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5Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, pp. 521-532. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230801
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 34D20, 34H15, 70F05, 70F07, 70F15
Keeping a Solar Sail near the Triangular Libration Point of a Dumbbell-Shaped or Binary Asteroid
The possibility of keeping a spacecraft with a solar sail near an unstable triangular libration point of a minor planet or a binary asteroid is studied under the assumption that only the gravitation and the solar radiation influence the spacecraft motion. The case where the solar sail orientation remains unchanged with respect to the frame of reference of the heliocentric orbit of the asteroid mass center is considered. This means that the angle between the solar sail normal and ecliptic, as well as the angle between this normal and the solar rays at the current point, does not change during the motion. The spacecraft equations of motion are deduced under assumptions of V.V.Beletsky's generalized restricted circular problem of three bodies, but taking into account the Sun radiation. The existence of a manifold of initial conditions for which it is possible to choose the normal direction that guarantees the spacecraft bounded motion near the libration point is established. Moreover, the dimension of this manifold coincides with that of the phase space of the problem at which the libration point belongs to the manifold boundary. In addition, some proposals for stabilization of the spacecraft motions are formulated for trajectories beginning in the manifold.
Keywords: solar sail, libration point, binary asteroid, three-body problem
1. Introduction
Minor planets are very interesting objects for humanity for a number of reasons. Firstly, some of them can be dangerous for the Earth. Secondly, they are potential sources of mineral raw materials. Thirdly, and it might be pointed out as the most important, sufficiently big asteroids are natural bases on the way to the deep space. Our goal here is not to formulate all advantages from the study of asteroids, but to show that a direct landing on an asteroid surface might be necessary. Such a process is very complex as minor planets have irregular shapes and, as a result, complex spin motions. Now only one or two missions that have included landing on the
Received May 21, 2023 Accepted July 20, 2023
Alexander V. Rodnikov [email protected]
Moscow Aviation Institute (National Research University) Volokolamskoe sh. 4, Moscow, 125993 Russia
A. V. Rodnikov
surface of a minor planet can be considered to be completely successful [10]. As a way out, one could place a space station near an asteroid. Moreover, at first one could place a light research apparatus only. To achieve the goals of an initial study, such an apparatus must be immovable with respect to at least a several points of the asteroid surface. In other words, the apparatus must be placed at a point of relative equilibrium in a rotating frame of reference attached to the asteroid in some way. Such equilibria are called libration points. In fact, the gravitational forces and forces of moving space compensate each other at the libration point. Thus, the number of libration points and its locations are determined by the gravitational potential and spin motion of the minor planet.
To date, dozens and dozens of papers have been published on the modeling of gravitational potentials of minor planets and on the existence and stability of their libration points. Without claiming to mention all relevant literature, we note publications concerning the existence and stability of libration points for asteroids of regular shapes [3, 5, 13] and on various approaches, including numerical ones, to solving the same problems, for asteroids of irregular shapes [1, 20, 21, 25-27].
In this paper, a different approach is taken. If a minor planet is like two balls stuck together, then one can analyze spacecraft motion in its vicinity using equations of motion of the generalized restricted circular problem of three bodies (GRCP3B), formulated by V. V. Beletsky [3]. In this problem an asteroid is presented as a dynamically symmetric rigid body whose motion with respect to its mass center is a regular precession. Also, the gravitational potential of the rigid body is approximated as a composition of the potential of two homogeneous balls or two particles. Examples of suitable asteroids are (4769) Castalia, (216) Kleopatra, (486958) Arrokoth, etc. [8]. A particular case of GRCP3B when a dumbbell-shaped asteroid moves at a right angle of nutation is considered. Note that the spin motion of a binary asteroid can be regarded as a variant of this case if the distance between the components does not change during the motion. Examples are 2017 YE5, (90) Antiopa, etc. [9].
As established in [3], a dumbbell-shaped asteroid has no more than two triangular libration points that are analogs of Lagrangian libration points of the classical restricted circular problem of three bodies (RCP3B) [14, 23]. We restrict ourselves to the case where there exist exactly two triangular libration points which are unstable. One can easily check that the degree of instability in this situation is 2 (see also [4]).
We consider the dynamics of a light spacecraft equipped with a perfect solar sail and placed near a triangular libration point of an asteroid that is dumbbell-shaped or binary. Our objectives can be formulated so as to keep the spacecraft near the unstable libration point using the solar radiation only, i.e., without fuel costs. At the same time, we strive to ensure that the angle between the normal to the solar sail and the direction of the solar rays at the normal application point remains constant throughout the motion, just like the angle between the same normal and the plane of the asteroid mass center orbit.
Estimating forces that influence the spacecraft motion in the asteroid frame of reference, one can check that, if the asteroid length is about 100 km, if its rotation is sufficiently fast, and if the ratio of the solar sail area and the spacecraft mass is close to the same ratio for the already implemented space missions [7, 11], then the influence of solar radiation turns out to be several orders of magnitude greater than the gravitational influence of the Sun and of other objects of the Solar system. Using this fact and taking into account that the force of moving space compensates the asteroid gravitation at the libration point, one can reduce the spacecraft equations of motion to a system of three inhomogeneous linear differential equations of second order with time-periodic right-hand sides that characterize the force of the solar radiation acting
on the solar sail. One of these equations can be solved separately and its solutions are always bounded. The characteristic equation for the system of the remaining two equations has two roots with positive real parts. Hence, the general solution of this system contains terms with exponents in positive powers. We claim that there exists a manifold of initial conditions of dimension 4, at the points of which the direction of the normal to the solar sail can be chosen in such a way that the terms with positive exponents vanish. So, the spacecraft trajectory beginning at the point of the manifold can be bounded in the vicinity of the libration point by using the solar radiation only. Let us remark that the libration point lies on the boundary of this manifold. Moreover, the necessary direction of the normal can be expressed in terms of two linear combinations of the initial conditions from a special system of two algebraic equations. We also note that, if one realizes the above-mentioned choice of the normal direction at any time, then the system of two remaining equations of motion will turn to a homogeneous one, and this new system will have no eigenvalues with positive real parts. Thus, the spacecraft bounded trajectories can be stabilized.
2. Notation and assumptions
Let A be a light spacecraft equipped with a solar sail that is a perfect reflective plane (see Fig. 1). Suppose that A moves in the vicinity of a triangular libration point L of an asteroid with the mass center C that describes an orbit around the Sun S. One can assume that this orbit is near-circular, but this is not critical. Also, suppose that the asteroid can be modeled as a dumbbell consisting of two homogeneous balls of mass m1 and m2 jointed by a weightless rod. Let the centers of the balls be M1 and M2, respectively. Without loss of generality one can also assume m1 ^ m2. Let Cxyz be a right-handed Cartesian coordinate system with axis Cx directed from C to M1. Let us restrict to the case where the asteroid rotates around Cz with an angular velocity w and Cz is permanently directed in the absolute space. Such assumption is acceptable if the asteroid rotation around Cz is sufficiently fast. Also, let R = S~C, pL = p = -A. Evidently, p < pL < R, where p = \p\ is the distance from the spacecraft to the libration point, pL = \pL\ is the distance between the libration point and the asteroid mass center (pL = const), R = \R\ is the current radius of the asteroid mass center orbit.
Let us remark that one can assume that the Sun and the asteroid influence the spacecraft motion, but the spacecraft does not influence the motions of the asteroid and the Sun. Also, the
R
S
Fig. 1. Notation and acting forces
asteroid does not influence the motion of the Sun. The main forces acting on the apparatus A and on the asteroid are
the force of the Sun's gravitation F1 acting on the ball with center M1,
the force of the Sun's gravitation F2 acting on the ball with center M2,
the gravitational force f1 acting on the spacecraft A from the ball with center M1,
the gravitational force f2 acting on the spacecraft A from the ball with center M2,
the force of the Sun's gravitation fs acting on the spacecraft,
the force of the solar radiation FS acting on the solar sail.
The last one can be computed as [15, 17, 24]
FS = PS(n, eS)2n, (2.1)
where n is the normal to the solar sail plane, |n| = 1, eS is a unit vector aligned with the solar rays i 6g ^ i P is a characteristic value of the solar pressure at the current distance from the Sun, S is the solar sail area. Of course, there are a number of other influences, but here we can assume them to be insignificant.
As in the situation under consideration, all conditions of GRCP3B [3] for the case of the right angle of nutation are fulfilled, one can write the coordinates of the triangular libration point L as
= yL = ±l\Ja2ri--A, zL = 0, (2.2)
where l is the distance between M1 and M2, and
a=G(m1+m2) = —mi— ui2l6 m1 + m2
are the dimensionless parameters of GRCP3B, G is the gravitational constant. Note that there are two triangular librat.ion points only if a > and they are unstable if [4]
A) >\, A0 = 9«-4/3At(l - fi) (V3 - i) . (2.4)
Here we restrict ourselves to the case of a positive yL. Note also that, if one replaces the dumbbell M1M2 with a binary asteroid, then a = 1, as in this case the classical RCP3B takes place. For example, the components of the binary (90) Antiopa rotate about its common mass center so that a & 1. At the same time, the bone-shaped asteroid (216) Cleopatra rotates so fast that a & 0.7. Formally, analogical estimation for the contact-binary (4769) Castalia is a & 0.5, which is impossible as in this case the components cannot be held together by its own gravitation. Presumably, the reason is the lack of complete data on the density distribution inside this small planet.
3. The equations of motion
Generally, the equations of motion of spacecraft A can be written as
m(p + Pl + R) = f1 + f2 + fS + Fs + £1, (3.1)
where £1 are perturbations that were not described above, particularly, gravitational influences of the big planets, etc. Moreover,
A(m1 + m2) = F1 + F2 + S2, (3.2)
where X2 has the same sense as E1. Also, note that, as L is the libration point,
mpL = /10 + /20 = -u2pL, (3.3)
where /10 = /1 |p=0, /20 = /2lp=0, i.e., /10 and /20 are values of /1 and /2, respectively, if the apparatus A is located directly at the libration point. Combining, one can rewrite the equations of motion of spacecraft A as
P = + + + + (3.4)
m m m1 + m2 m m1 + m2
4. The principal parts of the main influences
Evidently, the first term on the right-hand side of (3.4) characterizes the gravitational influence of the asteroid on the spacecraft motion with respect to the libration point. One can rewrite this term as
¿(/1 " /10 + /2 " /20) = w2 ("P + 3vepl (epl, p) + 3(1 - ß)ep2 (ep2, p) + pO (£)), (4.1)
where epl = j^py, ep2 = j^f-y Also, the next two terms on the right-hand side of (3.4) characterize the gravitational influence of the Sun. One can rewrite them as
n F1 + F2_ GMSPL ( _ 3eR, _ u0 [P ] + 0 / MX_
m m1 + m2 R3 \ pL R \ PL V \RJ \pL
where Ms is the mass of the Sun, epL = ^ is a unit vector having the same direction as pL,
and eR = ^ is a unit vector having the same direction as R.
Further, the next two terms on the right-hand side of (3.4) characterize other gravitational influences, which were considered as insignificant above.
Finally, the last term on the right-hand side of (3.4) is the influence of the Sun's radiation. As follows from the assumptions made above, it can be computed by (2.1), where es can be taken to be equal to eR.
5. The nondimensionalization and estimation of the gravitational and radiation contributions
Using l as a new unit of length, one can rewrite the equation of motion (3.4) in a dimen-sionless form (neglecting the contributions already assumed insignificant)
P" = -P + 3^epl (epl, P) + 3(1 - ^)e2 (ep2, P) + O (p2) +
+ A (epL - 3eB (epL, eR) + O + O (j-^ + A^n, ej2n, (5.1)
where the prime ( )' denotes the derivative with respect to dimensionless time r (dr = wdt). In this equation
_ GpLMs PS
A - ~ JuS2 ' m (5-2)
are dimensionless parameters that characterize the gravitational and radiation contribution of the Sun, respectively.
On can easily check that, if the asteroid length is about 100 km and its spin motion is sufficiently fast, then A ^ A1, i.e., the Sun's gravitational influence is negligible compared to its radiation influence. For instance, if ^ = 6 m2/kg, which is close to such ratio for the Light.Sail2 [11], then for the binary asteroid (90) Ant.iope we have & 3.8 • 10~5. Thus, the Sun's gravitational influence can be neglected, at least within the frame of the model problem.
6. Reduction of the equations of motion
Using A11 as a new dimensionless unit of length, taking into account the principal parts of the main contributions only and neglecting insignificant ones, one can reduce the equations of motion (3.4) to the form
p" = -P + 3№epi (epi, p) + 3(1 - ^)ep2 (ep2, p) + (n, eR fn. (6.1)
Let x, y, z be coordinates of p in a Cartesian coordinate system Lxyz with axes co-directed with the axes of Cxyz. Taking into account that Lxyz rotates, one can rewrite (6.1) in the coordinate form as
v" - 2y> = + ^«"2/3(l - 2 riJaW -\-y + (n, eRfnx, (6.2)
y" + 2x' = ^«-2/3(l " 2 ^«2/3 - I -x + \(A- a~2/3) y + (n, eR)\, (6.3)
z" = -z + (n, eR)2nz, (6.4)
where nx, ny, nz are the components of n in Lxyz. (ri% + n2 + n2 = 1 as n is a unit vector.)
Note that the last equation, (6.4), can be considered separately. Moreover, using (6.4), we obtain
^ (V2 + z2) = 2z'{z" + z)= 2z'(n, eR)2nz. (6.5)
Hence, if we choose the projection nz of the sign opposite to z', then the function z'2 + z2 will not increase and will be bounded by \jz'02 + Zq, where z'0 and z0 are the initial values of z' and z, respectively.
7. Computation of the normal n
Let Cx1y1z1 be a right-handed Cartesian coordinate system with axes invariably directed in the absolute space such that the plane Cx1 y1 coincides with the plane of the orbit of the asteroid mass center C and Cz lies in the plane Cy1z1 (see Fig. 2). Let n denote the angle between Cz1 and Cz. Evidently, 0 ^ n ^ n. Also, let nz1 be the projection of the normal to the solar sail n
Fig. 2. Computation of the solar sail normal
on Cz1 and nxy = n — nz1ez1, where ez1 is the unit vector of Czv Clearly, nxy lies in Cx-y^.. Let 7 denote the angle between eR (or es) and nxy. Physically, ^ 7 ^
It is clear that there exists an instant in time at which the direction of the axis Cx1 coincides with the direction to the Sun, i.e., ex1 = —eR. Let ^0 denote the angle between this direction
and CS at initial time t = 0. If the orbit of C can be assumed to be near-circular, then the angle between Cx1 and cSS at the current instant of time can be written as Q1t + ^0 or Qt + 00, where Q = Also, the angle between Cx1 and Cx can be written as cot + ip0 or r + ip0, where is the initial value of this angle. In this case, one can express the components of the normal n as
nx = -\A -«2lCOs(r + <pQ) COS(Qr + + 7)- sin(r + v?o) sin(Qr + 00 + 7) cos rj - nzl sin(r + tp0) sin rj, (7.1) ny = \J1 - n2zi sin(T + <Po) cos(Qr + 00 + 7)— \Jl — nzl cos(r + ip0) sin(Qr + 0o + 7)cos V — nzi cos(r + ip0) sin 77, (7.2) nz = \Jl — n2zl sin(Qr + 0o + 7) sin rj + nzl cos rj. (7.3)
Also, (n, eR) = \/l — nzl cos 7.
8. The simplest form of the spacecraft equations of motion
Now note that Q ^ 1 for the fast rotating asteroids. For instance, Q = 3.4 • 10_4 for (90) Antiope. Therefore, one can neglect terms with Qt, at least for sufficiently short intervals of time. (Such an interval for (90) Antiope can be estimated to be about a few months.) In this
case, the system (6.2)-(6.3) can be rewritten as
3 3 I l"
x" - 2y' - ja~'2/3x - -oT2,z(l - 2¿¿Wa2/3 ~j = ~A (l - ™2i) cos2 7sin(r + ip0 + k)
(8.1)
" + 2a;' - ^a~2/3(l - '2^)x\Ja2/3 - ^ 4~ a"2/3) = -A (l - n2zl) cos2 7cos(r + <p0 + k),
(8.2)
where k = arcsin if TV ^ 0 or n = ir — arcsin if TV < 0, M = \/l — ri2\ cos(0o + 7); — n2^ sin(0Q + 7) COS T] + sin 77, „4 — s/M2 + A/"2. Now, let 7 = const and nz1 = const during the motion. In this case, the system (8.1)-(8.2) is an inhomogeneous system of fourth-order linear differential equations with 2^-periodic right-hand sides. One can check that the eigenvalues of this system are X1 2 3 4 = ±a ± ib, where
1 1 / /—— 1 , N 2' 6 = 7fV^+2 (8"3)
and i is imaginary unit. Evidently, two of these eigenvalues have positive real parts, i.e., the degree of instability of the libration point L is 2.
Also, note that substituting (7.3) into (6.4) and taking into account the constancy of 7 and nz1, we obtain an equation that can be solved separately and its solutions will be always bounded. This means that, if the direction of the normal n is constant with respect to the direction of the solar rays and to the plane of the orbit of the asteroid mass center, then the variation of coordinate 2 is bounded.
Let us remark that in the case a = 1 and n = which is close to the binary (90) Ant.iope, we have \1 2 , 3, 4 = ±0.632 ± 0.948i.
9. Results
If 7 and nz1 are constants, then the general solution of the system (8.1)-(8.2) can be written as
£ \ a^ a2\ a3
= C1eaT sin br • 1 + C2eaT cos br • 2 + C3e-aT sin br J 3
yV U V2 U
+ C4e-aT cos br • (aM + A sin t • T 1 ) + B cos t • f2 ) , (9.1)
W W \q2j
where Cj are arbitrary constants and aj, (3j, pk, qk are constants that depend on 7, nz1 and the parameters of the problem under consideration. (Here j = 1, ..., 4, k = 1, 2.) Evidently, constants Cj can be expressed in terms of initial values of coordinates and velocities.
Note that if the initial conditions are so that C1 = C2 = 0, then the corresponding particular solution is bounded. Let us choose values of 7 and nz1 such that for the given initial values of coordinates and velocities, the constants C1 and C2 turn out to be zero. Using the standard
procedure, one can establish that suitable values of 7 and nz1 are solutions of the algebraic system
A(l- n2zl) cos2 7 = s/V2 + Q2, tg(v?0 + «) = (9.2)
where P and Q are linear combinations of initial conditions that can be written as
P = gi x(0) + g2 x'(0) + g3y(0) + g4y'(0), Q = h^(0) + h2 x'(0) + h3 y(0) + hAy'(0),
where • and /?.• are functions of a and ¿t. Particularly, if a = 1 and n = then
P = 0.2391526038x(0) + 0.6990859470x/(0) - 0.9787203258y(0) - 0.1151863547y/(0), Q = 0.4036174600x(0) - 0.02169479266x/(0) + 0.4803727097y(0) + 0.5650644440y/(0).
(9.3)
(9.4)
Fig. 3. Example of the area where the keeping is possible
Evidently, the system (9.2) has real solutions not for all values of P and Q, primarily due to the restriction —1 ^ nz1 ^ 1. It can be checked numerically that the area of admissible values of P and Q is some oval in the plane of these variables, and the origin P = Q = 0 belongs to this oval boundary (see the example in Fig. 3). This implies that the system (9.2) and equalities (9.3) assign a manifold in the phase space of (8.1)-(8.2) such that each point of this manifold can be the initial conditions for a bounded solution of (8.1)-(8.2). In other words, for any point of the above-mentioned manifold there exist values of 7 and nz1 such that the particular solution of (8.1)-(8.2) with initial values of coordinates and velocities from this point is bounded. One can easily check that the dimension of this manifold is 4. Moreover, taking into account that the variation of 2 is always bounded, one can say that there exists a 6-dimensional manifold of initial conditions for which the spacecraft motion can be made bounded. Note also that, as the point P = Q = 0 corresponds to the libration point, the libration point lies on the boundary of this manifold.
The above does not completely solve the problem of keeping the apparatus near the libration point, since the above bounded motions are unstable. The reason, as before, is the positive real parts of the two eigenvalues. To stabilize a bounded motion, one might choose the necessary values of 7 and nz1 not only at initial time, but at any time. To compute these values by (9.2),
one must replace with + t (and ^0 with Qt + ^0 if it is critical), and P and Q with P* and Q*, respectively, where
P * = 91% + §2X' + 93 V + 94 V, Q* = Kx + h,2x' + h:iv + h-4 V (9.5)
and x, x', v, V are values of the corresponding phase coordinates at a current instant of time. Trivially, in this case the system (8.1)-(8.2) is transformed to
x" - 2 y' - ^a-^x - ^a"2/3(l - 2 ^sja^ -\-y + P* = 0, y" + 2x' - ^a"2/3(l - a2/3 - i a"2/3) y + Q* = 0.
(9.6)
It is not hard to prove that the systems (8.1)-(8.2) and (9.6) have coinciding solutions under the same initial conditions if, of course, 7 and nz1 in (8.1)-(8.2) are computed from (9.2) and (9.3). But the system (9.6) is homogeneous and its eigenvalues are X1 2,3 , 4 = —a ± ib, ±i. Thus, the bounded motion of the spacecraft can be stabilized, at least linearly.
10. Discussion
Recently, hundreds of papers concerning the dynamics of spacecraft with solar sails and its applications in various space missions have been published. Without claiming to list all relevant literature, we mention the monographs [15, 17, 24] in which a large number of pertinent theoretical problems are formulated and solved. However, the problem of implementing prescribed motions with respect to a minor planet or an artificial space station by a controlled solar sail is still largely underexplored, although there have been attempts at solving it [18]. Also, there has been a large amount of research devoted to the study of the existence and stability of libration points of minor planets (see, for example, [20, 21, 25, 26], etc.). Moreover, some attention has been given in the literature to spacecraft flight near libration points of planet-moon systems like the Earth-Moon system [2, 22] or the Sun-Earth system [6, 16], in particular, using control based on solar radiation. But there have been no studies on possibilities of stabilization of an artificial object near an unstable libration point of a minor planet by using a controlled solar sail, especially if the degree of instability of the libration point is greater than one. In this paper, this gap is partially filled.
11. Conclusion
In this paper, the possibility of keeping a solar sail spacecraft near the unstable triangular libration point of a small planet is established. It is proved that, if the motion of the spacecraft starts from a certain manifold in the phase space of the spacecraft motions, then by choosing the orientation of the solar sail, which is invariable with respect to the direction of the solar rays and the plane of the orbit of the asteroid mass center, one can achieve that the motion of the spacecraft is bounded in the vicinity of the libration point. The dimension of this manifold is equal to that of the phase space of the problem. It is also shown that the bounded motions of the spacecraft can be stabilized if the specified choice of the solar sail orientation is made not only at the initial time, but also at any time. The results presented in this paper can be generalized to the cases of an imperfect solar sail, for instance, to the case of a sail with changing
or degrading reflectivity [12, 19] and to the case of inaccurate computation of its orientation, which is critically important for practice.
Acknowledgments
This work was carried out at the Moscow Aviation Institute (National Research University).
Conflict of interest
The author declares that he has no conflict of interest.
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