ISSN 1068798X, Russian Engineering Research, 2013, Vol. 33, No. 5, pp. 265–268. © Allerton Press, Inc., 2013. Original Russian Text © R.R. Nazirov, N.A. Eismont, E.N. Chumachenko, D.W. Dunham, I.V. Logashina, A.N. Fedorenko, 2013, published in Vestnik Mashinostroeniya, 2013, No. 2, pp. 43–46.

Control of Spacecraft Groupings near Sun–Earth Collinear Libration Points by Means of Solar Sails R. R. Nazirova, b, N. A. Eismonta, b, E. N. Chumachenkoa, b, D. W. Dunhama, c, I. V. Logashinaa, and A. N. Fedorenkoa a

Moscow Institute of Electronics and Mathematics National Research University “Higher School of Economics” bSpace Research Institute, Russian Academy of Sciences c KinetX, Inc., United States email: [email protected]

Abstract—The adjustment and maintenance of spacecraft configurations by means of solar sails is considered. DOI: 10.3103/S1068798X13050109

In space research, the focus is increasingly on groupings of spacecraft, rather than on individual vehicles. For example, that permits solution of the space–time indeterminacy of measured parameters when investigating plasma and the construction of telescopes with large focal lengths (tens of meters) for astrophysical research. One example is the XEUS project, a spacebased telescope. In that project, spec ified relative positions of the spacecraft bearing the telescope mirror and the vehicle with the Xray detec tor must be maintained. Solar sails with controllable reflective characteristics present an alternative here to traditional jet engines. The approach to constructing a superlongfocus telescope in space may also be applied to very large spacecraft. For the investigation of spacecraft, speci fied configurations must be maintained within group ings. That means that the line of observation to a par ticular point must pass through the optical centers of the detector spacecraft (DSC) and the mirror space craft (MSC). The precision of the vehicles’ relative position is determined by a sphere of radius 1 mm. As an exam ple, we consider the centers of mass of the spacecraft and consider the possibility of using various devices to maintain the required relative position of the space craft, taking account of the calculated perturbing forces. One such device is the solar sail, with control lable reflective characteristics [1, 2]. INFLUENCE OF GRAVITATIONAL FORCES At first, a relatively low circular nearEarth orbit was proposed for the XEUS project. However, an orbit in the vicinity of the collinear L2 libration point of the Sun–Earth System was then selected. That switch was prompted in part by the excessive perturbing gravita

tional forces on the XEUS spacecraft in the low circu lar nearEarth orbit, which are inversely proportional to the cube of the distance from the center of the Earth to the spacecraft and directly proportional to the dis tance between the centers of mass. The L2 libration point is on the Sun–Earth line, at a distance of about 1.5 million km from the Earth, on the opposite side from the Sun [3]. That means that the gravitational perturbations on the spacecraft at that point are about 10 million times less than in a cir cular orbit at a height of 600 km. More precise esti mates may be obtained by means of calculations for orbits in the vicinity of the L2 point with large ampli tude [4]. The mathematical model used to calculate the orbit takes account of the gravitational forces of the Earth (whose gravitational field is expressed in the form of a secondorder Legendre polynomial), the Sun, and the Moon. In the calculations, we assume a distance of 40 m between the centers of mass of the MSC and DSC. We also assume that the spacecraft are spherical objects. In the given conditions, the mirror–detector line—the target line (TL)—passes in the immediate vicinity (within about ±10°) of the plane orthogonal to the direction to the Sun. Hence, the most significant case for calculation of the gravitational perturbations corresponds to alignment of the MSC and DSC along the Y and Z axes in the solar elliptical coordinate sys tem with its origin at the center of the Earth. Calculations show that the relative acceleration is no more than 0.4 × 10–10 m/s2 and has nonzero com ponents in the direction of the X, Y, and Z axes. The calculated components of the acceleration are relatively small, but the observation line must be taken into account if the required positioning accuracy is to be

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obtained. For example, after 1000 s with constant accel eration equal to the limiting value 0.4 × 10–10 m/s2, the spacecraft moves by 2 mm relative to the calculated position. By linear extrapolation, the results obtained for a distance of 40 m between the centers of mass of the spacecraft may be extended to larger distances. In that case, the precision of the calculations may be regarded as acceptable for distances of several hundred km between the centers of mass of the MSC and DSC. INFLUENCE OF NONGRAVITATIONAL PERTURBATIONS Solar radiation acts by the reflection of photons, the absorption of photons, and the reemission of absorbed energy [5]. Specular reflection of the pho tons is possible, with equal angles of incidence and reflection. In that case, the pressure p created by the solar radiation depends on the intensity R of the radi ation passing through unit effective reflection area (orthogonal to the direction of the radiation) 2

p = 2R/ ( c2 cos ϕ ), where c is the velocity of light; ϕ is the angle between the normal to the surface and the direction of the radi ation. The pressure is aligned with the normal to the surface. With complete absorption, the incident photons cre ate a pressure R/c2cosϕ corresponding to the force on unit surface area in the direction of the velocity vector. If the reflection is diffuse, the sum of two vectors is considered: (1) R/c2cosϕ, which is in the direction of the radiation flux; (2) R/3c, which is normal to the surface. The energy absorbed by the surface is radiated back into space. That generates reactive forces. If the radia tion intensity of an elementary surface is denoted by Rb, the reactive pressure arising on account of the radiation is Rb/3c, on the assumption that the spatial distribution of the photons is the same as in diffuse reflection. The distribution of Rb over the surface is propor tional to the fourth power of the surface temperature. In turn, the surface temperature depends on various factors, such as the heat insulation and the power sup ply to the instruments. The electrical power of the solar batteries in the DSC is around 2.5 kW; in other words, the energy emitted by the instruments in the spacecraft is ther mal. We assume that the thermal energy is emitted solely by a single front surface (the limiting case). Then the reactive force is 2.78 × 10–6 N, and the accel eration due to the perturbing force for a 2200kg DSC is 1.26 × 10–9 m/s2; that is 31.5 times the perturbation due to the gravitational gradient. Besides the thermal radiation, radio emission may also generate reactive perturbing forces. For highly

directional antennas, the reactive force may be esti mated approximately as Rr/c, where Rr is the radiant power. For example, if Rr = 40 W (the best estimate for the XEUS project), the reactive force F = 0.129 × 10–6 N. Then the acceleration of the 2200kg DSC is 0.584 × 10–1 m/s2, which is 1.5 times the maximum force due to gravitational perturbation. Obviously, these perturbations are aggravated by the perturbations due to the difference in acceleration associated with the different ratio of the crosssec tional area and the mass for the DSC and MSC. That gives rise to the basic requirement on the design of the vehicles in the grouping: they should have the same radiant properties with specific acceleration, at least for calculations of the mutual position and angular orientation. Systems in the spacecraft such as the system for ori entation and control of the motor assembly or cooling devices produce further perturbation on account of leakage of the combustion gas. The corresponding acceleration may be estimated by analysis of preceding missions. These perturbations are conventionally regarded as negligible in comparison with the pertur bations due to solar radiation. CONTROL OF THE XEUS CONFIGURATION BY MEANS OF SOLAR SAILS Suppose that the orientation of the XEUS space craft is controlled by the traditional method—that is, by means of lowthrust hydrazine rocket motors and flywheels. The vehicles may be maintained in orbit around the libration point by periodically switching on the jet engines, with the application of a single pulse (1 m/s) once a year for each vehicle and corrective maneuvers once or twice a month [6]. That rotates the spacecraft to specified positions with known precision. To maintain and gradually change the interaction of the spacecraft, a different approach is employed (Fig. 1) [7]: three plane surfaces separated by angle Ψ, which act as a solar sail. Each surface consists of two layers (Fig. 2): the first layer is a liquidcrystal film, whose transparency varies under the action of the electrical voltage (the film may become opaque); the second layer (at a greater distance from the Sun) con sists of specularly reflecting foil. If voltage is applied to the first layer, it becomes opaque. In an ideal sail, the solar radiation is absorbed, and the resultant force vector runs along the velocity vector of the incident photons. In the absence of voltage (when the film is transparent), the foil is a reflecting surface; the resultant force vector is orthog onal to the sail surface. Suppose that Ψ = 45°. The situation in which the surface S1 is semitransparent, while surfaces S2 and S3 are opaque, is regarded as standard (with no control actions). Removing the voltage from surface S2 does not change the projection F2X, but the projection F2Y,

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CONTROL OF SPACECRAFT GROUPINGS (a) F2X

(a) F1

Transparent liquidcrystal film

X

S1

S2

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Y

Z

S3

F2Y ψ

2S'2

2F

2S'1 (b)

Foil

(b) Opaque liquidcrystal film

2S'2

Mirror

Opaque section 2S'2

F Foil

Fig. 1. Structure of solar sail (a) and position of transparent and opaque sections (b).

Fig. 2. Operation of solar sail.

which has previously been zero, will now take the fol lowing form

of the observation line in the ecliptic plane is possible in 3 h, with zero final velocity.

F 2Y = F 2X = – kS '2 ,

As already noted, the configuration of the spacecraft must be such that different accelerations due to solar radiation in the standard case. To that end, the cross sectional area of the MSC must be increased, since the ratio of the crosssectional area to the mass is initially half as large for the MSC as for the DSC (if the solar sails of the DSC are taken into account). That may be accomplished by installing solar sails on the MSC.

That results in acceleration along the Y axis. Obvi ously, application of that force to surface S3 may reverse the direction of the acceleration vector. If we need an acceleration vector in the direction +X, it is sufficient to adjust the ratio of the transparent and opaque sections of the surface S1—specifically, to increase the opaque area. The maximum force for the case in Fig. 1 is FX = 2 kS 2' . In other words, the supply of voltage to the whole surface S1 changes the projection of the force vector onto the X axis: FX = +2 kS '2 . With no voltage at sur face S1, we find that FX = –2 kS '2 . Increasing the number of inclined surfaces obtained by 90° rotation of the surfaces S2 and S3 around the X axis ensures acceleration along the Z axis. To determine the sail area, we assume that the error in determining the solar radiant force is ±10%. The area of the solar batteries in the DSC for the given volt age is assumed to be 12 m2, while the total crosssec tional area (orthogonal to the solar direction) is 15 m2. Then, on the assumption of complete photon absorp tion, the solar radiant force (in the direction away from the Sun) is 6.75 × 10–5 N. Hence, S1 = 3 m2 is suffi cient for compensation of the solar radiant force. Given the need for slow maneuvers and for some margin of strength, the sail area S1 must be increased. Thus, with 6m2 sails, in the ideal case, 0.25° rotation RUSSIAN ENGINEERING RESEARCH

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If the solar sails are manufactured by means of liq uidcrystal films, as for the DSC, the speed of maneu vering (change in direction of the observation line) in the grouping will automatically increase. The area of the reflecting sails for the MSC should be about 10 m2. Using the proposed sails for maneuvering, the rota tional speed of the observation line may be doubled. In terms of controllability, spherical sails with a liquid crystal film are also of great interest [8]. Thus, the perturbations due to the gradient of the gravitational force are relatively small, but must be taken into account in ongoing positional control of the spacecraft within the grouping. Perturbations due to solar radiation, including the reactive forces generated by the radiation of the spacecraft itself, may be 30 times greater than the gradient in the acceleration due to gravity, unless special measures are taken. Even the forces due to the telemetric transmissions of the highly directional antennas may be compared with the gravitational perturbations. Solar sails with controllable reflective properties may be used to maintain the required mutual position of the spacecraft within a grouping close to the col linear L2 libration point of the Sun–Earth System. They will be more effective if they are installed on both 2013

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spacecraft. The area of the sails for each spacecraft is no more than 10 m2. If the mass of each spacecraft is around 2200 kg, maintenance of the specified mutual position at dis tances up to 100 m requires the installation of a 10m2 sail on each spacecraft. The total costs of such perturbation are relatively small, which makes the proposed control method practicable. Work is executed according to the grant of the Government of the Russian Federation for support of scientific researches (the governmental order of the Russian Federation from 4/9/2010 no. 220.

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REFERENCES 1. Dunham, D.W., Chumachenko, E.N., Nazirov, R.R., and Eismont, N.A., International Collaboration on Piloted Space Flight for Investigation of the Solar Sys tem and Planetary Protection, Vestn. RAEN, 2012, vol. 12, no. 4, pp. 82–92. 2. Chumachenko, E.N., Kulagin, V.P., Nazirov, R.R., and Eismont, N.A., Information and Space Technologies in

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Studying the Motion of Celestial Objects, Inform. Obraz. Nauki, 2011, no. 3(11), pp. 178–187. Dunham, D.W. and Farquhar, R.W., Libration Point Missions, Conference on Libration Point Orbits and Applications, Gomez, G., Lo, M.W., and Masdemont, J.J., Eds., Aiguablava, Spain: World Scientific, 2002, pp. 45–73. Spacecraft Attitude Determination and Control, Wertz, J.R., Ed., Dordrecht, Holland: D. Reidel, 1978, pp. 129– 130, 570–573. Hechler, M. and Cobos, J., Herschel, Planck, and Gaia Orbit Design, Conference on Libration Point Orbits and Applications, Gomez, G., Lo, M.W., and Masdemont, J.J., Eds., Aiguablava, Spain: World Scientific, 2002, pp. 115–135. Eismont, N.A. and Nazirov, R.R., Solar Sails as a Tool for Spacecraft Motion Control near Solar–Terrestrial Libration Points, Eighteenth International Symposium on Space Flight Dynamics, Munich, 2004, ESASP 548, pp. 123–127. Nazirov, R.R. and Eismont, N.A., Small Solar Sails with Variable Reflectivity as a Tool for XEUS Formation Control, Moscow: SRI RAS, 2008, Pr2142. Chumachenko, E.N., Dunham, D.W., Nazirov, R.R., et al., Simulation and Control of Spherical Solar Sails, Vestn. TGTU, 2011, vol. 17, no. 4, pp. 1044–1052.

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