On a Sailed Spacecraft Motion along a Handrail Fixed to Two Heliocentric Space Stations Текст научной статьи по специальности «Физика»

Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 3, pp. 359-370. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230802

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 70F20, 70F15

On a Sailed Spacecraft Motion along a Handrail Fixed to Two Heliocentric Space Stations

V. S. Vaskova, A. V. Rodnikov

Motion of a particle modeling a spacecraft with a solar sail along a handrail joining two heliocentric space stations is considered under the assumption that the sail is a perfect reflecting plane that can be located at any angle with respect to the direction of solar rays, the particle does not leave the plane of the orbit of the stations, the handrail is a tether that realizes an ideal unilateral constraint whose boundary is some ellipse, and the particle motion is sufficiently fast with respect to the orbital motion of the stations to neglect noninertiality of the orbital frame of reference. The equations of particle motion are written in dimensionless form without parameters, and the existence of an energy integral for the case of the sail orientation depending only on the spacecraft location is established. This integral is used for complete integration of the equations of motion for the particle relocations along the constraint boundary. The optimal length of the tether for the fastest relocation of a particle between the most remote points of the constraint boundary is computed for the case of the sail being orthogonal to the solar rays throughout the motion. Such a relocation time is computed in dimensionless form and for some real and hypothetical situations. A set of pairs of points in the constraint boundary between which relocation along the constraint boundary with zero initial and final velocities and with the invariably oriented sail is possible is constructed depending on the eccentricity of the ellipse. The result is presented as several plots that illustrate the evolution of the pairs' regions as the eccentricity of the ellipse changes.

Keywords: space tether system, handrail constraint, unilateral constraint, solar sail, heliocentric space station

Received March 22, 2023 Accepted July 06, 2023

This research was carried out at the Moscow Aviation Institute (National Research University).

Varvara S. Vaskova [email protected] Alexander V. Rodnikov [email protected]

Moscow Aviation Institute (National Research University) Volokolamskoe sh. 4, Moscow, 125993 Russia

1. Introduction

Presently, limited fuel reserves are perhaps the main problem in planning space missions. Therefore, any way to save it is relevant. In particular, engines that do not require fuel expenditure at all can contribute to the achievement of the goals of a specific mission. One such engine is a solar sail. Apparently, F. A. Zander was the first to suggest using solar sails for long-range space flights [1]. To date, many theoretical problems concerning orbits correction and interplanetary flights using solar sails have been solved (see, for example, [2, 3]). It turned out that solar sails are useful for stabilization of some specific spacecraft orbits, in particular, unstable libration points [4, 5], and for many other purposes. Without claiming to cite all relevant literature, we also mention [6-8]. At the same time, due to the development of technologies for the production of ultralight film materials [9], spacecraft with a solar sail (SCSS) has become a space reality [10, 11]. Note that in all modern studies the influence of solar pressure on the long orbital motion is considered, whereas the solar sail can also be used for relative motions, for example, between closely located space stations. Of course, motions produced by a solar sail cannot be fast, but the modern solar sails can generate acceleration that allows SCSS to cover distances of several up to hundreds kilometers in a period of several hours up to one or two days, which is quite acceptable, given the lack of fuel consumption. The main inconvenience in planning SCSS relocation is the fact that the solar sail on its own cannot create an acceleration at an obtuse angle to the solar ray's direction. A similar problem for a sailing ship is solved by means of a keel preventing the ship motion in the transverse direction. Thus, to enable SCSS motion not only from the Sun, but also to the Sun, one needs some guides that are analogs of the keel. Such guides can be realized by a tether used as a kind of a handrail. (For space tether systems, see, e.g., [12].)

In this paper, we study SCSS motion along a tether joining two heavy heliocentric space stations moving in the same orbit. We assume that the tether realizes an ideal unilateral handrail constraint [13] that restricts the SCSS motion to the interior and the surface of an ellipsoid with foci in the tether ends fixed in the stations. We assume also that the solar sail is a perfectly reflecting flat mirror. Given the fact that rotation of the orbital frame of reference is quite slow compared to the SCSS motion along the handrail (see [14]), we can neglect noninertial effects. In this case one can claim that, if the normal to the solar sail's plane is always parallel to the plane of the stations' orbit, then there exists a manifold of the SCSS motions in this plane. Moreover, if the orientation of the normal depends on the spacecraft location only, then trajectories from the manifold can be found by a quadrature, at least for motions along the constraint boundary. Using this quadrature, we compute the time of the SCSS relocation between the most remote vertices of the ellipsoid for the case of the solar sail being orthogonal to the solar rays during the motion, and, analyzing this time as a function of the ellipsoid eccentricity, we find its minimum. Note that the SCSS will move along the constraint boundary only if some conditions are fulfilled. Analyzing these conditions, we construct the manifold of pairs of points in the constraint boundary between which motion with zero initial and final velocities is possible if the normal to the solar sail is invariably oriented during the motion. Also, we trace the evolution of this manifold when the ellipsoid eccentricity changes.

2. Statement of the problem and the equations of motion

Consider a space system consisting of two heavy space stations moving along one heliocentric orbit, nearby located and joined by a tether of length 2a. Assume that the tether is weightless,

inextensible and perfectly flexible. Let the tether ends be fixed at points F1 and F2, located at a distance of 2c from each other. Suppose that a particle A of mass m, modeling a light spacecraft, is placed on the tether and can move along it. In this case, the particle motion is restricted to an ellipsoid of revolution with foci F1 and F2, semimajor axis a and eccentricity e = = ^ (see Fig. 1). In fact, an ideal unilateral handrail constraint is realized. In our case, we can assume that F1F2 is always orthogonal to the solar rays. Let O be the midpoint of segment F1F2 and Oxyz be a right-handed Cartesian coordinate system with Ox having the same direction as the solar rays and Oy having the same direction as F1F2. Without loss of generality, we will assume that the focus F1 has a positive coordinate along Oy. Evidently, the coordinates x, y, z of particle A must obey the inequality

x2 + z2

b2

+

y2

^ 1,

(2.1)

where b = Va2 — c2.

y

Fig. 1. A space tether system with a solar sail

Further, let particle A be equipped with a solar sail that is a perfectly reflecting flat mirror of area S and n be the unit vector that is orthogonal to it. Also, let be direction

cosines of n in Oxyz. In the situation under consideration the force of the solar radiation acting on the sail can be written as Fs = PSn^n, where P is the solar radiation pressure. Note that P & 9 • 10"6 N/m2 at the distance 1 AU from the Sun. In the physical sense, the angle between n and Ox cannot be obtuse. This implies that nx must be nonnegative. One can check that, if the stations' orbit radius is about 1 AU, the distance between the stations does not exceed several kilometers and the relative velocity of particle A in the orbital frame of reference Oxyz does not exceed several meters per second and ^ ^ 1 m2/kg, then the resultant of the force of the Sun gravitation, the force of moving space and the Coriolis force is several order of magnitude less than the force Fs maximum. Hence, taking into account the main forces only, we can neglect this resultant and write the equations of motion for particle A by analogy with [14] as

mx = PSnt + A-S x dx

my = PSn2ny + A

fl 1dy:

(2.2)

Q J

mz = PSnlnr + A—, x z dz

2

a

where A is the Lagrange multiplier. Note that in our case A = 0 for motions inside the ellipsoid (2.1) and A ^ 0 for motions along the surface of the ellipsoid (2.1). Let a be the unit of

dimensionless length and r = tJ — be dimensionless time. Then one can rewrite (2.2) as

x" = nsx +

dx'

..//__2„ , ^дf

, = nxnv + A- ,

9V (2 3)

„ 2 xdf (3)

t" = n2xnz + A^,

/ \ x2 + Z , 2

f{X, V, Z) = --5~+y

1 — e2

where the prime ( )' denotes the derivative with respect to t, and A has the same sense as A.

Note that, if n depends only on the coordinates of particle A, then combining Eqs. (2.3), we obtain

^ (x/2 + y'2 + z'2) = J n2{nx dx + ny dy + nz dz) + h, (2.4)

where h is some constant. In fact, this equality presents the energy integral at least in two cases, namely, when the form nX(nx dx + ny dy + nz dz) is integrable (for instance, if n is invariable), and when the trajectory of particle A is known.

3. Integration of the equations of motion in the plane of the stations' orbit

By analyzing (2.3), it is easily shown that, if nz = 0, that is, the normal to the solar sail is always parallel to the orbit of the foci of the ellipsoid (2.1), then there exists an integral manifold z = 0 of the motions of particle A in the plane of this orbit. Evidently, such motions can begin only if z = z' = 0 at the initial instant of time. Here we restrict ourselves to motions from this manifold. Moreover, we study motions along the constraint boundary only, that is, along the ellipse

+ y2 = 1. (3.1)

1 - e2

If particle A moves in the plane Oxy along the ellipse (3.1), then one uniquely determines its current location by eccentric anomaly of this ellipse. Denoting this eccentric anomaly by

one can see that x = a/1 — e2 sin 0, y = cos 0. In this case, if the initial velocity of particle A is zero, then the energy integral (2.4) can be written as

where is the initial value of Further, we can complete the integration of the equations of motion via the quadrature

i> i-

1 - e2 cos2 £

-m^tr* (3-3)

Remember that in the case under consideration the condition A ^ 0 must be fulfilled. Using (2.3) and (3.1), we can prove that this inequality is equivalent to

+ (3.4)

Using this inequality, it can easily be shown that, if (n, v) ^ 0, where v is an external normal to the ellipse (3.1) in the current point, then Lagrange's multiplier A is nonpositive.

4. Motion with a sail orthogonal to the solar rays

Now, let the sail be invariably orthogonal to the solar rays. In this case nx = 1, ny = nz = 0 and the energy integral (3.2) is reduced to

1

- (l — e2 cos2 tp) 02 = \/l — e2(sin tp — sin 0O)-

2

(4.1)

Consider the motion of particle A between the most remote vertices of the ellipse, namely, V1 for which ^ = 0 and V2 for which ^ = n (see Fig. 1). Using (4.1), we can easily check that, if such motion starts with zero velocity, then it finishes with zero velocity too. Also, note that (3.4) is satisfied throughout the motion. Using (3.3), the flight duration T as a function of eccentricity e can be written as

n/2

T (e) = 2"*

— e

2\-I/4

1 — e2 cos2 ^

sin ^

d^.

(4.2)

This function is plotted in Fig. 2. Note that T(0) = 2K ^ 3.708, where K(k) is a complete

elliptic integral of the first kind with modulus k. Moreover, if e 1, then T(e) goes to infinity, as expected. Of course, both cases are of no interest for practice, but they allow us to verify (4.2).

0 0.2 0.4 0.6 0.8 1

Fig. 2. Graph of flight time versus eccentricity

Solving numerically the equation ^ = 0, we find that, the fastest flight between V1 and V2 will be at e = emin = 0.7906, at which T = Tmin = 3.557. If the radius of the stations orbit is about 1 AU, then this dimensionless value corresponds to Tmin = 4 h 7 min for a SCSS "LightSail2" [11] relocating on 2 km and Tmin = 1 day 17 h 10 min for a hypothetical SCSS with ^ = 100 m2/kg relocating on 200 km. Both situations can be assumed to be acceptable given the lack of fuel consumption.

5. Searching for pairs of points between which relocation with zero initial and final velocities is possible

Now, let n be invariable in Oxy. Moreover, let A1 and A2 be points on the ellipse (3.1), A1A2 ± n (see Fig. 3) and the motion of particle A starts in A1 with zero velocity. Then, if condition (3.4) is fulfilled throughout the motion, then the energy integral (3.2) reduces to

(l — e2 cos2 ip) 02 = nx (nx \/1 — e2(sin ip — sin 01) + ny(cos ip — cos ipi)j, (5.1)

where is the eccentric anomaly of point A1. By analyzing (5.1) it can easily be checked that particle A reaches A2 also with zero velocity. If the normal n remains invariable further, then particle A will return to point A1. In fact, librations between A1 and A2 occur. One should point out that, if condition (3.4) should be disturbed, then the tether weakens and particle A enters the ellipse. The consequences can be catastrophic, as this can lead to shock tension and breakage of the tether.

Fig. 3. Searching for pairs on the boundary of the ellipse

Let us find all pairs of points in the ellipse (3.1) between which the described librations are possible. Let (x1, y1) and (x2, y2) be the coordinates of A1 and A2, respectively. Evidently, y1 = y2, otherwise nx = 0 and the motion cannot begin. Clearly, if the pair A1, A2 is right, then A2, A1 is right too. Then, without loss of generality, we can assume that y2 > y1. Also, remember that nx > 0 in the physical sense.

Denote by ^2 the eccentric anomaly of point A2. As xi and yi are 2^-periodic functions of ^i (i = 1, 2), we can depict the set of pairs as a region in some square with a side 2^ in the plane of anomalies ip1} ip2. Let this square be —^ ^ ^f-

5.1. Conditions for the existence of librations

Denote by v1 and v2 the external normals to the ellipse (3.1) at points A1 and A2, respectively. The following two lemmas can be formulated.

Lemma 1. The inequalities

(n, vl) ^ 0, (n, v2) ^ 0

(5-2)

are necessary conditions for the existence of librations along the ellipse (3.1) between A1 and A2. Lemma 2. The inequalities

(n, vl) > 0, (n, v2) > 0

(5-3)

are sufficient conditions for the existence of librations along the ellipse (3.1) between A1 and A2.

Let us prove these lemmas.

First, if the motion between A1 and A2 with zero initial and final velocities along the ellipse (3.1) is possible, then condition (3.4) must fulfilled, in particular, in A1 and A2. But it can easily be checked that inequalities (5.2) are condition (3.4) written at these points for zero velocity.

Secondly, remember that n is invariable and orthogonal to segment A1A2. Points A1 and A2 divide the ellipse (3.1) into two arcs. One of these arcs contains a point M at which the external normal coincides with n. Note that the lengths of arcs A1M and MA2 are less than half the length of the ellipse. Further, consider a continuous function f (N) = (n, v(N)) of the ellipse point N where v(N) is the external normal to the ellipse at this point. Clearly, the equation f (N) = 0 has only two roots that are symmetric with respect to the center of the ellipse, O. Thus, there is not more than one root of this equation in each of the arcs AM and MA2. Also, if (5.3) are fulfilled, then f (A1) > 0 and f (A2) > 0. Moreover, f (M) = 1 > 0. Evidently, in this case the equation f (N) = 0 has no roots in arcs A1M and MA2. This implies that f (N) > 0 at each point of the arc A1MA2. But then the left-hand side of inequality (3.4) is positive at each point of this arc, that is, condition (3.4) is fulfilled for the motions of particle A between A1 and A2. This completes the proof.

Fig. 4. Proof of the lemmas

By the assumptions made, cos > cos and n is codirected with the vector

^(cos02 ~~ cos0i)) ( \A ~ e2(sin01 — sin02))-

Also, using (3.1), we can see that the vectors vi are codirected with the vectors ((1—e2)-1/2 sin cosaccordingly (i = 1, 2). Therefore, conditions (5.3) are equivalent to

(cos ^2 — cos sin + (1 — e2) (sin — sin ^2) cos > 0, (cos ^2 — cos sin ^2 + (1 — e2) (sin — sin ^2) cos ^2 > 0.

(5.4)

5.2. The limiting cases

Consider two limiting cases, namely, e = 0 and e ^ 1.

If e = 0 then the ellipse (3.1) turns into a circle and inequalities (5.4) are reduced to sin(^1 — — ) > 0. The region where the condition cos ^2 > cos is fulfilled simultaneously with this inequality is depicted in Fig. 5 by vertical hatching. However, given that, if the motion from A1 to A2 is possible, then so is the motion from A2 to A1, a region symmetric with respect to the line = ^2 should be added to this region (it is hatched horizontally in Fig. 5 and in the following figures). Thus, each internal point of the square ABCD corresponds to a pair of points of the ellipse (3.1) between which the librations described above are possible. As for this square boundary, an additional study is required. For example, the point D (|; —corresponds to the pair of the ellipse vertices between which librations are impossible, because in this case nx = 0, while the point E(n, 0) corresponds to the pair of the most remote vertices between which the librations are possible. Let us remark that this case is of no practical meaning for the problem under consideration, but corresponds to the situation studied in [14].

Fig. 5. The pairs set for e = 0

If e = 1 then, taking into account the inequality cos ^2 > cos ^1, (5.4) reduces to sin^1 > 0, sin ^2 > 0. These three inequalities fix a triangle AOEF hatched vertically in Fig. 6. As in the

previous case, we must add a triangle AOGF symmetric to AOEF with respect to ^1 = (this triangle is hatched horizontally in Fig. 6). Clearly, the case e = 1 is impracticable as in this situation the motion of particle A cannot begin because the ellipse degenerates to a segment that is orthogonal to the normal n. Nevertheless, we claim that the square OGFE with its boundary is included into the set of the "right" pairs for any e = 1. We illustrate below that if e ^ 1 then the set of "right" pairs becomes less and less different from this square.

Fig. 6. The residue of the set of pairs for e ^ 1

5.3. The general case 0 < e < 1

Conditions (5.4) can be rewritten as

e2 cos (sin — sin ) + sin(^1 — ^2) > 0) e2 cos ^2(sin ^2 — sin^1 ) + sin(^1 — ^2) > 0.

By analyzing these conditions and taking into account the possibility of changing the order of points in a pair, we can verify that each internal point of the area bounded by curves AG, GB, CE, ED and segments AD, BC in Figs. 7-10 corresponds to a pair of points between which librat.ions with the invariable n are possible. Specifically, Fig. 7 corresponds to 0 < e <

Fig. 8 to the case e = when the curved sections of the boundary have vertical or horizontal tangents at points A, B, C, D, Fig. 9 corresponds to ^ < e < 1, and Fig. 10 corresponds to e that is very close to 1.

Note that in all figures curves AG and GB, as well as curves EC and DE, are symmetric with respect to the line ^1 + ^2 = n, while curves AG and DE, as well as curves GB and EC, are symmetric with respect to the line ^1 = ^2, i.e., curves AG, GB, EC, DE are congruent.

Essentially, if e changes from 0 to 1, then the sought-for region evolves from the square ABCD to the square OGFE, but so that points A, B, C, D, G, F, E, O remain at the region boundary.

Fig. 7. The set of pairs for 0 < e <

Fig. 8. The set of pairs for e =

6. Conclusion

In this paper we study the relocation of a spacecraft between two heliocentric space stations along a tether joining these stations by using a solar sail. We establish that the duration of such a relocation is acceptable for practice even if the sail is invariably orthogonal to the solar rays.

The time-optimal ratio of the distance between the stations to the tether length is computed. Also, we find all pairs of points on the boundary of the ellipse restricting the spacecraft motion in the plane of the orbit of the stations between which relocation with the invariably oriented solar sail is possible.

Conflict of interest

The authors declare that they have no conflict of interest.

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