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We investigate solar sail in the circular restricted three-body problem, where the larger primary is a source of radiation and the smaller primary is an oblate spheroid in the system. Firstly, the differential equations of motion for solar sail in the system combined effects of radiation and oblateness of celestial bodies are built. Then the positions of the solar sail collinear Lagrange points are calculated as mass ratio or oblateness changes in certain extent. Linearization near the collinear equilibria of the system is applied. A linear quadratic regulator is used to stabilize the nonlinear system. Three different cases of solar sail equilibrium orbits are studied each with different choices for the weight matrices. The simulations reveal that solar sail equilibrium orbits can be stable under active control by changing three angles, incident angle, cone angle and clock angle of the solar sail.

The solar sail has become a hot topic of research during the last few decades [

The restricted three-body problem (RTBP) has been considered a basic dynamic model ever since the scientists have studied the solar sail [

This paper is organized as follows. Section 2 introduces the equations of motion of the system in the circular restricted three-body problem with the larger primary, a source of radiation and the smaller primary, an oblate spheroid. In Section 3, we investigate the equilibrium points of the system with the variations of lightness number of solar sail oroblateness of the smaller primary. Then, in Section 4, linearization near the collinear Lagrange points is taken into account, and the linear quadratic regulator (LQR) is developed to stabilize the nonlinear system. The simulation is given in Section 5. The conclusions are discussed in Section 6.

The restricted three-body problem with the larger primary a source of radiation and the smaller primary an oblate spheroid is investigated. We use a barycentric, rotating and dimensionless coordinate system Oxyz; the origin is at the barycenter of the primaries; the axis x is along the line joining with the primaries; the direction of the orbital angular velocity ω of the smaller primary defines the axis z; and the axis y completes the right-handed triad. We describe the circular restricted three-body problem in _{1} and m_{2} respectively, the mass of the infinitesimal body, the solar sail, is m_{3}. The distance between the primaries is

Then, in this system the masses of two primaries are

Considering the oblateness of the smaller primary, the equations of motion of the solar sail in the rotating coordinate system can be written as

where Ω is the pseudo-potential function [

Ω_{x}, Ω_{y}, Ω_{z} are the components of the partial derivative of the pseudo-potential function Ω on each coordinate

axis;

ber of the solar sail. _{E} and R_{P} are the equatorial and polar radii of the smaller primary, and R is the distance between the two primaries [_{x}, a_{y}, a_{z} are the projections of the acceleration produced by the solar radiation pressure force on the axis Ox, Oy, Oz. _{1} from larger primary; φ is the cone angle between the projection of n in xy plane; γ is the clock angle between the projection of n in xy plane and xz plane. The acceleration produced by solar radiation pressure force can be expressed [

The equilibrium points of the system are the solutions of the equations

where

We suppose that three collinear equilibrium points

According to Equations (13)-(15), we find that collinear equilibrium points will be get when φ and γ equal zero, that is to say, when the surface of solar sail is perpendicular to the radiate light, Equation (13) has effective solution. From Equation (13) we see that the position of collinear equilibrium points varies with the magnitude of the mass ratio μ, oblateness A and lightness number β. In the following, we set μ = 0.001, then the positions of L_{1}, L_{2}, L_{3} varying with the variations of A and β are shown in _{1}, L_{2} are obvious, but it has little impact on L_{3}.

To further investigate the characteristics of the solar sail orbit in the circular restricted three-body problem with oblateness, we need to linearize the system because the differential equations are nonlinear. Given that the collinear Lagrange points of the nonlinear system are

Substitute Equations (16), (17), and (18) into Equations (2), (3) and (4), and assume that the sail acceleration is constant under the small perturbation from the collinear equilibrium point [

where

Therein

where the six-dimensional state vector is defined

The LQR controller is developed to stabilize the nonlinear system in the neighborhood of the collinear libration point. We apply a linear feedback control

where the matrices Q and R represent the weights of the state and control, which are symmetric positive semidefinite and free to be chosen. We obtain the gain matrix

The closed-loop system is then obtained as

A necessary and sufficient condition for the collinear equilibrium points to be linearly stable is that the real part of the eigenvalues of the matrix A − BK are all less than or equal to zero [

In this section, we choose μ = 0.001, A = 0.001, β = 0.05 as basic parameters, and as an example, we set initial conditions as

_{1}. L_{1} is a sink of this system, small perturbation near the collinear Lagrange point L_{1} will be asymptotic stable. In

Different from Case A, _{1}, just like _{1} point. The maximum acceleration of solar radiation pressure is about 0.009906 units acceleration. The minimum _{1} varying with time are shown in

and (k)(l). The minimum and maximum quantitative values of them are displayed in

In

Eigenvalue of A − BK | ||||
---|---|---|---|---|

Case A | [11.9092, −1.8501, 0, 3.6136, 1.8541, 0; 6.7976, −1.0320, 0, 1.8541, 1.4213, 0; 0, 0, 0.01326, 0, 0, 0.3557] | −2.324090064634159 | ||

−2.177050606972909 | ||||

−0.26692728 + 1.81195149i | ||||

−0.26692728 − 1.81195149i | ||||

−0.17786086 + 1.93473213i | ||||

−0.17786086 − 1.93473213i | ||||

Case B | [11.3344, −1.9126, 0, 3.2339, 2.0254, 0; 7.0991, −1.1979, 0, 2.0254, 1.2686, 0; 0, 0, 0, 0, 0, 0.000035] | −2.251276210540859 | ||

−2.251261426017153 | ||||

−0.00002666 + 1.81924636i | ||||

−0.00002666 − 1.81924636i | ||||

−0.00001778 + 1.93947259i | ||||

−0.00001778 − 1.93947259i | ||||

Case C | [15.0850, −2.0293, 0, 6.0683, 0.7097, 0; 4.5272, 1.0991, 0, 0.7097, 3.8125, 0; 0, 0, 1.1526, 0, 0, 3.5078] | −1.624944077846913 | ||

−1.443576958130804 | ||||

−3.40622101 + 1.48524964i | ||||

−3.40622101 − 1.48524964i | ||||

−1.75394392 + 1.35568183i | ||||

−1.75394392 − 1.35568183i |

Axis | Position | velocity | |||
---|---|---|---|---|---|

Minimum | Maximum | Minimum | Maximum | ||

Case A | ξ | ||||

η | |||||

ζ | |||||

Case B | ξ | ||||

η | |||||

ζ | |||||

Case C | ξ | ||||

η | |||||

ζ |

In this paper, we investigate the solar sail equilibrium orbits in the circular restricted three-body problem with oblateness. We find that oblateness has little impact on the position of L_{3}. An LQR controller is used to obtain the numerical solution of the components of solar radiation pressure acceleration. We solve Equation (25) to get the changing laws of angles of the sail, which can make the system stable near the collinear Lagrange points. We can choose different weight matrices Q and R to obtain different solar sail orbits for different space mission requirements.

Ming Song,Xingsuo He,Yehao Yan,Dongsheng He, (2016) Solar Sail Equilibrium Orbits in the Circular Restricted Three-Body Problem with Oblateness. Open Access Library Journal,03,1-10. doi: 10.4236/oalib.1102620