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Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 537256, 11 pages
http://dx.doi.org/10.1155/2009/537256
Research Article

Orbital Dynamics of a Simple Solar Photon Thruster

1University of Beira Interior, 6201-001 Covilhã, Portugal
2University of Porto, 4169-007 Porto, Portugal
3Aeronautical Institute of Technology, 12.228-900 Sao José dos Campos, Brazil

Received 29 July 2009; Accepted 13 November 2009

Academic Editor: Maria Zanardi

Copyright © 2009 Anna D. Guerman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study orbital dynamics of a compound solar sail, namely, a Simple Solar Photon Thruster and compare its behavior to that of a common version of sailcraft. To perform this analysis, development of a mathematical model for force created by light reflection on all sailcraft elements is essential. We deduce the equations of sailcraft's motion and compare performance of two schemes of solar propulsion for two test time-optimal control problems of trajectory transfer.

1. Introduction

The use of solar pressure to create propulsion can minimize spacecraft on-board energy consumption during a mission [1, 2]. Modern materials and technologies made this propulsion scheme feasible, and many projects of solar sails are now under development, making solar sail dynamics the subject of numerous studies.

So far, the most extensively studied problem is the orbital maneuver of a Flat Solar Sail (FSS, Figure 1). In this case, the control is performed by turning the entire sail surface with respect to the Sun direction. This changes the radiation pressure and results in evolution of the vehicle trajectory. Some of the many missions studied are described in [313].

537256.fig.001
Figure 1: Flat Solar Sail (FSS).

The use of a compound solar sail, or Solar Photon Thruster (SPT), was proposed by Tsander long ago [1, 2], but the study of this spacecraft began quite recently [1418]. The SPT consists of a parabolic surface which concentrates the solar radiation pressure on a system of smaller mirrors. The control effort in such system is produced by displacement of a small mirror with respect to the parabolic surface. The sail axis is supposed to be oriented along the Sun-sailcraft direction. There exist several versions of compound solar sails. Forward [14] described two types of a compound sail, namely, Simple Solar Photon Thruster (SSPT) and Dual Reflection Solar Photon Thruster (DRSPT). The few existent studies on SPT dynamics consider the latter scheme.

In order to assess the dynamical characteristics and to compare the SPT performance to the most studied version, FSS, one should study the application of these propulsion schemes for orbital transfer and/or maintenance for various missions. To perform this analysis, a coherent mathematical model for force acting on such a structure due to solar radiation pressure is essential. There are many results concerning radiation pressure force and torque models for a sunlit body, and in many studies they are applied successively to develop a force model for an FSS. However, the usual approach cannot be applied for a Solar Photon Thruster due to multiple light reflections on the SPT elements.

Some attempts to develop a mathematical description for SPT force have been made before, mostly for a Dual Reflection Solar Photon Thruster. In [2] the model for an ideally reflecting DRSPT is described. In [19] this model is extended for nonideal DRSPT. Meanwhile, both of these models are based on the supposition that all the incoming light flux is reflected consequently on each one of the DRSPT elements and then leaves the system, which is not true [20]. Moreover, the results of [20] show that the existing shadowing and related energy dissipation diminish significantly the DRSPT efficiency, making dubious the advantages of this propulsion scheme compared to FSS.

In the present article, we focus on the other version of the compound scheme of solar propulsion, Simple Solar Photon Thruster (Figure 2). We develop a mathematical model for a solar radiation force acting on SSPT and provide a comparative study of trajectory dynamics and control for the FSS and SSPT schemes. In our analysis, we use the following assumptions.

537256.fig.002
Figure 2: Solar Photon Thruster (SPT).
(i)Solar radiation pressure follows inverse-square variation law.(ii)The only gravitational field is that of the Sun, and this field is central Newtonian.(iii)The sails are ideal reflectors (all photons are perfectly reflected).

We derive the equations of motion for the SPT and compare the orbital behavior of FSS and SPT studying two test problems: Earth-Mars transfer and Earth-Venus transfer.

2. Equations of Motion

To write down the equations of motion for a solar sail spacecraft, we introduce two right-oriented Cartesian frames with their origin in the center of mass of the spacecraft 𝑂 as follows:

(i)𝑂𝑥𝑦𝑧 is the coordinate frame attached to the spacecraft; the axes Ox, Oy, and Oz are the central principal axes of the spacecraft.(ii)OXYZ is the orbital frame, its axis OZ is directed along the radius vector of the point O with respect to the center of mass of the Sun, and the axis OY is orthogonal both to OZ and to the velocity of the point O.

We determine the position of the coordinate frame 𝑂𝑥𝑦𝑧 attached to the spacecraft with respect to the orbital frame using the transition matrix between these frames, 𝑎𝑖𝑗.

We use a set of canonical units which implies that the radius of the Earth’s orbit is 1 AU, and the period of its revolution is 2𝜋. The equations of orbital motion can be written in the form ..𝑟=𝑟𝑟3+𝑎𝑠,(2.1) where 𝑎𝑠 is the acceleration due to the radiation pressure.

To complete the equations of motion, we have to calculate also the force produced by the Sun radiation pressure.

2.1. Flat Solar Sail

The interaction of the solar radiation flow with a flat perfectly reflecting surface has been studied earlier [1, 2]. By the symmetry of falling and reflected flows, the total solar radiation force is directed along the symmetry axis of the sail and produces no torque with respect to any point of this axis, including the sail's center of mass (Figure 1). This force can be expressed as [1, 2] 𝑃=2𝑛𝜎,𝑛2𝑆Φ𝑟2,(2.2) where 𝑛 is the normal to the sail surface and points to the Sun, 𝜎 is the unit vector of the parallel light flow (i.e., it opposes the Sun-sailcraft direction), its coordinates in the Oxyz frame are 𝜎=(𝜎𝑥,𝜎𝑦,𝜎𝑧)=(𝑎31,𝑎32,𝑎33), 𝑆 is the total area of the sail, and Φ=4.563106 N/m2 is the nominal solar radiation pressure constant at 1 AU. The solar radiation force projections onto the spacecraft-connected and orbital coordinate frames are 𝑃𝑥=𝑃𝑦=0,𝑃𝑧Φ=2𝑆𝑟2𝑎233sign𝑎33,𝑃𝑋Φ=2𝑆𝑟2𝑎13𝑎233sign𝑎33,𝑃𝑌Φ=2𝑆𝑟2𝑎32𝑎233sign𝑎33,𝑃𝑍Φ=2𝑆𝑟2||𝑎33||𝑎233,(2.3) respectively.

If 𝑛 lies in the 𝑂𝑋𝑍 plane, then the components of the radiation force are 𝑃𝑋Φ=𝑆𝑟2||||sin2𝜃cos𝜃,𝑃𝑌=0,𝑃𝑍Φ=2𝑆𝑟2cos2𝜃||||cos𝜃,(2.4) where 𝜃 is the angle between the vector 𝑛 and the 𝑂𝑍 axis.

The above expressions are standard and appear in numerous studies of the propulsion effort of a Flat Solar Sail.

2.2. Solar Photon Thruster

We consider here another system that is shown in Figure 2. It consists of a parabolic collector and a control mirror (director). When reflection is ideal and the collector axis is exactly aligned with the Sun-sailcraft direction, the collector concentrates the sunlight in the center of the director. In order to minimize the solar radiation torque that causes perturbations of the sailcraft orientation, the director should be located at the sailcraft’s center of mass. This scheme of solar propulsion seems to be more reliable with respect to small misalignments of the sail axis than the DRSPT scheme studied in [1517] which uses a collimator.

In the analysis, we assume the control mirror small enough to disregard the influence of its shadow. We also suppose that the SPT axis is aligned exactly along the Sun direction.

We consider the parabolic surface described in the reference frame 𝑂𝑥𝑦𝑧 by the equation 𝑥2+𝑦2𝑥+2𝑎(𝑧𝑓)=0,2+𝑦2𝑅2,(2.5) where a is the parameter of the paraboloid, 𝑓 is the focal distance, and R is the radius of the sail’s projection on the plane Oxy. Since 𝑓=𝑎/2, equation (2.5) reduces to 1𝑧=𝑎2𝑎2𝑥2𝑦2.(2.6)

The sunlight is directed along the vector 𝜎=(0,0,1). Suppose that it is reflected on the element of the parabolic surface dS, containing the point 𝑘𝜉=𝑥𝑖+𝑦𝑗+𝑧 with z satisfying (2.6). The force produced by the falling light is given by 𝑑𝑃1=𝜌𝜎𝜎,𝑛𝑑𝑆,(2.7) where 𝑛 is the normal to the sunlit side of the sail surface 𝑥𝑘𝑛=𝑖+𝑦𝑗+𝑎𝑥2+𝑦2+𝑎2,(2.8) and ρ is the intensity of the light flow at the current point of the orbit Φ𝜌=𝑟2.(2.9) The ray reflected from the element dS of the collector’s surface has the direction 𝜎1 satisfying 𝜎1=𝜎2𝜎,𝑛𝑛.(2.10) Reflection of light from the element of surface dS produces the force 𝑑𝑃2=𝜌𝜎1𝜎,𝑛𝑑𝑆.(2.11) Finally, this ray is reflected at the focus on the director’s surface with the normal 𝜈=(𝜈𝑥,𝜈𝑦,𝜈𝑧). The force produced by the reflected light can be written as 𝑑𝑃3𝑃=𝑑2𝑃,𝑑4=𝜌𝜎2𝜎,𝑛𝑑𝑆.(2.12) Here 𝜎2=𝜎12𝜎1,𝜈𝜈(2.13) is the direction of the ray reflected from the control mirror.

The reflection of the light on the parabolic surface is unique if the normal to the director does not cross this surface, so the control angle must be greater than half the angular aperture, that is, 𝜃tan1𝑅𝑧𝑅.(2.14) Here 𝜃 is the angle between vector 𝜈 and the sail axis (cos𝜃=𝜈𝑧), and 𝑧𝑅=(1/2𝑎)(𝑎2𝑅2) is the 𝑧-coordinate of the collector’s border. Finally we arrive at the restriction ||||tan𝜃2𝑎𝑅𝑎2𝑅2.(2.15) Multiple reflections on the collector destroy the collector’s film and produce a considerable disturbing torque, and so have to be avoided. Therefore condition (2.15) has to be satisfied during the orbital maneuver.

The elementary force created by interaction of light with parabolic surface and mirror is 𝑑𝑃𝑃=𝑑1𝑃+𝑑2𝑃+𝑑3𝑃+𝑑4𝑃=𝑑1𝑃+𝑑4=𝜌𝜎𝜎,𝑛𝑑𝑆+𝜌𝜎2𝜎,𝑛𝑑=𝜌𝜎2𝜎𝜎,𝑛𝑑𝑆.(2.16) After integration, we obtain 𝑃𝑥Φ=2𝑟2𝜋𝑅2𝜈𝑥𝜈𝑧𝑎122𝑅2𝑅ln1+2𝑎2,𝑃𝑦Φ=2𝑟2𝜋𝑅2𝜈𝑦𝜈𝑧𝑎122𝑅2𝑅ln1+2𝑎2,𝑃𝑧Φ=2𝑟2𝜋𝑅2𝜈2𝑧+12𝜈2𝑧𝑎2𝑅2𝑅ln1+2𝑎2.(2.17) If the control mirror moves in the OXZ plane (𝜈𝑥=sin𝜃, 𝜈𝑦=0, 𝜈𝑧=cos𝜃), then the components of the light pressure force in the orbital coordinate frame are 𝑃𝑋=Φ𝑟2𝜋𝑅2𝑎122𝑅2𝑅ln1+2𝑎2sin2𝜃,𝑃𝑌𝑃=0,𝑍Φ=2𝑟2𝜋𝑅2cos2𝑎𝜃2𝑅2𝑅ln1+2𝑎2.cos2𝜃(2.18) If 𝜒=𝑅/𝑎1, then it is possible to simplify (2.18). One can use Taylor's formula to obtain the expressions 𝑃𝑋Φ=𝑟2𝜋𝑅2𝑅12𝑎2𝑅+𝑜3𝑎3sin2𝜃,𝑃𝑌𝑃=0,𝑍Φ=2𝑟2𝜋𝑅2sin2𝑅𝜃+22𝑎2𝑅cos2𝜃+𝑜3𝑎3.(2.19) If 𝜒2=𝑅2/𝑎2 is negligible (i.e., the sail is almost plane), we get 𝑃𝑋Φ=𝑟2𝑆sin2𝜃,𝑃𝑌=0,𝑃𝑍Φ=2𝑟2𝑆sin2𝜃,(2.20) where 𝑆=𝜋𝑅2 is the effective sail area, that is, the area of the sail projection on the plane Oxy. Formulas (2.20) are similar to those used in [1113] for a different scheme of SPT, so one can expect qualitative similarity of the results for small 𝜒, at least for the maneuvers that require control angles within limits (2.15).

3. In-Plane Orbital Motion

To compare the principal characteristics of SPT and FSS we studied two test time-optimal control problems of solar sail dynamics, namely, the time-optimal Earth-Mars and Earth-Venus transfers [3, 21] for both systems. Since our goal is to compare qualitative behavior of the above systems, we choose the simplest formulation for orbital transfer problem. In both cases, we assume that the planet orbits are circular and coplanar and that the spacecraft moves in the ecliptic plane, starting from the Earth-orbit at 1 AU with Earth-orbital velocity. We find the control law that guarantees the fastest transfer to the planet's orbit.

This model of orbital dynamics results in the following equations of motion in the orbit plane [3]: 𝑤̇𝑟=𝑢,̇𝜑=𝑟𝑤,̇𝑢=2𝑟1𝑟2+𝑎𝑠𝑍,̇𝑤=𝑢𝑤𝑟+𝑎𝑠𝑋.(3.1) Here 𝜑 is the polar angle, and u and w are the radial and transversal components of sail velocity, respectively.

For the FSS, the components of the light pressure acceleration onto the axis of orbital coordinate frame OXYZ are 𝑎𝑠𝑋=Φ𝑚𝑟2𝑆||||cos𝜃sin2𝜃,𝑎𝑠𝑍Φ=2𝑚𝑟2𝑆||cos3𝜃||.(3.2) For the SPT the light pressure acceleration is given by 𝑎𝑠𝑋=Φ𝑚𝑟2𝜋𝑅2𝑎122𝑅2𝑅ln1+2𝑎2𝑎sin2𝜃,𝑠𝑍Φ=2𝑚𝑟2𝜋𝑅2cos2𝑎𝜃2𝑅2𝑅ln1+2𝑎2.cos2𝜃(3.3) The control angle 𝜃 is limited by condition (2.15). In this case the sail surface has to follow the Sun direction.

4. Results

The time-optimal problems for Earth-Mars transfer and Earth-Venus transfer are studied numerically using the interactive software from [22]. This optimization software developed for personal computers running under MS Windows operating systems is based on the penalty function approach and offers to the user a possibility to effectively solve optimal control problems. During the interactive problem-solving process, the user can change the penalty coefficients, change the precision influencing the stopping rule, and choose/change the optimization algorithms. The system includes various gradient-free algorithms used at the beginning of the optimization, as well as more precise conjugate gradient and Newton methods applied at the final stage in order to obtain a precise solution.

The sail parameter is assumed to be Φ𝑆/𝑚=0.0843 for both systems (it corresponds to the acceleration due to solar radiation pressure equal to 1 mm/s2at the Earth-orbit).

The Earth-Mars transfer trajectories for the FSS and the SPT are shown in Figure 3. The continuous line corresponds to the SPT trajectory and the dot-dashed line to the FSS trajectory. We consider the ratio 𝜒=𝑅/𝑎=0.125. The best possible transfer time for the FSS is 𝑇𝑀FSS=2.87 (166.7 days), and for the SPT it is 𝑇𝑀SPT=2.71 (157.5 days), so SSPT maneuver is 5.6% faster than that of the FSS one. Figure 4 shows the variation of the SSPT control angle for the optimal transfer; for FSS the respective control is well known [3].

537256.fig.003
Figure 3: Earth-Mars transfer trajectories.
537256.fig.004
Figure 4: Earth-Mars transfer control effort for SSPT for 𝜒=0.125.

For the Earth-Mars problem, we also study the influence of SPT sail ratio 𝜒. The increase of 𝜒 results in longer maneuver time 𝑇SPT: for 𝜒=0.25 it is 𝑇𝑀SPT=2.76, and for 𝜒=0.5 the maneuver time is 𝑇𝑀SPT=2.96. For greater values of 𝜒, the control angle 𝜃 attains the limits described by restriction (2.15) more frequently.

Analyzing the maneuver to Venus orbit for these two sailcraft schemes (Figures 5 and 6), we have established that FSS reaches the objective in 181.2 days (𝑇𝑉FSS=3.12), while the SPT performs this maneuver in 158.6 days (𝑇𝑉SPT=2.73). In this case, the efficiency of SPT is more significant; SPT reaches Venus orbit 12.5% faster than FSS.

537256.fig.005
Figure 5: Earth-Venus transfer trajectories.
537256.fig.006
Figure 6: Earth-Venus transfer control effort for SSPT for 𝜒=0.125.

5. Conclusions

The problems of orbital dynamics and control are studied for two systems of solar propulsion: a Flat Solar Sail (FSS) and a Simple Solar Photon Thruster (SSPT). We develop a mathematical model for force acting on SSPT due to solar radiation pressure, taking into account multiple reflections of the light flux on the sailcraft elements. We derive the SSPT equations of motion. For in-plane motions of an almost flat sail with negligible attitude control errors, these equations are similar to those used in the previous studies of DRSPT.

For these two solar propulsion schemes, FSS and SSPT, we compare the best time response in two test problems (Earth-Mars transfer and Earth-Venus transfer). Our analysis showed a better performance of SSPT in terms of response time. The result was more pronounced for Earth-Venus transfer that can be explained by the greater values of the transversal component of the acceleration developed by SSPT compared to those of FSS.

Acknowledgments

The authors are grateful to Vladimir Bushenkov for fruitful discussions and his help in use of the optimization software [22]. This work was supported by FCT-Portuguese Foundation for Science and Technology (project CODIS-PTDC/CTE-SPA/64123/2006) and project CoDMoS (Portugal-Brazil collaboration program, FCT-CAPES).

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