 0. Equation (42) includes two time-varying variables, θ and u, which are determined by the asteroid’s rotation and revolution around the Sun, respectively. For a specific hovering point above an asteroid, the coefficients are constants. The right hand side of Eq. (42) always crosses zero when u varies over one orbit of the asteroid. Therefore, a permanent body-fixed hovering point does not exist for a solar sail. However, short-term body-fixed hovering is possible. Because u usually varies at much slower rate than θ , it can be regarded as being constant for a short-term problem. The problem then becomes one of searching for the range of u that satisfies aRx > 0.

(52) (53)

The first solution set is defined by a quadratic inequality. The coefficients of the quadratic polynomial are constants. The graph of the quadratic function is a parabola that opens upward if the leading coefficient c22 + c42 − c62 > 0 or that opens downward if c22 + c42 − c62 < 0. The discriminant of the quadratic equation can be used to check if the quadratic equation has two real roots. This is given by  Δ = 4 (c2 c5 − c1 c6 )2 + (c4 c5 − c3 c6 )2 − (c2 c3 − c1 c4 )2 (54) If the discriminant is positive, there are two real solutions (tan u1 and tan u2 (tan u1 < tan u2 )) for the quadratic equation. The solution set depends on the direction of opening of the parabola. If c22 + c42 − c62 > 0, the solution set is Π1 (u) = {u | tan u1 < tan u < tan u2 }

(45)

(51)

(55)

If c22 + c42 − c62 < 0, the solution set is Π1 (u) = {u | tan u > tan u2 or tan u < tan u1 }

(56)

If the discriminant is negative, there are no real solutions for the quadratic equation. The solution set also depends on the direction of the opening of the parabola. If c22 + c42 − c62 > 0, the solution set is empty. If c22 + c42 − c62 < 0, the solution set is the whole space. The inequality for the second solution set can be rewritten as  c5 c52 + c62 sin(u + us ) > 0 where tan us = . (57) c6 If the solution over one orbital loop of the asteroid is of interest, the second solution set is given by Π2 (u) = {u | −us < u < π − us }

(58)

It can be observed that body-fixed hovering is only allowed during half loops. The range may be further reduced by considering the first solution set.

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221

Fig. 4 Feasible body-fixed hovering region for 1992SK

Fig. 5 Feasible body-fixed hovering region for Ceres

First, three special families of hovering points are analyzed. They are hovering points above the asteroid equator (φ0 = 0), points above the north pole (φ0 = π/2), and points with an altitude equal to the synchronous orbit radius (h = hs ). For the point above the equator, the values of c5 and c6 are zero. The solution set of the inequality (50) is empty. However, the expression in Eq. (42) can be zero when the hovering altitude is equal to the synchronous orbit radius, i.e., h = hs . In this case, the SRP force is zero, and the problem degenerates to a hovering problem of a spacecraft with no SRP force. For the point above the north pole, the values of c1 ∼ c4 are zero. Inequality (50) degenerates to inequality (57), which means that hovering flight above the pole is possible during half periods over one orbital loop. For the point with the synchronous-orbit radius, the values of c1 ∼ c4 are zero, and the problem is similar to that of the point above the pole. Most importantly, the constraints on the SRP force (for the two cases) are independent of the self-rotation of the asteroid because the values of c1 ∼ c4 are zero. This means that the sail attitude does not need to rotate synchronously with the self-rotation of the asteroid. Such synchronous rotation would be difficult for the solar sail because the asteroid usually rotates very quickly. For the general case, the solution set given by Eq. (51) can be obtained for a given hovering altitude and latitude. Similarly, the asteroids 1992SK and Ceres are used to illustrate the feasible solutions for different hovering altitudes and latitudes. These are described by the shadowed areas in Figs. 4 and 5, respectively. The body-fixed hovering altitude is only feasible near the synchronous orbit radius except for the point above the pole. The range of the feasible hovering altitude increases as the hovering latitude increases. The range of feasible latitudes is always less than π , and it increases as the hovering altitude approaches the synchronous orbit radius.

3.2 Solar sail parameters in the feasible region The above analysis only considers the restriction on the direction of the SRP force. In fact, the magnitude of the SRP force is also restricted because the reflectivity rate is restricted to vary between 0 and υmax . Therefore, the solution set may be further reduced by considering the restriction on the reflectivity rate. The rotation angle θ and the latitude u are identified at a given moment in time. The reflectivity rate and two attitude angles can be obtained by combining Eqs. (12) and (32); namely a S = a R . First, the clock angle γ can be obtained from aRz (59) tan γ = aRy It is easy to obtain the reflectivity rate and the pitch angle numerically although it is difficult to obtain an analytical solution. The results indicate that the choice of the lightness number is important to obtain a feasible solution set. Different values of the lightness number lead to different feasible solution sets. It should be noted that the solution of the equation a S = a R can always be obtained numerically. However, if the lightness number is fixed, the reflectivity rate may be negative in certain cases. However, this is physically impossible. Hovering above 1992SK is used to illustrate this problem. The feasible solution space for a hovering point at a latitude of π/3 and an altitude of the synchronous orbit radius is solved numerically. It has been explained above that without any restriction on the lightness number the hovering problem will be feasible during half periods over one orbital loop. The corresponding solution set of the inequality aRx > 0 is 0 < u < π for this point. The solar sail parameters that are required to hover at these points are calculated for different lightness numbers. Only the feasible solution

222

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Fig. 6 Solar sail parameters in the feasible regions for different lightness numbers

is presented in Fig. 6. The clock angle is determined by the hovering point and is independent of the lightness number. The feasible solution set for a certain lightness number is always a subset of the solution set of aRx > 0. The solution sets for β = 0.1276, β = 0.06384, and β = 0.03192 are 0.5256 < u < 2.8176 rad, 0.8226 < u < 2.6466 rad, and 1.4076 < u < 2.2866 rad, respectively. There will be no feasible solution if the lightness number is further reduced to 0.02554. The time spans corresponding to the solution sets are 258.5603, 205.2121, and 94.1970 days, respectively. The bounds of the solution sets shrink as the lightness number decreases. Increasing the lightness number can increase the hovering time. It is known that increasing the lightness number is difficult in engineering practice. In addition, this leads to an increment of the pitch angle, as shown in Fig. 6. The pitch angles for the case of β = 0.1276 stay in the vicinity of 60 degrees. The efficiency of the solar sail will be greatly reduced at this pitch angle. A larger lightness number also requires a larger reflectivity rate and this means that more RCDs are required. Therefore, the lightness number should be carefully chosen to achieve a compromise between prolonging the hovering time and decreasing the pitch angle. The solar sail parameters are independent of the rotation angle of the asteroid because the hovering altitude is equal to the synchronous orbit radius. A different hovering altitude, h = 0.95 hs, is adopted to study the influence of the rotation angle on the feasible solution set and the solar sail parameters. The latitude is still π/3. The solution set of inequality (34) is 0.0006142 < u < 3.0556. Similarly, the feasible solution space will be reduced if the lightness number is fixed. The feasible solution sets for β = 0.1276 and β = 0.06384 are 0.6545 < u < 2.7105 and 0.9735 < u < 2.5145, respectively. There is no feasible solution for the case of β = 0.06384. The solar sail parameters for β = 0.1276 are listed in Fig. 7, and an enlarged view is also presented. It can be observed that the reflectivity rate and the attitude angles oscillate at the frequency of the rotation of the asteroid. This means that the attitude of the solar

Fig. 7 Solar sail parameters in the feasible region

sail has to be adjusted frequently to make hovering possible. Frequent attitude adjustments are difficult for a solar sail due to its large flexible structure. Therefore, hovering at an altitude that matches the synchronous orbit radius is advantageous (over other altitudes) because the required solar sail parameters are now independent of the asteroid’s selfrotation.

4 Conclusions In an elliptical restricted three body problem, a solar sail equipped with reflection control devices can equilibrate at certain points near the asteroid (in a non-uniformly rotating and pulsating frame). The allowed region for out-of-plane artificial equilibria can be approximately evaluated by using the average equilibrium equation. The average equation shows that the allowed region depends on the reflectivity rate, the mass of the asteroid and the semi-major axis of the asteroid’s orbit. The allowed region is almost independent of the eccentricity of the asteroid’s orbit. The solar sail

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lightness numbers that are required to generate different artificial equilibria are also identified. The results indicate that a low-performance solar sail can achieve certain equilibria near a small asteroid. Stability analysis shows that the equilibria are unstable and active control is always necessary to stabilize them. The body-fixed hovering problem is also discussed, using a solar sail equipped with reflection control devices. Permanent hovering is not possible in this situation. The results indicate that less than half of the orbit (of the asteroid around the Sun) is suitable for body-fixed hovering flight. Hovering flight above the equator is only feasible at an altitude that matches the synchronous-orbit radius, and no SRP is required in this case. The solar sail parameters are independent of the asteroid’s self-rotation if the hovering point is above the pole or if the hovering altitude is equal to the synchronous orbit radius. For the general case, the solar sail is only able to hover at an altitude that is near the synchronousorbit radius. The feasible region for hovering flight depends on the lightness number of the solar sail. The feasible region shrinks as the lightness number decreases, and no feasible region exists if the lightness number is reduced to certain value. Acknowledgements This work has been supported by the National Natural Science Foundation of China (No. 11272004 and No. 41174025).

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